CISM COURSES AND LECTURES
Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider...
373 downloads
1968 Views
12MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
CISM COURSES AND LECTURES
Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien
The Secretary General Bernhard Schrefler - Padua
Executive Editor Paolo Serafini - Udine
The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No.507
SIMULATION TECHIQUES FOR APPLIED DYNAMICS
EDITED BY MARTIN ARNOLD MARTIN LUTHER UNIVERSITY HALLE-WITTENBERG, GERMANY WERNER SCHIEHLEN UNIVERSITY OF STUTTGART, GERMANY
This volume contains 210 illustrations
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2008 by CISM, Udine Printed in Italy SPIN 12576876
All contributions have been typeset by the authors.
ISBN 978-3-211-89547-4 SpringerWienNewYork
/ )'/'3#- &:/#.+%#- 4:45'.4 %0/4+45 0( .'%*#/+%#- '-'%53+%#- *:
&3#6-+% #/& $+0-0)+%#- %0.10/'/54 #4 8'-- #4 %0/530- &'7+%'4 0(5'/ $#4'& 0/ %0.165'3 *#3&8#3' #/& 40(58#3' *' &'4+)/ 0( 46%* 4:4
5'.4 3'26+3'4 #&7#/%'& .0&'--+/) #/& 4+.6-#5+0/ 5'%*/+26'4 5*#5 #3' 5#+-03'& 50 .6-5+&+4%+1-+/#3: 130$-'.4 *' %061-+/) 0( .0&'-4 (30. &+ '3'/5 1*:4+%#- &0.#+/4 #/& 5*' '%+'/5 #/& 3'-+#$-' 4+.6-#5+0/ 0( %061-'& 1*:4+%#- 1*'/0.'/# +/ '/)+/''3+/) #11-+%#5+0/4 #3' 501
+%4 0( #%5+7' 3'4'#3%* +/ 7#3+064 '-&4 0( '/)+/''3+/) +/ 4+.6-#5+0/ 5'%*/0-0): #/& +/ 5*' &'7'-01.'/5 #/& #/#-:4+4 0( /6.'3+%#- 40-7'34 *' .'5*0&4 #/& 40(58#3' 500-4 0( .6-5+$0&: &:/#.+%4 +--6453#5' 5*' -#3)' 105'/5+#- 0( #&7#/%'& .0/0&+4%+1-+/#3: 4+.6-#5+0/ 5'%*/+26'4 +/ 5*' +/5')3#5'& &'4+)/ 0( *'5'30)'/'064 '/)+/''3+/) 4:45'.4 0/
.'%*#/+%#- 4:45'. %0.10/'/54 #3' '+5*'3 %0/4+&'3'& $: #113013+#5' '95'/4+0/4 0( %-#44+%#- .6-5+$0&: 4:45'. 4+.6-#5+0/ 500-4 03 $: 5*' %061-+/) 8+5* 45#/& 500-4 (30. 05*'3 &+4%+1-+/'4 -+,' *:&3#6-+%4 03 %0.165'3 #+&'& %0/530- '/)+/''3+/) +/ # %0 4+.6-#5+0/ (3#.'803, -5'3/#5+7'-: 5*' *'5'30)'/'064 '/)+/''3+/) 4:45'. .#: $' &'
4%3+$'& +/ # 6/+'& .0&'--+/) (3#.'803, (03 .6-5+&+4%+1-+/#3: #11-+
%#5+0/4 0/& )3#1* .0&'--+/) 03 .03' )'/'3#- 1035 $#4'& .0&'--+/) 5'%*/+26'4 #4 8'-- #4 )'/'3#- .0&'--+/) -#/)6#)'4 -+,' #/& 0&'-+%# #3' 5:1+%#- 3'13'4'/5#5+7'4 0( 5*+4 6/+'& #1130#%* *' &7#/%'& %*00- "+.6-#5+0/ '%*/+26'4 (03 11-+'& :/#.
+%4 8#4 *'-& #5 /5'3/#5+0/#- '/53' (03 '%*#/+%#- %+'/%'4 &+/' 5#-: (30. 5*' 5* 50 5*' 45 0( '15'.$'3 *' 0634' 8#4 #&&3'44'& 50 '/)+/''34 .#5*'.#5+%+#/4 #/& 1*:4+%+454 (30. +/&6453: #/& 3'4'#3%* +/45+565'4 8*0 #3' %0/%'3/'& 8+5* 4:4
5'. &:/#.+%4 %0/530- #/& %0.165'3 4+.6-#5+0/ 0( .'%*#/+%#- #/& .'%*#530/+% 4:45'.4 #/& 50 3'4'#3%* 4%+'/5+454 #/& 456&'/54 8+5* +/
5'3'454 +/ 5*' 5*'03'5+%#- $#%,)306/& #/& +/ 13#%5+%#- #11-+%#5+0/4 0( %0.165'3 4+.6-#5+0/ +/ #11-+'& &:/#.+%4 *' -'%563'34 #+.'& #5 1307+&+/) &'5#+-'& ,/08-'&)' 0/ .0&'--+/) #/& 4+.6-#5+0/ 0( .'%*#530/+% 4:45'.4 8+5* #11-+%#5+0/4 50 &:/#.+%##/#-:4+4 #/& .0&'- $#4'& %0/530--'3 &'4+)/ 5 +4 # 1-'#463' 50 4'' 5*#5 #-- 4'7'/ -'%563'34 *#7' %0/53+$65'& 50 5*+4 130%''&+/)4 70-6.' *' $00, 45#354 8+5* 40.' $#4+%4 +/ 5*' %-#44+%#- 5*'03: 0( 3+)+& .6-5+$0&: 4:45'.4 #/& # 5:1+%#- '95'/4+0/ 50 $+0.'%*#/+% 4:45'. 4+.
6-#5+0/ ! %*+'*-'/ #/& $'3*#3& *' &:/#.+% #/#-:4+4 0( '9+$-' .6-5+$0&: 4:45'.4 3'46-54 +/ %*#-
-'/)+/) 130$-'.4 (03 .0&'--+/) 4+.6-#5+0/ #/& %0/530- 5*#5 #3' &+4
%644'& +/ 5*' %0/53+$65+0/ 0( 3< 6-4 '5 #- *' .'5*0&4 #3' #11-+'& 50 4+.6-#5+0/ #/& %0/530- 0( # 7'*+%-' 8+5* 4'.+ #%5+7' 4641'/4+0/ #/& 50 5*' .05+0/ #/& 7+$3#5+0/ %0/530- 0( # -#3)' '9+$-' .#/+16-#503 0&'--+/) 4+.6-#5+0/ #/& %0/530- 0( .'%*#530/+% 4:45'.4 #3' 456&
+'& +/ )3'#5 &'5#+- $: #- #4', 8*0 45#354 8+5* # )'/'3#- +/530&6%5+0/ 50 .'%*#530/+% 4:45'.4 #/& %0/4+&'34 .'5*0&4 (03 .0&'- 4'561 4+.
6-#5+0/ #/& %0/530- 5*#5 #3' #11-+'& +/ 4641'/4+0/ #/& '/)+/' 456&+'4 +--6453#5+/) 5*' $'/'54 0( .0&'- 13'&+%5+7' %0/530- 453#5')+'4 035 $#4'& 5'%*/+26'4 (03 5*' 6/+'& .0&'--+/) 0( .6-5+&0.#+/ 1*:4+%#- 4:45'.4 +/ '/)+/''3+/) #11-+%#5+0/4 #3' +/530&6%'& +/ 5*' %0/
53+$65+0/ 0( 3''&7'-& *' 6/&'3-:+/) $0/& )3#1* /05#5+0/ +4 &0
.#+/ +/&'1'/&'/5 #/& %0.$+/'4 1*:4+%#- #/& %0.165#5+0/#- 4536%563' 8*+%* %0/53+$65'4 50 # 46$45#/5+#- 41''& 61 0( 5*' .0&'--+/) 130%'44 *' '%+'/5 #/& 3'-+#$-' 4+.6-#5+0/ 0( %0.1-'9 '/)+/''3+/) 4:4
5'.4 +4 # &'.#/&+/) 5#4, (03 /6.'3+%#- 40-65+0/ .'5*0&4 *' '95'/
4+0/ 0( *+)*-: &'7'-01'& /6.'3+%#- 40-7'34 (30. .6-5+$0&: &:/#.+%4 50 .6-5+&+4%+1-+/#3: %061-'& 130$-'.4 +4 &+4%644'& +/ 5*' %*#15'3 0/ /6.'3+%#- .'5*0&4 (03 4+.6-#5+0/ +/ #11-+'& &:/#.+%4 3/0-& #3&8#3' #/& *6.#/ +/ 5*' -001 #11-+%#5+0/4 3'26+3' &:/#.+%#4+.6-#5+0/4 +/ 3'#- 5+.' 6#&3#&0 '5 #- +/7'45+)#5' # 3'#- 5+.' %#
1#$-' %0.$+/#5+0/ 0( 015+.+;'& .6-5+$0&: (03.#-+4.4 8+5* #113013+#5' 5+.' +/5')3#5+0/ .'5*0&4 *'+3 #1130#%* +4 7#-+'& $: '91'3+.'/54 8+5* # 130505:1' %#3 #/& .#: $' 64'& #4 8'-- (03 %0/530--'3 &'4+)/ 5*306)* *6.#/ +/ 5*' -001 4+.6-#5+0/4 / # /#- %0/53+$65+0/ $'3*#3& '5 #- 13'4'/5 #&7#/%'& #1
1-+%#5+0/4 0( 5*' .'5*0&4 0( .6-5+$0&: &:/#.+%4 +/ 5'%*/+%#- &'4+)/ 130%'44'4 #5/'44 $#4'& %0/530- %0/%'15 #/& .6-5+ %3+5'3+# 015+
.+;#5+0/ 5'%*/+26'4 #3' #11-+'& 50 *+)*-: /0/-+/'#3 1#3#--'- ,+/'.#5+% .#%*+/'4 *' &:/#.+%4 0( )'#38*''-4 +4 #/#-:4'& 5#,+/) +/50 #%%06/5 5*' &'5#+-4 0( +.1#%5 -+,' %0/5#%54 +/ )'#353#+/4 (03 +'4'- '/)+/'4 5 8#4 5*' %0..0/ )0#- 0( #-- #65*034 50 1307+&' # %0.13'*'/4+7' +/530&6%5+0/ 50 45#5' 0( 5*' #35 #/& 3'%'/5 &'7'-01.'/54 +/ 5*' &:/#
.+%#- 4+.6-#5+0/ #/& %0/530- 0( *'5'30)'/'064 4:45'.4 #35+/ 3/0-& #/& !'3/'3 %*+'*-'/
'&"* *%& % ! ## *! % !!
"! '&"! ! "!&$" " ) '&"* *%& % & !( $" ! !
"! '&"! ! "!&$" " &$"! *% & % & "
"$&% "! " '&" ! *% *%& % ! $ % " "! $#% & !% ' $ &"% "$ '&"! ! ## *! % & !
'&"* *! % ! ##&"!% & $! $ & ''
(! ##&"!% & !! "# ! !
Multibody Systems and Applied Dynamics Werner Schiehlen and Peter Eberhard Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany Abstract Modern applied dynamics is characterized by systems consisting of mechanical, electrical, hydraulical and biological components subject to control. For the modelling and design of controlled systems the method of multibody systems is most appropriate due to the comparatively low number of degrees of freedom. Therefore, the method of multibody systems is briefly introduced in the following. Human locomotion is used as a typical example for a mechanical system controlled by redundant muscle actuators requiring optimization approaches for control design. The simulation techniques applied are based on inverse dynamics considering measurement results, too.
1
Multibody Dynamics
Multibody dynamics is based on analytical mechanics dealing with rigid bodies. Early applications of the dynamics of rigid bodies are related to gyrodynamics, mechanism theory and biomechanics as reviewed by Schiehlen (1997). However, the requirements for more complex models of satellites and spacecrafts, and the fast development of more and more powerful computers led to a new branch of mechanics: multibody system dynamics. The results of classical mechanics had to be extended to computer algorithms, the multibody formalisms. One of the first formalisms is due to Hooker and Margulies (1965). This approach was developed for satellites consisting of an arbitrary number of rigid bodies interconnected by spherical joints. Another formalism was published by Roberson and Wittenburg (1967). In addition to these numerical formalisms, the progress in computer hardware and software allowed formula manipulation with the result of symbolical equations of motion, too. First contributions are due to Levinson (1977) and Schiehlen and Kreuzer (1977). In the 1980s complete software systems for the modeling, simulation and animation were offered on the market as described by Schwertassek and Roberson (1986). The state-of-theart achieved by the end of the 20th century was documented in Schiehlen
2
W. Schiehlen and P. Eberhard
(1990). Reviews on multibody dynamics including analysis methods and applications were presented by Kort¨ um and Schiehlen (1985) and Huston. The progress achieved in flexible multibody systems was documented by Shabana (2005, 2003) who established the Absolute Nodal Coordinate Formulation (ANCF). Bauchau (2006) considered computational multibody dynamics approaches including impact problems. Today, software packages for multibody dynamics analysis are widely used in academia and industry, see e.g. http://real.uwaterloo.ca/∼mbody/#Software. Recent research topics cover theoretical and computational methods, flexible multibody systems, large deformation phenomena, parameter identification and parameter models, model reduction, contact and impact problems, control and mechatronics, nonholonomic systems, multiphysics problems, algorithms, integration codes and software, simulation and virtual reality, experiments and numerical verification, optimization of mechanical systems with applications to machines and mechanisms, vehicle dynamics, robotics and walking machines, biomechanical problems as well as education in multibody dynamics as presented by Schiehlen (2007). The method of multibody systems is reviewed in this section as one of the most efficient modelling approaches in applied dynamics. 1.1
Mechanical Modelling and Kinematics
First of all the engineering system has to be replaced by the elements of the multibody system approach: rigid and/or flexible bodies, joints, gravity, springs, dampers and position and/or force actuators. A system constrained by bearings and joints is disassembled as free body system using an appropriate number of inertial, moving reference and body fixed frames for the mathematical description. A system of p free rigid bodies holds 6p degrees of freedom characterized by translation vectors and rotation tensors with respect to the inertial frame as T , S i = S i (αi , βi , γi ), i = 1(1)p, (1) r i = ri1 ri2 ri3 see, e.g., Eberhard and Schiehlen (2006). Thus, the position vector of the free system can be written as x = [r11 r12 r13 r21 ... αp βp γp ]T
∈ IR6p .
(2)
Then, the free system’s translation and rotation remain as r = r (x),
S = S (x).
(3)
Multibody Systems and Applied Dynamics
3
Assembling the system by q holonomic, rheonomic constraints reduces the number of degrees of freedom to f = 6p − q. The corresponding constraint equations may be written in explicit or implicit form, respectively, as x = x(y, t)
or
Φ(x, t) = 0
(4)
where the generalized position vector y summarizes the f generalized coordinates of the holonomic system y(t) = [y1 y2 y3 ... yf ]T .
(5)
Then, for the holonomic system’s translation and rotation it remains r i = r i (y, t),
S i = S i (y, t).
(6)
By differentiation the absolute translational and rotational velocity vectors are found v i = r˙ i =
∂r i ∂r i = J T i (y, t)y˙ + v i (y, t), y˙ + ∂y T ∂t
(7)
ω i = s˙ i =
∂si ∂si = J Ri (y, t)y˙ + ω i (y, t) y˙ + ∂y T ∂t
(8)
where si means a vector of infinitesimal rotations following from the corresponding rotation tensor, see, e.g., Eberhard and Schiehlen (2006), and v i , ω i are the local velocities. Further, the Jacobian matrices J T i and J Ri for translation and rotation are defined by Eqs. (7) and (8). The system may be subject to additional r nonholonomic constraints which do not affect the f = 6p − q positional degrees of freedom, but they reduce the velocity dependent degrees of freedom to g = f − r = 6p − q − r. The corresponding constraint equations can be written explicitly or implicitly, too, ˙ y˙ = y(y, z, t)
or
Ψ(y, z, t) = 0
(9)
where the generalized velocities are summarized by z(t) = [z1 z2 z3 ... zg ]T .
(10)
For the system’s translational and rotational velocities it follows from Eqs. (7) to (9) v i = v i (y, z, t)
and ω i = ω i (y, z, t).
(11)
4
W. Schiehlen and P. Eberhard
By differentiation the acceleration vectors are obtained, e.g., the translational acceleration follows as ai =
∂v i ∂v i ∂v i = LT i (y, z, t)z˙ + ai (y, z, t) z˙ + y˙ + ∂z T ∂y T ∂t
(12)
where ai denotes the so called local accelerations. A similar equation yields the rotational acceleration. The Jacobian matrices LT i and LRi , respectively, are related to the generalized velocities, for translations as well as for rotations. 1.2
Equations of Motion of Rigid Body Systems
Newton’s equations and Euler’s equations are based on the velocities and accelerations from section 1.1 as well as on the applied forces and torques, and the constraint forces and torques acting on all bodies. The reactions or constraint forces and torques, respectively, can be reduced to a minimal number of generalized constraint forces also known as Lagrange multipliers. In matrix notation the following equations are obtained, see also Schiehlen (1997). One gets for free body system kinematics and holonomic constraint forces Mx ¨ + q c (x, x, ˙ t) = q e (x, x, ˙ t) + Qg,
Q = −ΦTx ,
(13)
for holonomic system kinematics and constraints M Jy ¨ + q c (y, y, ˙ t) = q e (y, y, ˙ t) + Qg,
(14)
and for nonholonomic system kinematics and constraints M Lz˙ + q c (y, z, t) = q e (y, z, t) + Qg.
(15)
On the left hand side of Eqs. (13) to (15) the inertia forces are characterized by the inertia matrix M , the global Jacobian matrices J , L and the vector q c of the Coriolis forces. On the right hand side the vector q e of the applied forces, which include control forces, and the constraint forces composed by a global distribution matrix Q and the vector of the generalized constraint forces g are found. Each of the Eqs. (13) to (15) represents 6p scalar equations. However, the number of unknowns is different. In Eq. (13) there are 6p + q unknowns resulting from the vectors x and g. In Eq. (14) the number of unknowns is exactly 6p = f + q represented by the vectors y and g, while in Eq. (15) the number of unknowns is 12p − q due to the additional velocity vector z and
Multibody Systems and Applied Dynamics
5
an extended constraint vector g. Obviously, the Newton-Euler equations have to be supplemented for the simulation of motion. The equations of motion are complete sets of equations to be solved by vibration analysis and/or numerical integration. There are two frequently used approaches resulting in differential-algebraic equations (DAE) or ordinary differential equations (ODE), respectively. For the DAE approach the implicit constraint equation (4) is differentiated twice and added to the Newton-Euler equations (13) resulting in qe − qc x ¨ M ΦTx . (16) = ˙t−Φ ˙ x x˙ g −Φ Φx 0 Equation (16) is numerically unstable due to a double zero eigenvalue originating from the differentiation of the constraints. During the last decade great progress was achieved in the stabilization of the solutions of Eq. (16). This is, e.g., documented in Eich-Soellner and F¨ uhrer (1998), and also discussed by Arnold (2008). The ODE approach is based on the elimination of the constraint forces using the orthogonality of generalized motions and constraints, J Q = 0, also known as d’Alembert’s principle, see D’Alembert (1743), for holonomic systems. Then a minimal number of equations remains M (y, t)¨ y + k(y, y, ˙ t) = q(y, y, ˙ t).
(17)
The orthogonality may also be used for nonholonomic systems, L Q = 0, corresponding to Jourdain’s principle, see Jourdain (1909), and Kane’s equations, see Kane and Levinson (1985). However, the explicit form of the nonholonomic constraints (9) has to be added, y˙ = y(y, ˙ z, t),
M (y, z, t)z˙ + k(y, z, t) = q(y, z, t).
(18)
Equations (17) and (18) can now be solved by any standard time integration code. 1.3
Equations of Motion for Flexible Systems
The equations presented can also be extended to flexible bodies as shown in Fig. 1. For the analysis of small structural vibrations often the relative nodal coordinate formulation (RNCF) with a floating frame of reference is used while for large deformations the absolute nodal coordinate formulation (ANCF) turned out to be very efficient, see, e.g., Melzer (1996) and Shabana (2005, 2003).
6
W. Schiehlen and P. Eberhard
Figure 1. Reference systems for flexible multibody systems
Within the RNCF the small number of ff relative coordinates describing the elastic deformations are added to the large number of fr rigid body coordinates of the reference frame moving with translation r(t) and rotation S(t) resulting in an extended position vector y(t) = [y Tr y Tf ]T
(19)
where the subvectors y r , y f summarize the corresponding coordinates. Then, the extended equations of motion read as M (y, t)¨ y + k(y, y, ˙ t) + ki (y, y) ˙ = q(y, y, ˙ t). In comparison to Eq. (17) the additional term 0 0 0 0 ki (y, y) ˙ = y+ y˙ 0 K 0 D
(20)
(21)
depends only on the stiffness and damping matrices K and D of the flexible bodies. Moreover, the inertia matrix shows the inertia coupling due to the relative coordinates M rr M rf . (22) M= M Trf M f f
Multibody Systems and Applied Dynamics
7
Within the ANCF for highly flexible bodies fa absolute coordinates are summarized in a vector y a characterizing the material points of the bodies by an appropriate shape function. Then, the equations of motion read as M y¨a + K a (y a )y a = q(y a , t)
(23)
where M is a constant mass matrix and the vector k of the generalized Coriolis forces is vanishing due to the absolute coordinates. This is true for standard finite elements like Euler beams or bricks. However, for Timoshenko beams with rotary inertia Eq. (20) may be found again as pointed out in von Dombrowski (2002). In any case, the stiffness matrix K a is highly nonlinear and requires special evaluation as shown by Shabana (2005). 1.4
Linear Motion Analysis
The dynamical analysis of multibody systems is closely related to vibration theory. For engineering applications mechanical vibrations of holonomic, rheonomic systems are most important. The dynamical phenomena are classified according to the linear and nonlinear equations of vibrations. For small vibrations the nonlinear Eq. (17) can be linearized resulting in M (t)¨ y + P (t)y˙ + Q(t)y = h(t).
(24)
This system may feature parametrically excited vibrations due to the timevarying, often periodic matrices. In the case of time-invariant matrices one gets after decomposition into the symmetric and skew-symmetric parts My ¨ + (D + G)y˙ + (K + N )y = h(t),
(25)
a system which performs forced vibrations due to the external excitation on the right hand side. In the case of h(t) = 0 only free vibrations remain. Furthermore, if the damping matrix D, the gyroscopic matrix G, and the circulatory matrix N are missing, the conservative system My ¨ + Ky = 0
(26)
with free undamped vibrations is found. On the other hand, nonlinear time-variant mechanical systems represented by Eq. (17), even with just one degree of freedom, may show chaotic vibrations. The special structure of Eqs. (25) and (26) simplifies the analysis. Marginal stability of Eqs. (26) is guaranteed if the stiffness matrix K is positive definite. Free damped vibrations due to Eq. (25) with G = N = 0 are asymptotically stable if both, the stiffness matrix K is positive definite and
8
W. Schiehlen and P. Eberhard
the damping matrix D is positive definite or pervasively positive semidefinite, respectively, see M¨ uller and Schiehlen (1985). Moreover, Eq. (25) is asymptotically stable if all eigenvalues have a negative real part. The general solution of Eqs. (26) reads as y(t) = Ψ1 (t)y 0 + Ψ2 (t)y˙ 0
(27)
where the transition matrices Ψ1 (t), Ψ2 (t) are found from a real eigenvalue analysis of dimension f . The general solution of Eqs. (25) can be written in state space form with the state vector x(t) summarizing the system’s state given by the generalized coordinates and their first time derivatives as y(t) x(t) = . (28) y(t) ˙ Then, the general solution reads simply x(t) = Φ(t)x0
(29)
where Φ(t) denotes here the state transition matrix following from a complex eigenvalue problem of dimension 2f . Matrix methods for linear systems with harmonic excitation h(t) lead to the concept of frequency response matrices while random excitation processes require spectral density matrices or covariance matrices, respectively. In the case of Eq. (24) with periodically time-varying coefficients Floquet’s theory allows closed form solutions, see M¨ uller and Schiehlen (1985).
2
Human Locomotion
Human locomotion research requires simulation techniques provided by applied dynamics. Therefore, human locomotion serves in this section as a typical application for applied dynamics. Dynamic simulation of human motion driven by physiological muscle models is increasingly used to understand human motion. Indeed, recent research has provided remarkable insights into the coordination of human walking. Because the neural excitations to the muscles are not known a priori, optimization techniques are used based on the assumption that the central nervous system (CNS) excites muscles in such a way as to optimize some physiological performance criteria. In walking, the metabolic cost of transport was shown to be an essential performance criterion, see e.g. Ralston (1976). The resulting neuromuscular optimal control problem is traditionally solved by transforming the problem in an optimization by means of discretizing the controls, i.e. the neural excitations. The approach consists of
Multibody Systems and Applied Dynamics
9
repeatedly solving the initial value problem using the reconstructed control histories while minimizing a user-defined cost function. This approach, called dynamic optimization in this contribution, is being increasingly used to study human walking, see e.g. Gerritsen et al. (1998), Neptune et al. (2001), Anderson and Pandy (2001), or Umberger et al. (2003). However, the prohibitive computational effort arising from the several integrations of the differential equations describing the dynamics of the musculoskeletal system prevents a broader use. Anderson and Pandy (2001), for instance, report 10000 hours of CPU time to find a solution for a 3-D musculoskeletal model. The successful application of dynamic simulation to the design of assistive devices, to the evaluation of rehabilitation therapies or to the planning of surgical interventions depends on the reduction of computational effort to acceptable levels. One promising alternative to avoid several integrations of the system equations is the use of inverse-dynamics approaches to solve the optimal control problem. For instance, Nagurka and Yen (1990) proposed an inversedynamics approach that consists of approximating the trajectories of the generalized coordinates of a nonlinear dynamic system by Fourier series, whose coefficients are incorporated to the design parameters. This allows the conversion of the optimal control problem into a parameter optimization problem, and consequently the use of standard and efficient nonlinear programming algorithms such as sequential quadratic programming (SQP). Furthermore, the controls are computed by inverse dynamics by means of the equations of motion from the generalized coordinates and their first and second time derivatives, which are obtained by analytical differentiation. As a result, numerical integrations of the system differential equations are not required and, thus, the computational effort is significantly reduced. Recently, Saidouni and Bessonnet (2003) and Bessonnet et al. (2005) applied a similar approach to determine near-optimal walking trajectories and joint moments for biped robots. Instead of Fourier series, Bessonnet et al. (2005) employed splines of class C3, i.e. approximating polynomials of fourth order, to approximate the time histories of the generalized coordinates. They divide the gait cycle into phases and model contact by means of kinematic constraints that are activated and deactivated along the walking cycle. The application of this approach to human locomotion requires, however, the introduction of physiological models of the muscles and their dynamics. Furthermore, the redundancy in the actuation resulting from the number of muscles being greater than the number of degrees of freedom has to be addressed. In this context, an inverse-dynamics approach is proposed, see Ackermann (2007a), that allows for the determination of near-optimal time histories of the generalized coordinates, muscle forces and neural exci-
10
W. Schiehlen and P. Eberhard
tations taking the contraction and activation dynamics of the muscles into account. As for the inverse-dynamics methods mentioned above, the trajectories of the generalized coordinates describing the motion of the skeletal system are approximated by parameterized functions. In addition, to deal with the redundancy caused by the introduction of muscles, also the time histories of the muscle forces are parameterized. The consistency between the generalized coordinates and the muscle forces is guaranteed by ensuring the fulfillment of constraints given by the equations of motion. The neural excitations, i.e. the controls, are computed by inverting the muscle contraction and activation dynamics. In this way, near-optimal time histories of the generalized coordinates and of the muscle forces are searched that optimize a time-integral cost function such as minimal metabolic cost of transport or maximal proximity to measured data. The method is described in detail in Ackermann (2007b). 2.1
Biomechanical Modelling
The model of the skeletal system adopted is shown in Fig. 2. The free model, i.e. without considering the ground contact constraints, is composed by 7 bodies, the HAT, the two thighs, the two shanks and the two feet, and has nine degrees of freedom. The motion of the model is performed in the sagittal plane and is described by f = 9 generalized coordinates as depicted in Fig. 2. The pelvis, upper part of the trunk, arms and head are modeled as a single rigid body referred to as HAT. The anthropometric parameters of the segments are obtained using the tables in de Leva (1996). The heels, the positions of the metatarsophalangeal joints, and the ankles are denoted by the capital letter A, B and C, respectively, followed by r for the right or l for the left foot. These points are used to define the constraints active at different walking phases. Each lower limb is actuated by 8 three-element Hill-type muscle units, see Fig. 3. During walking contacts occur between the feet and the ground. The contacts are modeled by means of kinematic constraints as implemented by Bessonnet et al. (2005). The constraints are activated and deactivated throughout the gait cycle leading to a time-varying kinematic topology. The kinematic constraints occur between the points Ar, Br, Al and Bl in Fig. 2 and the ground. The gait cycle is divided into eight phases based on the events observed during normal walking. The phases 1, 2, 3 and 4 correspond to the periods limited by the right heel strike, right toe contact (flat contact of right sole), left toe off, right heel off, and left heel strike, respectively. The phases 5, 6, 7 and 8 correspond to the periods limited by the left heel
Multibody Systems and Applied Dynamics
11
HAT
1
1 − Iliopsoas 2 − Rectus Femoris 3 − Glutei 4 − Hamstrings 5 − Vasti 6 − Gastrocnemius 7 − Tibialis Anterior 8 − Soleus
3 2 5 4 thigh
HAT
6 shank
thigh
7 8 foot
shank
foot
Figure 2. Model of the sceletal system
strike, left toe contact (flat contact of left sole), right toe off, left heel off, and right heel strike, respectively. The equations of motion of the free model, which are a set of ordinary differential equations as described by Eq. (17), are modified to account for the additional kinematic constraints modeling the contacts occurring
12
W. Schiehlen and P. Eberhard
CE
PE
SE
Figure 3. Hill-type muscle model
between the feet and the ground, see e.g. Schiehlen (2006), for each phase p as M (y)¨ y + k(y, y) ˙ = q r (y, y) ˙ + R(y)f m + C Tp (y)λp ,
(30)
cp = 0, p = 1, ..., 8,
(31)
where M is the symmetric, positive definite mass matrix, k is the vector of generalized Coriolis forces, q r is the vector of generalized forces other than the ones caused by the muscles, f m is the vector of 2 ∗ 8 = 16 muscle forces, R is the matrix that transforms the muscle forces into generalized forces, cp is the vector containing the kinematic constraints at the feet active in phase p, λ is the vector of Lagrangian multipliers, which may be interpreted as generalized constraint forces required to fulfill the kinematic constraints in cp , and C p represents the Jacobian matrix of the constraints active in phase p. The muscles generate moments at the joints as τ = Af m , where A is a matrix containing the muscles moment arms. The eight muscle groups considered are shown in Fig. 2 for one leg. The Hill-type muscle model, see e.g. Zajac (1989), is composed by a contractile element CE that generates force and represents the muscle fibers, and a series elastic element SE, while the force of the parallel elastic element PE is set to zero, see Fig. 3. All the structures in parallel to the CE and the SE are represented by total passive moments at the joints, which include the moments generated by all other passive structures crossing the joints, like ligaments. The model for the passive moments at the joints is adopted from Riener and Edrich (1999). A linear viscous damping is added to the hip and knee joints, whose values are determined by pendulum experiments
Multibody Systems and Applied Dynamics
13
Activation Dynamics
Muscle Contraction Dynamics
Skeletal System Dynamics
Figure 4. Muscoleskeletal system dynamics
performed by Stein et al. (1996). In order to account for the moments lmj,r and lmj,l applied at the metatarsophalangeal joints by the muscles flexor halucis longus and flexor digitorum longus, negative moments up to 10 Nm are allowed to act at points Br and Bl. Positive moments on the foot at the metatarsophalangeal joint pull the toes upwards having little or no influence on the dynamics of the model during walking. For this reason no positive moments are applied at the metatarsophalangeal joint in the model. Figure 4 shows a scheme of the dynamics of the complete musculoskeletal system having the neural excitations as controls. The vectors u and a are the vectors of the 16 neural excitations and muscle activations, respectively, of all muscle groups considered in the model. The vectors lm and ν m contain the lengths and lengthening velocities of the muscles, respectively. The activation dynamics is modeled by a first order differential equation as in He et al. (1991). The first order contraction dynamics arises from the presence of the series elastic element (CE) and from the muscle force-length-velocity relations, refer e.g. to Ackermann (2007b), and is based on the model of Nagano and Gerritsen (2001). In order to estimate the metabolic energy rate E˙ consumed by the muscles, phenomenological muscle energy expenditure expressions recently proposed in Umberger et al. (2003) are used as ˙ E˙ = E(u, a, ν ce , lce , f ce , pm )
(32)
where the muscle parameters are summarized in the vector pm . All muscle parameters are selected from Nagano and Gerritsen (2001), Menegaldo et al. (2003), and Umberger et al. (2003).
14 2.2
W. Schiehlen and P. Eberhard Optimization Approach
The approach proposed to solve the optimal neuromuscular problem is described in details in Ackermann (2007a,b). It consists in a conversion of the optimal control problem having neural excitations as controls into a parameter optimization based on the parameterization of states while the controls are determined by inverse dynamics. This dispenses with the need for numerically integrating the system differential equations, thus, reducing the computational cost. In order to deal with the fact that the musculoskeletal system is redundantly actuated, a parameterization of the muscle forces is proposed. The approach can be seen as a fusion of the approaches of Bessonnet et al. (2005) to determine near-optimal walking patterns for biped robots with the extended inverse dynamics (EID) approach of Ackermann and Schiehlen (2006) to determine optimal, physiological muscle forces and energetic requirements for a prescribed motion of the skeleton. The connection between both approaches occurs at the joint levels by imposing the agreement between joint moments applied by the muscles and joint moments required to generate the parameterized motion. The approach yields optimal motion patterns and joint moment histories that are compatible with muscle capabilities. Furthermore, more accurate estimations of metabolic cost can be assessed by using phenomenological muscle energy expenditure models from Eq. (32). Figure 5 shows a schematic representation of the approach proposed. The time histories of the generalized coordinates or kinematics, respectively, and of the muscle forces are approximated by splines that interpolate nodal values, which are incorporated to the optimization variables. The consistency between muscle forces and motion is guaranteed by satisfying the equations of motion (30) at discrete control points. From the reconstructed muscle force histories, the neural excitations are computed by inverting the muscle contraction dynamics and activation dynamics. The contraction dynamics is inverted by solving for the activation a from the muscle length ˙ and from the muscle force f m and lm (y) and lengthening velocity ν m (y, y), m ˙ its derivative f , obtained by numerical differentiation of f m . The activation dynamics is inverted subsequently by the corresponding expression for the neural excitation u from the muscle activation a and its derivative a, ˙ obtained by numerical differentiation of a. The optimization problem is formulated as a search for optimal motion and muscle forces that minimize a cost function, e.g., the metabolic cost of transport and the deviation from normal measured patterns, subject mainly to constraints that ensure neural excitations bounded by 0 and 1, fulfillment of kinematic constraints in Eqs. (31), and consistency between muscle forces and kinematics, i.e. fulfillment
Multibody Systems and Applied Dynamics Optimization Variables Muscle Forces Contr. Dyn.
15 Optimization Variables
Constraints Joint Moments
Muscle Moment Arms
Equations of Motion
Kinematics
Optimization Loop Muscle Activations
Act. Dyn.
Metabolic Cost of Transport
Cost Function
Neural Excitations
Motion Deviation
Measured Kinematics Constraints
Figure 5. Optimization approach
of the equations of motion throughout the walking cycle, see Eq. (30). The cost function adopted for the applications is characterized by the metabolic cost of transport E t . The expression for the metabolic cost of transport reads as, see Ackermann (2007a), Et =
16 i=1
tf
ti
E˙ i (t)dt
(33)
where E˙ i is the metabolic cost rate from Eq. (32) for muscle i. 2.3
Verification of Extended Inverse Dynamics Approach
One way to validate the extended inverse dynamics (EID) approach is to use the solution for the neural excitations obtained with the EID as controls for a numerical integration of the differential equations of the musculoskeletal model. If the kinematics obtained in this way agrees well with the measured one, used as input for the EID, it can be concluded that the approach works well and that the discretization adopted is sufficiently fine,
16
W. Schiehlen and P. Eberhard 80 Extended Inverse Dynamics
Time Integration Direct dynamics
Musculoskeletal Model
60
40 20
0
−20
−40 0
0.1
0.2
0.3
0.4
Figure 6. Schematic representation of the procedure to validate the EID approach on the left hand side.
i.e. the number n of nodes is sufficient to guarantee a proper modeling of the phenomenon. Fig. 6 shows a scheme of the described procedure on the left hand side. The right hand side shows a comparison between the measured kinematics y m (t) of the swing phase of subject 1 walking normally with the kinematics y f (t) computed by forward integrating the muscoleskeletal dynamics using the neural excitations obtained with the EID as controls. The kinematics y m (t) of the lower limb of a subject measured during the swing phase serves as input to the EID approach to compute the optimal neural excitations in uEID and the total metabolic cost EEID according to Eq. (33). The neural excitations at the nodes are then linearly interpolated and used as controls for a numerical forward integration of the differential equations with initial conditions for the states, muscle forces and muscle activations at t = 0, given by the values computed with the EID. The forward integration delivers the time history of the generalized coordinates y f (t). A new estimation of the total metabolic cost Ef is then computed. The results of the described procedure are shown in Fig. 6. On the left hand side, the metabolic cost estimations E t obtained with the EID and with the forward integration are shown, EEID = 46.1J and Ef = 44.3J, respectively, representing a small difference of about 4%. The comparison of the kinematics on the right hand side of Fig. 6 features a good agreement. The results indicate, therefore, that the method works well and that the discretization is sufficiently fine by showing a good reconstruction of the original measured kinematics and a reasonable match of the metabolic cost
Multibody Systems and Applied Dynamics
17
estimations. In fact, this comparison would show an even better agreement if the states at all nodes computed with the EID were used for a numerical integration throughout the inter-node times.
3
Conclusion
In the first section the historical development of multibody dynamics is reviewed briefly. The fundamental approaches of analytical dynamics are summarized for rigid and flexible bodies. Multibody dynamics is an excellent foundation for applied dynamics and sophisticated control design. Multibody dynamics offers challenging problems for computational methods required advanced simulation techniques. The second section illustrates the value of biomechanics based approaches to evaluate and analyze the potential of the inverse-dynamics based optimization framework. The results show how modelling and control of the sophisticated systems may be verified in applied dynamics by measurements and advanced simulation techniques.
Bibliography M. Ackermann. A novel optimization approach to generate physiological human walking patterns. In Proceedings of ASME, IDETC/CIE 2007, Las Vegas, USA, 2007a. Paper: IDETC35014. M. Ackermann. Dynamics and energetics of walking with prostheses. PhD Thesis, Shaker Verlag, Aachen, 2007b. M. Ackermann and W. Schiehlen. Dynamic analysis of human gait disorder and metabolical cost estimation. Archive of Applied Mechanics, 75(1012):569–594, 2006. F.C. Anderson and M.G. Pandy. Dynamic optimization of human walking. Journal of Biomechanical Engineering, 123(5):381–390, 2001. M. Arnold. Numerical methods for simulation in applied dynamics. In M. Arnold and W. Schiehlen, editors, Simulation Techniques for Applied Dynamics. Springer, Viena, 2008. Bauchau, editor. Comp. Methods Applied Mechanics Engineering. volume 195(50-51): Special issue on multibody dynamics analysis, 2006. G. Bessonnet, P. Seguin, and P. Sardain. A parametric optimization approach to walking pattern synthesis. The International Journal of Robotics Research, 24(7):523–536, 2005. J. D’Alembert. Trait´e de Dynamique. David, Paris, 1743. P. de Leva. Adjustments to Zatsiorsky - Seluyanov’s segment inertia parameters. Journal of Biomechanics, 29(9):1223–1230, 1996.
18
W. Schiehlen and P. Eberhard
P. Eberhard and W. Schiehlen. Computational dynamics of multibody systems: History, formalisms, and applications. Journal of Computational and Nonlinear Dynamics, 1(1):3–12, 2006. E. Eich-Soellner and C. F¨ uhrer. Numerical Methods in Multibody Dynamics. Teubner, Stuttgart, 1998. K.G.M. Gerritsen, A.J. van den Bogert, M. Hulliger, and R.F. Zernicke. Intrinsic muscle properties facilitate locomotor control – a computer simulation study. Motor Control, 2:206–220, 1998. J. He, W.S. Levine, and G.E. Loeb. Feedback gains for correcting small perturbations to standing posture. IEEE Transactions on Autonomic Control, 36:322–332, 1991. W. Hooker and G. Margulies. The dynamical attitude equations for an n-body satellite. J. Astronom. Science, 12:123–128, 1965. R.L. Huston. Multibody dynamics - modeling and analysis methods. Applied Mechanics Review, 44:109–117, 1991 and 49:35–40, 1996. P.E.B. Jourdain. Note on an analogue at Gauss principle of least constraint. Quarterly Journal on Pure Applied Mathematics, 40:153–197, 1909. T.R. Kane and D.A. Levinson. Dynamics: Theory and Applications. McGraw-Hill, New York, 1985. W. Kort¨ um and W. Schiehlen. General purpose vehicle system dynamics software based on multibody formalisms. Vehicle System Dynamics, 14: 229–263, 1985. D.A. Levinson. Equations of motion for multi-rigid-body systems via symbolical manipulation. J. Spacecraft Rockets, 14:479–487, 1977. F. Melzer. Symbolic computations in flexible multibody systems. Nonlinear Dynamics, 9:147–163, 1996. L.L. Menegaldo, A.T. Fleury, and H.I. Weber. Biomechanical modeling and optimal control of human posture. Journal of Biomechanics, 36(11): 1701–1712, 2003. P.C. M¨ uller and W. Schiehlen. Linear Vibrations. Martinus Nijhoff Publ., Dordrecht., 1985. A. Nagano and K.G.M. Gerritsen. Effects of neuromuscular strength training on vertical jumping performance - a computer simulation study. Journal of Applied Biomechanics, 17(2):113–128, 2001. M.L. Nagurka and V. Yen. Fourier-based optimal control of nonlinear dynamic systems. Journal of Dynamic Systems, Measurement, and Control, 112:17–26, 1990. R.R. Neptune, S.A. Kautz, and F.E. Zajac. Contributions of the individual ankle plantar flexors to support, forward progression and swing initiation during walking. Journal of Biomechanics, 34(11):1387–1398, 2001.
Multibody Systems and Applied Dynamics
19
H.J. Ralston. Energetics of human walking. In R.M. Herman et al., editor, Neural Control of Locomotion, pages 77–98. Plenum Press, New York, 1976. R. Riener and T. Edrich. Identification of passive elastic joint moments in the lower extremities. Journal of Biomechanics, 32(5):539–544, 1999. R.E. Roberson and J. Wittenburg. A dynamical formalism for an arbitrary number of interconnected rigid bodies, with reference to the problem of satellite attitude control. In Proceedings 3rd Congr. Int. Fed. Autom. Control, volume 1 number 3. Butterworth, London, 1967. T. Saidouni and G. Bessonnet. Generating globally optimised sagittal gait cycles of a biped robot. Robotica, 21:199–210, 2003. W. Schiehlen. Multibody Systems Handbook. Springer Berlin, 1990. W. Schiehlen. Multibody system dynamics: Roots and perspectives. Multibody System Dynamics, 1(2):149–188, 1997. W. Schiehlen. Computational dynamics: theory and applications of multibody systems. European Journal of Mechanics A/Solids, 25(4):566–594, 2006. W. Schiehlen. Research trends in multibody system dynamics. Multibody System Dynamics, 18(1):3–13, 2007. W. Schiehlen and E. Kreuzer. Aufstellen der Bewegungsgleichungen gew¨ohnlicher Mehrk¨ orpersysteme. Ingenieur-Archiv, 46:185–194, 1977. (in German). R. Schwertassek and R.E. Roberson. A perspective on computeroriented multibody dynamical formalisms and their implementations. In G. Bianchi and W. Schiehlen, editors, Dynamics of Multibody Systems, pages 263–273. Springer, Berlin, 1986. A.A. Shabana, editor. Nonlinear Dynamics. 4(1/2): Special Issue on Flexible Multibody Dynamics, 2003. A.A. Shabana. Dynamics of Multibody Systems. Cambridge University Press, Cambridge, 2005. R.B. Stein, M.K. Lebiedowska, D.B. Popovic, A. Scheiner, and H.J. Chizeck. Estimating mechanical parameters of leg segments in individuals with and without physical disabilities. IEEE Transactions on Rehabilitation Engineering, 4(3):201–211, 1996. B.R. Umberger, K.G.M. Gerritsen, and P.E. Martin. A model of human muscle energy expenditure. Computer Methods in Biomechanics & Biomedical Engineering, 6(2):99–111, 2003. S. von Dombrowski. Analysis of large flexible body deformation in multibody systems using absolute coordinates. Multibody System Dynamics, 8:409–432, 2002.
20
W. Schiehlen and P. Eberhard
E.F. Zajac. Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. CRC Critical Reviews in Biomedical Engineering, 19(4):359–411, 1989.
! * , ' J #, 9 ,2 - - - , @- ! - * - * $' ! % %%$ $ %$ + * %% * % % ''* " % * * $% !* %* %$
% %' %%$ 2 &($ '$( ', % ( # %$ %' * %%$ %' % 2% %' % % * % %$ '+ $% %' # %$, % 1 $ + ' * % % %$$# $ %%$ %' ' ' % 2 &($ ( * '% % $ $ ' , & $$% # ' + % &%' $ % %$ % $%,
"
#, 2 1 1#,- - ##-, --, -
- 1,1- ,; - 9 1#, 2 1 1#,- - 0, 2 -#1 '-,#- , 2-, : 1- 1#, 2 1 - 1# , - 0 1- -, 1 ; - 9 ##-, --, -,, + # , -,, 2-' # - ,- ##-, 1 - ( 1- - - ( #T , 1-,,; % + : ,', - - - , ,'- + ##-, 1 -
2-' - 2 - + 1- -' 1 M# , 2-, 2-' #; " +'9 #- : # 1 - # ' - #, 2 1- 1 -, - ; & 9 : -,, + 1 M0 2 , 1 19 - , 1- -, 2, + - '-- # - 9 , + + #1 - 2 - -2 , ; % -, 0-1, M0 2, 1- 1 - , + 1- #,- 9 ,- - ##9 : 1- 9 ,- - - + #2 ; # 19 , 2, , 1- --,
22
O. Brüls, A. Cardona and M. Géradin
, 2-, 1 # : 1# # '-, , - - , -, #: #-, --, ; &- 9 - - - - - -1 --, M0 2, 1#, 2 1; M0 2, 1#, 2 1 - 2 K - - ,, -
M0 2, 2 2 - M0 2, ( 1- $ 6;; ' ,#9 1- # '-, $ 7 - 2 ,1 6;; -17; % , 2-, 1 - 1- - , #,
1; !'-, 1 -' 2 , -# --, 2 K 1 - 1- - #,, -
+-9 3- - 6C??D7 - - , ' +; # 2 M # -, - -, -19 - -, -1 - M - -1 - - 1 ,, - ,- M0 2, 2 9 - ,,#- # A;
0. & -, -1 6,79 - -, -1 61
,7 - M - -1 6 7 - -; % %$ %( %% , , ,- , - K ,: 1 1 ; 2 ,# -, -,- - - - - : K + - # # -, -1; "9 2+ - ,- -; # 12 ,- -1, # 1 - 1- 9 ,- 2+ - - : -, - , -9 + # - 1#- ,- # - , - ,- ; % %%$ %( %% - 2 1, 1: #- ,- - , - K ,1 1 ,9 ,( - " 6AQLD7> - - - @- 6AQQA7> K, -,; 6AQQL7; % M0 2, 2 ,, 2 1 -2 ,# -, -9 2# - 1 - - -, -1 ,, + 1 - ,1 K ,1 1; % 1 , -#1 - , -, ,-1 ,1 - 1-,, + -1; & + 9 , 2-, 1 ,1 1 - ,-
Modelling, Simulation and Control…
23
-1, # 1 - -, -1 - - 1-,, -1, # ,- 1- ; "9 # -#1 K 1-, 1 1 ' -19 ,- - 2 1# # - -
, - ,- ; # - - -, 1#- - # -, -9 - - - @- 6AQQA7 - K, -,; 6AQQL7; -, -1 - - -, - - - 1 K ,1 11# ; & -9 1% %( %% 1- 2 - - 0 1#, 2 1-, 1 - # M0 2, 2 9 - $ .#2( 6AQLI7> !-2-- - 3: - 6AQOD7> ' - /+-( 6AQQA7> - -,; 6AQQA7; & , K - 1 - -1 - 2 ,',9 + 1 2 # :2 - 6;; - - ,- ' :
-7;
-, M0 2, - - # 1- 2 ; - -, - -9 ,- - 2 T , '-,#- M - -1 # - , - 1 ,9 2# 1#- - 1 ' ,' -1 #- - ,- - - , #, 2 1,0 , - - 1; % -#1 - , - ,- 2-' # + - M - -1 '-, -1 1 -- 2 1-,, 1- -
1-,, ', ; 1 -, #- 9 : 1 1- 2 ,# 1#,- -# 1 -,, , : - 2-' # 19 3-,,- - !+-( 6AQQA7; 0-1,9 #-, S S - , 2-1 + ,- - 9 + -0 -, # 2 - - 2 S; & M - -1 1#,-: 9 # 1- 1- 2 -# # 1,1- 1 S 1- 09 + T - - 2#
2 - : ' -, +-;
1- - -, -1 - M - -1 : - -9 , - K ,1 1 -,, + - # - -#-, +- 1- , - S - - 1 2 , - ,- 1 , 6;; 1 S S7 - ,- - , 1, 0: - ; - 9 ##-,, ,- - 2# - ,- #- - # - -, 1 1#- ,- ; "9 - 1 1 -' + - '-- - -+2-(; 3 2, ' - - # -2, ' 1 --, M0 2, 1#, 2 1 #, -,, + % ()% %% ; -,- ' 12 K ,1 - - + - 1 ,
24
O. Brüls, A. Cardona and M. Géradin
# -; & , - ##-, -1 9 1 :1 : # -,, + 1, - - , K ,1 1 , - 1,0 2 # 1-,, ,-1 -#1 ; % # # : #, - $ ( 9 + -#,- - 1- +- -1 2-' # + , M0 2, 2 #2##; ,, + + ( - - - @- 6AQQA79 + + ,, + - - -, -1 1 - 2 # , - 1#,- - #,1 + ,- -1, # 1 ; 1- M - -1 -: -9 1 K- S - #,1 0
1 - -, - - -, 1 -, -9 +- M - -1 1 , - 0, : -- 2+ - ,- -; "9 - 1#,- 2- -2 ,# -, - 6 K ,: 1 -7 S 2 , 12 K ,1 1 , - #,1 - 2 1 1 2+ -#- - 1#- -, T ; % K - - - # K ,1 - - 1#,- -1 1 1 2 9 M0 2, 2 - ( 1- $ ; % - + ,, 2 -2, K 1 - , -2 # - - 2 ( 2 @- - - - 6C??A7; # - 1 1,1 -, # +- 9 + - 2 ##,, 0,
' ,- C? - ,' # -, 2,1 #1 # K, -, - ; 0 K ,1 - - -, + 1- 1 + ,, -, 2 # ; ) 2, ( --1 ,-#-9 , #, -1 #- - 2- 1 - 1 #,- : 1; 1 , 1 - 1
,' #, -1 2,1; % - - - ,, +; ! C ( 1-
1#, 2 1 # K ,1 - -; %
-1 --, -
! D; %+ 1#,- : -1 M0 2, 2 - R , - K ,1 1 ! E - #,1 # ! F; ! I - 0 K ,1 1 , #,
--, 1- 1; #1 -, 1 - -, 1 -
# ! L; -,,9 -, - - 2 ! O - 1 ,# - -+ ! Q;
(
Modelling, Simulation and Control…
#
25
& 9 ( 1- - 1#, 2 1 2 # - K ,1 1 1 ,1 - ; ##-,
-1 9 2-19 ,, ' ,#1 ,1 - 2 K 1 - ,- -, K ,1 - # # M0 2, 2 ; & - + 9 - ,1 - -, - #21 - 1#, 2 19 # - - 2 9 - ( 1- $ - ,1; % #- 1 - # 1#,- # -2 ,# -, :
-9 + - ,, ' -, - ; # $ - :2 9 - - 2# -' - - ( 1- - @67 B /!
6A7
& ,, + 9 ( 1- 2 9 - - - #,- 1#, 2 1 - 2 ; ! 1 1 - -: - K ,1 - - + ,, # 2 ,,#- 2#
M0 2, 2 # , ! E; 1.0
)+( ,
1. / 1- 2 ; # - -2 - 2 + 1 ' + -, -1 9 # C; & - -, 1 - 2 # - - - - - - - -1 -- 2 ; % - -2 - 2 2
26
O. Brüls, A. Cardona and M. Géradin
- (% %$ ' 2 :-- -19 -
2 1:'- %%$ ' * -, -1; % ,- 2+ 2 ' ' 2 * B * 8
6C7
+ * ' -, -1 - D D - 1- 0; % -, 1 - -, - 2 1 + * B /; % - 1- 0 -,9 ;; B !
6D7
% 9 1 - 2# -' - I -, ; & ,9 --1 / 9 / - / - 2 # 2 - - - + - + 6/ / / 7!
6E7
.- # - 2, - --19 ;; #, -,9 - -,9 1-, - ' - - - ': ; 3 @- - - - 6C??A7 - -#-# - %- ,, 6C??D7 - # --1 - - ; 0-1,9 - - - ' ' --,,, - -0 + -1, # - -, 0; % - 1- 0 - 2 1# 1 # # 1#,67 B " 8
0 A 0 8 0 0
6F7
(+:11 1- 0 1 2 1 + ? 0 0 B 0 ? 0 ! 6I7 0 0 ? B # - ' #; -9 2 # 9 # # -, - 1- 0 - K 2 1- 0 5 B 6L7 (+:11 ; & - 2 ' K - 1- 0 1 2 1 ' 9 + -#,- ', -, -1; (+ 9 (+:11 1- 0 B 5 >
6O7
Modelling, Simulation and Control…
27
1 2 1 ' >9 + -#,- ', 2 :-- -1; % - - 1- 1 2 :-- -1 -, -1 , B >!
6Q7
% -#,- ', - 2 1# 1 1 '- ' - --1; 0-1,9 # - - - ' 9 + -' > B !675 6A?7 + !67 - - --1 - K 2 0 A 0 !67 B " 8 8 A ! 6AA7 0 0 0 T#, + -, + -2 - ,- - 1 1 -2 , 2-, --1 - 2- --1: ; & 9 - # --1 - -- 2 #,- ; -9 - - - ' ,- - #,- --1: - + 0 B C&; " +'9 - ,- ,# - 2, ' 1 T#,; ,9 1- # - , -, --1 - - + - - - 1 - - --1 1 : + - -9 ;; 1-, - - 2 - - -
@- 6AQOQ7; - # - , 2-, 2# # - --1: - 9 ;; # #, --1 9 - # 2 -
! 1- 6C??A79 1 ' ; & -9 --1 - , ( 2 -
-, - ; -,,9 1- # - 1 - -9 + , 1, , - --19 ;; ! 1 - .#:# 6AQOO7; - -9 # + ,, 2 2 1 - , ; 1.1
)+(
& # D9 + + 2 - 2 - $ ; % - + 2 :-- -1 - - 2 - - - ' - ; & K#- B B /9 + - - -, + -, -0 - -0 -, ; # 1 9 + -' B 6 7
B 6 7
6 B A C D7!
6AC7
% $ 1 1 - ,- ' 1 + 2 ; ,9 -,- - -0
28
O. Brüls, A. Cardona and M. Géradin
2. / 1- 60, ' +7;
-1 2 2 9 + + - - - -,- - * B * ! 6AD7 ! ,9 -0 - #, 1- -, -0 # 1 B ?
B ?
6AE7
+ ,- + -,- ( 1- - ' - B ?
B ?
6AF7
+ B 6 7 - B 6 7; 1.2
)+( + ', + = *(+'
3. % -, #2, #,#1; # -, #2, #,#1 # E; % 1 1 + 2-9 - $ - - -, $ ; %
Modelling, Simulation and Control…
29
2- A - C - -- 2 , - 9 1- -
9 - % - % - -' + - , - - ; % 1 1 ' -' K, B ;
4. ,1 -, #2, #,#1; % 1 1 ,1R + 2 - 9 - ,,#- # F; - 2 9 K - -' - -
-, K - --1 9 + ' - -, # 6 7; 6 B !!! 79 ' * 9 9 2 :-- -19 - 9 - - - ' ; & K#- 9 + -' * B * B H ? ?N 6AI7 * B H 8
?
?N
6AL7
- B / 6 B !!! 7; ! I - - K - 9 #, 0 CE -2: ,# -, -; " +'9 -,, - 2 - - '- -2, 1 , - + ,, 2 ,- ,, + ; ,9 -, $ + 1 # 1 ,, - - 2 # - * B /!
6AO7
! ,9 - 2 9 -' - 1- - I 2 - - K - -
-, ; 2- A9 -,- -, - * B * 8
6AQ7
30
O. Brüls, A. Cardona and M. Géradin
+ B + -' K#- ; % - -, - B !
6C?7
(+ 9 2- C9 + -' * B * 8
6CA7
B !
6CC7
+ B -
% ,9 $ 9 + - -, - - ' 2 #-: 6AF7 +- -,- -, - * B * !
6CD7
' -, ,- #- 6AO79 6C?79 6CC7 - 6CD7 , * 9 9 * - #, -- ' -, : -9 '- -2, - 2 , K 1 '-,#; % 1 # 2 # AC -, - ,, - ' B H* * N ! 6CE7 ' ,1 -, - - 2 K - ,: 1; 2 9 ' ,# -,- - - -' 9 - +,, - -,- - -
-,
B B
H* H*
* N * N !
6CF7 6CI7
9 ' -, - ,# - + B H N ! 6CL7 3 K 9 + + ,, - 2# - ,1 #- 1 - 2 1# 1 + ,1 -,
-;
' ,- 2+ ,1 - , 2-, -, : - ,- - ( , K ,1 -12, #> - 2 + - B 6 B A C D7 6CO7
Modelling, Simulation and Control…
31
+ - ,- , -, - 1- K 2 " / / / B / " / / / / / / / / " / B / / / " " / / / / " / / ! B / / / "
6CQ7
6D?7 6DA7
- 2 # - 6AO7 - +,, - K- ,- #- 6C?79 6CC7 - 6CD7 - 1, , -( - # ' ,- 6CO7; % ( 1- - #- 6AQ79 6CA7 - : 1#,- 6AF7 - 2 - ,1- 2# - @ 6 7
B * *
6DC7
@ 6 7 @ 6 7
B * * B H N !
6DD7 6DE7
+ B 6 79 B !!! ; % + , ( 1- - # @ 6 7 @67 B @ 6 7 B / 6DF7 @ 6 7 + - 2 0, , + - * 8 * * 8 @67 B
B /!
6DI7
& #11-9 ( 1- -, #2, #,#1
# AC -, - - O -,- ( 1- - 6C #2: D - - C -,- - 7; % 1 - # #
: : 19 + -1 + J #2, 1#,-;
$ 2.0
(' + ( +
% -, 1 #- 1 - M0 2, 1#, 2 1 + ' 1 ,- -, , 1- ;
32
O. Brüls, A. Cardona and M. Géradin
( 1- --, ' # 9 - M0 2, 1#, 2 1 - 2 # - -, - 9 + -' - - , 1 - @67 B /!
6DL7
% 1 -, - : , 1 - @6 5 7 B / 1- 2 - - 2 @- - - - 6C??A7; # -#1 - + ( + -, 0-, 67 6;; -, -' 79 1- ,- 2 67 - ( 6 7 5 B
A 5 675 C
6DO7
+ 11 1- 1- 0 1- -,
-; % -- - # '- ' 1 K
2 6 7 5 B 6 7 5 67 67! 6DQ7 1 0 9 #- 1 1 - 2
' # "-1 , ,; & - K 9 # - - 6 B ?7; & 1- -, 1 '- ' - -, - - 9 "-1 , , - -9 1 K - 1 - 9 -, - B 6 7 5 6E?7
- - -, -$ 1 2+ - ; & -, - - 6 % ?79 -$ 1 - ,# - 1 1 - 2,1 1
#2$
@67 B /!
6EA7
# # -- 1#, , 9 - - 1 : 1 - 2,1 ,- 2 # '-, # - 1 1
+ -#1 # -, B 66 7 5 @7!
6EC7
6ED7
Modelling, Simulation and Control…
33
- 2 -9 - - ' B ? 2 19 # #11- ' 9 . .@ . . ' 8 ' ' @ B ?! . G . G . . ! 1 K - - - 9 K 1 '- - 1 - . . .@ 8 ' 8 ' @ B ?! . G . . ) #- 6DQ79 0 2 1 . . . .@ . 8 8 8 ' 8 ' @ B ? . G . . . . + - - ' 6677 8 6 78 5 67 678@ 6778' @67 B ? 6EE7
+ 1 -, ,- H N B .". - 0-, H N B .". ; % - #- 6EE7 - ' # 0 ( #- 6DO7; 1 -,2- 1- #,- 9 ' - #-, ' 2 A .HN .HN .HN H N B G G ! 8 6EF7 C . . . -,,9 #- 1 -( -, 1 - S -,: -,2- 1 67 677 8 6 7 5 8 67 8 @ 67 @67 2.1
B 67
6EI7
B /!
6EL7
'+ (+,'
% K ,1 + 0, ' - 1- +- #- 1 2- ' # ; ! C;D9 ' - - ' ,- B
B A !!!
6EO7
34
O. Brüls, A. Cardona and M. Géradin
+ #12 ,1 K ,1 1; % -: - - 1 - 2 1# 2 #11 # 2# ,1 6 7 5 B 6 5 7 6EQ7
+ K 1 ( 9 -, 9 -
-, - ,1 6 5 7 B 6 5 7 6 7 6 7!
6F?7
% ( 1- - - -, 2 1 ,1- : 2# 9 ;; #- 6DF79 - -- 1#, ,
B ;;; !
@ 6 7 ;; @67 B ; @ 6 7
6FA7
"9 + -' @67 B
@ 6 7!
6FC7
% -#1 - -, #- 6ED7 , ,: 1- 2# B
6 6 5 7 @ 7
6FD7
- #- 6EE7 2 1
' 7 8 8
8 @ 8 ' @ B ?
6FE7
+ 9 9 -
- K 1 9 - ; & -
9 6@ 7 K 2 6@ 7 B
.@ .
!
6FF7
Modelling, Simulation and Control…
35
) #- 6EO7 - 6FE79 -12, ,- B @
B B B B
6FI7
6FL7
6FO7
6FQ7
6@ 7
6I?7
-,, + 1# #1 -,, - 1 #- 1 1 ,1- 1#,- ; ) - 1 ,- -12, #9 , - 1 , - #- 1 - 2 T , '-,#- 1 --, -, 0
K - ,1 ,',; 2.2
*(+'
& -, #2, #,#1 0-1, 6 # E - F79 K ,1 1 +- 1 ,1R + 2 - ; # 1# '- # , 2-, -1 #- 1 ,1 ' ; % -- - 1 1
6 7 5 B 6 5 7 8 6 5 7 8 6 5 7 6IA7 + ' -, - 9 9 - + K #- 6CEPCL7; ,9 -- - 2 1#
- 6 5 7 B 6 5 7 B
6 5 7 6 7 6 5 7 6 7!
6IC7 6ID7
& 9 6 7 B 6 7 B ? 1- 2 ; ! ,9 + -' 6 5 7 B ?
6IE7
36
O. Brüls, A. Cardona and M. Géradin
( 9 -, 1- - - + ,1; 1 2'- 9 - , 2-
"(
6 7 B 6 7 B
/ /
6 5 7 B 6 7 B
/ /!
6 B A C D7
6IF7 6II7 6IL7 6IO7
% ( - 2 1# - 6 5 7 B 6 5 7 B
A A *5 *5 8 > % > C C A A *5 *5 8 > % > C C
6IQ7 6L?7
+ -#,- ', ' > - > 2 -
- --1 - 1 '- ' 6 #- 6A?77 > B !6 75
> B !6 75 !
6LA7
% ,1 ( - 2 + - #- 6DO7 '
- ,1- 1- 1- - K - " " / / / / B / ! % ! / ! % ! / ! 6LC7 B / / / / / / / ) #- 6FI79 1- 1- 0 1 " / / / / ! % ! / / 67 B / / " / / / / ! % !
!
6LD7
% '- 1,1- - 9 + 2
- , 2 @- - - - 6C??A79 ,- / 5 5 7 B ! 6> % > 8 % ! 6LE7 / / 5 5 7 B ! 6> % > 8 % ! 6LF7 /
Modelling, Simulation and Control…
37
5 6 5 7 6 B + 1 '- ' - ! - # ! 7; ) #- 6FL79 1 -
/ 5 5 7 ! 6> % > 8 % ! 6 7 5 B / 5 5 ! 6> % > 8 % ! 7
6LI7
*(' % -, - 2 -' K, ' 2 * *
6 7 B 6 7 B
6LL7 6LO7
- ,1- 0-, - 1# - 6 7 B / /
6 7 B / ! /
6LQ7
) #- 6FQ79 , 2-, ' 0-,
/ 67 B ! /
6O?7
( % ,1 ( 1- - + K #-: 6DCPDE7; % ' -- 1#, , , -
B
6OA7
+ D A ' - - + D - 2- A9 D A ' + D - 2- C9 - C A ' + C - ; & 0 #2 9 -, - , + ,, 2 ', '-:
38
O. Brüls, A. Cardona and M. Géradin
,1- - ; % K-, #, 6@ 7 B ! 66 7 7 6OC7 6@ 7 B ! 66 7 7 6OD7
66 7H N 8 6 7H N 7 ! !6OE7 6@ 7 B ! 66 7H N 6 7H N 7 # , 2 #19 - #- - 2 # 1, ,- 0
! 6 H N 8 H N 7 6OF7 6@ 7 B ! 6 H N H N 7 & #- 6OC7 - 6OD79 1#, , - - #
- -, - 2 A - C9 +- H N -
H N - - -, - # -, -0 - ; #1 -, -12,9 + - # , 2 #1R ! 6* H N 8 H N 7 ! @ B 6OI7 ! 66* * 7 8 H N H N 7
+ ( '+ ( % #2 1 - #, #- 6OCPOE7; ,9 '-,#- - - 2# 6@ 7 9 0 .67". #, 2 '-,#- ; - # 9 , # 2' - .67 5 67 B .
6OL7
, - -, 1 67; - 9 # #- 6O79 - 11#- ' # - #- 6A?79 + -' 67 B > B !67 5 B > 5 B !
6OO7
Modelling, Simulation and Control…
39
K- 2+ #- 6OL7 - 6OO79 + # .67 B !67 B !67 .
6OQ7
B - 2 # ; % + -1 - 1- 1#,- #- ' ,, + ##, 1#,- .6 7 B ! .
6Q?7
.6 7 B ! B ! .
6QA7
-
B - B A C D; ) #,9 + 2- .@ . .@ . .@ .
B B B
" !
"
" " !
! ! !
!
6QC7 6QD7 6QE7
+ , 0 #- 6OCPOE7; 2.3
++(
& 9 -1 #- 1#, 2 1 -' 2 1#,- - -, ; 3 -' + - K ,1 -: - -,, + - 1- 1,1- #- 1 ; % K ,1 1 1- ,# 2 ,19 ( 1- $ -
M0 2, 2 9 + -,, + 2 1- 1 -2 - , -, 1,0 ; % #, #- 1 M0 2, 1#, 2 : 1 - 2- # K ,1 -12, #; & -
9 , - #- - -, 2 '-,#- 2 #1 -, -12, 1 --, -, 0 - ,1 ,',; % 0 + 2 1 - , 1#,- M0 2, 2 ; %+ S # - 9 -1,9 , - K ,1 # - #,1 #;
40
O. Brüls, A. Cardona and M. Géradin
% - ( , - K ,1 - -9 -2 ,# - - # 2 1 - 1- M0 2, 2 ; ! E;A -
K - - 1 K#- ; % - -, - # -2 ,# -, - # ! E;C - 1 -1 # , 2 #1 ',
! E;D; %+ -, ,1 - R ,-- ,- 2- 6! E;E7 - ,- 2-1 6! E;F7; 3.0
( (('
6. / 1- M0 2, 2 ; # - -, 1 - M0 2, 2 ,,#- : # I; & # 1 K#- 9 1- -, 2 1- -, ' 9 +- # K#- 6- 1 - 1- 79 -1 2 - -, ' *; % K#- 1- - K ,- 2+ - * - M0 2, 2 9 ;; * B -6 7
6QF7
% -,, ,- -, K D D - + - 2- D D *- 2 - 1- 0 K#- 1- HN B
.H*N ! .HN
6QI7
Modelling, Simulation and Control…
41
% - 1- - K 1-, 1- -, ' 1 # K#- - * B ; 1-,, ,-1 2,19 % % K
2 A B 6 8 7 "! 6QL7 C ,- ,-1 2,19 - 1 -, K #, 2 # 9 # - % K 2 B
A " ! C
6QO7
# - -# - - , - + +- - - , -; - #9 ' # - , - 1#,- ,- 2, 1-,, ,-1 2,19 - 1 -,, , - 1#,- 1- - ,- ,-1 2,1; % 1 , - -' - M# 1#,- ,- M0 2, 1#, 2 -1 ; - - 11 - # -- 2 I :
1 ; ' 9 - D D - 9 I A - ' K 2 B HH N 3.1
H N
H N
CH N
CH N
CH N N !
6QQ7
'+ (
9. - K ,1; K ,1 - -9 ' ,#1 M0 2, 2 -; & - K ,1 9 K#- 1- 0 1 - # - -, - * * B - 6 7 B
67* 67
6A??7
42
O. Brüls, A. Cardona and M. Géradin
+ #12 ; % ' -, - ,1 K - B H* ! ! ! * N !
6A?A7
% ,- 2+ - -, - - , - * B - 6 7 B 67 67 6A?C7 + B H " ! ! ! "N 1- 0 - # ,1; 3.2
+ (+ ', +
& ' #- 1 9 - #-: 6A?C7 # 0 ( 9 -, - -, ,- ; % ( 2 1 A A B )*5 * 5 B 5 ) 5 ! 6A?D7 C C % 1- 1- 0 # - 1 - ' 2 ) ! B
6A?E7
% '- - -, 0-, '* B ' , 8 '* , 8 ' B
6A?F7 + , ' 2 9 ' #- , - - 0-, #- ,1; % ' 0-, # ' 2
67 B
,
% ' #-, + ( -, ,- 0 - ' ' B
6A?I7
6A?L7
+ I A ' - 1 - I A ' 1 ; & -,9 ' ,- ' 2 - ,- ' B 6 7'
6A?O7
Modelling, Simulation and Control…
43
+ 6 7 :-,, - 1- 0; "9 #- 6A?L7 2 1 6A?Q7 ' B '
+ ,- -, ,- ! 6 7 B
6AA?7
- , - ,- 1- -,9 - - ,- 2 " ( ,-+ B
6AAA7
+ 9 1- 0 " ( 1- -, T ; % -, ,- #- 6AA?7 -,
- - , - +-; & # ' ', - -, : ,-1 ,- . 6 7 B )
8 ) . + 1- -, S 1- 0 ) K 2 B )
6AAC7
6AAD7
- 1 S 1- 0 ) K 2 1 H) N B
.HN
.H N
HN !
6AAE7
# 1-( - - , - 1#,- ,- + #, 2 2: - # -#1 K 1-, - 2 ; & -9 - 1- 0 -9 1 S 1- 0 '- - -, 1, # 6 7 B ) 6 7
6AAF7
+ ' -, - - K#- + 1 # 1 ; % , - 1#,- 9 +'9 '-, --, M0 2, 1#, 2 1 + ,- - ;
44
O. Brüls, A. Cardona and M. Géradin
;. ,- 2-; 3.3
+ '( ( '(( '( ,( '+
# ,-- 1#,- - ,- 2- ,1 : # O; % ,1 -- 2 + - E A ' -, - B H*
* N
6AAI7
+ * - * - C A ' -, ; 3 1-( - + ' -, - ,1 - + , 1; , # 1 K#: - 9 +- # , 1# 1 -, - B * * !
6AAL7
<. !- # ,- 2-; & K#- 9 2- -, -, - : - -2 - 1- -, 2 -2 - B !
6AAO7
1 9 -1 2 # - # ,1 * B 67* 8 67*
6AAQ7
Modelling, Simulation and Control…
45
+ , - - # 9 ,,#- # Q9 - K 2 67
B A "
6AC?7
67
B " !
6ACA7
% 0 6AAQ7 + - * B 6767 + E C 1- 0 - # ? 67 67 67 B ? 67 ? 1 #- 6A?E79 1- 1- 0 C ) ? B ) B I A ?
6ACC7
? 67
!
6ACD7
,1 1# - ? A ? C ? A 6ACE7 ? C ? A ? C
+ -- - + -#1 1-,,
1- +- # '-,#- -, K#- ; % -, 1- 1# 1 -0 -, 1-: 2 - ' B '2 * B '2 ! 6ACF7
& #- 9 #,- -0 -, B * 1- 2 1#
1 , - # ' ,- B 2
6ACI7
+ # 1 #,# 1- -,; & ,, + 9 0: -, ,- ' # -# - - ; % -# - ,- - , - ,- 1 , 2# + + ,, + - - 2 # - ,- - 2-; & -9 - ,- - 1 -,, , - 1 , + '-, -2 - 1 ; ( + '( % -# - --,: 1-,, ,-1 2,1; !- 1 #- 6QL79 1 2 B H N 1# - 2 B
.H*N A B H N A B A .HN
6ACL7
46
O. Brüls, A. Cardona and M. Géradin
+ '- ' - # 1- 0 + - - - 1- 0 B HA"
? A"
?N!
6ACO7
% -, #- 6ACF7 2 1
' B '
6 7 B ' 6 7
6ACQ7
- ' -, 2- - B 6 7
6AD?7
+ ' B H N ; + # A?6-79 : #, , 1-( - -, / K 1-,; ) #-: 6ACI7 - 6ACL79 + , - 1 , ,- B )6 7
6ADA7
+ S 1- 0 ) B 6 7 - K#-: B H? ? ?N ; # + - -# - - 2$ ' : # 1 2- # - # - - # ; & -9 + -' B H? ? / /N - -# - 2 B / A
6ADC7
S 1 ' # 1 #, ; - 9 , - 1#,- ,- 2- -# - '-, --, M0 2, 1#, 2 1 + ,- - ; '( + '( ), ( -# - 9 - - 2 # ' ,- ,-1 2,1; !- 1 #- 6QO79 1 2 B H N 1# - .* .* A A 2 B A B 6 7 A ! 6ADD7 C . . C ', 19 + A A * * * * 2 B 6" 7 A ! A B C C
6ADE7
Modelling, Simulation and Control…
47
0/. &-, ,- 2-R 6-7 , - 1#,- 9 627 : , - 1#,- ;
'- #- 6ADD7 + 9 - 1- 0 B 6 7 !
6ADF7
% -, ,- - 1# - B 6 7 !
6ADI7
- ' - B H N 9 + 1-( ' ,- - 9 # A?627; % ,- 2+ - , - - - S 1- 0 . .
. .6 7 8 . . B 6 7 8 6 7
B 6 7
6ADL7 6ADO7
+ K 1 1- -, S 1- 0 - 1 1 S 1- 09 + -, #,- -0 -, ; 1 #- 6ADE79 + 2' - 2 B ? - ,- -1, # 1 ; "9 , - 1#,- ,- 2- - -,, + - --, ,- ,-1 2,1 M0 2, 1#, 2 -1 ; 3.4
+ '( ( '( ,(+ '+
1 -,, 0- 1#,- 2 ! 1 6AQOF79 1 2-1 , -, -1; &
48
O. Brüls, A. Cardona and M. Géradin
-,- - - K, -, 2-1 -0 - # 2 1 - 1- -1 + % 1 ( : ; 1- - #,: #,, 1 ,9 % 1 ( 1 , - + 1- - '--R K,9 - 1- - 2 - 9 ,9 -1 ,- 1#,- - # - - -, ; "9 2-1 - M# K ,1 : ,- #9 + -,, + 1, -2, 1#- ;
00. 2-1 ( 1- ; # -#1 - 2-1 -,, - - - 2-1 1- ,- - 1 # ,- 1- 6 : # AA7; % ( 1- 2-1 2 2 #-, -0 * 67 - 2 - 1- -, -1 67 67 67 -- ; 3 -' 67 B 67
6 B A C D7
6ADQ7
+ 67 - - + - : ; "9 , - - 1- -, K#- ' 2 B 8 8 B 8 6AE?7 + B - B 8 ; 1 9 1 ' + * B * 67 8 67! 6AEA7 - 9 - K, 67 - 2 --1 # - - - ' 67; - #,9 K#- 2-1 2 - : 1 -, -,- K, * 67 - - :
1 -, - K, 67; ! 2 K, - - -
Modelling, Simulation and Control…
49
#- 9 : - , #,- #-, -0 - - 1- -,, + ; % 2-1 ,1 # 1 + - -, - - ' B * * ! 6AEC7 % K ,1 - -,- -, - - -, K, ,- * 67 67
B 67* 8 67* B 67 8 67
6AED7 6AEE7
+ , - - # - 2 # 6 # Q7; # - , - ,- - -, --1 #- 6AEE7 - 2 : 2 '-, -# 9 #, 1-# 1- #, '- - # - 1 ; " +'9 2,1 2$ ' - 2 , 1 - # - 1, K1 ,- #; 0-1,9 - #,, -1 2- ,- 2- 1-, - + - , K: #- 9 - - - @- 6AQOO79 2- ,- ' - 2+ - 9 K, - *, 6AQQQ7 - - - 6C??O7; % - -2 - 1- -, 2-1 1#
1 -, - # * B * 8 * 8 6 8 7!
6AEF7
3 # 9 + -,, 1, ' 0 ( - - 2-1; % ( ' 2 A B 6$*5 *5 8 > %>7 6AEI7 C + $ B ) 1- # ,9 % - - >67 -#,- ', 9 + - 5 2 1# 1 67 - 67 # #- 6A?7; # 2' 1 ,- 2+ ( 2-1 ,1 #- 6AEI7 - ( - 2 R - - 2 9 - 2-1 + ,, ' ,' - #-, 1; % - ' 2 ' B 6'A 8 ') 7! 6AEL7
50
O. Brüls, A. Cardona and M. Géradin
% 0 ' ,' ,, + 1 + - -,, '-,#- 1- -, -1 67 67 67 : R D A ' -0 -, - - 9 : R D A ' - 2 1 19 : AR D A ' -0 -, - - 1- 9 : )R D A ' - 2 1- ; % # ' ,- + - A ! 6AEO7 B ) - 1- -,9 I I 1- 0 ,- T
K 2
B -6
7
6AEQ7
+ -0 -, S9 - - - S -, -' -09 -, S9 - - - 2 S; 1 0 9 - - ,- - 2 ' 9 - 0,- @- - - - 6C??A7; % -, # - , ( 1 - # - # -
2 1,1 , 1 - 2,1; 3.6
++(
& 9 , - K ,1 - - +- # 1 ,, ,- 1 ; % 1#,- 2- -2 ,# -, - - - - , 0- 1 K#- 1- ; , - 1-# 1- 9 ;; # - 9 - 1 -, , - -- ,- 1 ,; % - - - , 1# # - - 1- 1- 09 - ,- 1#,- ,1 ' ,' - - -, K, ; & ,- 2-1 1#,- 9 #-, -0 : # - -,- -, K, 9 +- - # - - -, K, ; K, - : ,- , -, 1 -2 ,# -,- - - --1 + ,1; ,- - - 2 1,1 ' -, -; % - -, - ,- , - - #-, 2# ;
Modelling, Simulation and Control…
&
51
-
% K ,1 - - # - 1 -,, , - 1#,- ,- - ' ,' - - ,- #12 -, -; - #,9 #- 1 -( 1 - - 2# ,-
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
# : 1 ,; & -
- -, -9 -, 1 -, - - # #,1 -, -1 2 + - #T ,', -#-; & ,, + 9 , - # # - K ' + ; %9 , - - -, 1#,- #,1 ,, + 2
# 1#- ,- - - ;
52 4.0
O. Brüls, A. Cardona and M. Géradin ( +
# (%$$ %($ ( - -1 ## -: - 2 ( 9 -, 1- -
' #-, + ( -, ' B
A 5 5 C
B
A ) C
' B '
6AF?7
+ 9 A ' -, -9 9 1- 1- 09 )9 S 1- 09 - 9 ' -, ; 1 "-1 , ,9 #- 1 - ' 1 7 8 ) B !
6AFA7
& # -, 1 ,9 + -#1 - ' - 2 0- 1 1 -, - 6 # 7 - B ? 6AFC7 + ? 1- 0 1 :1 ; % '- - - 2 #2- - 2 1 :1 -
-#- # 1 , -2 , 1 : 1 2 -#-, 1 1; & 1#,- # #- 1 9 - - 1- # 0 9 - ' #-: 6AF?7 B
A 5 5 C
B
A ) C
' B '
6AFD7
+ # 1- 1- 09 S 1- 0 - # '-, ' B ? ?
) B ? )?
B ? !
6AFE7
% #- 1 # 1 ,, + - 7 8 ) B !
6AFF7
-, - # 1 9 #, 2 # - - -, - 9 + - -: 12, #,1 + 1 1; %: 9 -, ' -, - K -
B
6AFI7
Modelling, Simulation and Control…
53
+ ' -, - + -- : #; - 2 # - -, - 9 - - 2 # - 1 K - 1 -, - 1- #- 6AFC7 ,
- B ? 8 ?
6AFL7
+ ? 1- 0 - 2 # - 1 - ? 1- 0 -, 1 ; 3 -' ? B H? ? N - B H N 9 ;; -, - 2 # - 0, , -- #
1 ,; 4.1
)+( ( '+
+9 , # - #,1 # $% %($ (; % ' -, - ,1 K - B
6AFO7
+ -2 ,# - - - 9 - -, 1 ; % ' 1 -, * - -, - B *
!!!
*
!
6AFQ7
- -, - -9 ,- - 1#
- - -, -1 - ,, + 1 2 6 # AC7; % -1 2 - -,- ' * - - - 1- 0 9 + --1 # - ' ; % -1 - 1- 2+ -, - - -, -1 - 2 --, ; & - #,-9 ,- 2+ -2 ,# - - - 2 # - 6 B A !!! 7 - , -,
,-1 - , -, - + - -, -1 ' 2 *
B B
* 8 6 8 6 7!
7
6AI?7 6AIA7
8 ( (' (+ ' '-,# -2 ,# - - 2 # - 9 - -, -1
54
O. Brüls, A. Cardona and M. Géradin
01. / 1- - #,1 6 -
, -, -17;
-
K 2 , - 12 - *
B B
/ *
6AIC7
/
6AID7
+ / - -, + T - 1-,, -1, # 1-, - + - , K#- ; -9 - 2 K - ' #, 1# K: #- ; % 1-, - K # - 6 7 B 6 76 7!
6AIE7
% ,- #- + 1- 1 B
6AIF7
+ - 1 - ; - # 1-, - #- 6AID7 1- - 2-# - , - 12 - ,- - + #, 2 ;
Modelling, Simulation and Control… 4.2
55
( (' + ( '(
& - -, -19 -, 1- 1#
- A B ) 6AII7 C + ) # : S 1- 0 #,1 - ' , -, - ! 6AIL7 B ! ! ! 3 -' ' B ' )!
6AIO7
' - - -, ,- 2+ , -, - - -2 ,# - 1 ' B 67' 6AIQ7 -, 2 1 ' B ' )
6AL?7
+ ,- 0 -, B )!
6ALA7
% - - 2 '-,#- 1 #- 6AFOPAIA7 - 6AIL7; % #,, '- 9 + 2 @- - - - 6C??A79 # - -; & #11-9 '-,#- ,- - ' K: #- ' ,' # ' 1#- A; - -, -1 6#- 6AICPAID779 C; , -, - ' 6 #- 6AIL79 , -, ,-1 - - - 0- 1 #- 6AI?PAIA779 D; - -, - 67 6 ' 79 E; -, 6#- 6ALA77; 4.3
( (' + ( (
& ' # 9 + -' + - ,- - 2 1# # S 1- 0 ); (+ 9 - # 2
1# - # 1- 1- 0 , - # : 1 ,; % 1 '- ' -, - 5 -
56
O. Brüls, A. Cardona and M. Géradin
' - -, ', # - *5 *5 > 5 ;; ;; ; ; #B 5 B *5 *5 > 5 5 5
#
!
6ALC7
- 2# *5 - 1- - %)$ ', 0
- -, -1; > -#,- ', 1- -, -1 + 9 K - 0 1- 9 -, - -, -1; # 2' - - ', - : -2, #- ; % ,- 2+ # - 5 - 2 # 1 # B 675 + 2, ( - -, - -
67 B - !6 7 ! ! !
6ALD7
!6 7 " !
6ALE7
% ( '-,#- - -, -1; !-: 1 #- 6A?D7 2# 1 K ,1 ,- ', - -, -1 , B
A A # # B 5 ! 5 C C
6ALF7
"9 ,- ', #,, + : ,- ,-19 + 1 -, -1; 1 #- 6ALF79 : - 1- 1- 0 ' 2 67 B 67 67
6ALI7
- - 2 1# 1 #- 6EF7; - 9 + @- - - - 6C??A7 - 1 1 ' -1; 4.4
++(
& - ,- ,-1 2# 1-,, 1- 9 #,1 # -,, + 1, 1#- ,- - # #12 -, -; % - -, 1#- ,- - - , # # S - 1- 1-: + - 2- 2 - , - --, M0 2, 2 ; ,
Modelling, Simulation and Control…
57
%'
' /
' .
' 0
02. #,- - - , 1;
# #9 # - - - -1 1 9 ,- 1: - 1#,- ##-, 1- ; % - -, : - #,1 -,, + K ,1 -12, + 1- -, 1;
'
& 9 K ,1 1#,- 0 -
-1 --, 1- 1; % --, #, - # '- # , -, 1 R 1- -, ,19 , # 9 -: #- 9 9 ; 3 -' - K ,1 1 -,, + 1 ,, 1,0 M0 2, 1- 1; & 2 1- 19 + - )$! %%( $%% + K ,1 -1+ (; , - , - -:- 9 - - - '-- 1 #,- - -, + - ,-#- ' -1 , - , ; - 2, ( 2, ( --1 1 , - - - -, K ,1 - , #,
1- -, - - #- - 2 2- 2 #1 -, -12,; 6.0
+ '( ( ,' $ '+
,,#- # AD9 2, ( --1 ,-#- -,, + - 1 #,-
- , 1; 2, ( ,1 - 2 - -, K ,19 + - - -1 0
58
O. Brüls, A. Cardona and M. Géradin
1 # '- -2,
9
-, - '- -2, * - ## '- -2,
*5
B B
6 6
* 7
* 7!
6ALL7 6ALO7
% - - ## '- -2, - - + : - -1 1 ,9 +- ' # # 2 , -, - ,-
B 5 8 7 8
8
8 !
6ALQ7
+ ! - ,- 1- ; % #- 1- - - # '- -2, - -, ,-19 - ', 9 - -,- 9 - -, 6-- 1#, , 7 ## - 2, ( , 1; 6.1
"( + '' +( +
& #12 2, (9 , 2-, - - ## ' - K 2 * * B ;;; B ;;; 6AO?7 * - , 2-, - - #- + - *5 B B
6 5 7 * 7 6 5 7 * 7
6AOA7 6AOC7
+ # - 6 5 7 * 7 B
6 6
6 5 7 * 7 B
6 6
;; ;
* 7
6AOD7
* 7
;; ;
* 7
!
6AOE7
* 7
& #- 9 # '- -2, -' 2 0 1 9 9 5 79 - - #- 6ALQ7; 3 1-( - ## #- # 1, + ## '- -2, ;
Modelling, Simulation and Control…
59
% , 1 M# 2 -1 1- -, : 19 - ',9 1- -, 1 M# 2 , ; 3 - -, , -- 0, , #: # ' B 6AOF7 + -#- , -, - 1- 0; % , # 1- -, ' #-, + ( ' B ' - #, -1 #-: -( -, 1 677 8 6 7 5 8 67 8 @ B
6.2
67 8
6AOI7
@67 B *5 B
/ 6AOL7 6 5 7 * 7 6AOO7
6 5 7 * 7! 6AOQ7
B
*(+' = " ' (' ,' ' +
& -, #2, #,#1 0-1,9 , # + - 1 - - -,:&-,: '- ' 6&7 ,-+ G B
( (
6AQ?7
,
G 6( ( 7 (!
6AQA7
B
( '-,# $ -,9 ( $ -,9 -, 9 - , ## # 0 2 1 ; % ' #-, + ( 0 2 1 ' B '( , ; 8 '(# ( 8 '+ + '( % $ -, ( 0, , -'- ,-2, -2 ,# - : 1#,- 2 2 ; % 1#,- $ ,1 -9 +'9 2 , , - - - # ; & 9 $ -, -: K ( (
B
B
6AQC7
B
6AQD7
B
!
# # - -
-, -, - / + #, - - / B (; % #- - 2 ' , + - 6( /7 B ?
6AQE7
60
O. Brüls, A. Cardona and M. Géradin
+ 2 1 / / B ?!
6AQF7
"9 - ' ,1 K - @ 6 7 B / / +
B H
/N !
6AQI7
6AQL7
- , 2-, ' -, - - + AD 1: B H* * /N 6AQO7 +- ' -- 1#, , - Q 1 6 - -
1 7; # , 2 #19 ' - 2 19 @- - - - 6C??A79 ! 6* H N 8 H N H N 7 ! @ B ! 66* * 7 8 H N H N 8 H N 7 H N 6AQQ7
,-,9 H N - 1- - -, # -0 ; + '( " ,' $ ,9 #- 6AQ?PAQA7 - 2 - - #- 6ALLPALO7 + 2, ( # - ##
B H/ /N G
- # , -, - 1- ? !!! A B ? !!! ? B /
B H, N B
B /
? !!! ? ? !!! A
B /!
% -#- , -, - 1- 0 K 2 B ? ! ! ! A
6C??7 6C?A7 6C?C7
6C?D7
Modelling, Simulation and Control…
61
- -1 #- - - + -, :
- / , H N B , 6C?E7 + 1- - -, # -0 # , 2- 1 #;
)
& 9 #1 -, 1 ,' #- 1 1 - # ; % 1 - 1 - 2 ,- K -1 , R -, 1 K: 61#, - #:/#- 1: 79 -, 1 : 61 +1-( -1 ,7 - ##:' 1 6 ' 1 - '- -: -, - 7; & -9 + # -, 1 +1-( -1 ,9 +1-( 6AQFQ79 ""% 1 2 " ,2 -,; 6AQLL7 -
-, :/ 1 2 # - "#,2 6AQQD7; % 1 -' '-, +,,:( + - '-- 1#,- M0 2, 1#, 2 1 2 S S -,:-,2- #-: ; & - #,-9 -, :/ 1 12 - '-- : 1,1- 9 : -#-9 # -, -2 , 6:-2 , 7 - - $#-2, :# #1 -, -1 ; #1 : -, -1 -,, 1 - , 1 - # # :# S -# 2 : 1 - K ,1 1 ,; , -,; 6C??C7 -' # -, :/ 1 , - 19
- - - @- 6AQOQ7 -' # -, 1 : - 2,1 M0 2, 1#, 2 -1 - -' + 1 - #1 -, -1 09 - , - J #, 6C??L7 -' - ' - #,, ' --, -; 0: -, :/ 1 1- 2,1 +- --,
J #, - , '-, 6C??O7 - J #, - , 6C??O7; # 2 -, 1 ,- ; & 1#,- 9 -1 # , 2 #1 2 #- 6AOIPAOQ7 0-, - - 1 ; "9 -, 1 0, - #,- ## #- - 9 #, ( ,- -, -, : / 1 9 - 2 # ' # 1- 1- 0 - -
, 1 ,# , - -,- 2-(; 1 8 A9 #1 -, '- -2, 9 5 9 7 9 9 * 9 *5 - - #, #- 6AOIPAOQ79 +- ' ( -,- :, ( '- -2, K 2 # ,- 6A / 7( 8 / ( B 6A / 77 8 / 7
( B 7 !
6C?F7
62
O. Brüls, A. Cardona and M. Géradin
3 1- - ( - -#0 , - '- -2,9 + #-, # -,- 7; ! 9 #- - 2 - - - + 1 1- -, -1 # , 2 #1 #- 6AOI7; % -1 - '- -2, * - '- ' *5 1 ,- -1 ', 5 - '- ' 7; % 9 -#0 , - '- -2, & K 2 6A ' 7& 8 ' & B 6A ' 7*5 8 ' *5
& B *5 !
6C?I7
% -, :/ 1 2- # ( +1-( - 1#,- 5
B B
8 5 8 6?!F 17( 8 1( 5 8 6A +7( 8 +(
6C?L7 6C?O7
- & - - 1#,* B * 8 6A (7& 8 (& !
6C?Q7
!' -, 1 --1 / 9 / 9 19 +9 ' 9 ' - ( #, # 2 , 2- # -2, -#- - #1 -, -2 , : ; K / 9 / 9 1 - + - ,- -, -, :/ 1 - 1 ' B / 9 ' B / - + B (; " +: '9 + + ,, + - S - 1#,- - '- -2, - # - ' - 1-, 1 1 2+ -#- - -: 2 , ; % 9 -,, + ' B / 9 ' B / - ( B +9 - -,, -, 1 --1 - '- -2,; -, 1 ,' #- 6C?FPC?Q7 + -1 # , 2 #1 - - 1 2 # AE; & + - 9 ' ,' --1 1 B
A / / 7
16A
+ B
+ 1
( B
A ' (6A ' 7
+ - .7 B "1 .
. 5 B "+ .
% - 1- 0 ' 2 61 8 @ + 8 ) 7 @ / B 6 1 + 7 6 1 + 7
. *5 B "( ! .* / / / 6"( 7 6" 7
Modelling, Simulation and Control…
63
#
# # " / / / # ! 1 1 # + + # #
# " ' ' ' # ( ( # #
! !" # $ $ # $ $ # $ # + $ # 1 $ # $ # $ # ( $ # $ # / " /
# ' " ' 03. -, :/ 1 - 1 1- 1;
64
O. Brüls, A. Cardona and M. Géradin
+ - S 1- 0 ) B .67 8 8 8@ 7".; # 2 M # -, 1 --1; - 1- -, 1 2 : 9 -, : / -, 1 : -#- ' - 6 # - "#,2 6AQQD77 + B ?!F 8 / / ! 6CA?7 (+ 9 - , 1 2 K: 9 : 2 1 6*- -,; 6C???77 ( B ?!F 8 ' ' !
6CAA7
S 2,19 ,# #, 2 1# -#-, , , +:# -9 +- :# ,# #, - 2 -1 # 2 -, 1; % :# #1 -, -1 2 -, - # -, 1 - K ) R - # -1 1 -- 2 ) B A9 +- ) B ? 1- -1 - ,- :# ; & - # #, 1- -, -:- 19 1-, --1 - 2 K 1 - # 1- --, ; - #:
K '-,# ) H? AN9 # - "#,2 6AQQD7 -' : 1-, -, 1 --1 : 1- -, 1 / B
C) A ) 8 A
/ B
) ) 8 A
1 B ?!CF 6+ 8 ?!F7 !
6CAC7
K: ,,- -:- 1 ,9 *- -,; 6C???7 2-
A D) A ) ' B ! 6CAD7 ' B C ) 8 A ) 8 A % -, 1 K 2 #- 6CA?7 - 6CAC7 2 #- 6CAA7 - 6CAD7 - -2, + ? 6 :-2 , 7 - + 6-: 2 , 7; # 1- --, -, 1 #,, #,
1- 2,1 2 J #, - , '-, 6C??O7;
+ ;.0
( ( ((
% ,- - - 1 - ,- 6, 1- - ,- : -7 - , -, 1 1 ' , - ,-; & - # 1- # R -(: S 6 # ,- ,, 79 ,- 6 ,# 1- #+-79 -0 - 2-( 9 - #
Modelling, Simulation and Control…
65
-1 , - # - -; & 1# 2 --2,9 - - : ,, 2 - +#, -#, 1> ,- 1, - - - -1> -' - #,- - , - 1- -, : > M# ##-, M0 2 , 0 # ,- -
-(: S -9 ; % 1- + - - - 2 -( - # # #,1;
04. ,- - - ,- ' , - ,- - # +; # AF ,- - + -1 2-' # ,- - - ,- ' , - ,- # ,- 9 11 -, 2 - - 1-; & 1- , 0 2 1- - -2 2 , - #,- ## 6+ ( - 1, # :#-1 $ 7; ;.1
+-(# #'
% -, - - - # + - 1 :- ' #: ; % 21-( - 2 --, -1+ ( , - &:) ' - , '- - !1 6!: &F
"&$%'( ($$####$ $ $
66
O. Brüls, A. Cardona and M. Géradin
06. # I : -: , 1;
09. - -, 1 , -;
!
0;. !1 :- ' -1;
0<. !1 :- ' , -;
-#- 9 - - 1 , ,,; - , -#- - ,, 2 -#+ -,; 6C??E7; % :2 1 , - ,,#- # AL9 - : ,# -:2 9 # 1- 19 , :-( 1- 1 +,9 - - 1 , +,< # : - 6,--, 9 ' -, - -+ #7; % 1- -, 1 , ' ,' -2 # I?? : : 19 - #, 2 ,- 0 : ,# S # 2# 9 M0 2 , - 9 - - , # -, 1 , +,;
Modelling, Simulation and Control…
67
1/. - - 1-#' 6- - #-, K- 7;
# AO ,,#- , - 1 :- ' -#- ; % -#- 1 , -'- ,-2, , - -:- 1-R *5 B 6 B 6
* 7 * 7
+ #9 - - ##R
B H
" N
* B H
N
B H N!
-#- 0 > 9 , -, # '-,'> -
9 # 2 # - 1 -12> - 9 0 2 -1; % # - - ' 2 1-#-# (:-2 2 - :# ; % , ,-+9 -'- ,-2, - - 2, ( --1 1 ,9 - 6 # AQ7R - 2-( , - - 6 ' -#- 1 ,79 - 1- -#- - -,- 1-#1 1 -, - 6-'9 ,,9 - 79 - - , - -, ,; & K#9 ' ( -,- 1-# - # -:2 > ( 9 1 -, -,- > 9 1 -, > 9 -1 > '"9 0 - -1> - 9 , -, #; ,- - 1-#' - 2 1#,- 6 # C?7; % - - A? 1< -, ', 9 - - ' -, - :, 11- 9 + # - -,: 1 62, ' -#1 7; % 1 -
2-( # - # +,; % 1: 1#,- ?;?A - -, 1 --1 - K 1
-, - # ) B ?!Q; % 1#,- +- - # + - -2, 1#- -, 1 6- + 1 # - ( 1#7; # CA ,,#- -, -$ -; # ,-: -, , +, - :, -# ' 11- 9 - ,--, 2' -$ ; # CC - ,, -, -:2 ; % -1 2-' 1 :- ' ( -2 2 --, # CD - CE; !-#- 1-
68
O. Brüls, A. Cardona and M. Géradin 0.5
0.06 roll pitch
0 0.04
−0.5 −1
angle (rad)
0.02
y (m)
−1.5 −2 −2.5
0
−0.02
−3 −3.5
−0.04
−4 −4.5 0
10
20
30
40
−0.06 0
50
1
2
3
4
5
6
time (s)
x (m)
11. , -: 2 ;
10. " -, -$ -:2 ;
5
14
1.6
12 10
1.2
Pressure (Pa)
Electrical current (A)
1.4
1 0.8 0.6
8 6 4 2
0.4
0
0.2 0 0
x 10
Rear right Front right 1
2
3
4
5
time (s)
12. , -, # '-,';
Rebound chamber Compression chamber
−2 6
0
1
2
3
4
5
6
time (s)
13. " -#, # - -1;
1 - '- - , -, #; % # 1: -12 ' , + 0 -9 2# , # 2 # -12 1 -; ,, #, - -,, - - 2 - # - : - 1 ' , ,-+; ;.2
' ( *+(' :*,' +( '(
% 0-1, 1 - ' 2- , - 0 1-, 2 -1; % , :- 1- #,- -,9 + : # CF9 - 2 ', - - & # % , 9 3 ,:
Modelling, Simulation and Control…
69
14. -,; ( - ( - -#- : : 19 +- - 1-# -,- ;
6AQOI7> "# 6AQOO7; & - - -, - + - 9 -
S # - ' -,:+ , -, - ; " +'9 M0 2, : , ( -S -#-; -, - + ( 1-
: : 1 - ' -, ,-9 - -#- 2 + -#, , ; %+ , - - #9 K0 -#, , 9 1-# , 0 ; '9 ' 2- 1- 19 + -, 1 - ,- - 9 -, ; + # CI9 , ,-+ 2- - 1 + : 1: -, -9 + ' ,' 6 7 - , + ,, - ,,9 2, -$ -( 9 6 7 - - ,,9 + , 2 ,, -
1-#1 - -,- 1-#19 - ' -1 M0 2, 1 ; % ' 2- ,, 0, - -,: 1 1 , 1- 19 + 1#,- 0,- J #, -,; 6C??L7; % 1#,- 1 , 1- 1 ,# - 1 , M0 2, 1- 1 6, ( A9 , ( C - -#- , ( - 1 ,
- M0 2, 2-179 - , - -1 1 , -#- 2- 2 0 1-, K- 9 - - 2, ( --1 1 , , ,-+ 6# '-, 1 , 1,1 -,: 1 ,,7;
70
O. Brüls, A. Cardona and M. Géradin
)
$# $
)
%' )
% $
16. %+ : 1:-, ,R # ' ,- -, -#, -#- ;
! 1#,- #,
15
15
10
10
5
5
a (m/s2)
0
0
Y
aY (m/s2)
)
−5
−5
−10
−10
−15 0
0.5
1
1.5
2
−15 0
2.5
0.5
1
t (s)
1.5
2
2.5
2
2.5
t (s)
0 1-, #,
15
15
10
10
5
5
a (m/s2)
0
0
Y
aY (m/s2)
)
−5
−5
−10
−10
−15 0
0.5
1
1.5
t (s)
2
2.5
−15 0
0.5
1
1.5
t (s)
19. % - #2-;
Modelling, Simulation and Control…
71
% - ' -, --, # CL; % 1: 1#,- ?;?A 9 - -, 1 --1 - 2- -, - # ) B ?!L; 1 #,9 + ,# - 1#,- , -#-, -1 2-' # -#-, 1- 1;
,
& -9 - - - K ' + -1 --, M0 2, 1#, 2 19 ;; -, -19 - -, -1 - M - -1 - -; - 1 - + 1 - -+2-(> + 2, ' - - # -2, 1#,- ' 1 #, -,, + - 12 - 1 ; -, 1 ,, -1+ ( 2- K ,1 2 - ,; 3 -' + - -2 ,# -, - -,, + - 1- -12, '- # ,1R 2 9 ( : 1- $ 9 2-9 2-19 #,19 ; & -, 2, - # 1 - , 1 # - 2, ( --1 1#,- ; % K ,1 1 , 1- ' ,' - - ,- -, :
-; " +'9 #, #- 1 - ##-,, - -
- 2 ,' T , # - - ,'; % #,1 # -, -,, + 1 ' T 2 # 1 ,; % #9 + 2- - - -, 1#,- - # : 1 ,9 -, -2, ##-, 1 + ,- ,-1 2# 1-,, 1- ; % , - #- 1 - ' # #1 -, K: ,1 -12, #; % #, 2+ 1 9 1-: - ,, -1 - # - #,, +-; 1 , -, :/ 1 - 1 2 , #, 1#,- M0 2, 1#, 2 1 - 1- 1; -,,9 1#, 2 - 1- 1 - --, R - ,- : -9 - 1 :- ' - # - - M0 2, 1- #,- ; % 0-1, 1 - -2 , 1 -, + :
# -, 2,1;
, ' J #, # 2 , - - -, # ! K : - 6 !7 + -#,, -( +, ; % -# -( ; *; ; , '-, - ; ; # 0 1 ) ' =
72
O. Brüls, A. Cardona and M. Géradin
# -- + (; % - # ; ; 6) ' @ - , # #'- 9 , #17 - ; *; !+' 6/- , ( ) ' #'9 , #17 -' - - 1 : - ' - # 21-(> - -#,, -( +, ; 3 -, -( ; 3;*; ( - -1 6 - & # % , 9 );!;7 1-( -, # -'- ,-2, - 2# - 1,1- ' 2- ,,;
; , - ; J #,; ' -, :/ 1 : - 1- -, 1; $) ( %(9 AO6C7RAOFP C?C9 C??L; ; - - ; !#; 1 # ,-:-, : -1 -, 1 ' - - : , 1 ; "%$ 0
(9 CI6F7RAIQCPAL?Q9 C??F; ; -#-# - ; %- ,, ; % ' -, --1 - - ; $ % %(9 DCRLAPQC9 C??D; %; ,( - *; " ; :, - - K ,1 --, + ' : -; %%$ "%$ ( %$ 9 LRCFFPCLA9 AQLD; ; - ; ! 1-; - - 2 : -1 ; ( $ % % 9 AQAR EILPEOO9 C??A; ; J #, - ; , ; % -, :/ 1 - - , - 1#, - R % +- - -, 1- 1#,- ; "%$
(%%$ % $ % %(9 AD -9 - 9 C??O; ; J #, - *;: ; , '-,; #1 -, -1 1 -: #, 1- 1; ( $ % % 9 AQL6I:O7RFLLPFOO9 C??O; ; J #,9 ; # 09 - *;: ; , '-,; % , 2-, 1 -, --1 -: , - 1 ,: # M0 2, 1#, 2 -1 ; %%$ "%$ ( %$ 9 IQ6F7RQEOP QLL9 C??L; ; - - - ; @- ; 2-1 K ,1 :, - + K - ; %%$ "%$ ( %$ + 9 CIRCE?DPCEDO9 AQOO; ; - - - ; @- ; % 1 - #- 1 1- 1 --, ; ( % 9 DDRO?APOC?9 AQOQ; ; - - - ; @- ; #,1 1#,- 1- 1 --, ; %%$ "%$ ( %$ 9 DC6O7RAFIFPAFQE9 AQQA;
Modelling, Simulation and Control…
73
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
; -#+9 *; !+'9 - ; !-; , - 1 :- ' # - - -; & , .--/9 #'9 , #19 !12 C??E;
74
O. Brüls, A. Cardona and M. Géradin
;.; - ; - -; , - 2-1 ,1 1#,- -1+ ( - ' 1 - 1 : - 1#, 2 1 -1 ; ( % 9 OIR ELPID9 C??O; ; - -,; 2 1 1 1 ; ( % 9 ARFOAPI?A9 AQLA; ; ' - ;:/; /+-(; -, : 2- #2## M0 2, 1#, 2 -1 ; "%$9 CORAL?QPALAQ9 AQQA; ;; - !; ! '--; -,- - 1 , # : 1 , - 1; %% (% $9 EA 6AA7RAIDCPAIEE9 AQQI; ; ; ; -, 1 --, , - 1R ,,-2 , 9 2'-2 , - 1 , # ; %% (%
$9 CIRALPDC9 AQOA; ; +1-(; 1 1#- ##-, -1 ; "%$ % 9 OFRILPQE9 AQFQ; !; #2 ; &1 ' 1 1 - ##-, -1 --, ; "%$9 AD6O7RQQFPA??I9 AQLF; ;; !-2-- - ;; 3-; - # #
-1 --, - -, #2## + ,- -#,- - ; "%$ %$ %9 AARE?APEDA9 AQOD; *; ; ! 1 ; K - 2-1 1#,- ; % : 1 -, : -1 2,1; - &; ( $ % % 9 EQRFFPL?9 AQOF; *; ; ! 1 - ; .#:# ; -1 - # ,- 1 : - 1 -,, 0- - -; ( $ % % 9 IIRACFPAIA9 AQOO; %;*; !# - ;*; - ; , # - , M0 2, ## # /, ' ' ; "%$ % * $* % %(+ 9 AE6C7RCI?PCIL9 AQQA; ; 3-,,- - ; !+-(; - 1 S 1#, 2 1 1#,- ; %%$ "%$ ( %$ 9 DC6O7RAODD:AOF?9 AQQA; %;; 3- - ;/; ; 1#- -, - M0 2, 1#, 2 1; $ % #9 FI6I7RFFDPIAD9 C??D; %;; 3 , ; % - # M0 2, 1- #,- ; -5 9 - & # % , 9 AQOI;
! -HBG@DK 6@K@RDJ $ % !$ % $ " ! $ ! " $ ! $ $ # !
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
76
M. Valášek
4GDQDENQD SGHR BNMSQHATSHNM BNUDQR SGQDD L@HM SNOHBR 4GD QRS SNOHB HR SGD DBHDMS CDUDKNOLDMS NE RHLTK@SHNM LNCDKR NE LDBG@SQNMHB@K RXRSDLR 4GD RDBNMC SNOHB HR SGD LNCDK A@RDC BNMSQNK CDRHFM NE LDBG@SQNMHB@K RXRSDLR 4GD SGHQC SNOHB HR @ CDLNMRSQ@SHNM NE @OOKHB@SHNM NE SGD CDRBQHADC @OOQN@BGDR SNV@QCR SGD CDUDKNOLDMS NE HMSDFQ@SDC BG@RRHR BNMSQNK @MC SGD BNMSQNK NE BNLATRSHNM DMFHMDR NE UDGHBKDR
!
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
77
Modelling, Simulation and Control…
64B?2 -DBG@SQNMHBR @R RXMDQFHRSHB BNLAHM@SHNM NE SDBGMNKNFHDR
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
78
M. Valášek
BKNRD SGD LDBG@SQNMHB RXRSDL RSQTBSTQD 4GDQDENQD SGD A@RHR NE SGD NQHFHM@ SHNM NE SGD RL@QS RSQTBSTQDR HMSDKKHFDMS L@SDQH@KR LDBG@SQNMHB BNLONMDMSR LDBG@SQNMHB @OOKHB@SHNMR @QSDE@BSR VHSG HMSDKKHFDMS ADG@UHNTQ FDMDQ@KKX HR SGD HMUDMSHUD LNCHB@SHNM NE SGD DWHRSHMF NAIDBS OGXRHB@K RXRSDL RN SG@S SGD LDBG@SQNMHB RSQTBSTQD HR ONRRHAKD SN AD BQD@SDC 4GTR SGD LDBG@SQNM HBR G@R ADBNLD @ MDV RNTQBD NE HMUDMSHNMR @MC CDRHFM ! K@QFD @LNTMS NE SGD RNKTSHNMR MDV O@SDMSR MDV L@BGHMD @MC BNMRSQTBSHNMR HMMNU@SHNMR HR A@RDC NM SGD BQD@SHNM NE @M HMUDMSHUD LNCHB@SHNM NE SGD DWHRSHMF OGXRHB@K RXRSDL SN L@JD SGD HMSDKKHFDMS ADG@UHNTQ ONRRHAKD 4GHR HR SGD QD@RNM ENQ
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
Modelling, Simulation and Control…
79
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
! !
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
80
M. Valášek
SDL DWHRS +@QMNOO DS @K #DKKHDQ 6DHSK DS @K 6@K@RDJ DS @K 6@K@RDJ !QMNKC DS @K 4VN CHDQDMS BK@RRDR NE @O OQN@BGDR B@M AD CHRSHMFTHRGDC &HFTQD 6@K@RDJ DS @K 4GD QRS NMD HR SGD BNRHLTK@SHNM @MC SGD RDBNMC NMD B@M AD B@KKDC TMHENQL LNCDKHMF 4GD BNRHLTK@SHNM HR A@RDC NM HMSDQE@BHMF SGD QDRTKSDC LNCDKR HM CHDQDMS OGXRHB@K CNL@HMR @MC SGD TMHENQL LNCDKHMF HR A@RDC NM SGD LNCDKHMF NE SGD VGNKD LTKSHCHRBHOKHM@QX RXRSDL VHSGHM NMD TMHENQL K@MFT@FD
!
64B?2 !OOQN@BGDR SN -TKSHCHRBHOKHM@QX -NCDKHMF
<@6:B9.A6<;
4GD QRS @OOQN@BG HR SGD TR@FD NE SNNKR ENQ RODBH@KHYDC BNLONMDMS LNC DKKHMF HM NMD O@QSHBTK@Q OGXRHB@K CNL@HMR @MC SGDM HMSDQE@BHMF SGD QDRTKSHMF LNCDKR 6DHSK DS @K 4GD NTSOTS U@QH@AKDR EQNL NMD LNCDK @QD SGD HMOTS U@QH@AKDR NE @MNSGDQ LNCDK @MC UHBD UDQR@ 4GHR DWBG@MFD NE U@QH@AKDR HLHS@SDR SGD QD@K HMSDQ@BSHNM ADSVDDM SVN LNCDKR NE RTARXRSDLR &HFTQD
64B?2 0QHMBHOKD NE BNRHLTK@SHNM ,DS TR CDRBQHAD @ RHLOKD DW@LOKD NE CXM@LHB HMSDQ@BSHNM NE SVN L@RRDR
Modelling, Simulation and Control…
81
EQNL &HFTQD 6@K@RDJ DS @K @ 4GD HMSDQ@BSHNM NE SVN L@RRDR @MC AX SGD RSHMDRR @MC C@LOHMF DKDLDMSR HR CDRBQHADC AX RHLOKD RHLTKS@MDNTR DPT@SHNMR \ + \ +
+ + +> +> + + +> +> + +>
4GDRD DPT@SHNMR B@M AD ROKHS HMSN SVN FQNTOR M@STQ@KKX BNMBDMSQ@SDC @QNTMC SGD L@RRDR @MC \ + + + +> +> \ + +> + 4GD LNCDK @QNTMC SGD L@RR QDBDHUDR SGD U@KTDR NE U@QH@AKDR + +> EQNL SGD LNCDK @QNTMC SGD L@RR BNLOTSDR SGD ENQBD @MC SGD QHFGSG@MC RHCD NE SGD DPT@SHNM NE LNSHNM NE L@RR @MC SQ@MRLHSR SGD U@KTD NE SGD ENQBD SN SGD LNCDK @QNTMC SGD L@RR 4GD LNCDK @QNTMC SGD L@RR QDBDHUDR SGD U@KTD NE SGD ENQBD BNLOTSDR SGD QHFGSG@MC RHCD NE SGD DPT@SHNM NE LNSHNM NE L@RR @MC SQ@MRLHSR SGD U@KTDR NE U@QH@AKDR + +> SN SGD LNCDK @QNTMC SGD L@RR 4GD DPT@SHNMR NE LNSHNM B@M AD HMSDFQ@SDC AX NMD BNLLNM HMSDFQ@SNQ RNKUDQ NE CHDQDMSH@K DPT@SHNMR NQ RDO@Q@SDKX AX SVN HMSDFQ@SNQR @RRNBH@SDC VHSG SGD LNCDK @QNTMC SGD L@RR @MC @QNTMC SGD L@RR 4GHR DW@LOKD V@R TRDC @R @ SDRSHMF ADMBGL@QJ NE BN RHLTK@SHNM DMUHQNMLDMS ADSVDDM -@SK@A3HLTKHMJ @MC HMGNTRD CDUDKNODC &%! SNNK HM #BNCD &HFTQD 4GHR DW@LOKD NODMR @ KNS NE ONRRHAHKHSHDR GNV SN @QQ@MFD SGD BNRHLTK@SHNM
64B?2 3HLOKD DW@LOKD NE BNRHLTK@SHNM NE SVN LNCDKR 4GHR HR @ UDQX OQNLHRHMF @OOQN@BG ENQ LTKSHOGXRHB@K LTKSHCHRBHOKHM@QX LNCDKHMF @MC RHLTK@SHNM ADB@TRD SGD LDSGNCNKNFHDR @MC BNQQDRONMCHMF RNES
82
M. Valášek
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
Modelling, Simulation and Control…
83
64B?2 )MSDQE@BHMF#NTOKHMF NE #!% 4NNKR
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
## ## # ## #
# %# !# !## ## # $ # %# # "#$ # !#
!#
64B?2 6@QH@MSR NE BNRHLTK@SHNM HLOKDLDMS@SHNM
84
M. Valášek
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
64B?2 0QHMBHO@K BNRHLTK@SHNM RBGDLDR )M SGD CHQDBS RBGDLD ANSG RTARXRSDLR DWBG@MFD SGD HMENQL@SHNM @ANTS SGDHQ RS@SDR ITRS NMBD @MC SGDM SGDX @QD HMCDODMCDMSKX RHLTKS@MDNTRKX HMSD FQ@SDC TMSHK SGD MDWS L@BQNRSDO )M SGD @KSDQM@SDC RBGDLD ANSG RTARXRSDLR DWBG@MFD SGD HMENQL@SHNM @ANTS SGDHQ RS@SDR @MC SGD RTARXRSDLR @QD HMSD FQ@SDC BNMRDBTSHUDKX /MD RTARXRSDL HR HMSDFQ@SDC QRS VHSGHM SGD L@BQN RSDO A@RDC ITRS NM SGD HMENQL@SHNM @ANTS SGD RS@SD NE SGD NSGDQ RTARXRSDL HM SGD HMHSH@K SHLD HMRS@MBD NE SGD L@BQN RSDO @MC ONRRHAKD DWSQ@ONK@SHNM NE OQDUHNTR SHLD ADG@UHNTQ @MC SGDM SGD NSGDQ RTARXRSDL HR HMSDFQ@SDC TRHMF SGD VGNKD HMENQL@SHNM NE SGD HMSDFQ@SHNM NE SGD QRS NMD )M SGD HSDQ@SDC RBGDLD ANSG RTARXRSDLR DWBG@MFD SGD HMENQL@SHNM @ANTS SGDHQ RS@SDR HM SGD HMHSH@K SHLD HMRS@MBD NE SGD L@BQN RSDO SGDM SGDX @QD HMCDODMCDMSKX RHLTK S@MDNTRKX HMSDFQ@SDC TMSHK SGD MDWS L@BQNRSDO TRHMF ONRRHAKD DWSQ@ONK@SHNM NE OQDUHNTR SHLD ADG@UHNTQ @MC M@KKX SGD QDRTKSR @QD LTST@KKX HSDQ@SDC TMSHK QDPTHQDC DPT@KHSX NE HMSDFQ@SDC @MC DWSQ@ONK@SDC QDRTKSR @QD @BGHDUDC 4GD LNRS NESDM TRDC RBGDLD HR SGD QRS NMD SGD CHQDBS BNRHLTK@SHNM 6@K@RDJ DS @K @ &NQ SGD LDBG@SQNMHB@K RXRSDLR VGDQD NMD RTARXRSDL HR SGD OGXRHB@K OK@MS @MC SGD NSGDQ RTARXRSDL HR SGD BNMSQNK RXRSDL HS HR SGD M@STQ@K RHLTK@SHNM NE SGD BNMSQNK NE LDBG@SQNMHB@K RXRSDL 4GD L@BQNRSDO HM RTBG BNRHLTK@SHNM OQNBDRR BNQQDRONMCR SN SGD R@LOKHMF ODQHNC HM SGD QD@K RXRSDL 4GD RS@AHKHSX @BBTQ@BX @MC DBHDMBX NE SGD BNRHLTK@SHNM NE RTBG RXRSDL HR DPT@K SN SGD RS@AHKHSX @BBTQ@BX @MC DBHDMBX NE SGD BNMSQNK NE SGD
Modelling, Simulation and Control…
85
64B?2 4GQDD RBGDLDR NE BNRHLTK@SHNM HMSDFQ@SHNM OQNBDRR
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
86
M. Valášek
CDW ENQ LTKSHANCX RXRSDLR &HM@KKX SGD VD@J BNRHLTK@SHNM HMSDQE@BD VHSG RHLTKS@MDNTR HMSDFQ@SHNM HM ANSG RHLTK@SHNM O@BJ@FDR HR @U@HK@AKD
64B?2 %W@LOKD NE U@QHDSX NE BNRHLTK@SHNM HMSDQE@BDR 4GD @CU@MS@FD NE BNRHLTK@SHNM HR SG@S HS NDQR K@QFD RTOONQS NE KHAQ@QX DKDLDMSR CDUDKNODC ENQ O@QSHBTK@Q OGXRHB@K CNL@HMR @R VDKK @R HMSTHSHUD LNC DKKHMF @MC LDSGNCNKNFX ENQ SGD FHUDM RTAOQNAKDL VGDQD SGD CDRHFMDQ BQD@SDR SGD LNCDKR VHSGHM SGD O@QSHBTK@Q CNL@HM K@MFT@FD 4GD CHR@CU@MS@FDR @QD BNMMDBSDC VHSG SGD OQNAKDLR NE RS@AHKHSX @BBTQ@BX @MC DBHDMBX NE SGD BN RHLTK@SHNM
';63
4GD RDBNMC @OOQN@BG SQHDR SN CDRBQHAD SGD BNLOKDSD LTKSHCHRBHOKHM@QX RXRSDL AX TMHENQL K@MFT@FD 4Q@CHSHNM@KKX SGD L@SGDL@SHB@K LNCDKR @MC BNQQDRONMCHMF RHLTK@SHNM LNCDKR @QD ADHMF CDUDKNODC ENQ RXRSDLR EQNL NMD OGXRHB@K CNL@HM 4GD A@RHR NE TMHENQL LNCDKHMF HR SG@S SGD LNCDKR ENQ @KK O@QSHBTK@Q RTARXRSDLR @QD LNCDKDC AX SGD R@LD CDRBQHOSHNM HM SGD R@LD LNCDKHMF K@MFT@FD 4GD RTALNCDKR @QD CDRBQHADC AX SGD R@LD BNMBDOSR TR HMF SGD R@LD CDRBQHOSHNM 3TBG LNCDKR B@M AD M@STQ@KKX HMSDQBNMMDBSDC !R SGD TMHENQL K@MFT@FD SGD ENTQ CHDQDMS ENQL@KHRLR B@M AD TRDC DPT@SHNMR
Modelling, Simulation and Control…
87
CHDQDMSH@K @MC @KFDAQ@HB CXM@LHB AKNBJR LTKSHONKDR @MC ANMC FQ@OGR 4GD ETMC@LDMS@K TMHENQL LNCDKHMF HR SGD K@MFT@FD NE L@SGDL@SHBR !KK LNCDKR @QD CDRBQHADC AX SGD CHDQDMSH@K @MC @KFDAQ@HB DPT@SHNMR @MC SGD HM SDQBNMMDBSHNM NE DPT@SHNMR HR @KV@XR SGD M@K V@X NE BNMMDBSHMF SGD RTALNC DKR 4GD OQNBDCTQD HR QDK@SHUDKX RHLOKD 4GD DPT@SHNMR ENQ @KK RTALNCDKR @QD VQHSSDM SGD QDK@SHNMRGHOR ADSVDDM SGD HMOTSNTSOTS U@QH@AKDR HM SGDRD RTA LNCDKR @QD @CCDC @MC SGD QDRTKSHMF RDS NE DPT@SHNMR HR OQNBDRRDC HMSN @ ENQL RTHS@AKD ENQ MTLDQHB@K RNKTSHNM KHJD RS@SD RO@BD CDRBQHOSHNM 4GHR @OOQN@BG HR RTOONQSDC ENQ DW@LOKD AX SGD RHLTK@SHNM RNESV@QD !#3, !#3, VDA "DB@TRD SGD DPT@SHNMR @QD UDQX M@STQ@K BNLLNM K@MFT@FD ENQ @KK L@SGDL@S HB@K @MC SGTR RHLTK@SHNM LNCDKR QDBDMSKX SGD DPT@SHNMR G@UD ADBNLD SGD A@RHR NE SGD ONVDQETKK LNCDKKHMF K@MFT@FD VHSGHM 6($,!-3 K@MFT@FD @MC RS@MC@QC 6($,!-3 VDA 4GD CHR@CU@MS@FD NE SGHR @OOQN@BG HR SG@S DHSGDQ @ RXLANKHB DPT@SHNM FDMDQ@SNQ A@RDC NM @MNSGDQ TMHENQL LNCDKKHMF @OOQN@BG NQ SGD RODBHB RXLANKHB DPT@SHNM FDMDQ@SNQR ENQ D@BG OGXRHB@K CN L@HM VHSG OQDO@QDC LTKSHCNL@HM HMSDQE@BDR @QD MDDCDC "TS SGD FDMDQ@K OQNAKDL HR SG@S SGHR @OOQN@BG HR SDCHNTR LNQD NODM SN DQQNQR 4GD RHLHK@QHSX NE OGXRHB@K OQHMBHOKDR K@VR @MC QDRTKSHMF DPT@SHNMR @QD VDKK JMNVM @MC CDRBQHADC RDD DF +@QMNOO DS @K @MC G@UD KD@C SN SGD CDUDKNOLDMS NE GHFGDQ BNMBDOSR @MC LNCDK @RRDLAKX OQNBDCTQDR SG@S DM@AKD SN @RRDLAKD SGD LNCDK EQNL K@QFDQ O@QSR 4GD QRS FQNTO NE RTBG LNCDKR NE K@QFDQ O@QSR BQD@SDR AKNBJR NE CXM@LHB AKNBJ CH@FQ@LR ! AKNBJ QDOQDRDMSR DHSGDQ @ KHMD@Q CXM@LHB RXRSDL CDRBQHADC AX KHMD@Q NQCHM@QX CHE EDQDMSH@K DPT@SHNM NQ @M @KFDAQ@HB RS@SHB ETMBSHNM SG@S B@M AD KHMD@Q KHJD F@HM NQ MNMKHMD@Q KHJD RPT@QD QNNS "DB@TRD @MX RS@SD RO@BD CDRBQHOSHNM B@M AD SQD@SDC @R SHLD HMSDFQ@SHNM NE QHFGSG@MC RHCD BNMRHRSHMF NE MNMKHM D@Q @KFDAQ@HB RS@SHB ETMBSHNM SGDM @MX CXM@LHB RXRSDL B@M AD CDRBQHADC HM CXM@LHB AKNBJ CH@FQ@LR 4GD HLONQS@MS E@BS HR SG@S AKNBJR QDOQDRDMS B@TR@K HMOTSNTSOTS QDK@SHNMRGHO 4GDQDENQD SGD AKNBJ CH@FQ@L ENQ @ BDQS@HM RXRSDL B@M AD BQD@SDC NMKX @ESDQ BNLOKDSD B@TR@K @M@KXRHR NE SGD LNCDK NE SGD HMUDRSHF@SDC RXRSDL 4GD CXM@LHB AKNBJR @R SGD TMHENQL K@MFT@FD G@UD ADDM TRDC RHMBD SGD QRS RHLTK@SHNM K@MFT@FDR 4GDHQ CHR@CU@MS@FD HR SG@S QRS SGD RXLANKHB DPT@SHNMR NE SGD LTKSHCHRBHOKHM@QX LNCDK LTRS AD FDMDQ @SDC @MC RDBNMC SGDRD DPT@SHNMR LTRS AD RNQSDC @MC LNCHDC HMSN @ B@TR@K RDPTDMBD ANSG NTSRHCD SGD ENQL@KHRL NE CXM@LHB AKNBJR 4GD @OOQN@BG NE CXM@LHB AKNBJR HR RTOONQSDC HM @KLNRS @KK RHLTK@SHNM O@BJ@FDR 4GD AKNBJ CH@FQ@LR VDQD SGD SDBGMHPTD NE OQNFQ@LLHMF SGD @M@KNFTD BNLOTSDQR )S HR ENQ DW@LOKD SGD A@RHR NE RHLTK@SHNM RNESV@QD -@SK@A3HLTKHMJ -!4,!" VDA 4GD RDBNMC FQNTO NE RTBG LNCDKR NE K@QFDQ O@QSR BQD@SDR LTKSHONKDR 4GDX ADKNMF SNFDSGDQ VHSG ANMC FQ@OGR SN SGD MDSVNQJ @OOQN@BGDR "NSG SGDRD
88
M. Valášek
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
LDMSR HR CDRBQHADC AX SGD BNMRSQ@HMS % % SG@S B@M AD CDQHUDC @R SGD
DPT@KHSX NE RNLD ANTMC@QX BNMCHSHNMR % %
% % % %
Modelling, Simulation and Control…
89
SGDM SGD BNLOKDSD BNLAHMDC RXRSDL B@M AD CDRBQHADC AX SGD U@QH@SHNM@K OQHMBHOKD NE KD@RS @BSHNM NE SGD ,@FQ@MFD ETMBSHNM %> % %> %
%> % %> % %> % %> % % % VGDQD HR ,@FQ@MFD LTKSHOKHDQ BNQQDRONMCHMF SN SGD BNMRSQ@HMS % % 4GD K@RS DWOQDRRHNM B@M AD LNCHDC
% % % %
% % VGDQD MDV ,@FQ@MFD LTKSHOKHDQR @QD HMSQNCTBDC
4GDM SGD CXM@LHB DPT@SHNMR NE SGD BNLOKDSD BNLAHMDC RXRSDL CDQHUDC EQNL SGD ,@FQ@MFD ETMBSHNM @QD BNLONRDC EQNL SGD DPT@SHNMR ENQ D@BG RTARXRSDL RDO@Q@SDKX RTOOKDLDMSDC AX SGD BNMRSQ@HMS QDK@SHNMRGHO @MC %@BG RTARXRSDL @MC HR CDRBQHADC AX SGD U@QH@SHNM@K OQHMBHOKD NE KD@RS @BSHNM NE SGD ,@FQ@MFD ETMBSHNM %> % % @MC %> % % 4GD BNMRSQ@HMS QDK@SHNMRGHO @MC B@M AD CDRBQHADC @R SGQNTFG @MC @BQNRR U@QH@AKDR & & & &
% % % %
64B?2 -TKSHONKD LNCDK CDQHUDC EQNL U@QH@SHNM@K OQHMBHOKDR
90
M. Valášek
)M B@RD SG@S SGD HMSDQ@BSHNM B@MMNS AD CDRBQHADC @R SGD DPT@KHSX NE RNLD ANTMC@QX OQNODQSX NE HMSDQ@BSHMF DKDLDMSR HD SGD DWSDQM@K U@QH@AKD % SN SGD DKDLDMS B@MMNS AD ROKHS HM SGD BNMRSQ@HMS % % HMSN @M DWSDQM@K @CCHSHUD SDQL SGDM HS HR ONRRHAKD SN HMSQNCTBD @ MDV @TWHKH@QX U@QH @AKD R % SN SGD U@QH@AKDR % ! NE SGD DKDLDMS @MC HMRSD@C NE SGD HMCDBNLONR@AKD BNMRSQ@HMS @M DWSDQM@K BNMRSQ@HMS NE HMSDQ@BSHNM ADSVDDM SGD DKDLDMSR @MC HR HMSQNCTBDC
% %
@MC @M HMSDQM@K BNMRSQ@HMS HMRHCD SGD DKDLDMS HR HMSQNCTBDC
% %
4GD HMSDQLDCH@SD QDRTKS HR SG@S @ LTKSHONKD VHSG ! ONKDR VGHBG G@UD @RRN BH@SDC SGD SGQNTFG @MC @BQNRR U@QH@AKDR VHSG KNB@K OQNODQSX HR HMSQNCTBDC &HFTQD @ (NVDUDQ RTBG LTKSHONKDR CN MNS R@SHREX SGD FDMDQ@KHYDC +HQBG GNR K@VR ENQ SGQNTFG @MC @BQNRR U@QH@AKDR NE SGD VGNKD LTKSHONKD 3TBG R@SHRE@BSHNM HR @BGHDUDC AX SGD DWSDMRHNM NE SGD ONKDR AX SGD ! SG @TWHK H@QX ONKD VHSG SGD CDMHSHNM NE SGD SGQNTFG @MC @BQNRR U@QH@AKDR &HFTQD A
%
4GDM RTBG LTKSHONKD R@SHRDR SGD FDMDQ@KHYDC +HQBGGNR K@VR '
'
'
'
4GDM SGD RXRSDL ADHMF LNCDKKDC HR QDOQDRDMSDC AX SGD BNQQDRONMCHMF MDSVNQJ NE HMSDQ@BSHMF DKDLDMSR 4GD DKDLDMSR @QD CDRBQHADC AX JMNVM HMSDQM@K DPT@SHNMR @MC SGDHQ HMSDQ@BSHNMR @QD CDRBQHADC AX SGD FDMDQ@KHYDC +HQBGGNR K@VR ENQ SGQNTFG @MC @BQNRR U@QH@AKDR 3TBG LNCDKHMF @OOQN@BG DM@AKDR SN ATHKC @M@KNFX NE @MX OGXRHB@K RXRSDL @MC @M DKDBSQHB MDSVNQJ !M DW@LOKD NE RTBG @M@KNFX ENQ NMD LTKSHONKD HR HM &HFTQD 4GD HMSDQM@K CXM@LHB DPT@SHNMR @QD CDRBQHADC @R
%> %
4GD KNMFSHLD HMUDRSHF@SDC @M@KNFHDR ADSVDDM CHDQDMS OGXRHB@K CNL@HMR @QD HM &HFTQD
Modelling, Simulation and Control…
91
64B?2 !M@KNFX NE @ LTKSHONKD @MC @M DKDBSQHB MDSVNQJ
64B?2 4GQNTFG @MC @BQNRR U@QH@AKDR ENQ CHDQDMS OGXRHB@K CNL@HMR
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
92
M. Valášek
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
64B?2 "NMC FQ@OG LNCDKKHMF @OOQN@BG 4GD FDMDQ@K @CU@MS@FD NE MDSVNQJ @OOQN@BGDR HR SGD M@STQ@K LTKSHCN L@HM B@O@AHKHSX NE LNCDKKHMF VGHBG G@R ADDM ENQ KNMF SHLD @KLNRS SGD NMKX
Modelling, Simulation and Control…
93
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
64B?2 6@QH@AKD BK@RRHB@SHNM HM @ FDMDQ@KHYDC ANMC FQ@OG @OOQN@BG
4GD CDUDKNOLDMS OQNBDRR NE RHLTK@SHNM LNCDK HR NESDM TMCDQDRSHL@SDC 4GD QDRTKS NE SGD CDUDKNOLDMS RDDLR SN AD ITRS SGD BNCD NE SGD RHLTK@SHNM OQN FQ@L "TS CTQHMF SGD CDUDKNOLDMS OQNBDRR @ KNS NE RODBHB JMNVKDCFD HR CDQHUDC SG@S HR TRDC CTQHMF SGD RHLTK@SHNM LNCDK KHED BXBKD &HFTQD @MC CTQHMF JMNVKDCFD QDTRD HM SGD CDUDKNOLDMS NE MDWS RHLTK@SHNM LNCDK 4GD CDUDKNOLDMS NE RHLTK@SHNM LNCDK B@M AD BNMRHCDQDC @R @M @M@KNFX VHSG @ RNESV@QD OQNFQ@L CDUDKNOLDMS 6@K@RDJ 4GHR OQNBDRR G@R ADDM @M@K XRDC HM CDS@HKR VHSGHM SGD #KNBJVNQJ OQNIDBS :CQ@G@K DS @K 3SDHM A@TDQ DS @K 4GD CDUDKNOLDMS NE @ RHLTK@SHNM LNCDK SXOHB@KKX FNDR SGQNTFG @ MTLADQ NE S@RJR 4GDRD S@RJR @QD HKKTRSQ@SDC HM @ LNCDKKHMF @MC RHLTK@SHNM JMNVKDCFD EQ@LDVNQJ HM &HFTQD 4GD KHFGS @QQNVR QDOQDRDMS L@OOHMFR ADSVDDM SGD CHDQDMS DKDLDMSR @MC SGD AK@BJ @QQNVR QDOQDRDMS SQ@MRENQL@SHNM OQNBDRRDR ADSVDDM SGD S@RJR 4GD CDUDKNOLDMS NE @ RHLTK@SHNM @R RGNVM HM &HFTQD HMUNKUDR UD RSDOR
94
M. Valášek
64B?2 4Q@CHSHNM@K RHLTK@SHNM LNCDK KHED BXBKD
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
Modelling, Simulation and Control…
95
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
#"
#"
# #"
#"
# #
#"
# #
#
!
#
#
#
$ $
# #
#
64B?2 3HLTK@SHNM @MC LNCDKKHMF JMNVKDCFD EQ@LDVNQJ 4GD MDWS S@RJ HR OGXRHB@K LNCDKKHMF 4GD SGHQC RSDO HR SGD SQ@MRENQL@SHNM EQNL BNMBDOST@K VNQKC HMSN HCD@K LNCDK VNQKC )S BNMRHRSR NE QDOK@BDLDMS NE D@BG DKDLDMS NM SGD BNMBDOST@K KDUDK AX BNQQDRONMCHMF DKDLDMSR NM SGD OGXRHB@K KDUDK 4GD DKDLDMSR NE SGD OGXRHB@K LNCDK @QD HCD@K LNCDKKHMF DKD LDMSR DF QHFHC ANCX HCD@KKX DWHAKD ANCX HCD@K F@R HCD@K HMBNLOQDRRHAKD THC HCD@K B@O@BHSNQ DSB 4GD DMFHMDDQHMF RBHDMBDR CD@K OQDBHRDKX VHSG SGD HCD@K NAIDBSR -@SGDL@SHB@KKX CDRBQHADC DW@BS OQDCHBSHNM NE SGD ADG@UHNTQ HR @U@HK@AKD NMKX ENQ HCD@K NAIDBSR 4GD @QS NE DMFHMDDQHMF LNCDKKHMF @MC RHLTK@SHNM HR SGD B@O@AHKHSX SN SQ@MRK@SD SGD ADG@UHNTQ NE QD@K VNQKC NAIDBSR HMSN SGD ADG@UHNTQ NE @ RXRSDL BNMRHRSHMF NE HCD@K NAIDBSR !F@HM @R RNNM @R SGHR QDOK@BDLDMS HR BNLOKDSDC SGD OGXRHB@K LNCDKKHMF HR BNLOKDSDC )M SGHR RS@FD SGD L@INQHSX NE LNCDKKHMF @RRTLOSHNMR @QD Q@HRDC @MC L@MX LNCDKKHMF LNCHB@SHNMR @QD ADHMF CNMD )S HR SGD V@X NE LNC DKKHMF HS @MC QDOQDRDMSHMF SGD QD@K VNQKC RXRSDL HM @ OGXRHB@K LNCDKKHMF
96
M. Valášek
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
Modelling, Simulation and Control…
97
64B?2 )SDQ@SHUD OQNBDRR NE RHLTK@SHNM LNCDK CDUDKNOLDMS
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
98
M. Valášek
!
-TKSHANCX RXRSDLR @QD SGD LNRS NESDM TRDC LNCDKR NE LDBG@SQNMHB@K RXR SDLR 4GD BNMSQNK S@RJR ENQ LTKSHANCX RXRSDLR -"3 B@M AD RTLL@QHYDC @R 6@K@RDJ 4GD QRS ETMC@LDMS@K OQNAKDL HR SN RS@AHKHYD SGD LNSHNM NE LTKSHANCX RXRSDL @QNTMC HSR DPTHKHAQHTL )S HR SGD A@RHB BNMSQNK OQNAKDL !M HLONQS@MS RTAOQNAKDL HR ITRS SN BGDBJ SGD RS@AHKHSX NE RNLD OQNONRDC BNMSQNK 4GD MDWS OQNAKDL HR SN LNUD SGD LTKSHANCX RXRSDL EQNL NMD ONRHSHNM HMSN @MNSGDQ )S HR NMD NE SGD LNRS EQDPTDMS BNMSQNK OQNAKDLR NE -"3 4GHR OQNAKDL HR SQ@MRENQLDC HMSN NSGDQ OQNAKDLR 4GD OQNAKDL B@M AD RS@SDC @R SGD FKNA@K RS@AHKHY@SHNM NE SGD RXRSDL EQNL SGD HMHSH@K ON RHSHNM @QNTMC SGD M@K ONRHSHNM @R SGD RS@AHKHYDC DPTHKHAQHTL "X SGHR V@X SGD OQNAKDL HR SQ@MRENQLDC HMSN OQDUHNTR OQNAKDL 4GD NSGDQ OQNAKDL RS@SDLDMS HR SN RS@AHKHYD SGD RXRSDL @QNTMC SGD FHUDM SQ@IDB SNQX BNMMDBSHMF ANSG ONRHSHNMR )S HR VDKK JMNVM SQ@IDBSNQX SQ@BJHMF &NQ MNMKHMD@Q LTKSHANCX RXRSDLR HS B@M AD @ RDQHNTR OQNAKDL ITRS SN BNMRSQTBS @CLHRRHAKD SQ@IDBSNQX BNMMDBSHMF HMHSH@K @MC M@K ONRHSHNMR 4GD OQDUHNTR OQNAKDL B@M AD ENQLTK@SDC @R SGD CDSDQLHM@SHNM NE NO SHL@K SQ@IDBSNQX VHSG QDRODBS SN RNLD ODQENQL@MBD HMCDW 4GD OQNAKDL BNTKC AD ITRS SN MC @MX @CLHRRHAKD SQ@IDBSNQX NE SGD RXR SDL BNMMDBSHMF SVN ONRHSHNMR NE LTKSHANCX RXRSDL )S HR @ OQDO@Q@SNQX OQNAKDL ENQ BNMSQNK AX SQ@IDBSNQX SQ@BJHMF 4GD RS@AHKHYHMF OQNAKDLR @QD @CCQDRRDC AX RDUDQ@K FQNTOR NE LDSGNCR 4GDQD VDQD CDUDKNODC LDSGNCR SG@S DW@BSKX KHMD@QHYDC SGD MNMKHMD@Q RXRSDL @MC SGTR DM@AKD @ESDQ SGD KHMD@QHYHMF SQ@MRENQL@SHNM SN @OOKX RS@MC@QC BNMSQNK LDSGNCR ENQ KHMD@Q RXRSDLR !MNSGDQ @OOQN@BG HR SN OQNONRD @ BNMSQNK K@V A@RDC NM RNLD HMRHFGS HMSN SGD RXRSDL @MC SGDM SN B@QQX NTS SGD RXMSGDRHR NE BNMSQNK O@ Q@LDSDQR ENQ SGD MNMKHMD@Q OK@MS @R BNMSQNK F@HM NOSHLHY@SHNM AX SGD SDBGMHPTD NE LTKSHNAIDBSHUD O@Q@LDSDQ NOSHLHY@SHNM -/0/ 6DQX OQNLHRHMF LDSGNC HR SN QDVQHSD SGD MNMKHMD@Q RXRSDL HMSN SGD ENQL NE KHMD@Q RXRSDL @MC SGDM SN @OOKX SGD RS@MC@QC BNMSQNK LDSGNCR ENQ KHMD@Q RXRSDLR GNVDUDQ ENQ ONRHSHNM RS@SD CDODMCDMS RXRSDL )S HR @MNSGDQ V@X ENQ SQ@MRENQLHMF SGD MNMKHMD@Q RXRSDL HMSN KHMD@QKHJD NMD 4GD OQNAKDL NE BNMRSQTBSHMF RTHS@AKD SQ@IDBSNQX HR ENQLTK@SDC DHSGDQ CHQDBSKX @R SGD NOSHL@K BNMSQNK OQNAKDL SG@S HR FDMDQ@KKX MNS PTHSD D@RX SN AD RNKUDC
Modelling, Simulation and Control…
99
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
64B?2 /UDQUHDV NE LNCDKR TRDC ENQ BNMSQNK
100
M. Valášek .6; =?92:@ <3 0<;A?<9 <3 :B9A6/<1F @F@A2:@
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
Modelling, Simulation and Control…
101
@M DW@LOKD NE @ QHFHC OK@M@Q QNANS L@MHOTK@SNQ )E SGD RDBNMC INHMS HR MNS @BST@SDC SGDM SGD RXRSDL HR TMCDQ@BST@SDC @MC SGDQD @QD OQNAKDLR NE HSR BNMSQNK )E SGHR MNS@BST@SDC INHMS HR DPTHOODC VHSG ROQHMF @MC C@LODQ SGDM HS HR @ LNCDK NE DWHAKD QNANSHB @QL )E SGDQD HR MN ROQHMF HM SGHR MNS@BST@SDC INHMS SGDM HS HR @ LNCDK NE LDBG@SQNMHB SNX VHSG CHDQDMS BNMSQNK BG@KKDMFDR 4GD LTKSHANCX RXRSDL B@M AD CDRBQHADC AX SGD HMCDODMCDMS BNNQCHM@SDR DF @MC NQ AX SGD CDODMCDMS BNNQCHM@SDR DF SGD B@QSDRH@M BN NQCHM@SDR NE BDMSQD NE L@RR @MC @MFKDR NE KHMJR VHSG QDRODBS SN SGD EQ@LD NQ M@STQ@K #@QSDRH@M BNNQCHM@SDR NE DMCR NE KHMJR !BBNQCHMF SN SG@S SGD BNMSQNK HR ENQLTK@SDC @R SQ@CHSHNM@K BNMSQNK NE /$% NQ BNMSQNK NE $!%
64B?2 %W@LOKD NE RHLOKD LTKSHANCX RXRSDL VHSG CHDQDMS BNMSQNK OQNAKDLR
2@0?6=A6<; <3
<;96;2.? B9A6/<1F %F@A2:@
4GD MNMKHMD@Q LTKSHANCX RXRSDLR HMBKTCHMF SGD BNMSQNK @BSHNM @QD FDM DQ@KKX CDRBQHADC AX ,@FQ@MFD DPT@SHNMR NE LHWDC SXOD RDD DF 3SDIRJ@K @MC 6@K@RDJ @
@ @ @ # @ & @ B & & &
@
& VGDQD @ @QD SGD BNNQCHM@SDR CDRBQHAHMF SGD LTKSHANCX RXRSDL B HR SGD UDB SNQ NE HMOTS BNMSQNK U@QH@AKDR HR SGD L@RR L@SQHW # @QD SGD FDMDQ@KHYDC ENQBDR @QD SGD JHMDL@SHB GNKNMNLHB NQ MNMGNKNMNLHB BNMRSQ@HMSR HR SGD *@BNAH@M BNQQDRONMCHMF SN SGD JHMDL@SHB BNMRSQ@HMSR @QD SGD ,@ FQ@MFD LTKSHOKHDQR SGD BNMSQNK HMOTSR ENQBDR B @QD @BSHMF SGQNTFG SQ@MR LHRRHNMR & 4GHR E@BS SG@S SGD FDMDQ@K CDRBQHOSHNM NE LTKSHANCX RXRSDLR @
102
M. Valášek
TRDR CHDQDMSH@K@KFDAQ@HB DPT@SHNMR $!% HR NMD CHBTKS OQNAKDL NE BNM SQNK RXMSGDRHR ENQ LTKSHANCX RXRSDLR 4GDRD DPT@SHNMR NE LNSHNM ENQ SGD OTQONRDR NE BNMSQNK LTRS AD NESDM SQ@MRENQLDC HMSN SGD RS@SD RO@BD CDRBQHOSHNM 4GDQD @QD SVN ONRRHAKD V@XR GNV SN CN HS 4GD QRS @OOQN@BG OQDRDQUDR SGD R@LD U@QH@AKDR @ EQNL NQHFHM@K CDRBQHOSHNM @MC NMKX DKHLHM@SDR SGD ,@FQ@MFD LTKSHOKHDQR )E SGD JHMDL@SHB BNMRSQ@HMSR CN MNS CDODMC NM SGD UDKNBHSHDR HD SGD JHMDL@SHB BNMRSQ@HMSR @QD GNKNMNLHB NMDR SGDM SGDRD JHMDL@SHB BNMRSQ@HMSR HM @QD SVHBD CHDQDMSH@SDC VHSG QDRODBS SN SHLD
@ @ @ & & &
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
(DMBD SGD RS@SD RO@BD CDRBQHOSHNM VHSG RS@SD UDBSNQ ;@ CCS @= HM CDODMCDMS BNNQCHM@SDR @ B@M AD NAS@HMDC (NVDUDQ HM RTBG B@RD SGD QDRTKSHMF DPT@SHNMR @QD @ESDQ KHMD@QHY@SHNM MNS BNMSQNKK@AKD )S ENKKNVR EQNL SGD E@BS SG@S HM RTBG B@RD SGD CDODMCDMS BNNQCHM@SDR NM SGD KDUDK NE @BBDKDQ@SHNM @QD QDK@SDC AX SGD PT@CQ@SHB SDQLR HM UDKNBHSHDR @R ENKKNVR EQNL SGD RDBNMC CDQHU@SHUDR NE JHMDL@SHB BNMRSQ@HMSR HM 3TBG DPT@SHNMR @QD @ESDQ KHMD@QHY@SHNM MNS BNMSQNKK@AKD HM TRT@K ONRHSHNMR VHSG YDQN UDKNBHSHDR .DUDQSGDKDRR SGHR RHLOKD ENQLTK@SHNM HR RNLDSHLDR UDQX TRDETK DF ENQ NOSHL@K BNMSQNK 2NTAHBDJ @MC 6@K@RDJ 4GD NSGDQ @OOQN@BG HR A@RDC NM SGD CDRBQHOSHNM NE LTKSHANCX RXRSDLR HM HMCDODMCDMS BNNQCHM@SDR > 4GD BNNQCHM@SDR LTRS AD HMCDODMCDMS DF SGD
Modelling, Simulation and Control…
103
HMCDODMCDMS BNNQCHM@SDR > @QD RDKDBSDC BNMRSQTBSDC EQNL @ FDMDQ@KKX @ ? > @$ > & & @ $ > $ > & & & &
3TARSHSTSHMF EQNL HMSN SGD QRS SHLD CDQHU@SHUD NE SGD JHMDL@SHB BNM RSQ@HMS HS HR ENTMC NTS SG@S SGHR L@SQHW $ RO@MR SGD MTKK RO@BD NE SGD *@BNAH@M $ 4GTR LTKSHOKXHMF SGD DPT@SHNMR NE LNSHNM AX $ DKHLHM@SDR @F@HM SGD ,@FQ@MFD LTKSHOKHDQR 4GDM TRHMF SGD RS@SD UDBSNQ E;> C>CS= SGD DPT@ SHNMR NE LNSHNM B@M AD SQ@MRENQLDC HMSN SGD RS@SD RO@BD LNCDK > >& - > >- $ $ $ 4 $ $ - $ $ $ &B &
4GD SGDNQX NE LTKSHANCX RXRSDLR @MC SGD RNKTSHNM NE BNQQDRONMCHMF $!% CDRBQHADR RDUDQ@K LDSGNCR ENQ CDQHUHMF @MC NAS@HMHMF SGD MTKK RO@BD L@SQHW $ RDD DF 3SDIRJ@K @MC 6@K@RDJ .NV SGD CXM@LHBR NE LTKSHANCX RXRSDLR B@M AD CDRBQHADC HM SGD ENQL NE FDMDQ@K MNMKHMD@Q RXRSDL HM RS@SD RO@BD CDRBQHOSHNM CE 3 E 4 E B CS
VGDQD E ! HR SGD RS@SD UDBSNQ B HR SGD BNMSQNK UDBSNQ @MC FDMDQ@KKX 3 &NQ SGD OTQONRD NE BNMSQNK HS LTRS AD RS@SDC VGHBG U@QH@AKDR F B@M AD LD@RTQDC @MC TRDC @R SGD HMOTS SN SGD BNMSQNK RXRSDL )S HR RTOONRDC SGDHQ FDMDQ@K CDODMCDMBD F 5 E 4GTR SGDQD HR SGD NTSOTS U@QH@AKD UDBSNQ F #
2@0?6=A6<; <3 ;<;96;2.? =?2160A6C2 :<129@
4GD MNMKHMD@Q AK@BJ ANW SHLD RDQHDR LNCDKR @QD UDQX RTHS@AKD ENQ LNCDK A@RDC OQDCHBSHUD BNMSQNK 4GD BNMRHCDQDC BNMBDOS #45,/,)-/4 ,/B@K ,)MD@Q -/CDK 4QDD HLOKDLDMSDC AX #45 HM 0Q@FTD BNMRHRSR NE LTKSHOKD KHMD@Q CXM@LHB LNCDKR U@KHC ITRS HM KNB@K RTAQDFHNMR @MC RLNNSGKX RTLLDC
104
M. Valášek
HMSN NMD QDRTKSHMF MNMKHMD@Q LNCDK 3SDE@M DS @K 4GD HCDMSHB@SHNM @KFNQHSGL HR B@KKDC ,/,)-/4 @MC V@R CDUDKNODC HM .DKKDR 4GD MDV HLOKDLDMS@SHNM V@R CNMD @S #45 HM 0Q@FTD 3SDE@M DS @K 4GD KNB@K LNCDKR @QD CDRBQHADC @R KHMD@Q CHDQDMBD LNCDKR VHSG HMOTSR B ;B B B = B ;' ' ' = B ;' ' ' = B ;' ' ' = NE CDOSG NTSOTSR * NE CDOSG ! @MC BNDBHDMSR ( 4GDRD KNB@K LNCDKR @QD RTLLDC TRHMF SGD '@TRRH@M RLNNSGDMHMF ETMBSHNMR B VHSG BDMSDQR @MC U@QH @MBDR B
B
B
B 7 B 7 B 7
VGDQD 7
; = 4GD QDRTKSHMF #45,/,)-/4 LNCDK CDRBQHOSHNM VHSG RTALNCDKR HR B DWO
*[
*[ B
( ( ' ( '
( '
( ' ( '
( '
( ' ( '
( '
( * ( *
( * ! B
4GD @CU@MS@FD NE SGHR CDRBQHOSHNM HR SG@S SGD LNCDK HR KNB@KKX KHMD@Q ATS FKNA@KKX MNMKHMD@Q 4GD #45,/,)-/4 LNCDK B@M AD HCDMSHDC A@RDC NM RHLTK@SHNM LNCDK @R VDKK @R A@RDC NM DWODQHLDMS@K C@S@ EQNL QD@K RXRSDL
<129 /.@21 0<;A?<9 @F@A2:@
4GD LNCDK A@RDC BNMSQNK NE LDBG@SQNMHB@K RXRSDLR B@M AD CHUHCDC HMSN FQNTOR BNMSQNK VHSG HLOKHBHS NQ DWOKHBHS TR@FD NE OK@MS LNCDK ENQ KHMD@Q NQ MNMKHMD@Q OK@MS &HFTQD 4GD BNMSQNK @OOQN@BGDR @QD BNLAHMDC VHSG CHDQDMS JHMCR NE OK@MS LNCDKR 4VN LNCDK SXODR @QD CHRSHMFTHRGDC @M@KXS
Modelling, Simulation and Control…
105
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
! ! !! ! !!
! ! ! ! "! !
! ! "!# ! !!"! # $! "! !$! !!!
64B?2
-NCDK A@RDC BNMSQNK @OOQN@BGDR 4GDRD @OOQN@BGDR @QD CDRBQHADC ETQSGDQ HM CDS@HKR
221/.08 2E.0A 96;2.?6G.A6<; <3 ;<;96;2.? @F@A2:@
/MD NE SGD ETMC@LDMS@K @OOQN@BGDR SNV@QCR SGD BNMSQNK NE MNMKHMD@Q RXRSDLR HR SGD DW@BS EDDCA@BJ KHMD@QHY@SHNM )RHCNQH 3KNSHMD @MC ,H
106
M. Valášek
&KHDRR +G@KHK )SR HCD@ HR RHLOKD #NMRHCDQHMF SGD RXRSDL HS HR HMUDRSHF@SDC SGD DWHRSDMBD @MC SGD BNMRSQTBSHNM NE @ RTHS@AKD RS@SHB RS@SD EDDCA@BJ BNMSQNK B E E D @MC @ SQ@MRENQL@SHNM NE U@QH@AKDR G & E
SG@S SNFDSGDQ SQ@MRENQL SGD NQHFHM@K RXRSDL HMSN @ KHMD@Q NMD CG G D CS
4GD R@LD OQNAKDL B@M AD HMUDRSHF@SDC ENQ SGD DWHRSDMBD NE @ RTHS@AKD CX M@LHB RS@SD EDDCA@BJ BNMSQNK B E = E = D C= E E B CS
VHSG @ SQ@MRENQL@SHNM NE U@QH@AKDR SG@S @F@HM SQ@MRENQL SGD NQHFHM@K RXRSDL HMSN @ KHMD@Q NMD 4GDRD OQNAKDLR @QD B@KKDC HMOTSRS@SD DW@BS EDDCA@BJ KHMD@QHY@SHNM !MC M@KKX ANSG SGDRD OQNAKDLR B@M AD HMUDRSHF@SDC ENQ NTSOTS EDDCA@BJ VGDQD HMRSD@C NE RS@SD U@QH@AKDR E HM EDDCA@BJR NQ NMKX SGD NTSOTS U@QH@AKDR F EQNL @QD TRDC 4GDM SGDRD OQNAKDLR @QD B@KKDC HMOTSNTSOTS DW@BS EDDCA@BJ KHMD@QHY@SHNM 4GDM @ESDQ SQ@MRENQLHMF SGD NQHFHM@K RXRSDL HMSN SGD DPTHU@KDMS KHMD@Q NMD SGDQD @QD @OOKHDC SGD BNMSQNK SDBGMHPTDR ENQ KHMD@Q RXRSDLR KHJD ONKD OK@BDLDMS ,12 DSB 3DUDQ@K U@QH@MSR NE EDDCA@BJ DW@BS KHMD@QHY@SHNM B@M AD CHRSHMFTHRGDC )S HR HMOTSNTSOTS DW@BS EDDCA@BJ KHMD@QHY@SHNM SG@S ADFHMR EQNL SGD NTSOTS CDODMCDMBD SGDM SGD HMOTSRS@SD DW@BS EDDCA@BJ KHMD@QHY@SHNM SG@S HM UDRSHF@SDR SGD BNMRSQTBSHNM NE RTBG RTHS@AKD NTSOTS CDODMCDMBD SG@S HS HR DW@BS EDDCA@BJ KHMD@QHY@AKD 4GD RNKTSHNM NE SGD OQNAKDL NE HMOTSNTSOTS DW@BS EDDCA@BJ KHMD@QHY@SHNM HR RHLOKDQ 4GD BNMRSQTBSHNM NE ONRRHAKD KHMD@QHYHMF EDDCA@BJ ENQ SGD RXRSDL HR SGD QDBTQRHUD @OOKHB@SHNM NE SGD SHLD CHDQDMSH@SHNM QTKD 5 3 E 4 E B 3 E 4 E B E 5 5B G- G G 3 E 4 E B 3 E 4 E B F E G- G
F
Modelling, Simulation and Control…
107
4GDQD HR RTOONRDC ADRHCDR MDBDRR@QX RLNNSGMDRR SG@S 4 ENQ $ @MC 4 DM@AKHMF SN DWOQDRR 3 E 4 E B D B @MC SGHR KD@CR SN
D 3 E 4 E
F D
4GHR LD@MR SG@S SGD NQHFHM@K RXRSDL HR QRS SQ@MRENQLDC HMSN SGD "QTMNURJX B@MNMHB@K ENQL @MC SGDM HMSN SGD &QNADMHTR B@MNMHB@K ENQL 4GDRD B@MNMHB@K ENQLR OQNUD SGD BNMSQNKK@AHKHSX NE SGD RXRSDL @MC DM@AKD SN RS@AH KHYD SGD RXRSDL D@RHKX AX ONKD OK@BDLDMS 4GD SQ@MRENQL@SHNM HR
5 E
= G & E ; 5 E
5 E
(DQD @MC HM OQDUHNTR DWOQDRRHNMR SGDQD HR TRDC SGD ENQL@KHRL NE ,HD @KFDAQ@ 5 3 E 5 E 5 E E 3 4 3 E 4 E 4 E 4 E ;3 4= E E E 4 E ;3 4= E 4 E ;3 4= E
5 E
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
108
M. Valášek
)M @KK SGDRD CDQHU@SHNMR HS HR RTOONRDC SG@S SGD CHLDMRHNM NE F HR DPT@K SN SGD CHLDMRHNM NE B @MC D 4GHR HR BDQS@HMKX NESDM UHNK@SDC SG@S OQDUDMSR SN @OOKX SGHR LDSGNC ENQ SGD BNMSQNK NE TMCDQ@BST@SDC RXRSDLR
!"! B9A6720A6C2 ".?.:2A2? !=A6:6G.A6<;
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
= HM RTBG @ V@X SG@S SGD BTQQDMSKX 7/234 BQHSDQHNM VHSG SGD L@WHL@K U@KTD VHKK AD QDCTBDC NOSHLHY@ SHNM SQHDR SN CDBQD@RD SGD VDHFGSDC BQHSDQHNM =
= 4GD STMHMF O@Q@LDSDQR = @QD CDSDQLHMDC AX RNKUHMF @ LHML@WNOSHLHY@SHNM OQNAKDL VHSG BNMRSQ@HMSR @MC O@Q@LDSDQ QDRSQHBSHNMR LHM L@W
= E"$
#
#
#
Modelling, Simulation and Control…
109
64B?2 0@QDSNNOSHLTL ENQ SVN NAIDBSHUD ETMBSHNMR
4GD ODQENQL@MBD BQHSDQH@ NAIDBSHUD ETMBSHNMR L@X AD EQDD SN AD NOSH LHYDC NQ L@X AD O@QSH@KKX KHLHSDC AX @ RDS NE ODQENQL@MBD BNMRSQ@HMSR DF LDBG@MHB@K GXCQ@TKHB NQ DKDBSQNMHB QDRSQHBSHNMR KHJD ONVDQ OQDRRTQD BTQQDMS @MC UNKS@FD 4GD VDHFGSHMF E@BSNQR NQ CDRHFM O@Q@LDSDQR DM@AKD SGD TRDQ SN @C@OS SGD BQHSDQH@ SN @CDPT@SD RHYDR @MC SN CDSDQLHMD SGD CH QDBSHNM NE NOSHLHY@SHNM OQNBDRR AX VDHFGSHMF RNLD BQHSDQH@ LNQD HLONQS@MS =
= NSGDQR KDRR HLONQS@MS =
= =
=
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
110
M. Valášek
CDRHFM VHSG LHMHL@K QDPTHQDLDMSR NM SGD MNMKHMD@Q OK@MS SG@S HR BNLAHMDC VHSG SGD MDV @CU@MBDC BNMSQNK CDRHFM LDSGNCNKNFX HR SGD -TKSH/AIDBSHUD 0@Q@LDSDQ /OSHLHY@SHNM -/0/ @OOQN@BG 4GD L@HM RSDOR NE SGD -/0/ CDRHFM LDSGNCNKNFX @QD SGD ENKKNVHMF <129@ 4GD QDEDQDMBD RHLTK@SHNM LNCDK NE SGD OK@MS SN AD BNMSQNKKDC UDGHBKD RTRODMRHNM RXRSDL HR CDUDKNODC )S HR @R CDS@HKDC @R ONRRHAKD HM NQCDQ SN BNUDQ SGD CXM@LHB@K OGDMNLDM@ SN AD BNMSQNKKDC "DRHCDR SGHR QDEDQDMBD LNCDK SGD CDRHFM LNCDK HR @KRN CDUDKNODC )S HR RHLOKDQ SG@M SGD QDEDQDMBD LNCDK ATS HMBKTCDR @KK DRRDMSH@K ED@STQDR $/&R MNMKHMD@QHSHDR DSB <;A?<9 9.D @2920A6<; "@RDC NM SGD CDRHFM LNCDK SGD OGXRHB@K HMRHFGS HMSN SGD OK@MS SN AD BNMSQNKKDC HR CDUDKNODC 5RHMF SGHR OGXRHB@K HMRHFGS @ RTHS@AKD BNMSQNK K@V HR OQNONRDC )S TRT@KKX HMBKTCDR O@Q@LDSDQR SN AD CDRHFMDC ENQ SGD @OOQNOQH@SD ETMBSHNM NE SGD BNMSQNKKDC OK@MS <;A?<9 9.D 12@64; 4GD BNMSQNK K@V HR O@Q@LDSQHYDC 4GD O@Q@LDSDQR @QD CDSDQLHMDC AX SGD -/0/ @OOQN@BG ! ODQENQL@MBD HMCDW TRT@KKX HM SGD HMSDFQ@K ENQL HR RDKDBSDC )LONQS@MS @MC QDOQDRDMS@SHUD DWBHS@SHNM HMOTSR @QD S@JDM 4GD BGNHBD NE HMOTSR SGD BGNHBD NE ODQENQL@MBD HMCDBDR @MC SGD VDHFGSR NE SGDHQ HMCHUHCT@K BNLONMDMSR @QD SGD LNRS OQNAKDL@SHB O@QSR NE SGHR @OOQN@BG 4GD CDRHFM RHLTK@SHNM LNCDK HR HMSDFQ@SDC VHSG SGDRD HMOTSR @MC SGD ODQENQL@MBD HMCDW HR DU@KT@SDC 4GDM SGD ODQENQL@MBD HMCDW HR ITRS @ ETMBSHNM NE SGD TMJMNVM BNMSQNK K@V O@Q@LDSDQR 4GDX @QD CDSDQLHMDC AX HSR NOSHLHY@SHNM )S HR CDRHFMAXRHLTK@SHNM @MC CDRHFMAXNOSHLHY@SHNM (2?60.A6<; &HQRSKX SGD QDEDQDMBD LNCDK HR UDQHDC AX BNLO@QHRNM VHSG DWODQHLDMSR NM QD@K OK@MS 3DBNMCKX SGD CDRHFM LNCDK HR UDQHDC @F@HMRS SGD QDEDQDMBD LNCDK 4GDM SGD CDRHFMDC BNMSQNK K@V HR UDQHDC AX SGD RHLTK@SHNM NE SGD QDEDQDMBD LNCDK @KRN TRHMF @ K@QFDQ RDS NE HMOTS DWBHS@SHNMR &HM@KKX SGD QDRTKSHMF BNMSQNK K@V HR UDQHDC AX DWODQHLDMSR NM QD@K OK@MS %W@LOKD NE @OOKHB@SHNM NE -/0/ RXMSGDRHR ENQ UDGHBKD RTRODMRHNMR HR HM 6@K@RDJ DS @K
#$ <;96;2.? #B.1?.A60 $24B9.A .;1 ;<;96;2.? 96; 2.?6G.A6<; <3 ;<;96;2.? @F@A2:@
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
Modelling, Simulation and Control…
111
HR SGD HMMHSD GNQHYNM BNMSQNK * E #E B $B CS
)E SGDQD DWHRSR SGD CDBNLONRHSHNM NE SGD RXRSDL CXM@LHBR 3 E E E
@MC HE SGD ENKKNVHMF @RRTLOSHNMR "@MJR @MC -G@M@ @QD U@KHC 4GD L@SQHBDR # @MC $ @QD ONRHSHUD CDMHSD 4GD L@SQHBDR @MC 4 @QD @M@KXSHB U@KTDC ETMBSHNMR 4GD BNMSQNK ETMBSHNM B S DWHRSR @MC BNQQDRONMCHMF RS@SD SQ@IDBSNQX E & ENQ & VGHBG R@SHREX SGD RXRSDL CXM@LHBR @MC SGD ODQENQL@MBD HMCDW HR MHSD 4GD O@HQ NE L@SQHBDR E4E HR BNMSQNKK@AKD @MC RS@AHKHY@AKD ENQ D@BG E HM SGD KHMD@Q RXRSDL RDMRD HD Q@MJ;4 E E 4 E
E 4 E = !
4GD RS@SD UDBSNQ E HR ETKKX LD@RTQDC SGDM SGDQD DWHRSR SGD BNMSQNK B E E
VGHBG LHMHLHYDR SGD ODQENQL@MBD HMCDW (NVDUDQ HS G@R ADDM RGNVM SG@S SGHR BNMSQNK HR NMKX RTANOSHL@K 4GD MNMKHMD@Q F@HM L@SQHW E HR CDSDQLHMDC @R E $ 4 E " E
VGDQD "E HR SGD RNKTSHNM NE SGD 2HBB@SSH DPT@SHNMR
E " E " E E # " E 4 E $
4 E " E
4GHR BNMSQNK @OOQN@BG HR B@KKDC .NMKHMD@Q 1T@CQ@SHB 2DFTK@SNQ .12 6@K@RDJ @MC 3SDHMA@TDQ NQ 3S@SD $DODMCDMS 2HBB@SSH %PT@SHNM 3$2% BNMSQNK #KNTSHDQ 4GD JDX OQNAKDL HR SGD BNLOTS@SHNM NE SGD CDBNLONRHSHNM E 4GDQD G@UD ADDM CDQHUDC RNLD ENQLTK@R ENQ SGD BNLOTS@SHNM NE SGHR CDBNLONRHSHNM KHJD ,@MFRNM @MC !KKDXMD E
3 E
112
M. Valášek
"TS SGD BNLOTS@SHNM NE SGHR ENQLTK@ ENQ SGD LTKSHANCX RXRSDLR HR UDQX CHBTKS ADB@TRD HS QDPTHQDR SGD RXLANKHB L@MHOTK@SHNM VHSG SGD DPT@SHNMR NE LNSHNM VGHBG LHFGS AD UDQX K@QFD @MC BNLOKDW @MC LHFGS HMBKTCD Q@SHNM@K @MC SQHFNMNLDSQHB ETMBSHNMR 4GDQDENQD HS HR MNS QD@KHRSHB SN RTOONRD SN BNLOTSD ENQ LTKSHANCX RXRSDLR 4GDQDENQD SGDQD G@UD ADDM CDUDKNODC DBHDMS @KFNQHSGLR 6@K@RDJ @MC 3SDHMA@TDQ ENQ BNLOTS@SHNM NE SGHR CDBNLONRHSHNM SG@S LNQDNUDQ DM @AKDR SN TRD SGD MNMTMHPTDMDRR NE SGD CDBNLONRHSHNM ENQ BNMSQNK RXM SGDRHR 4GD RHLOKDRS OQNBDCTQD @LNMF SGDL HR ENKKNVHMF )SR HCD@ HR A@RDC NM SGD ENKKNVHMF CDBNLONRHSHNM CDRBQHADC ENQ RB@K@Q ETMBSHNM NE SGQDD U@QH @AKDR ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
) ) )
4GHR CDBNLONRHSHNM LTRS BDQS@HMKX RNKUD SGD CHUHRHNM AX YDQN @MC HS HR CDODM CDMS NM SGD NQCDQ NE U@QH@AKDR !BBNQCHMF SN K(NROHS@K QTKD SGD CHUHRHNM AX YDQN HR HM SGD KHLHS DPT@K SN SGD BNQQDRONMCHMF CDQHU@SHUD VGHBG ENQ RLNNSG ETMBSHNMR DWHRSR @MC B@M AD MTLDQHB@KKX BNLOTSDC AX CHUHRHNM ENQ L@FMH STCDR NE MTLDQHB@K @BBTQ@BX NE SGD BNLOTSDQ 4GD BNLOTS@SHNM NE SGHR CDBNLONRHSHNM HR MNS TMHPTD )S CDODMCR NM SGD NQCDQ NE U@QH@AKDR @MC ADRHCDR SG@S SGDQD @QD RNLD EQDD O@Q@LDSDQR ENQ CD BNLONRHSHNM O@Q@LDSQHY@SHNM 6@K@RDJ @MC 3SDHMA@TDQ 4GD HMTDMBD NE SGHR @LAHFTHSX B@M AD CDRBQHADC @MC HMUDRSHF@SDC AX SGD QDBTQRHUD CDBNL ONRHSHNM @OOKHDC SN D@BG @CCDMC NE SGD OQDUHNTR CDBNLONRHSHNM RSDO 4GDQD @QD RTBG @CCDMCR @S SGD DMC ENQ ! U@QH@AKDR VGHBG HR UDQX M@STQ@K ENQ ONKXMNLH@K ETMBSHNMR (NVDUDQ SGD OQNAKDL HR SG@S @MX @CCDMC HM @MX KDUDK NE CDBNLONRHSHNM B@M AD ROKHS HMSN BNMUDW RTL VGDQD B@M AD @MX ETMBSHNM 4GHR L@JDR SGD CDBNLONRHSHNM ETKKX @LAHFTH SHNTR 4GD KDUDK NE QDBTQRHUD CDBNLONRHSHNM @MCNQ SGD EQDD ETMBSHNMR B@M AD RDKDBSDC HM NQCDQ SN HLOQNUD BNMSQNKK@AHKHSX NE CDBNLONRDC RXRSDL VHSG 4GD L@HM OQNAKDL VHSG .12 HR SG@S QRS HS HR MNS SGD NOSHLTL BNMSQNK NE FDMDQ@K MNMKHMD@Q RXRSDL @R HS G@R ADDM RGNVM HM "@MJR @MC -G@M@ @MC RDBNMC SGDQD HR RSHKK LHRRHMF @ OQNNE NE FKNA@K RS@AHKHSX NE SGD BNMSQNK K@V CDROHSD O@QSH@K QDRTKSR %QCDL @MC !KKDXMD 4GDQD G@R MNS ADDM ENTMC @MX B@RD NE TMRS@AKD .12 BNMSQNK ATS FDMDQ@K OQNNE HR LHRRHMF 4GD OQ@BSHB@K DWODQHDMBD HR DWBDKKDMS @MC SGD RS@AHKHSX B@M AD BGDBJDC ENQ
Modelling, Simulation and Control…
113
HMCHUHCT@K B@RDR TRHMF SGD BNLOTSDC RNKTSHNM NE SGD 2HBB@SSH DPT@SHNM 4GD ,X@OTMNU ETMBSHNM HR OQNONRDC @R E " E E
)S HR ONRHSHUD CDMHSD @BBNQCHMF SN SGD OQNODQSHDR NE SGD RNKTSHNM NE 2HBB@SSH DPT@SHNM @MC SGD MDF@SHUDMDRR NE HSR SHLD CDQHU@SHUD B@M AD BGDBJDC RHLHK@QKX HM SGD R@LD FQHC @R SGD BNLOTSDC U@KTDR NE SGD F@HM E CDQHUHMF HMSN " E E #" E 4 E $ 4 E " E E 4 E E
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
4GHR @OOQN@BG NE .12 HR @OOKHB@AKD SN @KK TMCDQ@BST@SDC BDQS@HMKX DPT@K@BST@SDC RXRSDLR SG@S R@SHREX SGD BNMCHSHNM SG@S SGD KHMD@QHYDC RXRSDL @ANTS SGD DPTHKHAQHTL E HR BNMSQNKK@AKD %UDM SGHR BNMCHSHNM HR UHNK@SDC ENQ HLONQS@MS BK@RRDR NE TMCDQ@BST@SDC RXRSDLR DF RNLD MNMGNKNMNLHB LTKSHANCX RXRSDLR &NQ SGHR BK@RR NE LTKSHANCX RXRSDLR SGD BNMSQNK HR UDQX CHBTKS 4GD RS@AHKHYHMF BNMSQNK V@R CDUDKNODC ENQ RTBG RXRSDLR G@UHMF 3E ATS ENQ FDMDQ@K RXRSDLR VHSG 3E HS HR RSHKK @M NODM OQNAKDL
" <129 "?2160A6C2 <;A?<9
-NCDK 0QDCHBSHUD #NMSQNK -0# G@R ADBNLD UDQX RTBBDRRETK @MC VHCDKX @BBDOSDC @R @CU@MBDC BNMSQNK LDSGNCNKNFX )S RNKUDR L@MX SQ@CHSHNM@K BNM SQNK OQNAKDLR AX SGD CHQDBS @OOKHB@SHNM NE ETMC@LDMS@K OQHMBHOKDR )MRSD@C NE CHBTKS @MC SDCHNTR MCHMF SGD BKNRDC KNNO NOSHL@K EDDCA@BJ BNMSQNK HS HR A@RDC NM SGD QDOD@SDC RNKTSHNM NE NODM KNNO NOSHL@K BNMSQNK TRHMF TOC@SDC
114
M. Valášek
RS@SD @MC JMNVKDCFD NE SGD OK@MS OQDCHBSHUD LNCDK RDD DF 2NRRHSDQ #@L@BGN @MC "NQCNMR 4GD FDMDQ@K ENQLTK@SHNM NE -0# HR ENKKNVHMF 4GD MNMKHMD@Q RXRSDL E- 3 E B
HR RTAIDBSDC SN SGD NOSHL@K BNMSQNK OQNAKDL VHSG SGD BNRS ETMBSHNM LHM B
E &
LHM
E B
@MC ONRRHAKD MNMKHMD@Q BNMRSQ@HMSR 5 E B
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
* ' ( B ( B
4GDM SGD SQ@MREDQ ETMBSHNM HR SQ@MRENQLDC HMSN SGD BK@RRHB@K CHRBQDSD RS@SD CDRBQHOSHNM VHSG KNB@KKX BNMRS@MS RS@SD L@SQHBDR @MC E E B F E
Modelling, Simulation and Control…
115
64B?2 -NCDK OQDCHBSHUD BNMSQNK ENQ SGD OQDCHBSHNM GNQHYNM
64B?2
-NCDK OQDCHBSHUD BNMSQNK ENQ SGD LNUDC OQDCHBSHNM GNQHYNM
116
M. Valášek
VGDQD E @QD RXRSDL RS@SDR @MC F @QD HMUDRSHF@SDC NTSOTSR 4GD OQDCHB SHUD BNMSQNK B@M AD @OOKHDC SN SGD LNCDK BQD@SDC TRHMF SGD @ANUD CDRBQHADC OQNBDCTQD 7QHSHMF SGD QDK@SHNM ENQ RTARDPTDMS RSDOR @R ENKKNVR F , E , E E B F , E B
, E E B B F , E B B @MC QDVQHSHMF SGDRD DPT@SHNMR HMSN SGD L@SQHW ENQL SGD OQDCHBSHNM NE SGD NTSOTSR HR NAS@HMDC F , 3 B
3
E
VGDQD
4GD BNMSQNK HR CDQHUDC EQNL SGD NOSHLHY@SHNM NE @ PT@CQ@SHB ODQENQL@MBD HMCDW 4GD ODQENQL@MBD HMCDW HR NOSHLHYDC HM SGD RSDO TRHMF SGD , F , OQDCHBSHNM F , F ENQ F F D # F D B B B 3 D # B 3 D B B VGDQD SGD RTLL@SHNM HR NUDQ SGD OQDCHBSHNM GNQHYNM F HR SGD NTSOTS UDB SNQ D HR SGD CDRHQDC NTSOTS UDBSNQ # HR @ ODM@KHY@SHNM L@SQHW NE SGD NTS OTSR VHSG HCDMSHSX L@SQHW HR @ ODM@KHY@SHNM NE SGD HMOTSR @MC B B B HR SGD HMOTS UDBSNQ &QNL SGD QDPTHQDLDMS NE SGD LHMHLHY@SHNM NE SGD ODQENQL@MBD HMCDW LHM B
B # # D 3
SGD BNMSQNK K@V HR CDQHUDC
Modelling, Simulation and Control…
117
(NVDUDQ NMKX SGD QRS DKDLDMS NE SGD UDBSNQ B HR TRDC ENQ SGD MD@QDRS BNMSQNK @BSHNM 4GD BNMSQNK OQNAKDL UDQX NESDM HMBKTCDR SGD BNMRSQ@HMSR NE SGD U@KTDR NE CHDQDMS U@QH@AKDR )M NQCDQ SN HMBNQONQ@SD SGDRD BNMRSQ@HMSR SGD CHQDBS BNLOTS@SHNM OQNBDCTQD LTRS AD FDMDQ@KHYDC 1T@CQ@SHB OQNFQ@LLHMF @OOD@QR SN AD @ FNNC SNNK ENQ SGHR OTQONRD 6@K@RDJ DS @K 4GD TMBNMRSQ@HMDC NOSHLHY@SHNM OQNAKDL HR QDENQLTK@SDC @R @ PT@CQ@SHB NOSHLHY@SHNM OQNAKDL B B 3 B LHM VHSG @ BNMRSQ@HMS BNMCHSHNM B /
4GD O@QSHBTK@Q L@SQHBDR TRDC ENQ PT@CQ@SHB OQNFQ@LLHMF ENKKNV EQNL DPT@ SHNMR # 3 # D 3 4GHR NOSHLHY@SHNM OQNAKDL HR RNKUDC HM DUDQX R@LOKHMF HMRS@MS !F@HM NMKX SGD QRS DKDLDMS NE UDBSNQ B HR TRDC ENQ SGD MD@QDRS BNMSQNK @BSHNM
205.A?<;60 .;1 )?62;19F %B@=2;@6<;
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
118
M. Valášek
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
<:3
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
Modelling, Simulation and Control…
119
64B?2 3JX(NNJ Z HCD@K BNMBDOS @ @MC QD@KHY@SHNM A
4GD ETQSGDQ CDRHFM BNMRHCDQ@SHNMR @QD A@RDC NM SGD RHLOKD PT@QSDQB@Q LNCDK &HFTQD $DROHSD HSR RHLOKHBHSX HS BNUDQR SGD ETMC@LDMS@K RTR ODMRHNM CXM@LHB OQNODQSHDR NE SGD QD@K UDGHBKD &NQ SGD CDRBQHOSHNM NE SGD BNMSQNK BNMBDOS @ KHMD@Q PT@QSDQB@Q LNCDK HR TRDC GNVDUDQ SGD MNMKHMD@QH SHDR NE SGD ENQBD DKDLDMSR @QD S@JDM HMSN @BBNTMS HM ETQSGDQ CDRBQHOSHNMR 4GD C@LOHMF ENQBD NE SGHR BSHSHNTR C@LODQ QDCTBDR SGD ROQTMF L@RR UHAQ@SHNM )SR DPT@SHNMR NE LNSHNM @QD + +
+ + +> +> + + + +
VGDQD HR SGD TMROQTMF L@RR HR SGD ROQTMF L@RR HR SGD RSHMDRR NE SGD L@HM ROQHMF HR SGD RSHMDRR NE SGD SXQD HR SGD SXQD C@LOHMF BNMRS@MS @MC HR SGD ENQBD NE SGD O@RRHUD NQ RDLH@BSHUD C@LODQ NQ @BSHUD DKDLDMS 4GD LD@MHMF NE SGD JHMDL@SHB@K PT@MSHSHDR + + @MC + HR @KRN BKD@Q EQNL SGHR FTQD &HFTQD @ QDOQDRDMSR SGD HCD@K BSHSHNTR B@RD 4GD O@Q@KKDK BSHSHNTR C@LODQ SQHDR SN @CC SGD GHFGDQ C@LOHMF SN SGD ROQTMF L@RR &HFTQD A QDOQDRDMSR SGD QD@KHY@SHNM NE SGHR BNMBDOS 4GD BSHSHNTR ENQBD BNLOTSDC EQNL @CCDC 3JX(NNJ C@LODQ HR @OOKHDC AX SGD @BST@SNQ 4GD @BST@SNQ B@M AD @ ETKKX @BSHUD DKDLDMS @BSHUD ENQBD FDMDQ@SNQ NQ @ RDLH@BSHUD DKDLDMS U@QH@AKD RGNBJ @ARNQADQ 4GD HCD@K @BSHUD ENQBD NE SGHR DKDLDMS @BBNQCHMF SN SGD BNMSQNK K@V NE SGD 3JX(NNJ HR +>
120
M. Valášek
)E SGHR ENQBD HR CHQDBSKX @OOKHDC AX @BSHUD ENQBD FDMDQ@SNQ SGDM HS HR SGD @BSHUD 3JX(NNJ RTRODMRHNM &NQ RDLH@BSHUD RTRODMRHNM SGHR ENQBD LTRS AD KHLHSDC SN SGD Q@MFD NE ENQBDR @OOKHB@AKD AX SGD RDLH@BSHUD CDUHBDR &NQ SGD HCD@K RDLH@BSHUD C@LODQ SGHR SQ@MRENQL@SHNM EQNL SGD QDPTHQDC @BSHUD ENQBD SN SGD RDSSHMF NE SGD C@LOHMF Q@SD A RTBG SG@S SGD C@LOHMF ENQBD HR MD@QDRS SN SGD CDRHQDC NMD &NQ @M HCD@K KHMD@Q U@QH@AKD RGNBJ @ARNQADQ SGD C@LOHMF Q@SD HR RDS ENQ SGD HMSDQU@K @R
HE HE HE +> +>
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
$<.1?62;19F %B@=2;@6<;
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
Modelling, Simulation and Control…
121
64B?2 0@QDSN RDS NE @SS@HM@AKD UDGHBKD OQNODQSHDR BNLENQS Z G@MCKHMF
KHMD@Q PT@QSDQB@Q LNCDK HR TRDC GNVDUDQ SGD MNMKHMD@QHSHDR NE SGD ENQBD DKDLDMSR @QD S@JDM HMSN @BBNTMS HM ETQSGDQ CDRBQHOSHNMR 4GD A@RHB HCD@ NE SGD %WSDMCDC 'QNTMC(NNJ %'( HR CDOHBSDC HM &HF TQD
64B?2 %WSDMCDC 'QNTMC(NNJ HCD@K BNMBDOS @ @MC QD@KHY@SHNM A
122
M. Valášek
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
4GD O@Q@LDSDQR VDQD CDSDQLHMDC AX LD@MR NE CHDQDMS NOSHLHY@SHNM OQN BDCTQDR ANSG CDSDQLHMHRSHB @MC FDMDSHB 4GD O@Q@LDSDQR NE SGD %WSDMCDC 'QNTMC(NNJ VDQD NQHFHM@KKX BNMRHCDQDC BNMRS@MS ENQ SGD VGNKD RGNBJ @A RNQADQ UDKNBHSX HMSDQU@K +NQS\ TL @MC 6@K@RDJ 6@K@RDJ @MC +NQS\ TL "DB@TRD SGD BG@Q@BSDQHRSHBR NE SGD RGNBJ @ARNQADQ @QD MNMKHMD@Q SGD MNMKHMD@Q %WSDMCDC 'QNTMC(NNJ BNMSQNK VHSG RS@SDCDODMCDMS F@HMR
Modelling, Simulation and Control…
123
F@HM RBGDCTKHMF HR TRDC 4GD RSQNMF MNMKHMD@QHSX NE SGD BNMSQNKKDC RGNBJ @ARNQADQ DRODBH@KKX HSR @RXLLDSQX B@M AD S@JDM HMSN @BBNTMS ENQ SGD CDSDQ LHM@SHNM NE BNMSQNKK@V O@Q@LDSDQR 4GDQDENQD SGD MNMKHMD@Q %'( UDQRHNM VGHBG DM@AKDR SGD RS@SDCDODMCDMS BNDBHDMSR F@HMR NE SGD %'( BNMSQNK K@V V@R CDUDKNODC VHSG DWBDKKDMS QDRTKSR 6@K@RDJ @MC +NQS\ TL 4GDHQ CDODMCDMBD NM SGD QDK@SHUD UDKNBHSX HR CDSDQLHMDC AX SGD -/0/ NO SHLHY@SHNM @OOQN@BG
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
124
M. Valášek
64B?2 #NMSQNKK@AKD C@LODQ @MC HSR BNMSQNKK@AKD BG@Q@BSDQHRSHBR
NE SGDRD QDRTKSR L@XAD SG@S SGD @KKNVDC TRDETK O@XKN@C B@M AD HMBQD@RDC AX SNM NM SNMR
?6142?62;19F %B@=2;@6<;
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
%<69?62;19F %B@=2;@6<;
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
Modelling, Simulation and Control…
125
64B?2 #NLO@QHRNM NE SGD LD@RTQDLDMS NE HLOQNUDLDMS NE QN@C EQHDMCKHMDRR
RHLTK@SHNM RSTCX 4GD FTQD CDRBQHADR SGD CDUDKNOLDMS NE SGD NARS@BKD HM SGD ENQL NE SGD HMHSH@K ATLO NM SGD RNHK QN@C TOK@MC R@MCX KN@L @ESDQ O@RRDR VHSG O@RRDMFDQ NQN@C VHSG O@RRHUD RDLH@BSHUD NQ KHLHSDC@BSHUD RTRODMRHNM 6@K@RDJ DS @K A 4GD TR@FD NE SGD O@RRHUD RTRODMRHNM KD@CR SN SGD CDUDKNOLDMS NE SXOHB@K V@UX QN@C SGD TR@FD NE SGD BNMSQNKKDC RTRODMRHNM RDLH@BSHUD NQ @BSHUD B@M OQDUDMS RTBG QN@C CDSDQHNQ@SHNM
?.823?62;19F %B@=2;@6<;
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
126
M. Valášek
64B?2 4HLD ADG@UHNTQ NE QN@CSXQD ENQBDR NM BNRHMTR ATLO #NLO@QH RNM ADSVDDM BNMUDMSHNM@K @MC BNMSQNKKDC C@LOHMF EQNL QD@K LD@RTQDLDMSR
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
Modelling, Simulation and Control…
127
64B?2 #NLO@QHRNM ADSVDDM O@RRHUD @MC RDLH@BSHUD RTRODMRHNM NM @ AQHCFD AQHCFD CDDBSHNM
128
M. Valášek
64B?2 0@QDSN RDS NE RNHKEQHDMCKX RTRODMRHNM @ @MC CDS@HK NE QDRTKSHMF CDENQL@SHNM @ESDQ QDODSHSHUD KN@CHMF AX O@RRHUD RDLH@BSHUD NQ KHLHSDC @BSHUD RTRODMRHNMR A
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
Modelling, Simulation and Control…
129
64B?2 4GD RSNOOHMF CHRS@MBD ENQ CHDQDMS RTRODMRHNM BNMBDOSR NM @ FNNC RSNBG@RSHB QN@C
4GD UDGHBKD G@R @M HMHSH@K UDKNBHSX NE JLG 4GD DLDQFDMBX AQ@JHMF L@MNDTUQD HR RHLTK@SDC NM @ QN@C DWBHS@SHNM QDOQDRDMSHMF @ FNNC RSNBG@RSHB QN@C 4GD RHLTK@SHNM QDRTKSR OQDRDMSDC HM &HFTQD HMCHB@SD @ RGNQSDMHMF NE SGD UDGHBKD RSNOOHMF CHRS@MBD AX SN LDSDQR "DRHCDR SGD HMTDMBD NE SGD RXLLDSQHB BG@Q@BSDQHRSHBR NE SGD RDLH@BSHUD C@LODQ SGD BNMSQHATSHNM NE SGD QDCTBDC SHLD BNMRS@MS HR NARDQU@AKD 4GD KHLHSDC @BSHUD BNMBDOS AQHMFR ETQSGDQ QDCTBSHNM NE SGD RSNOOHMF CHRS@MBD AX L BNLO@QDC SN SGD ADRS RDLH@BSHUD BNMBDOS !KK SGDRD QDRTKSR @QD A@RDC NM SGD RHLTK@SHNM RSTCX 6@K@RDJ DS @K )M NQCDQ SN U@KHC@SD SGD RHLTK@SHNM QDRTKSR @ (@QCV@QDHMSGD,NNO (H, SDRS QHF G@R ADDM CDRHFMDC @MC L@MTE@BSTQDC RDD &HFTQD @ 3SDHMA@TDQ DS @K 4GD PT@QSDQ B@Q HR DPTHOODC VHSG @ L@FMDSNQGDNKNFHB@K C@LODQ 4GD SDRS QHF QDOQDRDMSR SGD UDQSHB@K UDGHBKD CXM@LHBR NE SGD UDGHBKD CTQHMF SGD RSNOOHMF L@MNDTUQD 4GD KNMFHSTCHM@K CXM@LHBR HR LNCDKDC HM @ QD@KSHLD BNLOTSDQ DMUHQNMLDMS SNFDSGDQ VHSG SGD C@LODQ BNMSQNK @KFNQHSGL 4GD QDRTKSR &HFTQD A @BGHDUDC NM SGD SDRS QHF HMCHB@SD SG@S ENQ SGD FHUDM QN@C BNMCHSHNMR SGD RSNOOHMF CHRS@MBD G@R ADDM CDBQD@RDC AX @ANTS LDSDQR ENQ SGD AQ@JDEQHDMCKX RTRODMRHNM
130
M. Valášek
64B?2 1T@QSDQ B@Q SDRS QHF VHSG @ L@FMDSNQGDNKNFHB@K C@LODQ ENQ SGD (H, RHLTK@SHNM @ @MC SGD BNLO@QHRNM NE SGD RSNOOHMF CHRS@MBD ENQ O@RRHUD @MC AQ@JDEQHDMCKX %'( UDGHBKD
<129 /.@21 =?2160A6C2 0<;A?<9 <3 0<:/B@A6<; 2;46;2
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
Modelling, Simulation and Control…
131
SGHR LNCDK SGD ETSTQD DMFHMD ADG@UHNTQ HM CDODMCDMBD NM ONRRHAKD BNMSQNK HMOTSR ETDK HMIDBSHNMR RTODQBG@QFDQ AK@CD ONRHSHNM DSB @QD O@Q@LDSQHYDC @MC TRHMF NOSHLHY@SHNM NE ODQENQL@MBD HMCDW SGD BNMSQNK HMOTS HR BGNRDM
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
132
M. Valášek
OQN@BG 4GHQCKX SGD DMFHMD OQDCHBSHUD BNMSQNK BNMBDOS A@RDC NM ,/,)-/4 HCDMSHB@SHNM HR HLOKDLDMSDC SN @ BNLATRSHNM DMFHMD LNCDK @MC NUDQ@KK HM UDRSHF@SDC 4GDQD V@R @ RDQHNTR OQNAKDL GNV SN CDRBQHAD SGD BNLOKHB@SDC CXM@LHB LNCDK NE BNLATRSHNM DMFHMD VHSG L@MX SHLD CDK@XR @MC ONRRHAKD BNMSQNK HMOTSR NMKX HM RL@KK SHLD HMSDQU@K NE ETDK HMIDBSHNM 4GHR G@R ADDM RNKUDC AX TRHMF SGD CHRBQDSD HMRS@MBDR NE ETDK HMIDBSHNM @R SHLD CHRBQDSHY@SHNM NE SGD CXM@LHB LNCDK 4GD QDRTKSHMF CXM@LHB LNCDK HR @ CHRBQDSD CXM@LHB LNCDK CDRBQHAHMF SGD DMFHMD ADG@UHNTQ EQNL NMD ETDK HMIDBSHNM TMSHK SGD MDWS NMD 4GD RHLTK@SHNM QDRTKSR HMCHB@SD @ FNNC OQDCHBSHUD B@O@AHKHSX NE SGD @O OKHDC @OOQN@BG RDD &HFTQD @MC BNMSQNK B@O@AHKHSX NE OQDCHBSHUD BNMSQNK LDSGNC %6:B9.A6<; :<129 <3 0<:/B@A6<; 2;46;2 4GD DMFHMD RHLTK@SHNM LNCDK G@R ADDM CDUDKNODC HM NQCDQ SN BQD@SD @ QD@KHRSHB ATS BNLOTS@SHNM@KKX D BHDMS RHLTK@SHNM LNCDK -@BDJ DS @K 4GD OGXRHB@K RHLTK@SHNM LNCDK TRDR @KK RODBH@K ED@STQDR NE BNLATRSHNM DMFHMD RHLTK@SHNM 4GD LNCDK HR A@RDC NM @BBDOS@AKX RHLOKHDC RHLTK@SHNM LNCDK NE CHDRDK BNLATRSHNM DM FHMD &HFTQD
64B?2 #NMBDOST@K LNCDK NE $ CHDRDK BNLATRSHNM DMFHMD )S HR @ RNB@KKDC $ LNCDK SG@S QDOK@BDR SGD SGDQLNLDBG@MHB@K OQNBDRRDR
Modelling, Simulation and Control…
133
VHSG OTQD SHLD CDK@XR NE @OOQNOQH@SD U@QH@AKDR 4GD RHLTK@SHNM LNCDK G@R RS@SD U@QH@AKDR BQ@MJRG@ES @MFKD ;= ! BQ@MJRG@ES QDUNKTSHNMR ODQ LHMTSD ;QOLHM= ! QDUNKTSHNMR NE STQANBG@QFDQ ;QOLHM= @HQ L@RR HM HMKDS STAD ;JF= @HQ L@RR HM DWG@TRS OHOD ;JF= @HQ SDLODQ@STQD HM HMKDS STAD ;+= @HQ SDLODQ@STQD HM DWG@TRS OHOD ;+= @MC BNMSQNK U@QH@AKDR L@RR NE HMIDBSDC ETDK HMSN BXKHMCDQ ;JF= ! QDEDQDMBD @QD@ NE STQANBG@QFDQ ;L =
64B?2 -NCDK OQDCHBSHUD BNMSQNK NE DMFHMD QOL Z RHLTK@SHNM QDRTKSR KNVDQ BNMRSQ@HMS NE K@LAC@
<129 /.@21 2;46;2 =?2160A6C2 0<;A?<9 4GD LNCDK A@RDC OQDCHBSHUD BNMSQNK NE BNLATRSHNM DMFHMD HR A@RDC NM SGD #45,/,)-/4 OQDCHBSHUD LNCDK SG@S DM@AKDR SN BNLOTSD SGD RXRSDL L@SQHBDR ENQ -0# AX CHQDBS V@X 4GDM SGD SQ@MREDQ ETMBSHNM HR SQ@MRENQLDC HMSN SGD RS@SD
134
M. Valášek
RO@BD CDRBQHOSHNM "@RDC NM SG@S SGD OQDCHBSHUD BNMSQNK B@M AD @OOKHDC 4GD RNKTSHNM OQNBDCTQD HR FDMDQ@KHYDC TRHMF PT@CQ@SHB OQNFQ@LLHMF HM NQ CDQ SN D@RHKX HMBNQONQ@SD U@QHNTR BNMRSQ@HMS BNMCHSHNMR 4GD @KFNQHSGL NE LNCDK OQDCHBSHUD BNMSQNK A@RDC NM HCDMSHDC #45,/,)-/4 LNCDKR @MC PT@CQ@SHB OQNFQ@LLHMF B@M AD AQHDX RTLL@QHYDC @R ENKKNVR 6@K@RDJ DS @K
64B?2 -NCDK OQDCHBSHUD BNMSQNK NE DMFHMD QOL Z RHLTK@SHNM QDRTKSR SVN RHCDC BNMRSQ@HMS NE K@LAC@ 4GD MNMKHMD@Q OQDCHBSHUD #45,/,)-/4 LNCDKR NE LNSNQ @MCNQ BNLOKDW RHLTK@SHNM LNCDK NE LNSNQ @QD HCDMSHDC @RRDLAKDC @MC SDRSDC AX #45,/,)-/4 OQNFQ@L 4GD LD@RTQ@AKD PT@MSHSHDR NE RXRSDL @QD BGNRDM @R NTSOTSR NE HMCHUHCT@K #45,/,)-/4 LNCDKR 4GD @CCHSHNM@K #45,/,)-/4 LNCDKR @QD @RRDLAKDC @MC SDRSDC ENQ BNMSQNKKDC PT@MSHSHDR ENQ DW@LOKD K@LAC@ RTAIDBSDC SN BNMRSQ@HMSR CTQHMF BNMSQNK 4GD MNMKHMD@Q OQDCHBSHUD #45,/,)-/4 LNCDKR @QD HMBNQONQ@SDC HMSN BNMSQNKKDQ RHLTK@SHNM LNCDK NE BNMSQNKKDQ 4GD NTSOTSR NE L@HM
Modelling, Simulation and Control…
135
#45,/,)-/4 LNCDKR LD@RTQ@AKD PT@MSHSHDR @QD BNMRHCDQDC @R RS@SDR NE RTARSHSTSHNM@K KHMD@Q LNCDK TRDC ENQ OQDCHBSHUD BNMSQNK 4G@S LD@MR SGD #45,/,)-/4 LNCDKR @QD KNB@KKX ENQ LD@RTQDC @BST@K U@KTDR NE RS@SDR KHMD@QHYDC @MC HMRDQSDC HMSN L@SQHW ENQL 4GD KNB@KKX KHMD@QHYDC @CCHSHNM@K #45,/,)-/4 LNCDKR @QD @RRDLAKDC SN DPT@KHSX @MC MNMDPT@KHSX BNMRSQ@HMSR !BST@KKX SGD TOODQ @MC KNVDQ KHLHSR NE K@LAC@ G@UD ADDM BNMRHCDQDC 4GD BNMSQNK HMOTS HR BNLOTSDC AX RNKTSHNM NE PT@CQ@SHB OQNFQ@LLHMF OQNAKDL /MKX SGD QRS DKDLDMS NE UDBSNQ B HR TRDC ENQ SGD MD@QDRS BNMSQNK @BSHNM 4GD PT@KHSX NE #45,/,)-/4 OQDCHBSHUD LNCDKR HR CDLNMRSQ@SDC HM &HFTQD 4GD RHLTK@SHNM QDRTKSR NE BNMSQNK @QD NM &HFTQD 4GD MTLDQHB@K DWODQHLDMSR RGNV SG@S SGD TRDC OQDCHBSHNM GNQHYNM RGNTKC AD SQTMB@SDC HM SGD UHBHMHSX NE K@LAC@ KHLHSR RDD SGD KHMD NM &HFTQD A 4GD @OOKHB@SHNM NE KHLHSR @DBSR @SS@HM@AKD U@KTDR NE OQHL@QHKX BNMSQNKKDC PT@MSHSX QOL 3HLHK@Q QDRTKSR @QD ENQ SVN RHCDC BNMRSQ@HMS NE K@LAC@ HM &HFTQD 4GD QD@K DWODQHLDMS VHSG HLOKDLDMSDC -0# VHSG #45,/,)-/4 OQDCHBSHUD LNCDKR HR HM &HFTQD )S HR SGD -0# BNMSQNK NE RTODQBG@QFDC F@R DMFHMD GDKC NM BNMRS@MS RODDC VGDQD SGD DMFHMD SNQPTD HR BNMSQNKKDC AX LD@MR NE F@R @O NODQ@SHNM
64B?2 )LOKDLDMS@SHNM NE -0# BNMSQNK NM RTODQBG@QFDC F@R DMFHMD
136
M. Valášek
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
)M SGHR BNMSQHATSHNM SGD V@XR NE DBHDMS LNCDKHMF RHLTK@SHNM @MC LNCDK A@RDC BNMSQNK CDRHFM G@UD ADDM CDRBQHADC 4GDX @QD DRRDMSH@K ENQ SGD CDRHFM @MC CDUDKNOLDMS NE LDBG@SQNMHB@K RXRSDLR 4GD CDRBQHADC LDSGNCR VDQD @OOKHDC ENQ SGD CDRHFM NE HMSDFQ@SDC BG@RRHR @MC BNLATRSHNM DMFHMD BNMSQNK NE UDGHBKDR 4GD @OOKHB@SHNMR G@UD CDLNMRSQ@SDC SG@S SGD OQNONRDC LDSGNCR @QD ETKKX @OOKHB@AKD HM OQ@BSHBD @MC UDQX DBHDMS
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
-&2*$". /.42/, /.'&2&.$& !KATPTDQPTD O@FDR * 2 #KNTSHDQ $ 4 3S@MRADQX @MC - 3YM@HDQ /M SGD QDBNUDQ@AHKHSX NE MNMKHMD@Q RS@SD EDDCA@BJ K@VR AX DWSDMCDC KHMD@QHY@SHNM BNMSQNK SDBG
Modelling, Simulation and Control…
137
MHPTDR 2/$ /' 4)&
-&2*$". /.42/, /.'&2&.$& 3@M $HDFN O@FDR $)6).% 2/$&&%*.(3 /' /.$,5%*.( /.'&2&.$& 2NSSDQC@L $9-/,! GSSOVVVCXM@RHLRD %- %KADGDHQX $# +@QMNOO -% %K@Q@AX @MC !- !ACDKQ@@NTE !C U@MBDC 'QNTMC 6DGHBKD 3TRODMRHNM 3XRSDLR @ #K@RRHDC "HAKHNFQ@GX &)*$,& 834&- 8."-*$3 %" %QCDL @MC ! !KKDXMD 'KNA@KKX RS@AHKHYHMF RDBNMC NQCDQ MNMKHMD@Q RXR SDLR AX 3$2% BNMSQNK 2/$ /' 4)&
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
138
M. Valášek
2 +\ TAKDQ @MC 7 3BGHDGKDM 4VN -DSGNCR NE 3HLTK@SNQ #NTOKHMF "4) &-"4*$", ".% /-054&2 /%&,,*.( /' 8."-*$", 834&-3 7 ,@MFRNM @MC ! !KKDXMD )MHMSD GNQHYNM NOSHL@K BNMSQNK NE @ BK@RR NE MNMKHMD@Q RXRSDLR 2/$ /' 4)& -&2*$". /.42/, /.'&2&.$&
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
Modelling, Simulation and Control…
139
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
"@KJDL@ 2NSSDQC@L O@FDR - 6@K@RDJ @MC 7 +NQS\ TL 3DLH@BSHUD 3TRODMRHNM 3XRSDLR )) )M & $)".*$", &3*(. ".%#//+ #2# 0QDRR - 6@K@RDJ 7 +NQS\ TL : 3HJ@ , -@FCNKDM @MC / 6@BTKHM $DUDKNOLDMS NE 3DLH!BSHUD 2N@C&QHDMCKX 4QTBJ 3TRODMRHNMR /.42/, .(*.&&2*.( 2"$4*$&
140
M. Valášek
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
-=L=J J==