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Decentralized Control of Uncertain Systems Via Sensitivity Models
Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993
DECENTRALIZED CONTROL OF UNCERTAIN SYSTEMS VIA SENSITIVITY MODELS D.A. Schoenwald* and U. Ozguner** ./nstrumentation and Controls Division, Oak Ridge National Laboratory, P.O. Box 2009, Oak Ridge, IN 37831-8066, USA ••The Ohio State University, Department of Electrical Engineering, Columbus, OH 43210, USA
Ab8tract. In this paper, we present a decentralized strategy for optimal control of interconnected systems exhibiting parametric uncertainty. First, we demonstrate that sensitivity models for linear interconnected systems can be generated at each subsystem U8ing only locally available information. Second, we present an optimal control law that incorporates sensitivity functions in the feedback path. The control scheme is completely decentralized and is proposed as a means of making the closed loop system less sensitive to parameter deviations. Finally, we give an example of an interconnected system and show how this control strategy is implemented.
Key Word8. Decentralized control; interconnected systems; optimal control; parametric uncertainty; sensitivity analysis
classes of linear systems. Ultimately, the importance of this paper is to show that any control law that utilizes sensitivity functions (e.g. adaptive or optimal control laws) can be done in a decentralized framework .
1 INTRODUCTION The use of sensitivity functions in control theory to make the closed loop system less susceptible to changes in plant parameters has been studied for several decades. In particular, methods of generating sensitivity functions such that they can be utilized on-line in a control system have been extensively researched. Sensitivity models are a means of generating these sensitivity functions from the nominal plant model. However, very little effort has been spent in generating sensitivity models for coupled subsystems in a decentralized manner.
2 DECENTRALIZED SENSITIVITY MODELS Consider two interconnected MIMO linear systems. The outputs, Yi, are of dimension Pi, i = 1,2. The inputs, Ui, are of dimension mi. The unknown parameter vectors, ai, Pi, ,i, are of dimensions ni, ri, and qi, respectively. The transfer matrices, Qi, Wi, Wij, are of compatible dimensions. All vectors and matrices are functions of the Laplace Transform complex variable, s. The transfer matrices, Qi, represent dynamic feedback from the outputs to the inputs. The transfer matrices, ",'ij, represent coupling terms between the subsystems. The transfer matrices, Wi, represent the primary dynamics between plant inputs Ui and plant outputs Yi .
As an additional tool for decentralized control, it is of interest to determine the feasibility of generating these models using only local signals for interconnected systems. That is, we wish to investigate the possibility that the ith subsystem's output sensitivity function can be generated using only plant signals from the ith subsystem. This is important if one wants to use sensitivity functions in a decentralized control environment. For instance, one could use decentralized sensitivity functions for self-tuning control (tuning the gains of a control law when some of the plant parameters are unknown) as is done in the work of Hung (1985).
The block transfer matrix of the entire system can be obtained by writing input-output relationships as follows Yi = WdUl - Q1Yi) + W 21 (U2 - Q21'2) 1'2 = W2 (U2 - Q21'2) + W 12 (Ul - Q1Yi) .
(1) (2)
Letting ~l = 1+ W1Ql - W 21 Q2[I + W 2Q2]-lW12 Ql 1 ~2 = 1+ W 2Q2 - W12QdI + W 1Qd- W 21 Q2
In this paper, decentralized sensitivity models are suggested for robust optimal control of certain 539
where I is the identity matrix, we obtain the input-output description of the system (after some algebraic manipulation)
with linear couplings similar to that of (Siljak 1991) but utilizing sensitivity functions to address parametric uncertainty in an optimal manner. A version of the centralized case appears in (Frank 1978). The performance index consists of a sum of N local cost criterions where N is the number of subsystems.
with
Formally, we have
Fu Fn
= =
F21 F22
= =
ai'"l[Wl - W:nQl(I + W2Q2)-lWn ] ai'"l[W21 - W21 Q2(I + W2Q2)-lW2] a;l[W12 - W12Ql(I + W1QJ)- l W1] a;l[W2 - W12Ql(I + W1 QJ)- l W21 ].
N
Xi
= Ai(cri)Xi+ LAiiXi+BiUi ,j = 1, . . . , N(8) ;=1
i~i
We are interested in output sensitivity vectors of the system with respect to the unknown parameter vectors, ai, Pi, and "Yi. It is sufficient to study sensitivity vectors of the first subsystem since the two subsystems are symmetric as is easily observed.
with :J:i E JlRi, Ui E Rmi, and ai E Rpi where ai are the vectors of unknown parameters. It is desired to minimize the quadratic cost criterion J
=
t la{'Xl (xT
Qi:J:i + uT R;Ui
i=1
We proceed with the sensitivity ofYl with respect to al which begins by utilizing (1)
N
+
(9)
"AT.S··A L..J 'J 'J 'J·· )dt j=1
via local state feedback where Qi ,Sij are positive semidefinite matrices, R; are positive definite matrices, and Aij = ~ lai=ai are the sensitivity vectors associated with the ith subsystem with a~ the nominal parameter values. As was demonstrated in the previous section, the crosssensitivity functions Aij, j :f:. i can be generated at the ith subsystem with only local information needed.
(4) where ali is the ith component of the unknown parameter vector. Continuing with the analysis, one obtains (5)
The sensitivity models are described by the following linear time-invariant differential equations
which leads to
~ii =
BABa>i
N
+ AiAi; + LA;jA;j
•
Noting that the term in parentheses is merely (1), we have the result
~ij
(10)
j=1
jeti
= AjAij + LAj}Ai}
(11)
.=1 }etj
(7)
which again demonstrate the decentralized nature of sensitivity models for the system (8). Define the augmented state vector
which is a completely decentralized result. The output sensitivity vector depends only on the output signal itself plus some transfer matrices. The remaining sensitivity functions can be obtained in a likewise manner, but the analysis is omitted here for brevity (see (Schoenwald 1992) for more details).
(12) which leads to the full augmented state space description
4 DECENTRALIZED OPTIMAL CONTROL VIA SENSITIVITY FUNCTIONS
(13) where
= [iT···i~f, A = block-diag[A 1 . .. AN], Ae = [A;j] , j:f:. i, B= block-diag [B 1 ··.BN ],
i
It is now of interest to apply the above decentralized sensitivity models to performance issues associated with decentralized control. The framework pursued here is that of linear subsystems
and U = 540
[uT ·· . u~] .
The performance criterion (9) can be reexpressed in these new coordinates as
The control law (17) is completely decentralized, u = -KjZi, which means that each subsystem can be regulated with only locally available information even in the presence of parametric uncertainty. Indeed, it is the use of locally generated sensitivity functions that sets this strategy apart from others. Fig. 1 illustrates this method.
(14)
where Qi = block-diag[Qj Sjj Sjj]. This is written in the full state space as
3 EXAMPLE
We consider a system consisting of two inverted penduli coupled by a spring subject to two independent torque inputs as shown in Fig. 2. Physically, this system is analogous to two one-link manipulators joined together by a string, cable, or other spring-like medium. The deflections from vertical are assumed to be small enough such that the gravity term can be linearized. The equations of motion are (Siljak 1991)
(15)
Q = block-diag [Q1' .. QN] and R = block-diag [RI' .. RN]' Note that Q and R will
where
be positive semidefinite and positive definite, respectively, if Qi ,Sij and Ri are positive semidefinite and positive definite, respectively.
2"
ml1 (11 2" ml2 (12
It is assumed that the decoupled subsystems
= =
ka «(11 - (12) + U1(19) 2 mgl2(12 - ka «(12 - (It) + U2 2
mgl1 (11 -
where all parameters are defined in Fig. 2 except for 9 which is the gravitational constant. The state vector is chosen as Xl = «(l1,0 1 )T, the input vector is u = (U1, U2)T , and the uncertain parameters are cri = f. These parameters physically represent the squ~res ofthe natural frequencies of oscillation of the de coupled penduli. We assume some uncertainty from their nominal values.
(16)
are controllable, i.e., (A, B) are a controllable pair. Thus, the cost criterion (15) can be minimized by solving the linear quadratic regulator separately for each subsystem. That is, let (17)
u=-Kz
where K = block-diag[K1 ·· ·KN]. These Ki are computed by solving the algebraic Riccati equation
for the unique positive definite matrix P = block-diag [PI' .. PN]. Because of the structure of A, iJ, Q, R, the solution P will automatically be in this block-diagonal form.
Fig. 2. Inverted penduli coupled by a spring. Utilizing (10)-(11), the sensitivity vectors
u
ith Subsystem
~
ii, ij Sensitivity
~
~ll
A.ii,A.1j
_ [ 0 1
~
+[
on
+ [ ;; mll
~
] Xl
~
~12
=[
~2l
= [ 0 n_O~ ~ ml 1
Models
n 0~
O2 -
ml~
~22
LQRLaw I.
_ [ 0 1
~
+ [ ;ml.;
~
] All
] A2l
+ [ ;mll;
~
] A22
on
+[ ~•
~
0
Fig. 1. Decentralized optimal control with sensitivity models. 541
] A12
+[
] X2
] All
mll
] A12
1
1
~~ ~
1
2
~~ ~ ml.
] A2l
] A22
closed loop system once inaccuracies in the parameters are introduced. In fact, even with no uncertainty in the parameters, one of the closed loop poles is at the origin once the coupling matrix is included. As the uncertainty is increased this pole moves further into the right half plane. But for both cases of uncertainty (O, 20%), the higher order design with sensitivity models maintains closed loop stability. The price paid is a higher order system, but the gain is a significant amount of stability robustness with respect to parametric uncertainty.
are generated. The augmented state vector % is formed as follows (20) which is 12th order. We choose ka 2 = IN'm, ml~ = lkg·m2 , ml~ = 0.5kg.m2 , Q~ = =1 and Q~ = f; = 2 ~.
t
i2 ,
The quadratic cost criterion is chosen such that all states and sensitivity functions are weighted equally, i.e., Q is a 12x12 identity matrix. Likewise, R is selected to be a 2x2 identity matrix. Analysis simulated on MATLAB solves the Riccati equation (18) and implements the decentralized control strategy of the last section. For comparison purposes, a decentralized design is carried out on the same system without using sensitivity models. That is, the nominal parameter values were taken as exact in the control design. The uncertainties tested were 10% and 20%, respectively, i.e., the true values were Q1 = 1.1, Q2 = 2.2, and Q1 = 1.2, Q2 = 2.4 for the two simulation runs. For both designs, the controllability assumption is satisfied.
Finally, some comments are in order concerning the relationship of the above problem to the question of transient stability in power systems. The above system is mathematically similar (though not exact) to a two-machine swing equation model of a power system. The equations for a single machine are quite similar to that of a simple pendulum as noted in (Lefebvre et al. 1983). Adding a second machine results in coupling close to the spring connection in the above example. The forces exerted on the penduli correspond to the electrical power delivered by the machines. Thus, some transient stability results can be studied from this example, however in a realistic setting, constraints would have to be imposed on the inputs to the machines .
TABLE 1 Closed loop poles of simulation runs with no uncertainty.
No iahrco •• ectio ••
I.t erco •• ectio ••
TABLE 2
-0 .88e%j O.5 -1 .414 _1.414
.. O.:J3,:i:j 1 .255 _0.824%jO .582 ..1.374, .. LOst _0 .2'%j1.20e _1.198%jO . 41
- 1.118%jl .207 -2.'29 0 .0
.. 1.'79 .. 1.0"6 -2.U7%j1.52 _0.U%jl.724 - 0 . 138%j1.259 _0 . 102%)1.04 -0 .22%jO .'01 -1.1 . . % · 0.00.
ACKNOWLEDGMENT The authors wish to acknowledge the support of the AFOSR under contract F49620-89-C-0046. The first author gratefully acknowledges the support of the Ohio Aerospace Institute through a doctoral fellowship.
Closed loop poles of simulation runs with 20% uncertainty. 0 .109
No iaterco •• ectio ••
-1 .• 4
0.135 -2 .983
REFERENCES
Se •• it.i.ity model -0 .24.:t:j 1.4U -0 . 17'%jO .&41 -2 .7.e. - ' .707 -0 .281%jl .683 -0 . 11'%jO.90&
Frank, P. M. (1978) . Introduction to System Sensitivity Theory, New York: Academic Press . Hung, S. T . (1985) . Multi-input/multi-outputsensitivity points tuning. M. S. Thesis, University of Illinois, Urbana, IL, May. Lefebvre, S., S. Richter, and R . DeCarlo (1983) Decentralized Variable Structure Control Design for a Two-Pendulum System. IEEE Transactions on Automatic Control, vol. AC-28, no. 12, pp. 1112-1114. Schoenwald, D. A. (1992). Analysis and Design of Robust Decentralized Controllers for Nonlinear Systems. Ph.D. Dissertation, The Ohio State University, Columbus, OH, June. Siljak . D. D. (1991) . Decentralized Control of Complex Systems. New York: Academic Press.
.. 1.1 . . . . 1.11'1 .. 3 .45:1jO .• 59
laterco •• ecdo ••
-' . 114
- 0 .lU%j1.784
0.e01 _0 .1'11
_0.18.%jl . 341 _0 .043:l:.jO . 17. .. O, 113:1::jO .•••
-1.177
-1.18&% · 0 .001
The results are summarized in Tables 1-2. The first column of numbers of each table represent the closed loop poles for the 4th order decentralized design without sensitivity models. The second column of numbers of each table represent the closed loop poles for the 12th order decentralized design with sensitivity models. The first row of each table corresponds to the case of ignoring the interconnection terms whereas the second row includes these terms in the closed loop analysis. The results show that the lower order design without sensitivity models fails to stabilize the 542
DECENTRALIZED CONTROL OF UNCERTAIN SYSTEMS VIA SENSITIVITY MODELS D.A. Schoenwald* and U. Ozguner** ./nstrumentation and Controls Division, Oak Ridge National Laboratory, P.O. Box 2009, Oak Ridge, IN 37831-8066, USA ••The Ohio State University, Department of Electrical Engineering, Columbus, OH 43210, USA
Ab8tract. In this paper, we present a decentralized strategy for optimal control of interconnected systems exhibiting parametric uncertainty. First, we demonstrate that sensitivity models for linear interconnected systems can be generated at each subsystem U8ing only locally available information. Second, we present an optimal control law that incorporates sensitivity functions in the feedback path. The control scheme is completely decentralized and is proposed as a means of making the closed loop system less sensitive to parameter deviations. Finally, we give an example of an interconnected system and show how this control strategy is implemented.
Key Word8. Decentralized control; interconnected systems; optimal control; parametric uncertainty; sensitivity analysis
classes of linear systems. Ultimately, the importance of this paper is to show that any control law that utilizes sensitivity functions (e.g. adaptive or optimal control laws) can be done in a decentralized framework .
1 INTRODUCTION The use of sensitivity functions in control theory to make the closed loop system less susceptible to changes in plant parameters has been studied for several decades. In particular, methods of generating sensitivity functions such that they can be utilized on-line in a control system have been extensively researched. Sensitivity models are a means of generating these sensitivity functions from the nominal plant model. However, very little effort has been spent in generating sensitivity models for coupled subsystems in a decentralized manner.
2 DECENTRALIZED SENSITIVITY MODELS Consider two interconnected MIMO linear systems. The outputs, Yi, are of dimension Pi, i = 1,2. The inputs, Ui, are of dimension mi. The unknown parameter vectors, ai, Pi, ,i, are of dimensions ni, ri, and qi, respectively. The transfer matrices, Qi, Wi, Wij, are of compatible dimensions. All vectors and matrices are functions of the Laplace Transform complex variable, s. The transfer matrices, Qi, represent dynamic feedback from the outputs to the inputs. The transfer matrices, ",'ij, represent coupling terms between the subsystems. The transfer matrices, Wi, represent the primary dynamics between plant inputs Ui and plant outputs Yi .
As an additional tool for decentralized control, it is of interest to determine the feasibility of generating these models using only local signals for interconnected systems. That is, we wish to investigate the possibility that the ith subsystem's output sensitivity function can be generated using only plant signals from the ith subsystem. This is important if one wants to use sensitivity functions in a decentralized control environment. For instance, one could use decentralized sensitivity functions for self-tuning control (tuning the gains of a control law when some of the plant parameters are unknown) as is done in the work of Hung (1985).
The block transfer matrix of the entire system can be obtained by writing input-output relationships as follows Yi = WdUl - Q1Yi) + W 21 (U2 - Q21'2) 1'2 = W2 (U2 - Q21'2) + W 12 (Ul - Q1Yi) .
(1) (2)
Letting ~l = 1+ W1Ql - W 21 Q2[I + W 2Q2]-lW12 Ql 1 ~2 = 1+ W 2Q2 - W12QdI + W 1Qd- W 21 Q2
In this paper, decentralized sensitivity models are suggested for robust optimal control of certain 539
where I is the identity matrix, we obtain the input-output description of the system (after some algebraic manipulation)
with linear couplings similar to that of (Siljak 1991) but utilizing sensitivity functions to address parametric uncertainty in an optimal manner. A version of the centralized case appears in (Frank 1978). The performance index consists of a sum of N local cost criterions where N is the number of subsystems.
with
Formally, we have
Fu Fn
= =
F21 F22
= =
ai'"l[Wl - W:nQl(I + W2Q2)-lWn ] ai'"l[W21 - W21 Q2(I + W2Q2)-lW2] a;l[W12 - W12Ql(I + W1QJ)- l W1] a;l[W2 - W12Ql(I + W1 QJ)- l W21 ].
N
Xi
= Ai(cri)Xi+ LAiiXi+BiUi ,j = 1, . . . , N(8) ;=1
i~i
We are interested in output sensitivity vectors of the system with respect to the unknown parameter vectors, ai, Pi, and "Yi. It is sufficient to study sensitivity vectors of the first subsystem since the two subsystems are symmetric as is easily observed.
with :J:i E JlRi, Ui E Rmi, and ai E Rpi where ai are the vectors of unknown parameters. It is desired to minimize the quadratic cost criterion J
=
t la{'Xl (xT
Qi:J:i + uT R;Ui
i=1
We proceed with the sensitivity ofYl with respect to al which begins by utilizing (1)
N
+
(9)
"AT.S··A L..J 'J 'J 'J·· )dt j=1
via local state feedback where Qi ,Sij are positive semidefinite matrices, R; are positive definite matrices, and Aij = ~ lai=ai are the sensitivity vectors associated with the ith subsystem with a~ the nominal parameter values. As was demonstrated in the previous section, the crosssensitivity functions Aij, j :f:. i can be generated at the ith subsystem with only local information needed.
(4) where ali is the ith component of the unknown parameter vector. Continuing with the analysis, one obtains (5)
The sensitivity models are described by the following linear time-invariant differential equations
which leads to
~ii =
BABa>i
N
+ AiAi; + LA;jA;j
•
Noting that the term in parentheses is merely (1), we have the result
~ij
(10)
j=1
jeti
= AjAij + LAj}Ai}
(11)
.=1 }etj
(7)
which again demonstrate the decentralized nature of sensitivity models for the system (8). Define the augmented state vector
which is a completely decentralized result. The output sensitivity vector depends only on the output signal itself plus some transfer matrices. The remaining sensitivity functions can be obtained in a likewise manner, but the analysis is omitted here for brevity (see (Schoenwald 1992) for more details).
(12) which leads to the full augmented state space description
4 DECENTRALIZED OPTIMAL CONTROL VIA SENSITIVITY FUNCTIONS
(13) where
= [iT···i~f, A = block-diag[A 1 . .. AN], Ae = [A;j] , j:f:. i, B= block-diag [B 1 ··.BN ],
i
It is now of interest to apply the above decentralized sensitivity models to performance issues associated with decentralized control. The framework pursued here is that of linear subsystems
and U = 540
[uT ·· . u~] .
The performance criterion (9) can be reexpressed in these new coordinates as
The control law (17) is completely decentralized, u = -KjZi, which means that each subsystem can be regulated with only locally available information even in the presence of parametric uncertainty. Indeed, it is the use of locally generated sensitivity functions that sets this strategy apart from others. Fig. 1 illustrates this method.
(14)
where Qi = block-diag[Qj Sjj Sjj]. This is written in the full state space as
3 EXAMPLE
We consider a system consisting of two inverted penduli coupled by a spring subject to two independent torque inputs as shown in Fig. 2. Physically, this system is analogous to two one-link manipulators joined together by a string, cable, or other spring-like medium. The deflections from vertical are assumed to be small enough such that the gravity term can be linearized. The equations of motion are (Siljak 1991)
(15)
Q = block-diag [Q1' .. QN] and R = block-diag [RI' .. RN]' Note that Q and R will
where
be positive semidefinite and positive definite, respectively, if Qi ,Sij and Ri are positive semidefinite and positive definite, respectively.
2"
ml1 (11 2" ml2 (12
It is assumed that the decoupled subsystems
= =
ka «(11 - (12) + U1(19) 2 mgl2(12 - ka «(12 - (It) + U2 2
mgl1 (11 -
where all parameters are defined in Fig. 2 except for 9 which is the gravitational constant. The state vector is chosen as Xl = «(l1,0 1 )T, the input vector is u = (U1, U2)T , and the uncertain parameters are cri = f. These parameters physically represent the squ~res ofthe natural frequencies of oscillation of the de coupled penduli. We assume some uncertainty from their nominal values.
(16)
are controllable, i.e., (A, B) are a controllable pair. Thus, the cost criterion (15) can be minimized by solving the linear quadratic regulator separately for each subsystem. That is, let (17)
u=-Kz
where K = block-diag[K1 ·· ·KN]. These Ki are computed by solving the algebraic Riccati equation
for the unique positive definite matrix P = block-diag [PI' .. PN]. Because of the structure of A, iJ, Q, R, the solution P will automatically be in this block-diagonal form.
Fig. 2. Inverted penduli coupled by a spring. Utilizing (10)-(11), the sensitivity vectors
u
ith Subsystem
~
ii, ij Sensitivity
~
~ll
A.ii,A.1j
_ [ 0 1
~
+[
on
+ [ ;; mll
~
] Xl
~
~12
=[
~2l
= [ 0 n_O~ ~ ml 1
Models
n 0~
O2 -
ml~
~22
LQRLaw I.
_ [ 0 1
~
+ [ ;ml.;
~
] All
] A2l
+ [ ;mll;
~
] A22
on
+[ ~•
~
0
Fig. 1. Decentralized optimal control with sensitivity models. 541
] A12
+[
] X2
] All
mll
] A12
1
1
~~ ~
1
2
~~ ~ ml.
] A2l
] A22
closed loop system once inaccuracies in the parameters are introduced. In fact, even with no uncertainty in the parameters, one of the closed loop poles is at the origin once the coupling matrix is included. As the uncertainty is increased this pole moves further into the right half plane. But for both cases of uncertainty (O, 20%), the higher order design with sensitivity models maintains closed loop stability. The price paid is a higher order system, but the gain is a significant amount of stability robustness with respect to parametric uncertainty.
are generated. The augmented state vector % is formed as follows (20) which is 12th order. We choose ka 2 = IN'm, ml~ = lkg·m2 , ml~ = 0.5kg.m2 , Q~ = =1 and Q~ = f; = 2 ~.
t
i2 ,
The quadratic cost criterion is chosen such that all states and sensitivity functions are weighted equally, i.e., Q is a 12x12 identity matrix. Likewise, R is selected to be a 2x2 identity matrix. Analysis simulated on MATLAB solves the Riccati equation (18) and implements the decentralized control strategy of the last section. For comparison purposes, a decentralized design is carried out on the same system without using sensitivity models. That is, the nominal parameter values were taken as exact in the control design. The uncertainties tested were 10% and 20%, respectively, i.e., the true values were Q1 = 1.1, Q2 = 2.2, and Q1 = 1.2, Q2 = 2.4 for the two simulation runs. For both designs, the controllability assumption is satisfied.
Finally, some comments are in order concerning the relationship of the above problem to the question of transient stability in power systems. The above system is mathematically similar (though not exact) to a two-machine swing equation model of a power system. The equations for a single machine are quite similar to that of a simple pendulum as noted in (Lefebvre et al. 1983). Adding a second machine results in coupling close to the spring connection in the above example. The forces exerted on the penduli correspond to the electrical power delivered by the machines. Thus, some transient stability results can be studied from this example, however in a realistic setting, constraints would have to be imposed on the inputs to the machines .
TABLE 1 Closed loop poles of simulation runs with no uncertainty.
No iahrco •• ectio ••
I.t erco •• ectio ••
TABLE 2
-0 .88e%j O.5 -1 .414 _1.414
.. O.:J3,:i:j 1 .255 _0.824%jO .582 ..1.374, .. LOst _0 .2'%j1.20e _1.198%jO . 41
- 1.118%jl .207 -2.'29 0 .0
.. 1.'79 .. 1.0"6 -2.U7%j1.52 _0.U%jl.724 - 0 . 138%j1.259 _0 . 102%)1.04 -0 .22%jO .'01 -1.1 . . % · 0.00.
ACKNOWLEDGMENT The authors wish to acknowledge the support of the AFOSR under contract F49620-89-C-0046. The first author gratefully acknowledges the support of the Ohio Aerospace Institute through a doctoral fellowship.
Closed loop poles of simulation runs with 20% uncertainty. 0 .109
No iaterco •• ectio ••
-1 .• 4
0.135 -2 .983
REFERENCES
Se •• it.i.ity model -0 .24.:t:j 1.4U -0 . 17'%jO .&41 -2 .7.e. - ' .707 -0 .281%jl .683 -0 . 11'%jO.90&
Frank, P. M. (1978) . Introduction to System Sensitivity Theory, New York: Academic Press . Hung, S. T . (1985) . Multi-input/multi-outputsensitivity points tuning. M. S. Thesis, University of Illinois, Urbana, IL, May. Lefebvre, S., S. Richter, and R . DeCarlo (1983) Decentralized Variable Structure Control Design for a Two-Pendulum System. IEEE Transactions on Automatic Control, vol. AC-28, no. 12, pp. 1112-1114. Schoenwald, D. A. (1992). Analysis and Design of Robust Decentralized Controllers for Nonlinear Systems. Ph.D. Dissertation, The Ohio State University, Columbus, OH, June. Siljak . D. D. (1991) . Decentralized Control of Complex Systems. New York: Academic Press.
.. 1.1 . . . . 1.11'1 .. 3 .45:1jO .• 59
laterco •• ecdo ••
-' . 114
- 0 .lU%j1.784
0.e01 _0 .1'11
_0.18.%jl . 341 _0 .043:l:.jO . 17. .. O, 113:1::jO .•••
-1.177
-1.18&% · 0 .001
The results are summarized in Tables 1-2. The first column of numbers of each table represent the closed loop poles for the 4th order decentralized design without sensitivity models. The second column of numbers of each table represent the closed loop poles for the 12th order decentralized design with sensitivity models. The first row of each table corresponds to the case of ignoring the interconnection terms whereas the second row includes these terms in the closed loop analysis. The results show that the lower order design without sensitivity models fails to stabilize the 542