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Decentralized Sampled-Data Control of Interconnected Systems
Copyright © IFAC Large Scale Systems, Rio Patras . Greece, 1998
DECENTRALIZED SAMPLED-DATA CONTROL OF INTERCONNECTED SYSTEMS
M. Erol Sezer and Ozay Oral
Department of Electrical and Electronics Engineering Bilkent University, Ankara, Turkey
Abstract : This paper presents a decentralized adaptive stabilization scheme for a class of interconnected systems using high-gain adaptive controllers. The nominal subsystems are assumed to satisfy some mild conditions required by standard adaptive control schemes, and the interconnections certain structural conditions. The decentralized controllers operate on sampled values of local outputs. The sampling frequencies of the controllers also serve as their gains. The controllers are synchronized in that their sampling frequencies are integer multiples of a common base frequency. The base frequency (hence the common gain) is adjusted by a simple adaptation rule that results in a stable closed-loop system with bounded controller gains. Copyright @ 1998 [FAC Keywords: Interconnected system , Decentralized control , High-gain feedback , Adaptive control , Sampled-data control.
mechanism to increment the gain parameter step wise at discrete instants. Ocal! and Sezer (1992) proposed a sampled-data state-feedback controller for robust stabilization of systems under time-varying additive perturbations of a certain class. The controller, which simulates high-gain continuous-time feedback in the absence of perturbations , guarantees stability for a sufficiently small sampling period which depends on the bounds of perturbations.
1. INTRODUCTION High-gain feedback control is a standard tool for robust stabilization in the presence of modeling uncertainities (see, for example, Zames and Bensoussan ,1983; Saberi and Sannuti, 1990). In the case of a single-input/single--output (SISO) system , design of such a controller requires that the system have stable zeros and its relative degree , the sign of its high frequency gain and the bounds of the system parameters or perturbations be known . Similar information is needed for multiinput/multi-output (MIMO) systems. It has been shown by Byrnes and Willems (1984) that for systems with relative degree one robust stability can be achieved without the need to know the bounds of the perturbations by tuning the gain parameter adaptively. A similar result has been obtained for systems with higher relative degree by Khalil and Saberi (1987) , who employed an adaptation
High-gain feedback finds a natural application in decentralized control of interconnected systems, where the essential uncertainity lies in the interconnections among the subsystems (Ikeada and colleagues, 1976, 1980; Sezer and Hiiseyin , 1978; Sezer and Siljak, 1981 ; Hiiseyin and colleagues, 1982; Saberi and Khalil, 1985; Yu and colleagues, 1992). As in high-gain control of a single system , adaptive techniques have been used (Gavel and Siljak ,1989; Shi and Singh, 1992 ; Yu and col-
255
[Yl ... YN V is the overall output, and e",i and eyi satisfy
leagues, 1993, 1994, 1996) to eliminate the need for a priori information about the bounds of interconnections.
N
Ile"'i(t , x)11
In this paper we summarize an approach that brings together several concepts and techniques of robust stabilization, adaptive control, and multirate decentralized sampled-data control. We first consider the robust stabilization problem of an interconnected system by using decentralized continuous-time output-feedback controllers. The controller structure is determined completely off-line by making use of the structural properties of the system. Then we provide a simple adaptation rule to adjust the feedback gain parameters to eliminate the need to know the interconnection bounds. Finally, we consider the case where the controllers are allowed to operate on sampled values of local outputs only, rather than continuous-time measurements. We propose that robust stability can be achieved for sufficiently small sampling intervals, and then present an adaptation mechanism which decreases the common sampling interval slowly until it is small enough . In this scheme, the sampling frequency has a double role: it also determines the controller gains. This way we extend our previous results.
j=l N
leYi(t, y)1
Our purpose is to stabilize the interconnected system in (1) by using decentralized sampled-data controllers. We note that without any restrictions on the bounds aij ' a;j in (2) , assumption (b) above is necessary for stabilizability of the interconnected system (Yu and colleagues, 1996) .
3. A CONTINUOUS-TIME DECENTRALIZED CONTROLLER
To stabilize the interconnected system in (1) , we use (Yu and colleagues, 1996) decentralized dynamic output feedback controllers of the form
Vi v
{ 2v + 1, v/(mi - 1),
IT (mi -
1)
mi mi
=1 #
1 (4)
m;;tl
where Xi(t) E nn. is the state of Si , and ei(t, x(t» stands for additive non linear interconnection effects on a nominal subsystem represented by the triple (Ai, bi, with x = [xT . . . x~]T representing the overall state. We make the following assumptions concerning the nominal subsystems and the interconnections.
This choice of the controllers was based on the observation that without the interconnection terms, Si in (1) behaves like a nominal subsystem with a transfer function TJiO/ Sm ; at high frequencies. If fi + hT(sI - Fi)-lgi = Pi(S)/TJiOqi(S), then for a sufficiently high constant gain Pi(t) = Pi, nominal closed-loop transfer function becomes Hi(S, p;) = TJiOqi(S/ p;)/ p'(" [(si Pi)m;qi(S/ p;) - Pi(S/ Pi)] . This allows us ~o choose the decentralized controllers to make the ith nominal closed-loop subsystem have stable poles of the order of Pi (Brasch and Pearson, 1970). With the relative magnitudes of the gains properly adjusted as in (4), their stabilizing effect can be shown to overcome any possible destabilizing effects of the interconnections.
en,
(a) (Ai , bi,en is controllable and observable. (b) With Hi(s) = eT(sI - A;)-lbi = TJi(s)/di(s), zeros of the numerator polynomial TJi (s) = TJioSn.-m. + TJilSn;-m;-l + ... + TJi,n;-m; are in the open left-half complex plane. (c) The high frequency gain TJiO and the relative degree m; of Hi(s) above are known .
+
(3)
=
AiXi(t) + biUi(t) + ei(t, x(t)) eT Xi(t) (1)
ei(t, x) = bie"'i(t, x)
Pi(t)Fiz i + p'(';(t)giYi(t) pi(t)hT Xci(t) + p'('; (t)fiYi(t)
where Zi E n m ;, and the time-varying gain parameters Pi(t) are generated from a common gain as Pi(t) pv·(t) , where
We consider an interconnected system that consists of N single-input/single-output subsystems Si described as
interconnections
(2)
for some (unknown) constants aij ' a;j > O.
2. SYSTEM STRUCTURE
(d) The
L a;jlYjl
<
j=l
Zi(t) Ui(t)
Xi(t) Yi(t)
< Laijllxjll
In the above scheme, how high the common gain pet) should be depends on the bounds of the interconnections. To eliminate the need to know these
are
of the form eyi(t, y), where y = 256
bounds, we employ (Yu and colleagues, 1996) a simple centralized adaptation rule:
where {3y , (3z > 0 are arbitrary numbers, and Z = [zi , ... z'J: f is the overall controller state. This allowed the controller gains to increase to finite levels high enough to guarantee closed-loop stability.
Figure 1: Three coupled inverted penduli
high frequency behavior of the discrete model of the nominal subsystem S i is given by a pulse transfer function H.(z) = r;"T/i( Z)/( z where T Z = e· • . As in the continuous-time case , if fi + hT( z I -Fi)-lg. = Pi( Z)/TliOqi(Z), then the closedloop characteristic polynomial of the ith nominal subsystem becomes (z - It'dd( z) - np(z )nc(z). Thus the decentralized controllers can be chosen to stabilize the nominal closed-loop subsystems for sufficiently small 1i.
4. A DECENTRALIZED SAMPLED-DATA CONTROLLER
lr·,
We construct the decentralized sampled-data controllers by discretizing the continuous-time cont roll ers in (3) . In doing so, we make the following observations. (a) Faster controllers (larger Pi) require faster sampling. To simplify the analysis we use a single parameter to determine both the gain and the sampling rate of the controllers.
Finally, to increase the common gain Tt 1 to values as required by the sterngth of interconnections , we choose the adaptation rule as
Mk + Mk-1uk
(b) Adaptation of the gains (hence, the sampling rates) of the controllers require some sort of synchronization.
int (J.Lk+d
Based on these observations we choose the kth common sampling interval of all controllers to be tk+l - tk Tk I/Mk , and let the subsampling interval of the ith controller be Tik = 1/M;' . (Thus the ith controller operates on M;·-l samples in the kth common sampling interval.) Defining Zi k (I) to be the I th subsample of the discrete state of the ith controller in the kth common sampling interval, we construct the decentralized controllers as
=
(7)
5. AN EXAMPLE
=
Fizik(l) + ~~-m'g'Yi(tk ~k 1 hT Zik(l) + ~km. !;y.(tk + mk)
Consider three coupled inverted penduli shown in Figure 1, with the first two forming a subsystem controlled by a torque applied to the first one, and the third another subsystem with its own control torque. Defining the states as
+ mk) (6) and writing down the state equations in the usual way, we observe that the high frequency behavior of the subsystems are described by the transfer functions K 1c / J 1 S3 and 1/ hs 2 respectively, where K 1c is the spring coefficient coupling the first two penduli, and Ji are the inertias, and that all interaction terms can be grouped under ei (z) terms which satisfy (2).
for tk + mk ::; t < tk + (/ + 1)1ik with the convention that Zik(M;·-l) = Z. ,k+1(O). Note that all controllers are synchronized at the beginning of each common sampling period during which each controller operates on samples taken at a uniform rate. The reciprocal of the common sampling interval T 1 = M" acts as the common gain p(t) of the continuous-time controllers of (3), and the reciprocals of the subsampling intervals Tii;1 M;' as the local gains P.(t) .
t
Since ml = 3 and m2 = 2, we find v = 2, VI = 1, V2 = 2, and choose the decentralized sampleddate controllers as in (6) with Ta = I/Mk ,T 2 k 1/Mf. We design the controllers to place the poles of the nominal closed-loop subsystems at {O, 0 A =F jO.2, 0.8 =F jOo4 } and { 0, 0.8 =F jOo4 } respectively.
=
=
As in the previous section, we observe that under uniform sampling with 1ik = 1i (constant), the 257
0.4 r-----------------------------------~
xlI xl3 x21
0.2
4.0
ul
.......
2.0
u2
0.0
0 .0
I - _ - -__~___:.___¥L...~~::::::::: .. >.~,.-=-.--==-==----2.0
i
.!
' .. -0.2 L-________________ ________ ______ -4 .0 0 .0 2.0 4 .0 6.0 8 .0 0 .0 , /
~
~
~
2.0
4 .0
6.0
8.0
Figure 2: Simulation results: States
Figure 3: Simulation results : Control inputs
The simulation results corresponding to some arbitrary choice of the system parameters and initial conditions are shown in Figures 2 and 3. We observe that the proposed scheme stabilized the whole system with the controller gains and sampling rates converging to reasonable values (limk_oo Mk = 6 so that sampling intervals converge to Tl = 0.167 and T2 = 0.028 respectively).
Saberi, A. and H. K. Khalil (1985) . Decentralized stsbilization of interconnected systems using output feedback, Int . 1. Control, 41, 14611475. Saberi, A. and P. Sannuti (1990) . Observer-based control of uncertain systems with non-linear uncertainities, Int. J. Control, 52 , 1107-1130. Sezer, M. E. and 6. Hiiseyin (1978) . Stabilization of linear, time-invariant interconnected systems using local state feedback, IEEE Trans . Syst. Man Cybern ., SMC-S , 751-756.
REFERENCES Brasch , F . M. Jr. and J . B. Pearson (1970) . Pole assignment using dynamic compensators, IEEE Trans. Autom. Control, AC-15, 34-43. Byrnes , C . I. and C. J . Willems (1984). Adaptive stabilization of multi variable linear systems Proc. 23rd CDC, Las Vegas , NV , 15741577. Gavel, D. T. and D. D. Siljak (1989). Decentralized adaptive control: Structural conditions for stability, IEEE Trans . Autom. Control, 34 , 413-426. Hiiseyin , 6., M. E. Sezer and D. D. Siljak (1982) . Robust decentralized control using output feedback, Proc. IEE-D, 129, 310-315. Ikeda, M., O. Omefuji and S. Kodama (1976). Stabilization of large-scale linear systems, Systems, Computers, Control, 7, 34-45. Ikeda, M. and D.D. Siljak (1980) . On decentrally stabilizable large-scale systems, Automatica, 16,331-335. Khalil , H. K. and A. Saberi (1987). Adaptive stabilization of a class of nonlinear systems using high-gain feedback, IEEE Trans. Autom. Control, AC-32, 1031-1035. OCall, O . and M. E. Sezer Robust sampled-data control, IEEE Trans. Autom. Control, 37, 1591-1597.
258
Sezer, M. E . and D. D. Siljak (1981) . On decentralized stabilization and structure of linear largescale systems, Automatica, 17, 641-644 . Shi, 1. and S. K. Singh (1992) . Decentralized adaptive controller design for large-scale systems with higher order interconnections, IEEE Trans. Autom. Control, 37, 1106-1118. Yu, R. and M. E. Sezer (1992). Decentralized stabilization of interconnected systems: Structural conditions, Information and Decision Technologies, IS , 333-345. Yu, R. , O . Ocah and M. E. Sezer (1993) . Adaptive robust sampled-data control of a class of systems under structured perturbations, IEEE Trans. Autom. Control, 3S, 1707-1713. Yu, R. and M. E . Sezer (1994). On decentralized adaptive control of interconnected systems with a priori knowledge of isolated subsystems, Control Theory and App., 11(5),545-556. Yu, R. , M. E. Sezer and O . Ocah (1996). Robust adaptive stabilization of a class of systems under structured nonlinear perturbations with application to interconnected systems, Int . J. Control, 63 , 183-194. Zames, G. and D. Bensoussan (1983) . Multivariable feedback, sensitivity and decentralized control, IEEE Trans . Autom. Control, AC28, 1030-1035.
DECENTRALIZED SAMPLED-DATA CONTROL OF INTERCONNECTED SYSTEMS
M. Erol Sezer and Ozay Oral
Department of Electrical and Electronics Engineering Bilkent University, Ankara, Turkey
Abstract : This paper presents a decentralized adaptive stabilization scheme for a class of interconnected systems using high-gain adaptive controllers. The nominal subsystems are assumed to satisfy some mild conditions required by standard adaptive control schemes, and the interconnections certain structural conditions. The decentralized controllers operate on sampled values of local outputs. The sampling frequencies of the controllers also serve as their gains. The controllers are synchronized in that their sampling frequencies are integer multiples of a common base frequency. The base frequency (hence the common gain) is adjusted by a simple adaptation rule that results in a stable closed-loop system with bounded controller gains. Copyright @ 1998 [FAC Keywords: Interconnected system , Decentralized control , High-gain feedback , Adaptive control , Sampled-data control.
mechanism to increment the gain parameter step wise at discrete instants. Ocal! and Sezer (1992) proposed a sampled-data state-feedback controller for robust stabilization of systems under time-varying additive perturbations of a certain class. The controller, which simulates high-gain continuous-time feedback in the absence of perturbations , guarantees stability for a sufficiently small sampling period which depends on the bounds of perturbations.
1. INTRODUCTION High-gain feedback control is a standard tool for robust stabilization in the presence of modeling uncertainities (see, for example, Zames and Bensoussan ,1983; Saberi and Sannuti, 1990). In the case of a single-input/single--output (SISO) system , design of such a controller requires that the system have stable zeros and its relative degree , the sign of its high frequency gain and the bounds of the system parameters or perturbations be known . Similar information is needed for multiinput/multi-output (MIMO) systems. It has been shown by Byrnes and Willems (1984) that for systems with relative degree one robust stability can be achieved without the need to know the bounds of the perturbations by tuning the gain parameter adaptively. A similar result has been obtained for systems with higher relative degree by Khalil and Saberi (1987) , who employed an adaptation
High-gain feedback finds a natural application in decentralized control of interconnected systems, where the essential uncertainity lies in the interconnections among the subsystems (Ikeada and colleagues, 1976, 1980; Sezer and Hiiseyin , 1978; Sezer and Siljak, 1981 ; Hiiseyin and colleagues, 1982; Saberi and Khalil, 1985; Yu and colleagues, 1992). As in high-gain control of a single system , adaptive techniques have been used (Gavel and Siljak ,1989; Shi and Singh, 1992 ; Yu and col-
255
[Yl ... YN V is the overall output, and e",i and eyi satisfy
leagues, 1993, 1994, 1996) to eliminate the need for a priori information about the bounds of interconnections.
N
Ile"'i(t , x)11
In this paper we summarize an approach that brings together several concepts and techniques of robust stabilization, adaptive control, and multirate decentralized sampled-data control. We first consider the robust stabilization problem of an interconnected system by using decentralized continuous-time output-feedback controllers. The controller structure is determined completely off-line by making use of the structural properties of the system. Then we provide a simple adaptation rule to adjust the feedback gain parameters to eliminate the need to know the interconnection bounds. Finally, we consider the case where the controllers are allowed to operate on sampled values of local outputs only, rather than continuous-time measurements. We propose that robust stability can be achieved for sufficiently small sampling intervals, and then present an adaptation mechanism which decreases the common sampling interval slowly until it is small enough . In this scheme, the sampling frequency has a double role: it also determines the controller gains. This way we extend our previous results.
j=l N
leYi(t, y)1
Our purpose is to stabilize the interconnected system in (1) by using decentralized sampled-data controllers. We note that without any restrictions on the bounds aij ' a;j in (2) , assumption (b) above is necessary for stabilizability of the interconnected system (Yu and colleagues, 1996) .
3. A CONTINUOUS-TIME DECENTRALIZED CONTROLLER
To stabilize the interconnected system in (1) , we use (Yu and colleagues, 1996) decentralized dynamic output feedback controllers of the form
Vi v
{ 2v + 1, v/(mi - 1),
IT (mi -
1)
mi mi
=1 #
1 (4)
m;;tl
where Xi(t) E nn. is the state of Si , and ei(t, x(t» stands for additive non linear interconnection effects on a nominal subsystem represented by the triple (Ai, bi, with x = [xT . . . x~]T representing the overall state. We make the following assumptions concerning the nominal subsystems and the interconnections.
This choice of the controllers was based on the observation that without the interconnection terms, Si in (1) behaves like a nominal subsystem with a transfer function TJiO/ Sm ; at high frequencies. If fi + hT(sI - Fi)-lgi = Pi(S)/TJiOqi(S), then for a sufficiently high constant gain Pi(t) = Pi, nominal closed-loop transfer function becomes Hi(S, p;) = TJiOqi(S/ p;)/ p'(" [(si Pi)m;qi(S/ p;) - Pi(S/ Pi)] . This allows us ~o choose the decentralized controllers to make the ith nominal closed-loop subsystem have stable poles of the order of Pi (Brasch and Pearson, 1970). With the relative magnitudes of the gains properly adjusted as in (4), their stabilizing effect can be shown to overcome any possible destabilizing effects of the interconnections.
en,
(a) (Ai , bi,en is controllable and observable. (b) With Hi(s) = eT(sI - A;)-lbi = TJi(s)/di(s), zeros of the numerator polynomial TJi (s) = TJioSn.-m. + TJilSn;-m;-l + ... + TJi,n;-m; are in the open left-half complex plane. (c) The high frequency gain TJiO and the relative degree m; of Hi(s) above are known .
+
(3)
=
AiXi(t) + biUi(t) + ei(t, x(t)) eT Xi(t) (1)
ei(t, x) = bie"'i(t, x)
Pi(t)Fiz i + p'(';(t)giYi(t) pi(t)hT Xci(t) + p'('; (t)fiYi(t)
where Zi E n m ;, and the time-varying gain parameters Pi(t) are generated from a common gain as Pi(t) pv·(t) , where
We consider an interconnected system that consists of N single-input/single-output subsystems Si described as
interconnections
(2)
for some (unknown) constants aij ' a;j > O.
2. SYSTEM STRUCTURE
(d) The
L a;jlYjl
<
j=l
Zi(t) Ui(t)
Xi(t) Yi(t)
< Laijllxjll
In the above scheme, how high the common gain pet) should be depends on the bounds of the interconnections. To eliminate the need to know these
are
of the form eyi(t, y), where y = 256
bounds, we employ (Yu and colleagues, 1996) a simple centralized adaptation rule:
where {3y , (3z > 0 are arbitrary numbers, and Z = [zi , ... z'J: f is the overall controller state. This allowed the controller gains to increase to finite levels high enough to guarantee closed-loop stability.
Figure 1: Three coupled inverted penduli
high frequency behavior of the discrete model of the nominal subsystem S i is given by a pulse transfer function H.(z) = r;"T/i( Z)/( z where T Z = e· • . As in the continuous-time case , if fi + hT( z I -Fi)-lg. = Pi( Z)/TliOqi(Z), then the closedloop characteristic polynomial of the ith nominal subsystem becomes (z - It'dd( z) - np(z )nc(z). Thus the decentralized controllers can be chosen to stabilize the nominal closed-loop subsystems for sufficiently small 1i.
4. A DECENTRALIZED SAMPLED-DATA CONTROLLER
lr·,
We construct the decentralized sampled-data controllers by discretizing the continuous-time cont roll ers in (3) . In doing so, we make the following observations. (a) Faster controllers (larger Pi) require faster sampling. To simplify the analysis we use a single parameter to determine both the gain and the sampling rate of the controllers.
Finally, to increase the common gain Tt 1 to values as required by the sterngth of interconnections , we choose the adaptation rule as
Mk + Mk-1uk
(b) Adaptation of the gains (hence, the sampling rates) of the controllers require some sort of synchronization.
int (J.Lk+d
Based on these observations we choose the kth common sampling interval of all controllers to be tk+l - tk Tk I/Mk , and let the subsampling interval of the ith controller be Tik = 1/M;' . (Thus the ith controller operates on M;·-l samples in the kth common sampling interval.) Defining Zi k (I) to be the I th subsample of the discrete state of the ith controller in the kth common sampling interval, we construct the decentralized controllers as
=
(7)
5. AN EXAMPLE
=
Fizik(l) + ~~-m'g'Yi(tk ~k 1 hT Zik(l) + ~km. !;y.(tk + mk)
Consider three coupled inverted penduli shown in Figure 1, with the first two forming a subsystem controlled by a torque applied to the first one, and the third another subsystem with its own control torque. Defining the states as
+ mk) (6) and writing down the state equations in the usual way, we observe that the high frequency behavior of the subsystems are described by the transfer functions K 1c / J 1 S3 and 1/ hs 2 respectively, where K 1c is the spring coefficient coupling the first two penduli, and Ji are the inertias, and that all interaction terms can be grouped under ei (z) terms which satisfy (2).
for tk + mk ::; t < tk + (/ + 1)1ik with the convention that Zik(M;·-l) = Z. ,k+1(O). Note that all controllers are synchronized at the beginning of each common sampling period during which each controller operates on samples taken at a uniform rate. The reciprocal of the common sampling interval T 1 = M" acts as the common gain p(t) of the continuous-time controllers of (3), and the reciprocals of the subsampling intervals Tii;1 M;' as the local gains P.(t) .
t
Since ml = 3 and m2 = 2, we find v = 2, VI = 1, V2 = 2, and choose the decentralized sampleddate controllers as in (6) with Ta = I/Mk ,T 2 k 1/Mf. We design the controllers to place the poles of the nominal closed-loop subsystems at {O, 0 A =F jO.2, 0.8 =F jOo4 } and { 0, 0.8 =F jOo4 } respectively.
=
=
As in the previous section, we observe that under uniform sampling with 1ik = 1i (constant), the 257
0.4 r-----------------------------------~
xlI xl3 x21
0.2
4.0
ul
.......
2.0
u2
0.0
0 .0
I - _ - -__~___:.___¥L...~~::::::::: .. >.~,.-=-.--==-==----2.0
i
.!
' .. -0.2 L-________________ ________ ______ -4 .0 0 .0 2.0 4 .0 6.0 8 .0 0 .0 , /
~
~
~
2.0
4 .0
6.0
8.0
Figure 2: Simulation results: States
Figure 3: Simulation results : Control inputs
The simulation results corresponding to some arbitrary choice of the system parameters and initial conditions are shown in Figures 2 and 3. We observe that the proposed scheme stabilized the whole system with the controller gains and sampling rates converging to reasonable values (limk_oo Mk = 6 so that sampling intervals converge to Tl = 0.167 and T2 = 0.028 respectively).
Saberi, A. and H. K. Khalil (1985) . Decentralized stsbilization of interconnected systems using output feedback, Int . 1. Control, 41, 14611475. Saberi, A. and P. Sannuti (1990) . Observer-based control of uncertain systems with non-linear uncertainities, Int. J. Control, 52 , 1107-1130. Sezer, M. E. and 6. Hiiseyin (1978) . Stabilization of linear, time-invariant interconnected systems using local state feedback, IEEE Trans . Syst. Man Cybern ., SMC-S , 751-756.
REFERENCES Brasch , F . M. Jr. and J . B. Pearson (1970) . Pole assignment using dynamic compensators, IEEE Trans. Autom. Control, AC-15, 34-43. Byrnes , C . I. and C. J . Willems (1984). Adaptive stabilization of multi variable linear systems Proc. 23rd CDC, Las Vegas , NV , 15741577. Gavel, D. T. and D. D. Siljak (1989). Decentralized adaptive control: Structural conditions for stability, IEEE Trans . Autom. Control, 34 , 413-426. Hiiseyin , 6., M. E. Sezer and D. D. Siljak (1982) . Robust decentralized control using output feedback, Proc. IEE-D, 129, 310-315. Ikeda, M., O. Omefuji and S. Kodama (1976). Stabilization of large-scale linear systems, Systems, Computers, Control, 7, 34-45. Ikeda, M. and D.D. Siljak (1980) . On decentrally stabilizable large-scale systems, Automatica, 16,331-335. Khalil , H. K. and A. Saberi (1987). Adaptive stabilization of a class of nonlinear systems using high-gain feedback, IEEE Trans. Autom. Control, AC-32, 1031-1035. OCall, O . and M. E. Sezer Robust sampled-data control, IEEE Trans. Autom. Control, 37, 1591-1597.
258
Sezer, M. E . and D. D. Siljak (1981) . On decentralized stabilization and structure of linear largescale systems, Automatica, 17, 641-644 . Shi, 1. and S. K. Singh (1992) . Decentralized adaptive controller design for large-scale systems with higher order interconnections, IEEE Trans. Autom. Control, 37, 1106-1118. Yu, R. and M. E. Sezer (1992). Decentralized stabilization of interconnected systems: Structural conditions, Information and Decision Technologies, IS , 333-345. Yu, R. , O . Ocah and M. E. Sezer (1993) . Adaptive robust sampled-data control of a class of systems under structured perturbations, IEEE Trans. Autom. Control, 3S, 1707-1713. Yu, R. and M. E . Sezer (1994). On decentralized adaptive control of interconnected systems with a priori knowledge of isolated subsystems, Control Theory and App., 11(5),545-556. Yu, R. , M. E. Sezer and O . Ocah (1996). Robust adaptive stabilization of a class of systems under structured nonlinear perturbations with application to interconnected systems, Int . J. Control, 63 , 183-194. Zames, G. and D. Bensoussan (1983) . Multivariable feedback, sensitivity and decentralized control, IEEE Trans . Autom. Control, AC28, 1030-1035.