- Email: [email protected]
An Application of Aggregation Methods to Optimal Control of Large-Scale Discrete Dynamical Systems
AN APPLICATION OF AGGREGATION METHODS TO OPTIMAL CONTROL OF LARGE-SCALE DISCRETE DYNAMICAL SYSTEMS F. Capkovic Institute of Technical Cybernetics, Slovak Academy of Sciences, Dubravska cesta 5, 80931 Bratislava, Czechoslovakia
Abstract. An unusual application of aggregation methods to optimal control of large-scale linear discrete time dynamical systems is presented. In a first step the original system is aggregated by usual aggregation methods to obtain a simplified one, to be utilized as a first subsystem of a new global system. In a second step a residual subsystem of this global system is established in such a way to guarantee one-way interconnections between both sUbsytems. The relation between the original system and the new one is reciprocal one-to-one. The new global system existence conditions are examined and the optimization of the original large-scale system with respect to quadratic performance criterion is indicated in this way to avoid the usual iterative coordination process. Keywords. Aggregation, decomposition, discrete time systems, disaggregation, large-scale systems, optimization, optimal control, systems theory. INTRODUCTION In many areas of control theory, the mathematical models of large-scale plants are used. These models contain a large number of variables and parameters. As a matter of the fact, the requirements to numerically handle such models lead to large-scale problems. The most critical situation is in the case of the need to optimize large-scale dynamical systems. Present exact approaches to computing the optimal control of large-scale systems are based on hierarchical multilevel systems theory, the foundations of which having been monographically treated by Mesarovic, Macko, and Takahara (1970). The original large-scale problem under consideration is decomposed into several interconnected optimization subproblems which are solved relativly separatly. To get an expected solution to be the same as that which would be obtained by simultaneous handling the original problem as a whole, a coordination principle must be used. Because this coordination process is an iterative one, there are several supplementary obstacles (e.g. convergency problem, the associated extension of the operation time) which must be surmounted. Often, if the precise solution of any 34 9
engineering problem is quite complicated in nature, it is more useful by far to have a Simple apl,roximate solution than the actual exact onc. One way of cutting down on both the operation time and the memory size required for the computational solution of the optimization problem mentioned above, is an application of the noniterative in nature aggregation methods. The state-space aggregation method related to optimal control was defined by Aoki (1968). The time-domain aggregation method was mentioned by Suchorukov and Gorbunov (197·1 ). 30th aggregation methods also can be successfully combined into space-time aggregation. The first author to notice the general features of the aggregation related to nonlinear controlled systems was Pavlovskyi (1974). However, the main disadvantage of using the aggregation approaches (particularly the state-space aggregation) is represented by difficulties concerning the impossibility of a disaggregation process required in order to obtain the exact solution of the originjl problem. In this paper the simplification of the large-scale discrete time linear dynamical system is carried out by aggregation, a residual subsystem is established and the utilization of a new global system in optimal control
F. Capkovic
350
computation is examined. i=O,l,...
AGGREGATION Consider the following optimization problem to be that of the optimization of the large-scale discrete time dynamical system S: x(i+!) = AxeD + Bu(i) i=O,l, •••
en
N-l
L
f[xCi) , u(i)]
where Xl' x2 have foregoing importance, and Z(.) is an aggregated m-dimensional state vector. Proceeding in a purely formal fashion, let us define an p x r real constant matrix (6)
with A,B being, respectively, n x n and n x r real constant matrices, x (.), u C.) being, respectively, n-dimensional state, and r-dimensional control vectors, and i being the symbol for discrete time; with respect to the performance criterion J =
(5)
(2)
i=O where N is an integer and f is a scalar function of the vectors x and u. Suppose that both the state vector x and the control vector u are decomposed into two parts - subvectors. Let x = (Xl' x 2 ) T and u = (u l ,u2 ) T, where Xl' x 2 are, respectively, m-dimensional, and (n-m)-dimensional subvectors of the vector x; u l ' u2 are, respectively, p-dimensional, and Cr-p)-dimensional subvectors of the vector u. T symbolizes the vector or/and matrix transposition. Hence, the system S may be written in the form
where D·l is an p x p nonsingular ma trix, and D2 is an p x (r-p) arbitrary one. By means of the matrix D the additional aggregation transformation
i
=
0,1, ••.
may be defined, with v C.) being a new aggregated p-dimensional control vector. The subvectors u l ' u2 have foregoing importance. Since the matrices Cl' Dl are nonsingular, the following expressions can be written (8) (9)
Let us establish the aggregated model of the form Sa'. z(i+l> = Fz(n + Gv(i)
(10)
where F, G are, respectively, m x m , and m x p real constant matrices. It can be easily proved that the following conditions must be fulfilled to obtain the model Sa FC = CA GD = CB
(ll) (12)
Time-Domain Aggregation i=O,l, •••
(3)
where Akj , Bkj for k,j = 1,2 are, respectively, corresponding submatrices of the matrix A, and B. Consider the matrix C of the form
where Cl is an m x m real constant nonsingular matrix, and C is an m x(n-m) real constant arnitrary one. After aggregation defined by Aoki (1968) can be written
Let us point out an application of the time-domain aggregation. Consider the transformation of the form k = Ki, i
= 0, 1, •••
where K > 1 is a positive integer. From the foundations of discrete time systems theory we have
CH)
Generally speaking,
An application of aggregation methods
x(Ki> = AKx[KU-l)) Ki-l
+ ~
+
KO ° 1 A 1-J- Bu(j)
j=K (i-D Take the number K as a unit of the new time k. In this new time-scale k = 0, 1, ••• , while in the old one k = 0, K, 2K, •••• After the transition to the new time-scale we obtain time-domain aggregated system of the form x(k) = Ax(k-l)
+
Bu(k-l)
(16)
where A = AK -B = [K-l AB, ••• , AB, B ] lieo) = [U(O), •.• , ueK-l>] T li(k=Ki> = [uCK(i-D) , ••• ••• , u(K1-1)] T °
The system (16) may represent the original one to which the proposed approach mentioned above can be applied. The transformation (13) may be called the time-domain aggregation. THE NEW GLOBAL SYSTEM Substituting (8), (9) into the or1g1na1 system (3) and premu1tiplying the obtained vector equation by the matrix
with I n-m being the en-m) x (n-m) identity matrix we have a new global system of the form
fZ(i+uJ [FA A0J[zu> J L x
Sg : x (i + 1> = 2
+
351
21
22 •
[~21 ;2J[:~::J
2
with qar(i) = A21 zU) + i321 ·V(i) , is introduced. There exists the reciprocal one-to-one relation between the original global system S and the new global one, based on the relations (8), (9), and (17). The one-way interconnection q ar will guarantee the noniterativity of the coordination process Cno coordinator is required) and the existence of the residual subsystem will guarantee the exact disaggregation. OPTIMIZATION APPLICATION TO LQ-PROBLEM In the case called LQ-problem, the proposed approach leads to a very simple relation between performance subcriteria understood in terms of the hierarchical multilevel approach. LQ-prob1em is known as that of the optimization of the system S with respect to performance criterion of the form
with Q, R being, respectively, n x n real symmetric positivly semidefinite, and r x r real symmetric positivly definite matrices. Regarding the decomposition of both the state vector x and the control vector u, introduced above, we have J
where
( il +
(18)
where A21 = A21 C-1 l _ A C-1 C A22 = A22 21 l 2 821 = B21 0 -1 1 822 = B22 - B21 0 1-1 02 From Ua)it is clear that the system Sg consists of two one-way coupled subsystems Sa and Sr' where the residual subsystem
N-l
=
T
Lrx kCD Qk .x. (0 i=O
+
J J
with Qkj' Hkj being, respectively, the corresponding submatrices of the matrices Q and R. After substituting the relations (8), (9) into (20), the following favourable form of performance criterion is obtained (23) where
N-l J a = C[z T (i)Q z(i) i=O a
+
v T (i) Hav(i)]
F. Capkovic
352
disaggregation problem is also brought to a solution.
J
ar
=
N-l
Crz T (:ilQ i=O
x,Ji) ar -
+
v TCD R
u (D]
ar 2
with -l)T -1 Qa = (Cl QllC l -1
T
An application of this approach to LQ-optimization problem is studied in the end of this paper.
-1
Qr =-(C l C2 ) Q12 - Q2l Cl C2 + +
Q22
+
-1
T
-1
(-1
Rr =- Dl O2 +
REFERENCES
(Cl C2 ) QllC l C2
Qar = (CilC2)TQllCil - Q2l Ci -1 T -1 Ra = (0 1 ) RllO l )T
-1
R12 - R21 0 l O2
l
+
R22 + (Oi l02) TR110il02
Rar = (0- 1 0 )TR 1
2
0- 1 11 1
The approach seems to be suitable also for disturbed linear dynamical systems or/and for the systems upon which the additional restrictions are imposed. It is, of course, suitable also for continuous linear dynamical systems.
R 0- l 2l l
since the matrices Q, R are symmetric, the term J is vanishing when the ar conditions (24)
are fulfilled. If this is the case, the performance criterion (23) will be divided into two independent parts belonging to the corresponding subsystems of the new global system. The conditions (24), (25) are not as hard as to be unsatisfyable. CONCLUSIONS The importance of the presented contribution lies in the fact that it enables us to by-pass rigorous difficulties concerning the iterative coordination process. No coordinator is required in this case. The optimization problem of the new global system linked with the original one by reciprocal one-to-one transformation i~ solved. The new global system con: s1sts of both the subsystem obtained by aggregation of the original system and the residual subsystem supplementing the former one to fill up the information interspace between the original system and its aggregated representation. Between the two subsystems there is a one-way linkage. By implementing this approach the
Aoki, M. (1968). Control of largescale dynamic systems by aggregation. IEEE Trans. Autom. Control, AC-ll, 246-2;)3. Aoki, M. <'I97TJ. Aggrega tion. In D. A. Wismer (Ed.) , Optimization Methods for Large-Scale Systems (with Applications), Mc Graw Hill, New York. Chap. 5, pp. 191-232. Capkovic, F. (1977). Optimal control of one type of technological process by space-time aggregation method. In Proceedings of the All-State Conference ASRTP'77 on Process Control. Vratna dolina. pp. :2 3 1-2 ·17 • Cirak, J., F. Capkovic, S. Kozak and F. Varga (1978). Control of Static and Slowly Changing Complex Systems. Hesearch Hep. Institute of Technical Cybernetics, Slovak Academy of Sciences, Bratislava. pp. 198-247. Mesarovic, M. D., O. Macko, and Y. Takahara (1970). Theory of IIierarchical Multilevel Systems. Academic Press, New York. 294 pp. Pavlovskyi, J. N. (1974). Aggregation of complex models and hierarchical control systems synthesis. Issled. Oper., 1, Vychisl. Centr AN SSSH, Moscow. pp. 3-18. Suchorukov, G. A., S. D. Gorbunov U974). Synthesis of hierarchical systems for technological process control. Tekh. Kibern., 12, pp. 22-33. Ulicny, J. (976). Control of Hierarchical Production Systems under Cond1t10ns of Uncertaintl. Doctoral d1ssertat10n. Inst1tute of technical Cybernetics, Slovak Academy of Sciences, Bratislava. 217 pp.
Abstract. An unusual application of aggregation methods to optimal control of large-scale linear discrete time dynamical systems is presented. In a first step the original system is aggregated by usual aggregation methods to obtain a simplified one, to be utilized as a first subsystem of a new global system. In a second step a residual subsystem of this global system is established in such a way to guarantee one-way interconnections between both sUbsytems. The relation between the original system and the new one is reciprocal one-to-one. The new global system existence conditions are examined and the optimization of the original large-scale system with respect to quadratic performance criterion is indicated in this way to avoid the usual iterative coordination process. Keywords. Aggregation, decomposition, discrete time systems, disaggregation, large-scale systems, optimization, optimal control, systems theory. INTRODUCTION In many areas of control theory, the mathematical models of large-scale plants are used. These models contain a large number of variables and parameters. As a matter of the fact, the requirements to numerically handle such models lead to large-scale problems. The most critical situation is in the case of the need to optimize large-scale dynamical systems. Present exact approaches to computing the optimal control of large-scale systems are based on hierarchical multilevel systems theory, the foundations of which having been monographically treated by Mesarovic, Macko, and Takahara (1970). The original large-scale problem under consideration is decomposed into several interconnected optimization subproblems which are solved relativly separatly. To get an expected solution to be the same as that which would be obtained by simultaneous handling the original problem as a whole, a coordination principle must be used. Because this coordination process is an iterative one, there are several supplementary obstacles (e.g. convergency problem, the associated extension of the operation time) which must be surmounted. Often, if the precise solution of any 34 9
engineering problem is quite complicated in nature, it is more useful by far to have a Simple apl,roximate solution than the actual exact onc. One way of cutting down on both the operation time and the memory size required for the computational solution of the optimization problem mentioned above, is an application of the noniterative in nature aggregation methods. The state-space aggregation method related to optimal control was defined by Aoki (1968). The time-domain aggregation method was mentioned by Suchorukov and Gorbunov (197·1 ). 30th aggregation methods also can be successfully combined into space-time aggregation. The first author to notice the general features of the aggregation related to nonlinear controlled systems was Pavlovskyi (1974). However, the main disadvantage of using the aggregation approaches (particularly the state-space aggregation) is represented by difficulties concerning the impossibility of a disaggregation process required in order to obtain the exact solution of the originjl problem. In this paper the simplification of the large-scale discrete time linear dynamical system is carried out by aggregation, a residual subsystem is established and the utilization of a new global system in optimal control
F. Capkovic
350
computation is examined. i=O,l,...
AGGREGATION Consider the following optimization problem to be that of the optimization of the large-scale discrete time dynamical system S: x(i+!) = AxeD + Bu(i) i=O,l, •••
en
N-l
L
f[xCi) , u(i)]
where Xl' x2 have foregoing importance, and Z(.) is an aggregated m-dimensional state vector. Proceeding in a purely formal fashion, let us define an p x r real constant matrix (6)
with A,B being, respectively, n x n and n x r real constant matrices, x (.), u C.) being, respectively, n-dimensional state, and r-dimensional control vectors, and i being the symbol for discrete time; with respect to the performance criterion J =
(5)
(2)
i=O where N is an integer and f is a scalar function of the vectors x and u. Suppose that both the state vector x and the control vector u are decomposed into two parts - subvectors. Let x = (Xl' x 2 ) T and u = (u l ,u2 ) T, where Xl' x 2 are, respectively, m-dimensional, and (n-m)-dimensional subvectors of the vector x; u l ' u2 are, respectively, p-dimensional, and Cr-p)-dimensional subvectors of the vector u. T symbolizes the vector or/and matrix transposition. Hence, the system S may be written in the form
where D·l is an p x p nonsingular ma trix, and D2 is an p x (r-p) arbitrary one. By means of the matrix D the additional aggregation transformation
i
=
0,1, ••.
may be defined, with v C.) being a new aggregated p-dimensional control vector. The subvectors u l ' u2 have foregoing importance. Since the matrices Cl' Dl are nonsingular, the following expressions can be written (8) (9)
Let us establish the aggregated model of the form Sa'. z(i+l> = Fz(n + Gv(i)
(10)
where F, G are, respectively, m x m , and m x p real constant matrices. It can be easily proved that the following conditions must be fulfilled to obtain the model Sa FC = CA GD = CB
(ll) (12)
Time-Domain Aggregation i=O,l, •••
(3)
where Akj , Bkj for k,j = 1,2 are, respectively, corresponding submatrices of the matrix A, and B. Consider the matrix C of the form
where Cl is an m x m real constant nonsingular matrix, and C is an m x(n-m) real constant arnitrary one. After aggregation defined by Aoki (1968) can be written
Let us point out an application of the time-domain aggregation. Consider the transformation of the form k = Ki, i
= 0, 1, •••
where K > 1 is a positive integer. From the foundations of discrete time systems theory we have
CH)
Generally speaking,
An application of aggregation methods
x(Ki> = AKx[KU-l)) Ki-l
+ ~
+
KO ° 1 A 1-J- Bu(j)
j=K (i-D Take the number K as a unit of the new time k. In this new time-scale k = 0, 1, ••• , while in the old one k = 0, K, 2K, •••• After the transition to the new time-scale we obtain time-domain aggregated system of the form x(k) = Ax(k-l)
+
Bu(k-l)
(16)
where A = AK -B = [K-l AB, ••• , AB, B ] lieo) = [U(O), •.• , ueK-l>] T li(k=Ki> = [uCK(i-D) , ••• ••• , u(K1-1)] T °
The system (16) may represent the original one to which the proposed approach mentioned above can be applied. The transformation (13) may be called the time-domain aggregation. THE NEW GLOBAL SYSTEM Substituting (8), (9) into the or1g1na1 system (3) and premu1tiplying the obtained vector equation by the matrix
with I n-m being the en-m) x (n-m) identity matrix we have a new global system of the form
fZ(i+uJ [FA A0J[zu> J L x
Sg : x (i + 1> = 2
+
351
21
22 •
[~21 ;2J[:~::J
2
with qar(i) = A21 zU) + i321 ·V(i) , is introduced. There exists the reciprocal one-to-one relation between the original global system S and the new global one, based on the relations (8), (9), and (17). The one-way interconnection q ar will guarantee the noniterativity of the coordination process Cno coordinator is required) and the existence of the residual subsystem will guarantee the exact disaggregation. OPTIMIZATION APPLICATION TO LQ-PROBLEM In the case called LQ-problem, the proposed approach leads to a very simple relation between performance subcriteria understood in terms of the hierarchical multilevel approach. LQ-prob1em is known as that of the optimization of the system S with respect to performance criterion of the form
with Q, R being, respectively, n x n real symmetric positivly semidefinite, and r x r real symmetric positivly definite matrices. Regarding the decomposition of both the state vector x and the control vector u, introduced above, we have J
where
( il +
(18)
where A21 = A21 C-1 l _ A C-1 C A22 = A22 21 l 2 821 = B21 0 -1 1 822 = B22 - B21 0 1-1 02 From Ua)it is clear that the system Sg consists of two one-way coupled subsystems Sa and Sr' where the residual subsystem
N-l
=
T
Lrx kCD Qk .x. (0 i=O
+
J J
with Qkj' Hkj being, respectively, the corresponding submatrices of the matrices Q and R. After substituting the relations (8), (9) into (20), the following favourable form of performance criterion is obtained (23) where
N-l J a = C[z T (i)Q z(i) i=O a
+
v T (i) Hav(i)]
F. Capkovic
352
disaggregation problem is also brought to a solution.
J
ar
=
N-l
Crz T (:ilQ i=O
x,Ji) ar -
+
v TCD R
u (D]
ar 2
with -l)T -1 Qa = (Cl QllC l -1
T
An application of this approach to LQ-optimization problem is studied in the end of this paper.
-1
Qr =-(C l C2 ) Q12 - Q2l Cl C2 + +
Q22
+
-1
T
-1
(-1
Rr =- Dl O2 +
REFERENCES
(Cl C2 ) QllC l C2
Qar = (CilC2)TQllCil - Q2l Ci -1 T -1 Ra = (0 1 ) RllO l )T
-1
R12 - R21 0 l O2
l
+
R22 + (Oi l02) TR110il02
Rar = (0- 1 0 )TR 1
2
0- 1 11 1
The approach seems to be suitable also for disturbed linear dynamical systems or/and for the systems upon which the additional restrictions are imposed. It is, of course, suitable also for continuous linear dynamical systems.
R 0- l 2l l
since the matrices Q, R are symmetric, the term J is vanishing when the ar conditions (24)
are fulfilled. If this is the case, the performance criterion (23) will be divided into two independent parts belonging to the corresponding subsystems of the new global system. The conditions (24), (25) are not as hard as to be unsatisfyable. CONCLUSIONS The importance of the presented contribution lies in the fact that it enables us to by-pass rigorous difficulties concerning the iterative coordination process. No coordinator is required in this case. The optimization problem of the new global system linked with the original one by reciprocal one-to-one transformation i~ solved. The new global system con: s1sts of both the subsystem obtained by aggregation of the original system and the residual subsystem supplementing the former one to fill up the information interspace between the original system and its aggregated representation. Between the two subsystems there is a one-way linkage. By implementing this approach the
Aoki, M. (1968). Control of largescale dynamic systems by aggregation. IEEE Trans. Autom. Control, AC-ll, 246-2;)3. Aoki, M. <'I97TJ. Aggrega tion. In D. A. Wismer (Ed.) , Optimization Methods for Large-Scale Systems (with Applications), Mc Graw Hill, New York. Chap. 5, pp. 191-232. Capkovic, F. (1977). Optimal control of one type of technological process by space-time aggregation method. In Proceedings of the All-State Conference ASRTP'77 on Process Control. Vratna dolina. pp. :2 3 1-2 ·17 • Cirak, J., F. Capkovic, S. Kozak and F. Varga (1978). Control of Static and Slowly Changing Complex Systems. Hesearch Hep. Institute of Technical Cybernetics, Slovak Academy of Sciences, Bratislava. pp. 198-247. Mesarovic, M. D., O. Macko, and Y. Takahara (1970). Theory of IIierarchical Multilevel Systems. Academic Press, New York. 294 pp. Pavlovskyi, J. N. (1974). Aggregation of complex models and hierarchical control systems synthesis. Issled. Oper., 1, Vychisl. Centr AN SSSH, Moscow. pp. 3-18. Suchorukov, G. A., S. D. Gorbunov U974). Synthesis of hierarchical systems for technological process control. Tekh. Kibern., 12, pp. 22-33. Ulicny, J. (976). Control of Hierarchical Production Systems under Cond1t10ns of Uncertaintl. Doctoral d1ssertat10n. Inst1tute of technical Cybernetics, Slovak Academy of Sciences, Bratislava. 217 pp.