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Decentralized Control Design by Exploitation Structural Properties
DECENTRALIZED CONTROL DESIGN BY EXPLOITATION STRUCTURAL PROPERTIES
R.l. Widodo ' Electrical Engineering Department, Institute Technology Bandllng Emai l : [email protected]@lsI
ABSTRAC : Decentralization by exploitation strllctllral properties is the method based on the structural properties of the system rather than the composition into weakly coupled subsystems. A considerable reduction of the design problem can also be gained from specific structures of the controller to be designed Simplificat ions of the design problem can be derived if the control law is restricted to be of proportional-integral character. The application of the decentralized proportional-integral control/er is a direct consequence of the requirement of asymptotic regulation for step'wise disturbance and command Signals. According 10 Ihe internal model principle, each control stalion has 10 have several integrators. Keywords: decentralized control, structural properties, proportional-integral controllers.
I.
INTRODUCTION
= Ax(t) + 8u(t) yet) = Cx(t) + Du(t) x(t)
The theory of large-sca le systems is devoted to the problems that arise from the large size of systems to be controlled. Agricultural systems rely on the corporat ion of many different parts. If such systems are to be controlled, their analysis and control problem becomes very comp lex. In simplified way, this problem can be stated as the task to control at different places all the agricultural area. A Decentralized control scheme is appropriate because the variables to be controlled and the control inputs are many kilometers away from each other. Using faster computers with larger memories cannot solve this task simply.
(I)
Most of the difficulties of analytical and control by the complete problem are raised interdependence of the subsystems, there are links between arbitrary pairs of subsystems . The link may be mediated by one or more other subsystems. Conceptual simpl ification of analytical and control prob lem can be obtained if some subsystems have only a one-way effect on some ot hers.
2. DECENTRALIZED DESIGN
Decentralized design by exploitation of structural properties is based on the structura l properties of the system. 'Structural' properties are not strongly depending on the system parameters. In many practical application the plant cannot be describe by some precise linear model. Uncerta inties may resul t from incomp lete knowledge of the plant, from non-linear or timevarying properties, which cannot be reflected by some linear model, or from approximation made to reduce the model size. The system structure can defined on the basis of a cl ass ificati on of matrices A, B, C and D of the model :
In decentralized contro l, the overall plant is no longer controlled by single controller, but by several independent controllers. The controllers are designed in different steps by means of models that describe on ly the relevant parts of plant. Coordination in decentralized structures is restricted in accordance with the simplifications in the practical implementation of coupled decision makers, which are gained from the absence of information links, decentral ized structures are often used reduced the qual ity of the solut ion. Th e so lution is obtained by means of comp lete ly independent decis ion-makers .
140
station. Algebraical ly the hierarc hical structure of the system:
Each decision-maker needs only a limited information. [f the properties of the subsystem change, only model of decision-makers of this subsystem has to be adapted (figure J) .
Local unit I
= Ax + diagB Y = diagCjx X
A
(8)
/,
Y =Cx Where: A = A - B( diagK i )C has the block triangular form. Hence, (9)
Yj
=CjX j
The system has a hierarchical structure can be stabilized by decentra li zed control if and only if the isolated subsystem are comp letely controllable and completely observable through the input UI and output YI.
u j = K)'jy j (4) or decentra lized proportional-integral controller is (5)
With KII is integral gain and K p1 is proportional gain. The different control station has to be designed independently of each other on the basis of partia l models of the system.
2. I
A
x=Ax+Bv
u j = KjY; (2) The decentralized contro l state feedback is u 1 = KjX j (3) And decentralized control output feedback is
V;)
(7)
This property is preserved under de centralized control: U i ::: - K j (y j - V j ) independently of whether the control stations have dynam ical parts or not. In the model of system on the equation (6):
A decentralized control structure has to be chosen for reliability reason . If one decentralized controller fa ils to operate, only one part fails to do its task. A decentralized controller is a feedback controller, which consist of independent contro l stations, each of which receives the measurement data y, and influences the control input Ui only of attached subsystem, where the loca l units represent control stations
= - K Ii X,.i - K Pi (y i
(6)
o o
Fig. I. Decentralized control structure.
j
U
is reflected by the property of A to be block triangular:
Local unit 2
Subsystem I
U
j
2.2
Decentralized Design of ProporlionalIntegral (P I) Controllers
The control law of decentralized PI control ler is given by equation (5), where:
Decentralized Design ol Hierarchically Structured Systems
X
ri
=Y
j
-
V
j
(10)
The problem of designing decentralized PI controllers will be considered for the additional requi rements that the controller should ensure closed-loop stability even if some control stations are disconnected from the plant, that the contro l laws shou ld be obtained without setting up a complete model of the plant by sequential on-l ine tuning, or that design of the control stat ions should be carried out completely independently
The system is said to have a hierarchical structure since the cluster of subsystems can be grouped in different levels where the information fl ow is unidirectiona l from clusters at higher levels towards clusters at lower levels. The hierarchical structure system is stable if and only if isolated closed- loop subsystems are stable. Control lable and observable isolated subsystem can be stab ilized by attached control
141
A linear system with stable linear plant as defi ned by equat ion (I ) is decentralized integral controllable if and only if all principal minors of K, are positive. The static transmission matrix is:
Ks 2.3
= D CA - IB
Decentralized Design Composite Systems
eventually cooperative structure if there is an integer N such that the system is stable whenever the number N of subsystems exceeds
(1 1)
of
Controllability and observability of the symmetric composite system is not just necessary but even sufficient for the absence of fix modes under decentra lized control. These properties can be tested by the pairs (As, B) and (Au , B) are controllab le and the pairs (As , C) and (Ao , C) are observable, where:
Sym metric
Sym metric composite systems, whose structural properties are the identity of the subsystem and the symmetry of the dynamics intercon nections. It will be shown that sym metry within the whole system gi ves rise to substanti al simplifications of the modeling, analytical and desi gn problems. Since these resu lts are based on the structural properties of the overall system, albeit identical dynamic properties of the subsystems, arbitrarily strong interactions and an unrestra ined number of subsystems.
A. Ao
Yi
Z;
= AX I + BujEs i =Cx i =Czx ,
= A+E(Ld -Lq)C z = A+E[L d +(N - l)Lq ~z
( 15)
Symmetric composite systems make it possible to re formulate the problem of decentra lized control of the overall system as a prob lem of robust decentralized control of an auxiliary plant. The resu lts on symmetric system s can be extended to systems that consist of similar rather than identical subsystems. The structural properties of the system to be nearly symmetric can be used to develop a method fo r analyzing the stab ility of co mpos ite systems. The investigations of symmetric composite systems are relevant to technological processes whose subsystems behave si mi larly as, fTom a technological point of view; they participate in doing the same task
A system is called a symmetric composite syste m, while it can re presented by a model of the form : Xi
-
~
N (N > N) .
(1 2)
The interconnectio ns are de scribed by:
s = Lz
(13)
3.
where L is matrix symmetric :
Ld Lq
In practical application, plant cannot be described by linear model. It may result from non-li near, time-varying properties or incomplete knowledge of the plant. The fact that certain system properties exist as a resu lt of the system structure, and can be fou nd without knowing th e prec ise system parameters.
( 14)
Lq
Lq
STRUCTURAL PROPERTIES OF DECENTRA LIZED DESI GN
Lt!
Equation (1 2), (13) and (14) reflect the assu mpt ion that the subsystems are identical and coupled in a symmetric way. No restrictions are required on the dynamical pro perties of the subsyste ms . The stabil ity of the subsystem matrix A is necessary condition for the stability of the symmetric composite system, if the intercon nection matrix L has the property Ld = Lq The simplici ty of the stabi lity analysis is based on the structure of symmetric composite systems and doesn' t depend on some weakness of interactions.
Structura l propert ies are properties of large variety of parameter value, and not strongly dependent upon certain parameter value. Matrices A, B, an d C in the eq uation (I) are transformed into structure matrices by replacin g all non-vanishing elements by asterisks. Structural investigations of systems are based on the properties of structu ral matri ces . Structure matrices S., Sb, Se defines a class of structurally equivalent systems . The relation of matrix Aand a structure matrix S. is:
A symmetric composite system is said to have an asym ptotically cooperative structure or
142
AEp(Sa)={A:[A]=Sa}
(J6)
A structurally complete system is both structurally controllable and structurally observab le. Structurally controllability and structurall y observability are really structural properties. An uncertain system can be described by general model with the structure shown in figure 2.
Fig. 2. General model structure The set of decentralized fixed modes cons ists of the sets of the centralized fixed modes of the isolated subsystems. Subsystem in the equation (12) and (13) are completely controllable and completely observable, so the class of structurally equivalent systems has no structurally fixed modes.
the analysis of nth order matrices A, and Ao in equation (15) .
CONCLU SION Decomposition method is used for decomposing the overall system utilize structural properties. Instead of deri ving the models of the subsystems from the overall system description, such new methods should select that part or those properties of the overall system that have to be referred to in the analysis of a certain subsystem or design of a certain control station. Models that describe these properties can be set up at the subsystem level by corresponding control authorities. Method of decentralization by exploitation of structural properties starts from the overall system model. Most of the results can be applied witho~l~ using this complete model, but by explOltmg structural properties of whole system. The existence of which can be checked or empirically assumed to be satisfied.
The problem of designing a decentralized contro lieI':
Xri = FXri +GYi + HVj 'U j
(17)
=-K,x ri -K)'Yi +K,Vi
whe re i '" I ,2,3, ...... N, with N control stations that stabilizes the overall system is equivalent to designing a robust centralized controller that simultaneously stabilizes the plant. Th~ 110 behavior of the system is approximately describes by centralized system. Exploiting the structural propel1ies of the interaction can reduce the overall design problem. The reduction is possible for arbitrarily strongly couple subsystems and is particularly usefu l for plants with many subsystems. The structural property of the overall system to be nearly symmetric can be used to develop a method for analyzing the stability composite system. Conceptual and numerical simplifications of stability analysis of the composite system are not gained from breaking down the subsystem interconnection as in the com posite system method . Simplifications result from interaction of an approximate model, which protect structural properties of the overall system and whose stability analysis can be reduced to
143
REFERENCES Bahnasawi, AA , AI-Fuaid, A.S. and Mahmoud M.S. (1990). Decentralized and hierarchicai of interconnected uncertain control systems. Proc lEE, 0-137, 311-321. Davison, E.J. (1977b). Connectability and structural controllability of composite systems. Automatica, 13, 109- 113. Findeisen, W. (J 982). Decentralized and hierarchical control under consistency or disagreement of interest. Automatica. ) 8, 647-664 .
Grosdidier, P.and Morari, M. (1987) A Computer aidet methodo logy for the design of decentralized contro llers. Compul. Chem. Fundam., 11, 423-432. Jamshidi, M. (1983). Large Scale Systems. Modelling and Control, North- Holland. New York. Li, R.H. and Singh, M.G. (1983). Information Structures in deterministic decentralized control problem. IEEE Transaction System, Man. Cybern.,SMC-J3, 1162-1166. Lunze, 1. (1989a) Decentralized Control of Strongly Coupled Symmetric Composite Systems. IFAC Symp. On Large Scale Systems. Berlin. Vol. J, pp. ! 72-177.
Lunze, J. (I 989b). Stability Analysis of Large Scale System Composed of Strongly Coupled Similar Subsystem. Automatica, 25,561-570. Lunze, J. (1989c) Model Aggregation of Large Scale Systems with Symmetry Properties. System Analysis Model Simulation.,6, 749760. Mesarovic, M.D., Macko, M. and Takahara, Y (1970a) TheOlY of Hierarchical Multi/eve/ Systems, Academic Press: New York. Sezer, M.E. and Siljak. 0.0. (I98Ib). Structurally Fixed Modes, System Control. Lell., 1,60-64.
144
R.l. Widodo ' Electrical Engineering Department, Institute Technology Bandllng Emai l : [email protected]@lsI
ABSTRAC : Decentralization by exploitation strllctllral properties is the method based on the structural properties of the system rather than the composition into weakly coupled subsystems. A considerable reduction of the design problem can also be gained from specific structures of the controller to be designed Simplificat ions of the design problem can be derived if the control law is restricted to be of proportional-integral character. The application of the decentralized proportional-integral control/er is a direct consequence of the requirement of asymptotic regulation for step'wise disturbance and command Signals. According 10 Ihe internal model principle, each control stalion has 10 have several integrators. Keywords: decentralized control, structural properties, proportional-integral controllers.
I.
INTRODUCTION
= Ax(t) + 8u(t) yet) = Cx(t) + Du(t) x(t)
The theory of large-sca le systems is devoted to the problems that arise from the large size of systems to be controlled. Agricultural systems rely on the corporat ion of many different parts. If such systems are to be controlled, their analysis and control problem becomes very comp lex. In simplified way, this problem can be stated as the task to control at different places all the agricultural area. A Decentralized control scheme is appropriate because the variables to be controlled and the control inputs are many kilometers away from each other. Using faster computers with larger memories cannot solve this task simply.
(I)
Most of the difficulties of analytical and control by the complete problem are raised interdependence of the subsystems, there are links between arbitrary pairs of subsystems . The link may be mediated by one or more other subsystems. Conceptual simpl ification of analytical and control prob lem can be obtained if some subsystems have only a one-way effect on some ot hers.
2. DECENTRALIZED DESIGN
Decentralized design by exploitation of structural properties is based on the structura l properties of the system. 'Structural' properties are not strongly depending on the system parameters. In many practical application the plant cannot be describe by some precise linear model. Uncerta inties may resul t from incomp lete knowledge of the plant, from non-linear or timevarying properties, which cannot be reflected by some linear model, or from approximation made to reduce the model size. The system structure can defined on the basis of a cl ass ificati on of matrices A, B, C and D of the model :
In decentralized contro l, the overall plant is no longer controlled by single controller, but by several independent controllers. The controllers are designed in different steps by means of models that describe on ly the relevant parts of plant. Coordination in decentralized structures is restricted in accordance with the simplifications in the practical implementation of coupled decision makers, which are gained from the absence of information links, decentral ized structures are often used reduced the qual ity of the solut ion. Th e so lution is obtained by means of comp lete ly independent decis ion-makers .
140
station. Algebraical ly the hierarc hical structure of the system:
Each decision-maker needs only a limited information. [f the properties of the subsystem change, only model of decision-makers of this subsystem has to be adapted (figure J) .
Local unit I
= Ax + diagB Y = diagCjx X
A
(8)
/,
Y =Cx Where: A = A - B( diagK i )C has the block triangular form. Hence, (9)
Yj
=CjX j
The system has a hierarchical structure can be stabilized by decentra li zed control if and only if the isolated subsystem are comp letely controllable and completely observable through the input UI and output YI.
u j = K)'jy j (4) or decentra lized proportional-integral controller is (5)
With KII is integral gain and K p1 is proportional gain. The different control station has to be designed independently of each other on the basis of partia l models of the system.
2. I
A
x=Ax+Bv
u j = KjY; (2) The decentralized contro l state feedback is u 1 = KjX j (3) And decentralized control output feedback is
V;)
(7)
This property is preserved under de centralized control: U i ::: - K j (y j - V j ) independently of whether the control stations have dynam ical parts or not. In the model of system on the equation (6):
A decentralized control structure has to be chosen for reliability reason . If one decentralized controller fa ils to operate, only one part fails to do its task. A decentralized controller is a feedback controller, which consist of independent contro l stations, each of which receives the measurement data y, and influences the control input Ui only of attached subsystem, where the loca l units represent control stations
= - K Ii X,.i - K Pi (y i
(6)
o o
Fig. I. Decentralized control structure.
j
U
is reflected by the property of A to be block triangular:
Local unit 2
Subsystem I
U
j
2.2
Decentralized Design of ProporlionalIntegral (P I) Controllers
The control law of decentralized PI control ler is given by equation (5), where:
Decentralized Design ol Hierarchically Structured Systems
X
ri
=Y
j
-
V
j
(10)
The problem of designing decentralized PI controllers will be considered for the additional requi rements that the controller should ensure closed-loop stability even if some control stations are disconnected from the plant, that the contro l laws shou ld be obtained without setting up a complete model of the plant by sequential on-l ine tuning, or that design of the control stat ions should be carried out completely independently
The system is said to have a hierarchical structure since the cluster of subsystems can be grouped in different levels where the information fl ow is unidirectiona l from clusters at higher levels towards clusters at lower levels. The hierarchical structure system is stable if and only if isolated closed- loop subsystems are stable. Control lable and observable isolated subsystem can be stab ilized by attached control
141
A linear system with stable linear plant as defi ned by equat ion (I ) is decentralized integral controllable if and only if all principal minors of K, are positive. The static transmission matrix is:
Ks 2.3
= D CA - IB
Decentralized Design Composite Systems
eventually cooperative structure if there is an integer N such that the system is stable whenever the number N of subsystems exceeds
(1 1)
of
Controllability and observability of the symmetric composite system is not just necessary but even sufficient for the absence of fix modes under decentra lized control. These properties can be tested by the pairs (As, B) and (Au , B) are controllab le and the pairs (As , C) and (Ao , C) are observable, where:
Sym metric
Sym metric composite systems, whose structural properties are the identity of the subsystem and the symmetry of the dynamics intercon nections. It will be shown that sym metry within the whole system gi ves rise to substanti al simplifications of the modeling, analytical and desi gn problems. Since these resu lts are based on the structural properties of the overall system, albeit identical dynamic properties of the subsystems, arbitrarily strong interactions and an unrestra ined number of subsystems.
A. Ao
Yi
Z;
= AX I + BujEs i =Cx i =Czx ,
= A+E(Ld -Lq)C z = A+E[L d +(N - l)Lq ~z
( 15)
Symmetric composite systems make it possible to re formulate the problem of decentra lized control of the overall system as a prob lem of robust decentralized control of an auxiliary plant. The resu lts on symmetric system s can be extended to systems that consist of similar rather than identical subsystems. The structural properties of the system to be nearly symmetric can be used to develop a method fo r analyzing the stab ility of co mpos ite systems. The investigations of symmetric composite systems are relevant to technological processes whose subsystems behave si mi larly as, fTom a technological point of view; they participate in doing the same task
A system is called a symmetric composite syste m, while it can re presented by a model of the form : Xi
-
~
N (N > N) .
(1 2)
The interconnectio ns are de scribed by:
s = Lz
(13)
3.
where L is matrix symmetric :
Ld Lq
In practical application, plant cannot be described by linear model. It may result from non-li near, time-varying properties or incomplete knowledge of the plant. The fact that certain system properties exist as a resu lt of the system structure, and can be fou nd without knowing th e prec ise system parameters.
( 14)
Lq
Lq
STRUCTURAL PROPERTIES OF DECENTRA LIZED DESI GN
Lt!
Equation (1 2), (13) and (14) reflect the assu mpt ion that the subsystems are identical and coupled in a symmetric way. No restrictions are required on the dynamical pro perties of the subsyste ms . The stabil ity of the subsystem matrix A is necessary condition for the stability of the symmetric composite system, if the intercon nection matrix L has the property Ld = Lq The simplici ty of the stabi lity analysis is based on the structure of symmetric composite systems and doesn' t depend on some weakness of interactions.
Structura l propert ies are properties of large variety of parameter value, and not strongly dependent upon certain parameter value. Matrices A, B, an d C in the eq uation (I) are transformed into structure matrices by replacin g all non-vanishing elements by asterisks. Structural investigations of systems are based on the properties of structu ral matri ces . Structure matrices S., Sb, Se defines a class of structurally equivalent systems . The relation of matrix Aand a structure matrix S. is:
A symmetric composite system is said to have an asym ptotically cooperative structure or
142
AEp(Sa)={A:[A]=Sa}
(J6)
A structurally complete system is both structurally controllable and structurally observab le. Structurally controllability and structurall y observability are really structural properties. An uncertain system can be described by general model with the structure shown in figure 2.
Fig. 2. General model structure The set of decentralized fixed modes cons ists of the sets of the centralized fixed modes of the isolated subsystems. Subsystem in the equation (12) and (13) are completely controllable and completely observable, so the class of structurally equivalent systems has no structurally fixed modes.
the analysis of nth order matrices A, and Ao in equation (15) .
CONCLU SION Decomposition method is used for decomposing the overall system utilize structural properties. Instead of deri ving the models of the subsystems from the overall system description, such new methods should select that part or those properties of the overall system that have to be referred to in the analysis of a certain subsystem or design of a certain control station. Models that describe these properties can be set up at the subsystem level by corresponding control authorities. Method of decentralization by exploitation of structural properties starts from the overall system model. Most of the results can be applied witho~l~ using this complete model, but by explOltmg structural properties of whole system. The existence of which can be checked or empirically assumed to be satisfied.
The problem of designing a decentralized contro lieI':
Xri = FXri +GYi + HVj 'U j
(17)
=-K,x ri -K)'Yi +K,Vi
whe re i '" I ,2,3, ...... N, with N control stations that stabilizes the overall system is equivalent to designing a robust centralized controller that simultaneously stabilizes the plant. Th~ 110 behavior of the system is approximately describes by centralized system. Exploiting the structural propel1ies of the interaction can reduce the overall design problem. The reduction is possible for arbitrarily strongly couple subsystems and is particularly usefu l for plants with many subsystems. The structural property of the overall system to be nearly symmetric can be used to develop a method for analyzing the stability composite system. Conceptual and numerical simplifications of stability analysis of the composite system are not gained from breaking down the subsystem interconnection as in the com posite system method . Simplifications result from interaction of an approximate model, which protect structural properties of the overall system and whose stability analysis can be reduced to
143
REFERENCES Bahnasawi, AA , AI-Fuaid, A.S. and Mahmoud M.S. (1990). Decentralized and hierarchicai of interconnected uncertain control systems. Proc lEE, 0-137, 311-321. Davison, E.J. (1977b). Connectability and structural controllability of composite systems. Automatica, 13, 109- 113. Findeisen, W. (J 982). Decentralized and hierarchical control under consistency or disagreement of interest. Automatica. ) 8, 647-664 .
Grosdidier, P.and Morari, M. (1987) A Computer aidet methodo logy for the design of decentralized contro llers. Compul. Chem. Fundam., 11, 423-432. Jamshidi, M. (1983). Large Scale Systems. Modelling and Control, North- Holland. New York. Li, R.H. and Singh, M.G. (1983). Information Structures in deterministic decentralized control problem. IEEE Transaction System, Man. Cybern.,SMC-J3, 1162-1166. Lunze, 1. (1989a) Decentralized Control of Strongly Coupled Symmetric Composite Systems. IFAC Symp. On Large Scale Systems. Berlin. Vol. J, pp. ! 72-177.
Lunze, J. (I 989b). Stability Analysis of Large Scale System Composed of Strongly Coupled Similar Subsystem. Automatica, 25,561-570. Lunze, J. (1989c) Model Aggregation of Large Scale Systems with Symmetry Properties. System Analysis Model Simulation.,6, 749760. Mesarovic, M.D., Macko, M. and Takahara, Y (1970a) TheOlY of Hierarchical Multi/eve/ Systems, Academic Press: New York. Sezer, M.E. and Siljak. 0.0. (I98Ib). Structurally Fixed Modes, System Control. Lell., 1,60-64.
144