Decentralized Multirate Control of Interconnected Systems

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1 Decentralized Multirate Control of Interconnected Systems LUBOMIR BAKULE JOSEF BOHM Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Prague CZECH REPUBLIC Abstract: - The objective of the paper is to present a decentralized control approach to interconnected multirate linear systems with synchronous sampling. Each subsystem is assigned a single sampling rate. It simplifies local controllers design. The overall discrete-time system eponential convergence is achieved at the common sampling period via the concept of the M-matrices. The overall original continuous-time system stability is assured using the stability of the corresponding multi-rate discrete-time system. A numerical eample is supplied. The control strategy is natural for plants with widely different characteristic frequencies of subsystems or when sensor measurements are performed at different sampling rates. Key Words: Decentralized systems large scale systems multirate systems control systems Introduction Decentralized control approaches to comple dynamic systems are attractive in practice with a view to their facility mainly in implementation low compleity and structural properties. Considering a comple system as an continuous-time interconnected system having subsystems with widely different bandwith and characteristic frequencies multirate sampling is a logical strategy for designing discrete-time controllers. Both frequency and state-space description methods are utilized. A state-space description of centralized multirate systems developed Araki and Yamamoto 3 sampling rate selection proposed Berg et al.. Recently centralized multirate sampleddata control system received a great deal of attencion e.g. in optimal control 5 predictive control two-time scale systems or 9. Though decentralized control systems have been considered mainly for continuous-time systems papers dealing with decentralized multirate control are rare Sequential loop closure (SLC) for multirate control systems described for instance Franklin et al. 7. In contrast to previous works this paper deals with the problem of decentralized stabilization of multirate interconnected systems in the state-space where first local subsystems control is designed at each subsystem s rate for equivalent discrete-time system and then the overall discrete-time closed-loop system stability is tested using M-matrices at the common system rate. Eponential stability is assured by this test. Moreover the relation of structural perturbations of interconnections between the original system and its sampled equivalent system is given in terms of connective stability. Problem Formulation Denote (P c HLS) an interconnected multirate sampled-data system: a continuous-time plant and a digital controller. P c denotes a finite dimensional linear time-invariant system as follows: ẋ c A c c B c u c () y c C c c where c ( ct ct ) T u c (u ct u ct ) T y c (y ct y ct ) T are n m p dimensional state input output vectors consisting of the corresponding subsystem vectors. A c (A c ij ) Bc diag(b c B ) C c diag(c c C ) are appropriately dimensioned matrices. HLS is a linear causal multirate decentralized sampled-data controller. H is a zero-order singlerate hold as follows: H diag(h H ) u ci H i u i i u ci u ci (kt t) u i kt () < t T ; k... where T denotes a common sampling period. Denote T m mt where m is an integer.

2 S is a multirate sampler defined by S diag(s S ) y i S i y ci i y i (3) y kt y c (kt ) y kmt y c (kmt ) k... L is a discrete-time controller in the form u (kmj)t K y (kmj)t r (kmj)t L (kmj)t r (kmj)t () u (kmj)t K y(km)t r(km)t L (km)t r(km)t j... m ; k... where r r are given reference signals and K i L i Ci c for i. We consider a static controller only. Define a state of the system ()-(). A state at time t is sufficient information for the computation of all future values of all signals i.e. the continuous signals c u c and the discrete signals u y. Denote u (.) (u T (.) ut (.) )T. The information c (t) at the sampling time t kmt o and u T (km )T o is sufficient to compute the future values of all signals. To get c (t) u c (t) y c (t) for kmt o < t (km )T o we obtain a candidate state for the hybrid system in an analogous way of reasoning as in Denote A e A e dii e A c T o e A c ii To To B e e Acν dνb c I C e C c To Bi e e Ac ii ν dνb c i I C ie C ic i. Lemma. If in the continuous time (A c B c ) (A c B) c (A c B) c are stabilizable and the matrices A c A c A c satisfy the assumptions A A A3 respectively so in discrete time are (A e B e ) (A e d Be ) (A e d Be ). Similarly if (C c A c ) (C A c ) (C A c ) are detectable in the continuous time so are (C e A e ) (C e A e d ) (Ce A e d ) respectively in the discrete time. Assertion analogous to Lemma can be simply given when changing T by T m. Theorem. Suppose A c A c A c satisfy the assumptions A A A3 respectively. If hf hf hf are eponentially convergent so are h (t) h (t) h (t) respectively. T h (t) ( ct (t) u T (km )T o )T kmt o < t (km )T o. (5) Definition. h (t) is eponentially convergent if there eist constants a b > such that for every initial time t o and initial state h (t) h (t) h (t o ) be a(t to) t t o () where h ( T h h) /. We summarize sufficient conditions for stabilizability and detectability of the above state model. Assumptions. Suppose that the following hold: A. No point α jπk/t α jπk/t m α is an eigenvalue of A c. A. No point α jπk/t α jπk/t m α is an eigenvalue of A c. A3. No point α jπk/t α jπk/t m α is an eigenvalue of A c. The Problem. The goal is to derive stability relations for the system ()-() by using disjoint decomposition approach for interconnected systems and supply an illustrative numerical eample of their usage. 3 Solution The way of the solution proceedes as follows: First independent local controllers are designed each at its own rate to assure the eponential stability of each closed-loop system. Then the overall closed-loop system is constructed using the designed decentralized controllers. This system is further rewritten at the slow rate. Such system is a time-invariant system which is more convenient for stability tests. Its eponential stability is tested using the M-matrices. The eponential convergence of the hybrid system then follows the eponential convergence of the corresponding discrete-time system.

3 3. Local subsystems control design Denote where B d (km)to A d kmto B d u kmto y kmto C c kmto (7) A d ep(a c dt o ) diag(a d A d ) To A c d diag(a c A c ) ep(a c dν)b c dν diag(b d B d ). Suppose that the assumptions A A3 are satisfied. The closed-loop systems have the form (km)t o (A d B d L ) kmt o B d r kmt o A f kmt o B d r kmt o () (km)t o A m d (A d B d L ) kmt o B d r kmt o A s kmt o B d r kmt o. The stability of A f A s is evaluated by their eigenvalues. 3. Overall system control Denote the sampled-data system corresponding to Eq.() with the sampling period T o as follows: (km)to (km)t o   kmt   kmt ˆB ˆB u kmt ˆB ˆB u (9) kmt where the matrices are defined as follows  ep(a c T )  ep(a c T )  ep(a c T )  ep(a c T ) ˆB ˆB ˆB ˆB T T T T ep(a c ν)dνb c ep(a c ν)dνb c ep(a c ν)dνb c () ep(a c ν)dνb c. The closed loop system has two local loops as follows: u (kmj)t L (kmj)t r (kmj)t u (kmj)t L kmt r kmt () j... m ; k... The resulting closed-loop system may be rewritten for j in the form (km)t (km)t kmt kmt  ˆB L   ˆB L  ˆB L ˆB L ˆB ˆB r kmt ˆB ˆB rkmt () kmt We proceed in deriving the overall closed-loop system at the slow rate to get a time-invariant system for j m as follows: (kmm)t (kmm)t  ˆB L   ˆB (kmm )T L  (kmm )T ˆB ˆB r (kmm )T ˆB ˆB rkmt ˆB L ˆB L kmt  ˆB L   ˆB L   ˆB L   ˆB L  (kmm )T (kmm )T  ˆB L  ˆB L  ˆB L  ˆB L  ˆB L  ˆB ˆB  ˆB L  ˆB ˆB r (kmm )T rkmt ˆB ˆB r (kmm )T ˆB ˆB rkmt ˆB L kmt. kmt ˆB L... (3) This equation may be considered in the global form as follows: or equivalently (k)mt (k)mt Ā (kmt ) Br e () (k)mt kmt kmt Ā Ā Ā Ā Br e (5)

4 where m Ā A m A i ˆBo i B A m ˆB A m ˆB... ˆB Â A ˆB L Â Â ˆB () L Â ˆB ˆB ˆB ˆB ˆB ˆB L ˆB o ˆB L r T e (r kt r kt r (k )T r kt... r (km )T r kt ). (7) Consider further the closed-loop subsystem description at the slow rate. It has the form or equivalently (k)mt (k)mt (k)mt Ād kmt B d r e () A m f A s A m f B f B s kmt (9) kmt r e r e. The matri A I Ā Ād (A Iij ) defines interconections i j. Denote further ζ ij λ / M (AT Iij A Iij) λ / M (.) is a maimal eigenvalue of the matri (.). σ M (.) is the maimal singular value of (.) P i is a unique positive definite solution to the Lyapunov equation Ā T diip i Ā dii P i I i. () The eponential stability test is made using the following theorem. Theorem. The system () is eponentially stable if the matri W (w ij ) is an M-matri where w ii s i ζ ii w ij ζ ij () s i σ M (P i ) σ / M (P i)σ / M (P i I i ). Let us deal with structural perturbations now. Denote ε ε(t) ε ε is an arbitrary time function. Introduce A Iε εa I and prove the following. Theorem 3. Suppose the closed-loop overall system described by the matri Ā satisfies the condition of Theorem. Then the system described by the matri Āε A d A Iε is connectively stable. Note only that a singlerate case is considered in. Consider the eponential convergence now. Denote the hybrid state for the slow rate as hs (t) T ( c (t) u (k )mto ) for kmt o < t (k )mt o. Theorem. Suppose that A c satisfies the assumption A. If hs is eponentially convergent so is h (t). Eample Formulation: Consider the mechanical system consisting of two masses m and M connected by a spring with a constant k. Two forces acts independently on each mass. Position and velocity of each mass is available for measurements. The scheme of the eperiment is drawn in the Fig.. u M k m u Fig. Two-mass spring interconnected system The system is described by the following state space model: ẋ ẋ ẋ 3 k m k m 3 k ẋ M k M m u () u M y where k m M. Suppose a given sampling with T o. and T (sec). The eigenvalues EV (.) of the matrices A c A d A d are: EV (A c ) ± j.; EV (A d ) ± j. and EV (A d ) ± j.3. The matrices satisfy all required assumptions. We

5 use the reference inputs r (t) 5sign(sin(.t)) and r (t) 5sign(sin(.t)). The goal is to design local gain matrices L L using the derived methodology stabilizing the resulting closed-loop system and satisfying dynamic requirements on the closed-loop system responses. These requiremets are as follows: maimum point deviation from the reference signal ±3 for the fast deviation ± for the slow deviation and no vibrations. Results: First continuous-time centralized controller is designed to get reference responses satisfying design requirements. Then multirate decentralized controller is designed to approach in the best way the reference responses of the centralized control design case. Pole placement technique has been used. Denote these two different cases for the given continuous-time system as follows:. Centralized continuous-time controller. The global closed-loop system response of positions serve as a reference response.. Decentralized multirate controller for the overall system. Case. L c We choose the gain matric L c as follows: (3) The corresponding closed-loop eigenvalues EV c are EV c () The responses of positions are in Fig.. Case. The fast discrete-time subsystem matrices sampled at the fast rate are as follows: A f (5) Choosing B f.5.99 T. L () we get the closed-loop eigenvalues EV f EV f (7) The slow discrete-time subsystem matrices sampled at the slow rate are A s () B s Choosing L (9) we get the closed-loop eigenvalues EV s as follows: EV s.5.3. (3) The closed-loop subsystems are stable. X X Centralized fast state 5 5 Multirate fast state 5 5 X3 X3 Centralized slow state 5 5 Multirate slow state 5 5 Fig. Two-mass spring interconnected system The closed-loop overall discrete-time system with the gain matrices L L is described at the slow rate by the matri Ā as follows: Ā (3) The matri A I has the form A I (3) The solution of the Lyapunov functions P P has the form.59.3 P (33) P. (3) The corresponding M-matri is W.79.9 with the eigenvalues (35)

6 The responses of positions are in Fig.. The conditions on the stability as well as on the dynamic requirements are satisfied. 5 Conclusions The paper solves the problem of decentralized stabilization of interconnected multirate linear systems where each subsystem has assigned a single sampling rate. The control design of local controllers is performed at the subsystem level and at the subsystems rates. The overall system must be rewritten at the slow rate to get an time-invariant system which is more suitable for stability testing. Stability of such system is tested using M-matrices. It enables simultanesously to consider and evaluate the influence of structural perturbations of interconnections on the system stability. The eponential convergence of the hybrid system is assured using this approach. A numerical eample of control design for the interconnected mechanical system is supplied. The presented methodology refers mainly to the generality of generation of gain matrices using different methods. Two interconnected subsystems are considered only. The generalization to more subsystems is straightforward. Acknowledgment This work was supported in part by the ASCR under Grant A75. References Al-Rahmani H.M. and Franklin G.F. A new optimal multirate control of linear periodic and time-invariant systems. IEEE Transactions on Automatic Control AC pp. 5. Apostolakis I.S. and Jordan D. A time invariant approach to multirate optimal regulator design. International Journal of Control pp Araki M. and Yamamoto K. Multivariable multirate sampled-data systems: state-space description transfer characteristics and Nyquist criterion. I.E.E.E. Transactions on Automatic Control AC-3 9. pp Berg M. Amit N. and Powell J.D. Multirate digital control system design. I.E.E.E. Transactions on Automatic Control AC-33 9 pp Colaneri P. Scattolini R. and Schiavoni N. LQG optimal control of multirate sampled-data systems. I.E.E.E. Transactions on Automatic Control AC pp. 75. Francis B.A. and Georgiou T.T. Stability theory for linear time-invariant plants with periodic digital controllers I.E.E.E. Transactions on Automatic Control AC-33 9 pp Franklin G.F. Powell J.D. and Workman M.L. Digital Control of Dynamic Systems (Addison-Wesley NJ) 99. Ito H. Ohmuri H. and Sano A. Stability analysis of multirate sampled-data control systems. IMA Journal of Mathematical Control and Information 99 pp Ito H. Ohmuri H. and Sano A. A subsystem design approach to continuos-time performance of decentralized multirate sampled-data systems. International Journal of Systems Science 995 pp Lee J.H. Gelormino M.S. and Morari M. Model predictive control of multi-rate sampleddata systems: a state-space approach. International Journal of Control pp Lennartson B. Multirate sampled-data control of two-time-scale systems. I.E.E.E. Transactions on Automatic Control AC-3 99 pp.. Sezer M.E. and Šiljak D.D. Robust stability of discrete systems. International Journal of Control 9 pp Sezer M.E. and Šiljak D.D. Decentralized multirate control. I.E.E.E. Transactions on Automatic Control pp.. Šiljak D.D. Decentralized Control of Comple Systems N.York Academic Press Xu X.-M. Xi Y.-G. and Zhang Z.-J. 9 Decentralized predictive control (DPC) of large scale systems. Information and Decision Technologies 9 pp