STABILITY ANALYSIS TOOL FOR TUNING UNCONSTRAINED DECENTRALIZED MODEL PREDICTIVE CONTROLLERS Massimo Vaccarini Sauro Longhi M. Reza Katebi D.I.I.G.A., Università Politecnica delle Marche, Ancona, Italy Industrial Control Centre, University of Strathclyde, Glasgow, Scotland, UK Abstract: Some rocesses are naturally suitable to be controlled in a decentralized framework: centralized control solutions are often infeasible in dealing with large scale lants and they are technologically rohibitive when the rocesses are too fast for the available comutational resources. In these cases, the resulting control roblem is usually slit in many smaller subroblems and the global requirements are guaranteed by means of a roer coordination. The unconstrained decentralized case is here considered and a coordination strategy is roosed for imroving the global control erformances. This aer resent a tool for setting u and tuning a nominally stable decentralized Model Predictive Controller. Numerical examles are roosed for testing and validating the develoed technique. Keywords: Decentralized Control, Predictive Control, Communication Networks, Stability Analysis, Otimal Regulators. INTRODUCTION The current diffusion of networks allows the control technologies and methodologies to fully exress their otentials in many alication fields. By means of the fast communication technologies, nowadays different indeendent controllers distributed in a wide area can exchange information in order to imrove both the local and the global control erformances. Robotic alications with multile autonomous agents ursuing a common goal, control alications in manufacturing and rocess industry where multile units cooeratively make a roduct, suly chains in which multile actors that influence each other are involved, and large scale ower systems reresent some tyical situations. A decentralized control solution based on indeendent agents is here considered for the regulation of different interacting rocesses. Global objectives, such as closed-loo stability and erformance requirements for the global rocess require coordination among the control agents. Different solutions have been develoed in literature. The required coordination can be introduced by a hierarchical decentralized control scheme, where a suervisor comutes the global otimum and coordinates all the control agents (Šiljak, 996). A decentralized filtering and control architecture managed by a global coordinator has been roosed by Katebi and Johnson (997) for guaranteing steady state otimality of the control actions. A different solution based on the team theory and asynchronous teams has been roosed in Camonogara (). A decomosition of the global rocess into subsystems, based on the structural roerties and hysical constraints, has been develoed in Jia and Krogh (), Cheng and Krogh (), Camonogara et al. (), Jia (), where each agent controls a subsystem making use of local model, objectives and constraints. Gudi et al. (4) investigated the identification rocedure oriented to decentralized MPC. The effectiveness of the communication based MPC and the imortance of the interaction analysis for imroving the control erformances has been validated in Venkat et al. ().
A artitioning of both actuators and sensors has been roosed by Motee and Sayyar-Rodsari () as a solution for the roblem of comutational comlexity for the use of MPC technology in large scale dynamic systems. Adoting a roer decentralized solution, the global comutational effort can be reduced without significant degradations of the control erformances and the fault tolerant issues can be imroved (El-Farra et al., 4). The functionalities of a Networked Decentralized Model Predictive Control solution (ND-MPC) have been recently tested on real lants with relatively strong interactions. The results obtained with the decentralized MPC of a gasifier seem to be satisfactory if comared with those obtained by the classical centralized MPC (Longhi et al., ). Stability analysis tools for testing the stability of the ND-MPC architecture have been also roosed by Vaccarini et al. (6). This aer summarizes the results on the stability analysis develoed by the authors and resent a set of simulation tests to validate the roosed tool and to illustrate its use for the synthesis of decentralized controllers with communication networks. After a descrition of the adoted ND-MPC strategy rovided in Section, the stability analysis tool is resented in Section. Based on the former considerations, the control algorithm is outlined in Section 4 and the tool is alied to the synthesis of decentralized controllers for a set of testing lants in Section. In order to simlify the mathematical exressions, some notations are here introduced. Given the numbers k Z, h Z, m N, j N, i N, n N such that k h, j, m h k +, i =,...,n: v A A T va is the norm of vector v induced by matrix A; λ j {A} is the j-th eigenvalue of a square matrix A. ˆx(k h) E [ x(k) Y h ] is the (h k)-ste ahead rediction of x, given the measurements Y h u to time k; u(k h) is the value of u(k), comuted at time h; X i (k,m h) is a stacked vector made by the vectors x i (k h),...,x i (k + m h); X(k,m h) is a stacked vector made by the vectors X (k,m h),...,x n (k,m h);. NETWORKED DECENTRALIZED MPC For achieving global erformance objectives, a coordination scheme based on communication among control agents is develoed. The control actions are comuted by a set of subcontrollers which are indeendent agents able to dynamically exchange a restricted set of information. In the roosed control architecture, each agent A i imlements an MPC algorithm for the subsystem S i using both local information acquired on S i and the estimate of the interactions among S i and the other subsystems S j, j =,...,n, j i. The resulting otimal sequence and the future rediction of the state over the rediction horizon, have to be exchanged among subsystems through a Local Area Network (LAN). As well known, Model Predictive Control acts according to the receding horizon rincile: at each samling time, using a redictive model of the system dynamics, the resonse of the rocess to changes in maniulated variables over a fixed horizon is redicted. Based on a roer cost function, a finite-horizon otimal control roblem is solved to obtain current and future moves of the maniulated variables. Only the first comuted move is alied to the real system. The same rocedure is reeated at the next control ste based on the new measurement. Although this comutation is an oen-loo control roblem, the receding horizon rincile allows MPC to generate a feedback-control law. Let consider a linear, discrete-time reresentation of a lant P: x(k + ) =Ax(k) + Bu(k) y(k) =Cx(k). (a) (b) and denote with x R n x, u R n u, y(k) R n y and y d R n y, its state, control inut, outut and desired outut, resectively. Suose that P is comosed by n subsystems S i whose state-sace reresentation is: x i (k + ) =A ii x i (k) + B ii u i (k) + w i (k) y i (k) =C ii x i (k) + v i (k). (a) (b) where vectors x ii, u i, y i and y d i are the local state, control inut, outut and desired outut resectively, and vectors w i and v i, named state and outut interaction vectors, are given by: v i (k) w i (k) n j=( j i) n j=( j i) A i j x j (k) + C i j x j (k). n j=( j i) B i j u j (k), (a) (b) Definition. (ND-MPC). Given the lant P comosed by n subsystems S i, i =,...,n the unconstrained Networked Decentralized Model Predictive Control roblem with rediction horizon and control horizon m consists of finding, at time k, a set of indeendent agents A i, i =,...,n, such that each A i minimizes the local cost function J i l= ŷ i (k + l k) y d i (k + l k) + Qi + m l= u i (k + l k), Ri subject, for l =,...,, to the model constraint: (4a)
ŷ i (k + l k) = ˆv i (k + l k ) +C ii A l ii ˆx i (k k)+ + l s= + C ii A s ii B ii u i (k + l s k)+ l s= C ii A s ii ŵ i (k + l s k ). (4b) Definition. (Interaction). When a changes in inut, state or outut variables of a subsystem S i roduces variations of inut, state or outut variables of a subsystem S j, it is said that S i interacts with S j. Definition. (Connection). When the agent A i sends information about the future behavior of subsystem S i to the agent A j, it is said that A i is connected with A j. Definition 4. (Neighboring Agent). If the agent A i is connected with A j, A i is called inut neighboring agent of A j and A j is called outut neighboring agent of A i. A i and A j are said neighboring agents. Definition. (Neighborhood of an Agent). The inut (outut) neighborhood of an agent A i is the set of its inut (outut) neighboring agents. Each agent A i, solution to the ND-MPC roblem, can be decomosed in three arts: an otimizer, a state redictor and an interaction redictor. At time k, based on the exchanged information, the interaction rediction together with the local measurement is used by the otimizer to solve the MPC otimization roblem. Once comuted the otimal sequence { u i (k k),..., u i (k + m k)}, which minimize the cost function (4a), the first element u i (k k) is selected and u i (k) = u i (k ) + u i (k k) is alied as control inut to S i. Then the state redictor comutes an estimate of the future state trajectory and broadcasts the otimal control sequence over the control horizon and the state redictions over the rediction horizon to its outut neighborhood. The following assumtions are here considered: Assumtions. rediction and control horizons are the same for each agent; control agents are synchronous; control agents communicate only once within a samling interval; the communication channel introduces a delay of a single samling time. Lemma. (Quadratic Program). Under the Assumtions, at ste k, each agent A i solution to the ND- MPC roblem, has to solve the following otimization roblem: min J i = U i (k,m k) T H i U i (k,m k) G T i U i (k,m k). U i (k,m k) () where H i =Ni T Q i N i + R i, (6a) G i =Ni T Q i [Yi d (k +, k) L i ˆx i (k k) M i u i (k )+ S i Ŵ i (k, k ) T i ˆV i (k, k )]. (6b) The introduced matrices are used for comuting the outut redictions: L i, M i, N i, S i deend only on the system matrices A i, B i and C i, matrix T i introduces a unit delay in the interaction vector and matrices Q i, R i are made by a block relication of the local weighting matrices Q i, R i. Equation () defines an unconstrained Quadratic Program which has to be solved on-line at each samling instant. For the sake of brevity, all the roofs of the results stated in this aer are are omitted. Refer to Longhi et al. () and Vaccarini et al. (6) for further details.. STABILITY ANALYSIS OF ND-MPC The main idea is to find an exlicit solution for the ND-MPC roblem and to use it for obtaining a mathematical reresentation of the closed-loo system. Once the closed-loo dynamic is known, the stability condition can be verified by analyzing the dynamic matrix. For this urose, denote with à i, B i, C i, Ā i and B i roer local interaction matrices and with Ã, B, C, Ā and B their corresonding global stacked versions. The gain of each MPC agent for the control effort movement is reresented by K i whereas matrix κ i reresents the gain for the magnitude of the local control effort over the whole control horizon m. Define I as the matrix for selecting the first comuted movements from the otimal control sequence. Denote with L i, M i two state rediction matrices comosed by system matrices, and introduce I i as an auxiliary identity block matrix. Assume that L, M, L, M, S, T, I, κ are diagonalizations of the local matrices L i, M i, L i, M i, S i, T i, I i, κ i resectively and define: θ κl, φ κ ( Sà + T C ), (7a) ρ I I κ ( MI + S B ). (7b) Lemma. (Interaction Predictor). Under the Assumtions, at ste k, for each agent A i solution to the ND-MPC roblem, the redictions of the interaction vectors are given by: Ŵ i (k, k ) =à i ˆX(k, k ) + B i U(k,m k ), (8a) ˆV i (k, k ) = C i ˆX(k, k ), (8b) and the global rediction vectors take the following form: Ŵ(k, k ) =à ˆX(k, k ) + BU(k,m k ), (9a) ˆV (k, k ) = C ˆX(k, k ). (9b)
The stacked vectors X(k, h) and U(k,m h) are built with both the local estimations and the information collected from the inut neighborhood by agent A i. Null entries will corresond to the subsystems which don t belong to the inut neighborhood. Note that at time k, agent A i uses the redictions comuted and broadcasted at time k { ˆv i (k k ),..., ˆv i (k + k )} and {ŵ i (k k ),...,ŵ i (k+ k )}. Lemma. (State Predictor). Under the Assumtions, at ste k, for each agent A i solution to the ND- MPC roblem, the decentralized state rediction for the agent A i is exressed by: ˆX i (k +, k) = L i ˆx i (k k) + M i U i (k,m k)+ + Ā i ˆX(k, k ) + B i U(k,m k ). () and the decentralized rediction equation for the overall system is given by the matrix form: ˆX(k +, k) = L ˆx(k k) + MU(k,m k)+ + Ā ˆX(k, k ) + BU(k,m k ). () Lemma 4. (Exlicit Solution). Under the Assumtions, at ste k, for each agent A i solution to the ND-MPC roblem, the exlicit form of the control action alied by the agent A i to the subsystem S i is given by: u i (k) = (I K i M i )u i (k )+ [ ] + K i Yi d (k +, k) L i ˆx i (k k) + K i S i Ŵ i (k, k ) K i T i ˆV i (k, k ). () Lemma. (Otimal Control Sequence). Under the Assumtions, at ste k, for each agent A i solution to the ND-MPC roblem, the exression of the otimal control sequence U i (k,m k) is: U i (k,m k) = I iu i (k )+ [ κ i Yi d (k +, k) L i x i (k) M i u i (k )+ ] S i Ŵ i (k, k ) T i ˆV i (k, k ). () and its global exression is: U(k,m k) = θ ˆx(k k) + φ ˆX(k, k ) + ρu(k,m k )+ + κy d (k +, k). (4) Theorem. (ND-MPC Stability). The closed-loo system given by the feedback connection of the lant P with a solution of the ND-MPC roblem, comosed by a set of indeendent agents A i, i =,...,n, is asymtotically stable if and only if: λ j A BI L Ā M B θa + φ L φā ρ + θbi + φ M φ B I mnu <, j [,...,n ND ], n ND = n x + n x + mn u, () where the n ND is the order of the global closed-loo system. The first two block rows of the global closed loo dynamic matrix in Equation () are formed by elements of matrix A (in the first two block columns) and matrix B (in the remaining two block columns). The Third block row is made by all the rocess matrices A, B and C and the weighting matrices Q i and R i and deends also on the horizons and m. 4. CONTROL ALGORITHM At samle time k, each agent A i, solution to the ND- MPC roblem: (i) Acquires the measures. (ii) Acquires the redicted future state trajectories X j (k, k ) and control inuts U j (k,m k ) from the neighboring agents and, once combined with the local state trajectory X i (k, k ) and the control inut U i (k,m k ), it builds X(k, k ) and U(k,m k ) and comutes the corresonding redictions of the interactions (8a). (iii) Comutes the otimal control sequence (). (iv) Alies the first element u i (k) = Ii U i(k,m k) of the otimal sequence U i (k,m k) as control inut to S i. (v) Comutes the future state trajectory () of the subsystem S i over the rediction horizon where ˆx i (k k) = x i (k) is given by the measures. (vi) Broadcasts the otimal sequence U i (k,m k) and the redicted state trajectory X i (k +, k) to the neighboring agents. (vii) Iterates. In the revious equations, the state rediction ˆx i (k k) has been relaced with the actual state x i (k) because of the hyothesis of fully accessible state. The desired outut Y d i (k +, k) for the agent A i is rovided by a roer reference generator R i that can assume either known or unknown future desired outut.. NUMERICAL TESTS In this section the roosed distributed control strategy is alied to a testing rocess. Although different weighting matrices can be used for each controller and the control horizon can be smaller than the rediction horizon, for simlifying the grahical reresentations, equal rediction and control horizons ( = m) and weighting matrices R = I nu and Q = I ny will be used. The following unstable, non-minimum hase lant P is considered for testing the resented tools. [ ].7 α [ ] y (s) (s )(s + ) = (s + ) u (s) y (s) α.7(s ) u (s) (s + ) (s + ) (6)
MPC ND MPC MPC ND MPC Log Log Log Log 4 4.. maximum eigenvalue.. maximum eigenvalue... Log. Log (a) (a) y y y y u u u u 4 time (s) (b) Fig.. Stability regions and maximum closed-loo eigenvalues for α =. (a) and control erformances with =. and = (b). The corresonding discrete-time state-sace realization is decomosed in two SISO subsystems. Assuming that inut u controls outut y and inut u controls outut y, the coefficient α reresents a measure of the interactions. The maximum eigenvalues comuted by the dynamic matrices obtained with the revious stated theorems are lotted in the three dimensional grahs of Figures (a), (a) and (a). The Z axis of these lots reresents the maximum eigenvalue for the centralized MPC (dark gray surface) and the decentralized MPC with communication (light gray surface). The X and Y axis reresent the logarithm of the weight and the rediction horizon, resectively. The corresonding stable region (white surface) is reresented in the uside art of these lots both for the centralized and the decentralized case. For each lant, the control erformances of ND-MPC (black lines) are comared with the centralized MPC (gray lines) for a given combination of the arameters, as shown in Figures (b), (b) and (b). 4 time (s) (b) Fig.. Stability regions and maximum closed-loo eigenvalues for α =. (a) and control erformances with =. and = (b). In the roosed examles, which are a set within the erformed numerical tests, the stability erformances deend on the choice of the tuning arameters and. In articular the stability of the closed-loo system is guaranteed for different combinations of the tuning arameters. A wider range of tuning arameters is available for weak interactions whereas a smaller stability region characterizes lants with strong interactions. At the same time, more weak are the interactions, more ND-MPC shows control erformances similar to centralized MPC. Often, when MPC is stable, ND-MPC has the same maximum eigenvalues; in some cases ND-MPC seems to be even better. There are, of course, situations in which the closed loo cannot be stabilized by the roosed decentralized strategy. For examle, when the interactions become strong as in Figure, ND-MPC cannot stabilize the rocess (for this reason it is not lotted in Figure ) whereas MPC rovide an accetable set of stable tuning arameters.
maximum eigenvalue y y u u MPC Log Log (a) ND MPC Log 4 time (s) (b) Fig.. Stability regions and maximum closed-loo eigenvalues for α = (a) and control erformances with =. and = (b). 6. CONCLUSIONS AND FUTURE WORKS In this aer a stability analysis tool for tuning ND- MPC architectures has been resented. The rovided examles show that the roosed ND-MPC can often stabilize the rocess with a roer choice of the tuning arameters. In these regions the maximum eigenvalues aroach that ones of the centralized case. However, in some cases, ND-MPC is not able to stabilize the rocess and other strategies must be considered. In this work the unconstrained MPC for linear rocess with accessible state has been considered. Future works are needed for extending these conclusions in the more general case of constrained MPC. Techniques for obtaining exlicit solutions in the constrained case have already been investigated in literature (Tøndel et al., ). Therefore, these techniques can otentially be used for extending the roosed aroach to the constrained ND-MPC. Furthermore also robustness results (Bemorad and Morari, 999) with resect to model uncertainties and disturbances are needed. For instance, a study about the effect that erturbations on the model matrices roduce on the closed-loo eigenvalues can be erformed. 7. REFERENCES Bemorad, A. and M. Morari (999). Robust model redictive control: A survey. In: Robustness in Identification and Control, Lecture Notes in Control and Information Sciences (A. Garulli, A. Tesi and A. Vicino, Eds.). Vol. 4.. 7 6. Sringer. London. Camonogara, E. (). Controlling networks with collaborative nets. PhD thesis. Carnegie Mellon. Camonogara, E., D. Jia, B. H. Krogh and S. Talukdar (). Distributed model redictive control. IEEE Control Syst. Mag., 44. Cheng, X. and B. H. Krogh (). Stabilityconstrained model redictive control. IEEE Trans. on Aut. Cont. 46, 86 8. El-Farra, N. H., A.Gani and P. D. Christofides (4). Fault-tolerant control of multi-unit rocess systems using communication networks. In: 7th IFAC Symosium on Dynamics and Control of Proc. Syst.. Prague, CZ. Gudi, Ravindra D., James B. Rawlings, Aswin Venkat and N. Jabbar (4). Identification for decentralized mc. In: DYCOPS. Boston, MA. Jia, D. (). Distributed coordination in multiagent control systems through model redictive control. PhD thesis. Carnegie Mellon. Jia, D. and B. H. Krogh (). Distributed model redictive control. In: Proceedings of the A.C.C.. Vol. 4.. 767 77. Katebi, M. R. and M. A. Johnson (997). Predictive control design for large-scale systems.. Automatica, 4 4. Longhi, S., R. Trillini and M. Vaccarini (). Alication of a networked decentralized MPC to syngas rocess in oil industry. In: 6th IFAC World Congress. Prague, CZ. Motee, Nader and Bijan Sayyar-Rodsari (). Otimal artitioning in distributed model redictive control. In: Proceedings of the American Control Conference. Denver, Colorado.. 866 87. Tøndel, P., T. A. Johansen and A. Bemorad (). An algorithm for multi-arametric quadratic rogramming and exlicit mc solutions. Automatica 9(), 489 497. Vaccarini, M., S. Longhi and M.R. Katebi (6). State sace stability analysis of unconstrained decentralized model redictive control systems. In: Proceedings of the American Control Conference. Minneaolis, Minnesota. To be ublished. Venkat, A. N., J. B. Rawlings and S. J. Wright (). Stability and otimality of distributed model redictive control. CDC-ECC Joint Conference. Šiljak, D. D. (996). Decentralized control and comutations: status and rosects. Annual Reviews in Control, 4.