Multiobjective Robust Dynamic Output-feeback Control Synthesis based on Reference Model

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1 49th IEEE Conference on Decision and Control December 5-7, 2 Hilton Atlanta Hotel, Atlanta, GA, USA Multiobjective Robust Dynamic Output-feeback Control Synthesis based on Reference Model Wagner Eustáquio Gomes Bachur, Eduardo Nunes Gonçalves, Reinaldo Martinez Palhares, and Ricardo Hiroshi Caldeira Takahashi Department of Electrical Engineering, Centro Federal de Educação Tecnológica de Minas Gerais, Av. Amazonas 7675, Belo Horizonte, MG, Brazil. eduardong@des.cefetmg.br Department of Electronic Engineering, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, Brazil. palhares@cpdee.ufmg.br Department of Mathematics, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, Brazil. taka@mat.ufmg.br Abstract This paper presents a new strategy for robust dynamicoutput-feedbackcontrol synthesisfor uncertain continuous-time linear time-invariant systems represented by polytopic models. It is considered a multiobjective optimization problem to guarantee tracking response specifications, disturbance rejection, and noise attenuation. The controller design criterion is to minimize theh -norm of the error between a reference model and the closed-loop transfer function to attain thetrackingresponse specifications. Theproposed control synthesismethodologyisbased on a two step iterative procedure: synthesis, based on a non-linear optimization algorithm, and analysis, based on a branch-and-bound algorithm combined with LMI analysis formulations. A numeric example is presented to illustrate the effectivenessof the proposed approach. Index Terms Robust control, dynamic outputfeedback control, reference model, polytopic systems. I. Introduction Procedures for analysis and synthesis of robust control systems basedon linear matrixinequality (LMI) became popular because they are convex optimization problems that can be solved efficiently by means of available free or commercial softwares []. However, there are robust control problems that result in bilinear matriz inequality (BMI) problems that are non-convex in general. This is the case of the robust dynamic output-feedback control synthesis. There are several works that present methods to transform BMI problems in LMI problems. Some of these works are based on linearizing change of variables [2], [3], [4], [5], [6], [7] and others are based on transforming the BMI problem in two LMI problems with a non-convex coupling relation between them [8], [9], [], []. In these formulations, the controller matrices are functions of the system matrices which means that they can not be applied to uncertain systems based in polytopic models. This work was supported in part by CNPq, Capes, and FAPEMIG, Brazil This paper deals with the multiobjective robust dynamic output-feedback control synthesis problem to handle the tracking response specifications, disturbance rejection, and noise attenuation. Most of the works in the field of robust control consider robust regional pole placement constraints to guarantee the tracking transient response specifications [2]. The main contribution of this paper is to present a new strategy to guarantee the tracking response specifications based on an H -optimal model reference strategy, that attains the transient response specifications, and the closed-loop transfer function between the reference signal and the plant output [3]. This strategy requires the inclusion of an additional state variable related with the tracking error integral that helps to attain steady state tracking error specification. In [3], is considered only the tracking response specifications based on a BMI formulation. The BMI problem is solved applying a two step iterative procedure where, in each step, part of the variables are optimized while the other part is fixed to transform the BMI problem into an LMI problem. In this paper, the control design includes both the disturbance rejection and noise attenuation. Besides the new objectives, a procedure based on a two steps procedure is used to solve the control problem, namely: ) synthesis basedonan optimizationalgorithm that is suitable to tackle non-linear problems, where the controller parameters are the optimization variables, considering a finite set of points of the uncertain domain, and 2) analysis based on a branch-and-bound algorithm considering LMI analysis formulations as upper bounds. The motivation to use this synthesis procedure is the fact that it was already applied with success to other robust control problems such as robust state-feedback control synthesis [4], robust PID control synthesis [5], robust filter synthesis [6], and robust model reduction [7]. The proposed procedure has the drawback of the increased complexity to its implementation but, once implemented, it presents several advantages in relation to LMI synthesis formulations such as to achieve less conservative results, to find feasible solutions where LMI //$26. 2 IEEE 233

2 formulations fail, and the possibility to consider controllers of any order and structure with rather arbitrary additional constraints. An illustrative example is presented to demonstrate the efficiency of the proposed synthesis procedure to achieve robust dynamic output-feedback controllers that attain the tracking response specifications with disturbance rejection and noise attenuation. The notation in this paper is standard. The compact notation: A B G(s) = C D is applied to denote the transfer function G(s) = C(sI A) B + D. II. Problem Formulation Consider a continuous-time linear time-invariant system described by ẋ(t) = Ax(t) + B u u(t) + B w w(t) z(t) = C z x(t) + D zu u(t) + D zw w(t) y(t) = C y x(t) + D yw w(t) () where x(t) R n is the state vector (including the tracking error integral, v(t) [r(t) c(t) n(t)]dt), u(t) R nu is the control signal vector (manipulated variables), w(t) R nw is the exogenous input vector (reference signal, r(t), disturbances, d(t), and measurement noises, n(t)), z(t) R nz is the controlled output vector (plant output, c(t), and control signals, u(t)), and y(t) R 2 is the measured output vector (plant output, c(t), and tracking error integral, v(t)), that are the inputs to the dynamic output-feedback controller. To simplify the notation, the system matrices in Eq. () are gathered in the matrix: A B u B w S C z D zu D zw (2) C y D yw that can include uncertain parameters belonging to a known convex compact set, or polytope, defined by its vertices: { } N P(α) S : S = α i S i ; α Ω (3) Ω { i= α : α i, } N α i = i= (4) [ with S i, i =, ]...,N, the polytope vertices and α = α... α N the vector that parameterizes the polytope. The dependence of the system matrices of α will be omitted from now on. In this paper it is considered a dynamic outputfeedback controller represented by: Ac B K(s) = c (5) C c D c Let T cr (s) = C(s)/R(s) be the closed-loop transfer function related with the tracking response, T cd (s) = C(s)/D(s) be the closed-loop transfer function related with the disturbance rejection, and T un (s) = U(s)/N(s) be the closed-loop transfer function related with the noise attenuation. The closed-loop system matrices, with f corresponding to the subscripts cr, cd, or un, in the compact notation: Af B T f (s) = f (6) C f D f are given by A + Bu D A f = c C y B u C c B c C y A c Bw + B B f = u D c D yw B c D yw C f = C z + D zu D c C y D zu C c D f = [ D zw + D zu D c D yw ] (7) with B w, C z, Dzu, Dzw, and D yw corresponding to submatrices of matrices B w, C z, D zu, D zw, and D yw, such that specific channels are selected from some components of w to some components of z. Consider a reference model that attains the tracking transient response specifications (overshoot, settling time, etc.): Am B T m (s) = m (8) C m D m The error between this reference model and the closedloop transfer function, E(s) T m (s) T cr (s), can be represented by the following state-space model: E(s) = A m B m A cr B cr C m C cr D m D cr (9) The multiobjective robust control problem considered inthis papercanbe statedas: givena polytope-bounded uncertain, continuous-time, linear time-invariant system, P(α), α Ω, and a reference model, T m (s), find a dynamic output-feedback controller, K(s), that minimizes the maximum H -norm of the error between the reference model and the closed-loop transfer function, E(s), the maximum H -norm of the transfer function T cd (s) related with the disturbance rejection, and the maximum H 2 -norm of the transfer function T un (s) related with the noise attenuation, in the uncertainty domain: K = argmin K subject to: K F max α Ω E(s, α, K) max α Ω T cd(s, α, K) max α Ω T un(s, α, K) 2 () with F the set of controllers such as the closed-loop system is robustly stable. 233

3 To apply the proposed controller synthesis presented in the next section, the multiobjective optimization problem is transformed in a scalar optimization problem as: K = argmin K ( λ max α Ω T cd + λ 2 max α Ω T un 2 subject to: max α Ω E ǫ m K F () where λ, λ 2, and ǫ m can be selected to result different solutions to the multiobjetive problem. This choice of the scalar optimization problem considers that there is an ǫ m such as max α Ω E ǫ m, which guarantees that the tracking transient response will be closer to the one specified by the reference model. It is verified that the two first objectives in () are not totally conflicting when the reference model is chosen appropriately. If λ 2, it is required that D c =. III. Proposed multiobjective Robust Control Synthesis Procedure The proposed synthesis procedure to tackle the multiobjective optimization problem () is based on a nonconvex optimization problem considering the controller parameters as optimization variables. A necessarystep of the proposed non-convex formulation is the computation of an upper bound for the objective functions in the uncertain domain, Ω. perform this task, it is employed here an optimization procedure based on two steps: synthesis and analysis. In the synthesis step it is applied an optimization algorithm to solve the scalar optimization problem () with the infinite set Ω replaced by a finite set of points Ω Ω. This finite set is initially the set of vertices of the polytope as considered in convex formulations. To consider only the polytope vertices is not sufficient to guarantee the robust stability of the closedloop system and the minimization of E, T cd, and T un 2 for all α Ω. To verify the controller computed in the first step, in the second step it is applied an analysis procedure based on a combination of a branch-and-bound algorithm and LMI formulations [8]. If the analysis procedure finds an instance of an unstable system in the uncertain domain or if it is verified that the maximum value of E, T cd, or T un 2 does not occur in a point belonging to Ω, then this point is included in Ω and it is necessary to execute the two steps of the procedure again. The procedure ends when it is verified that the closed-loop system is robustly stable and the maximum values of the objective functions are in points belonging to Ω (or near then accordingly to a specified accuracy). In the synthesis step, the scalar optimization problem () can be solved by means of the cone-ellipsoidal algorithm [9]. Let x R d be the vector of optimization parameters (in this case the controller parameters), f(x) : R d R be the objective function to be minimized, ) and g i (x) : R d R, i =,...,s, be the set of constraint functions. Consider the ellipsoid in the iteration k described as E k = { x R d (x x k ) T Q k (x x k) }, where x k is the ellipsoid center and Q k = Q T k is the matrix that determines the direction and the dimension of the ellipsoid axes. Given the initial values x and Q, the ellipsoidal algorithm is described by the following recursive equations: with x k+ = x k d + Q k m Q k+ = d 2 d 2 ( Q k 2 ) (2) d + Q k m m T Q k m = m k / m T k Q km k. where m k is the sum of the normalized gradients (or subgradients) of the violated constraint functions, g i (x) >, when x k is not a feasible solution, or the gradient (or sub-gradient) of the objective function, f(x), when x k is a feasible solution. In the analysis step, it is required to compute the α i Ω, i =,...,3, corresponding to the maximum of each objective function in () or to find an α Ω that corresponds to an unstable system. The basic strategy of the branch-and-bound algorithm is to partition the uncertainty domain, Ω, such as lower and upper bound functions converge to the maximum value of the norm. This algorithm ends when the difference between the bound functions is lower than the prescribed relative accuracy. The algorithm is implemented considering as lower bound function the H (or H 2 ) norm computed in the vertices and as upper bound function the H (or H 2 ) guaranteed cost computed by means of LMI formulations, both functions calculated for the original polytope and its subdivisions [8]. In this paper, the guaranteed cost computations are based on: Lemma presented in [2] for H guaranteed cost and combination of Lemmas and 2 presented in [2] for H 2 guaranteed cost. A partition technique based on simplicial meshes [22] is applied to tackle uncertainty domains not restricted to the hyper-rectangle case. This partition technique allows this algorithm be applied to both affine parameterdependent as well as polytopic models with improved efficiency. More details about the proposed synthesis procedure can be found in [4], [6], [5], [7]. The proposed procedure has required only one iteration to solve the control problem in most of the cases that have been investigated up to now, but when it was necessary more than one iteration, the proposed procedure converged effectively [6]. IV. Illustrative Example Consider the level control of the interacting tank system presented in Fig.. A linear model of an operating point is considered here. The variables in capital letters 2332

4 are the operating point values: Qu = Q =,4m 3 /s, Q d =,m 3 /s, Q2 =,5m 3 /s, H = 2m, and H 2 = m. The variables in lower case are the deviations around the operating point. The state vector is defined as x(t) [h (t) h 2 (t) v(t)] T, with v(t) [r(t) h 2 (t) n(t)]dt, the plant output is the level of the tank 2, h 2 (t), the control signal is the inlet flow-rate of the tank, u(t) = q u (t), and the disturbance is the inlet flow-rate of the tank 2, d(t) = q d (t). The state space model of the system is k k A A d dt h h 2 v = + z = y = k k + k 2 A 2 A 2 A q u + A 2 [ [ ] x + u ] [ x + h h 2 v r q d n ] r q d n (3) One of the advantages of the proposed procedure is the capability to chose any desired structure for the controller matrices. It is considered a controller in a canonical form with two decoupled transfer functions, the first one of second order and the second one of first order (6 optimization parameters). Initially, it will be considered λ = and λ 2 =, to minimize just max T un 2 with ǫ m varying in the range.2 ǫ m.. The candidate Pareto curve is presented in Fig. 2. max T un u u d d max E Fig. 2. Candidate Pareto curve considering the objective functions max E and max T un 2. H +h (t) H +h (t) k 2 2 A A2 k 2 It is chosen the controller achieved with ǫ m =.6: 2 2 Fig.. Interacting tank system. The cross-sectional areas of the tanks are A = m 2 and A 2 = 5m 2. Let k and k 2 be uncertain parameters varying in the ranges:.5 k.25 and.2 k 2.3. The uncertain system is represented by a polytopic model with 4 vertices corresponding to combinations of the extreme values of the two uncertain parameters. The design goals are: to achieve a tracking response similar to a specified reference morel; to reject the influence of the disturbance, q d (t), over the plant output, h 2 (t), and to attenuate the effect of the sensor noise, n(t), over the control signal, q u (t). It is required that q u (t) be constrained in acceptable bounds for the following test signals: r(t) =.(t), q d (t) =.(t 2), and n(t) a random signal with uniform distribution on the interval:. n(t).. The reference model is the balanced realization of: ω 2 n T m (s) = s 2 + 2ζω n s + ωn 2, ζ =.9, ω n =.5 (4) K 2 (s) = C c (si A c ) B c = = [ 6.922(s +.962) s s s E.6, T cd.74, T un The frequency responses of T m (s) and T cr (s), considering K 2 (s), are presented in Fig. 3. One can verify that, in the bandwidth range, the achieved closed-loop frequency response is closer to the one specified by means of the reference model. This controller presents a reasonable disturbance rejection and noise attenuation, with an acceptable control effort, reproducing perfectly the specified tracking response, as shown in Figs. 4 and 5. Considering only the minimization of max T cd with λ =, λ 2 =, and ǫ m =.6, it is achieved the ] 2333

5 Bode Diagram.2. Magnitude (db) 5 5 q u (m 3 /s) Phase (deg) Frequency (rad/sec) Fig. 3. Frequency responses of the reference model (dashed) and of the 4 vertices (solid) for K 2. Fig. 5. Transient responses of the control signal, q u(t), of the 4 vertices for K 2..2 Bode Diagram..8 Magnitude (db) h 2 (m) Phase (deg) Frequency (rad/sec) Fig. 4. Transient responses of the plant output, h 2 (t), of the reference model (dashed) and of the 4 vertices (solid) for K 2. Fig. 6. Frequency responses of the reference model (dashed) and of the 4 vertices (solid) for K. following controller: K (s) = C c (si A c ) B c = , [, (s ) = s s s + 55 E.45, T cd.3, T un 2 4, The frequency responses of T m (s) and T cr (s), considering K (s), are presented in Fig. 6. This controller reproduces perfectly the specified tracking response and presents an excellent disturbance rejection but the control effort is too high with low noise attenuation as shown in Figs. 7 and 8. ] Of course, by choosing the values of λ and λ 2 appropriately, it is possible to achieve controllers with a better compromise between noise attenuation and disturbance rejection in relation to the controllers K 2 and K. V. Conclusions A new strategy for robust dynamic output-feedback control synthesis that guarantees the tracking response specifications, disturbance rejection and noise attenuation has been presented. An objective function representing the H -norm of the error between a reference model and the closed-loop transfer function relating the reference signal and the plant output has been shown to constitute an an effective control design criterion to attain the tracking response specifications as verified in the illustrative example. This idea can be extended to deal with robust control synthesis of non-linear systems 2334

6 h 2 (m) Fig. 7. Transient responses of the plant output, h 2 (t), of the reference model (dashed) and of the 4 vertices (solid) for K. q u (m 3 /s) Fig. 8. Transient responses of the control signal, q u(t), of the 4 vertices for K. or systems with time-delay. The proposed approach to tackle the multiobjective non-convex optimization problem based on a synthesis step, considering a finite set of points of the uncertain domain, followed by an analysis step, considering the whole set of the uncertain domain, has presented good results for the problem considered in this paper as already observed in previous applications of the same general idea. The parameters of the proposed scalar optimization problem allow the designer to achieve several solutions to the multiobjective optimization problem considering different compromises between the three objectives. References [] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox: For Use with MATLAB R, The MATH WORKS Inc., Natick, 995. [2] C. W. Scherer, From single-channel LMI analysis to multichannel mixed LMI synthesis: a general procedure, Selected Topics on Identification, Modelling and Control, vol. 8, pp. 8, 995. [3] C. Scherer, P. Gahinet, and M. Chilali, Multiobjective output-feedback control via LMI optimization, IEEE Transactions on Automatic Control, vol. 42, no. 7, pp , 997. [4] I. Masubuchi, A. Ohara, and N. Suda, LMI-based controller synthesis: an unified formulation and solution, International Journal of Robust andnonlinear Control, vol. 8, pp , 998. [5] P. Apkarian, H. D. Tuan, and J. Bernussou, Continuoustime analysis, eigenstructure assignment and H 2 synthesis with enhanced LMIcharacterizations, IEEE Transactions on Automatic Control, vol. 46, no. 2, pp , 2. [6] M. C. de Oliveira, J. C. Geromel, and J. Bernussou, Extended H 2 and H norm characterizations and controller parametrizations for discrete-time systems, International Journal of Control, vol. 75, no. 9, pp , 22. [7] Y. Ebihara and T. Hagiwara, New dilated LMIcharacterizations for continuous-time multiobjective controller synthesis, Automatica, vol. 4, pp , 24. [8] K. M. Grigoriadis and R. S. Skelton, Low-order control design for LMI problems using alternating projection methods, Automatica, vol. 32, no. 8, pp. 7 25, 996. [9] T. Iwasaki, The dual iteration for fixed-order control, IEEE Transactions on Automatic Control, vol. 44, no. 4, pp , April 999. [] T. Shimomura and T. Fujii, Multiobjective control design via sucessive over-bounding of quadratic terms, Proceedings of the 39th Conference on Decision and Control, pp , 2. [] M. C. de Oliveira, J. C. Geromel, and J. Bernussou, Design of dynamic output feedback decentralized controllers via a separation procedure, International Journal of Control, vol. 73, no. 5, pp , 2. [2] M. Chilali, P. Gahinet, and P. Apkarian, Robust pole placement in LMI regions, IEEE Transaction on Automatic Control, vol. 44, no. 2, pp , December 999. [3] L. A. Rodrigues, E. N. Gonçalves, V. J. S. Leite, and R. M. Palhares, Robust reference model control with LMI formulation, in Proceedings of the IASTED International Conference Control and Applications. Cambridge, UK: IASTED, July 29, pp [4] E. N. Gonçalves, R. M. Palhares, and R. H. C. Takahashi, Improved optimisation approach to robust H 2 /H control problem for linear systems, IEE Proceedings Control Theory & Applications, vol. 52, no. 2, pp. 7 76, 25. [5], A novel approach for H 2 /H robust PID synthesis for uncertain systems, Journal of Process Control, vol. 8, no., pp. 9 26, January 28. [6], H 2 /H filter design for systems with polytopebounded uncertainty, IEEE Transactions on Signal Processing, vol. 54, no. 9, pp , 26. [7] E. N. Gonçalves, R. M. Palhares, R. H. C. Takahashi, and A. N. V. Chasin, Robust model reduction of uncertain systems maintaining uncertainty structure, International Journal of Control, vol. 82, no., pp , November 29. [8] E. N. Gonçalves, R. M. Palhares, R. H. C. Takahashi, and R. C. Mesquita, H 2 and H 2 ε-guaranteed cost computation of uncertain linear systems, IET Control Theory and Applications, vol., no., pp. 2 29, January 27. [9] R. H. C. Takahashi, R. R. Saldanha, W. Dias-Filho, and J. A. Ramírez, A new constrained ellipsoidal algorithm for nonlinear optimization with equality constraints, IEEE Transactions on Magnetics, vol. 39, no. 3, pp , 23. [2] P. J. de Oliveira, R. C. L. F. Oliveira, V. J. S. Leite, V. F. Montagner, and P. L. D. Peres, H guaranteed cost computation bymeans ofparameter-dependent Lyapunov functions, Automatica, vol. 4, pp , April 24. [2], H 2 guaranteed cost computation by means of parameter-dependent Lyapunov functions, International Journal of Systems Science, vol. 35, no. 5, pp. 53 6, 24. [22] E. N. Gonçalves, R. M. Palhares, R. H. C. Takahashi, and R. C. Mesquita, Algorithm 86: SimpleS - an extension of Freudenthal s simplex subdivision, ACM Transactions on Mathematical Software, vol. 32, no. 4, pp ,