Synthesis of output feedback controllers for a class of nonlinear parameter-varying discrete-time systems subject to actuators limitations

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1 21 American Control Conference Marriott Waterfront Baltimore MD USA June 3-July 2 21 ThC9.5 Synthesis of output feedback controllers for a class of nonlinear parameter-varying discrete-time systems subject to actuators limitations Eugênio B. Castelan Valter J. S. Leite Márcio F. Miranda and Vitor M. Moraes Abstract We present a methodology for computing output feedback control laws for a class of nonlinear systems subject to input saturations. This class of systems consists of a L ure type nonlinear system with some time-varying parameters which are assumed to be real-time available. Based on some tools from the absolute stability theory on a modified sector condition to take into account input saturation effects and on the concept of contractive sets applied to a level set obtained from a particular parameter dependent Lyapunov function an LMI framework is proposed to design dynamic output feedback compensators. Two local stabilization strategies are considered: i) with saturation avoidance and ii) with saturating actuators. In both cases convex optimization problems are proposed to compute the controllers matrices aiming at the maximization of the basin of attraction. I. INTRODUCTION The design of most practical control systems requires to consider the presence of certain dynamical characteristics and practical constraints that are inherent to the plant to the actuators and to the sensors. To cope with the presence of the nonlinearities due to the plant dynamics or induced by the actuators limits among the various existing nonlinear control systems approaches absolute stability theory has been considered in the literature for analysis and synthesis of the so-called Lur e system 11 Furthermore timevarying parameters can also be considered using robust control techniques for uncertain systems; in particular if the time-varying parameter is real-time available parameterdependent control laws can be implemented. Besides that the control designer is faced to the necessity of using an output feedback control law which in most applications imply the use of a dynamic output feedback controller. More recently the research on these topics has been intensified mainly due to the possibility of using the Linear Matrix Inequality (LMI) framework and the existing efficient numerical tools for computations 1. Considering linear systems with saturating inputs a large amount of works can be found in the literature as for instance (see also references therein): 9 and 15 considering state feedback control laws; and 18 regarding dynamic output feedback controller synthesis; 7 and 14 addressing the anti-windup synthesis. In the same This work was partially supported by CAPES CNPq and FAPEMIG Brazil. Eugênio B. Castelan and Vitor M. Moraes are with Departamento de Automação e Sistemas Universidade Federal de Santa Catarina Florianópolis (S.C.) Brazil {eugeniorattus}@das.ufsc.br Valter J. S. Leite is with CEFET MG / Campus Divinópolis Rua Monte Santo Divinópolis MG Brasil. valter@ieee.org Márcio F. Miranda is with COLTEC / UFMG Av. Antônio Carlos Belo Horizonte MG Brazil. fantini@coltec.ufmg.br line LMI conditions have also been proposed to synthesize stabilizing control laws for nonlinear systems subject to actuator amplitude limitations and for which the dynamics can be decomposed into the feedback interconnection of a linear system with a sector bounded nonlinearity: 3 4 for precisely-known systems and 2 1 for some uncertain nonlinear systems. Still in the context of nonlinear systems subject to control saturations the synthesis of a dynamic output feedback controller has been considered in 5 for continuous and time-invariant Lur e systems. Inspired on the previous works the controller structure was composed by a time-invariant compensator presenting the following inputs: the plant output an anti-windup term (related to the input saturation) and the value of the sector bounded nonlinearity associated to the plant dynamics. Differently from the BMI results proposed in 12 the approach in 5 consider simultaneously the two sector bounded nonlinearities and furthermore the proposed stabilization conditions are stated in LMI forms. In the this work we present a methodology for the synthesis of output feedback control laws for L ure type discretetime nonlinear systems with some time-varying parameters which are assumed to be real-time measurable and subject to actuators limitations. Thus based on some tools from the absolute stability theory and on a modified sector condition to take into account input saturation effects and using the concept of contractive sets applied to a level set obtained from a particular parameter dependent Lyapunov function an LMI framework is proposed to design the controller. The obtained controller is called partially dependent of the timevarying parameters because only the matrices associated to the controller state equation depend on the considered parameters. Two local stabilization strategies are considered: i) with saturation avoidance and ii) with saturating actuators and convex optimization problems are proposed to compute the controller matrices aiming at the maximization of the basin of attraction. Notice that a similar technique has been used in 18 but in a different context where the plant is considered to be linear time-invariant and a polytopic model is used to model the saturations. The paper is organized as follows. Section II presents the problem. Preliminary concepts and results are presented in section III. Section IV presents the local stabilization conditions and two convex optimization problems for synthesizing the compensators. The paper finishes with a numerical example and a conclusion. Notations. For two symmetric matrices A and B A>B means that A B is positive definite. A denotes the transpose of A. A (i) denotes the i th row of matrix A. v (i) is the i th component of vector v. stands for symmetric /1/$ AACC 4235

2 blocks; stands for an element that has no influence on the development. I n denotes an n-dimensional identity matrix. diag(ab) is the block-diagonal A matrix. B II. PROBLEM FORMULATION Consider a nonlinear discrete-time system with timevarying parameters represented by: x k+1 = A(α k )x k + G(α k )ϕ(z k )+B(α k )sat(u k ) z k = L(α k )x k (1) y k = Cx k where x k R n u k R m z k R p y k R q ϕ(.) : R p R p and α k is an available time-varying vector parameter belonging to the unit simplex: Ξ={α k R N ; N α k(i) = 1 α k(i) }. The structure of the system matrices are assumed to be of the form: A(αk ) B(α k ) G(α k ) L (α k ) = (2) N α ki Ai B i G i L i and C R q n where the matrices A i B i G i and L i have appropriate dimensions. The nonlinearity ϕ(.) verifies the cone bounded sector condition ϕ(.) Ω z R p 11 i.e.: there exists a symmetric positive definite matrix Ω = Ω R p p such that ϕ (z k ) ϕ(z k ) Ωz k z k R p ϕ()= (3) where R p p is any positive diagonal matrix: = diag{δ l } δ l > l = 1... p. By definition the nonlinearity ϕ(.) globally satisfies the sector condition (3). Note that Ω is given by the designer and assumed to be known in the sequel. On the other hand from the definition of we see that (3) is verified if p independent classical conditions ϕ (l) (z k)ϕ(z k ) Ωz k (l) are also verified. Thus represents a degree-of-freedom (then an optimization parameter). The control inputs are bounded in amplitude and the standard saturation function is considered: sat(u (l) (t))=sign(u (l) (t)) min(ρ (l) u (l) (t) ) (4) l=1...m where ρ (l) > denotes the symmetric amplitude bound relative to the l-th control. Throughout this work we assume that the parameter vector α k is real-time available. Thus the considered output feedback nonlinear controller has the following structure that partially depends on the parameter varying vector: x ck+1 = A c (α k )x ck + B c (α k )y k + G c (α k )ϕ(z k )+E c (α k )Ψ(u k ) (5) u k = C c x ck + D c y k + F c ϕ(z k ) where x ck R n n and Ψ(.) : R m R m is the dead-zone nonlinearity given by: Ψ(u k )=u k sat(u k ) (6) The controller parameter-varying matrices are structured as follows: Ac (α k ) B c (α k ) G c (α k ) E c (α k ) = N α k(i) Aci B ci G ci E ci (7) where: A ci R n n B ci R n q G ci R n p and E ci R n m. Furthermore the real constant matrices associated to the output of the controller are such that: C c R m n D c R m q and F c R m p. Notice that with G c (α k ) k and F c the above feedback controller requires either the knowledge of ϕ(.) or its availability as a signal. Also E c (α k ) is a parameter-varying anti-windup gain matrix that helps to mitigate the effects of the control inputs saturation. Notice that the anti-windup term acts only when saturation occurs i.e. Ψ(u k ). Let us define the augmented state vector ξ k = x k x ck R 2n and the augmented state feedback matrix K R m 2n : K= D c C C c Thus from (1)-(8) we represent the closed-loop system by: ξ k+1 = A(α k )ξ k +G(α k )ϕ(z k ) B(α k )Ψ(u k ) z k = L(α k )ξ k (9) u k = Kξ k + F c ϕ(z k ) where: A(α k ) G(α k ) B(α k ) L (α k ) = N α k(i) Ai G i B i L i with by definition: Ai + B A i = i D c C B i C c Gi + B G B ci C A i = i F c ci G ci Bi B i = L E i = L i. ci In the sequel we propose solutions for the following constrained control problem: Problem 1: Determine feedback gain matrices A ci B ci C c D c E ci F c G ci and a region S R n as large as possible such that the origin of the closed-loop system (9) is asymptotically stable for any initial condition x S and for any ϕ(.) verifying the sector condition (3). For practical purposes the designer can also be interested in guaranteeing a certain degree of performance to be defined in the sequel for the closed-loop trajectories emanating from S. Thus it is due the presence of limits on the control variables the parameter varying nature of the considered nonlinear system and the performance requirement that the closed-loop stability properties are studied in the Local Asymptotic (LA-) Stability context. Thus two types of solutions for Problem 1 will be studied and compared in the sequel: 1st) to obtain LA-Stability with saturation avoidance i.e. the control constraint u k ρ has to be respected for any trajectory emanating from S ; and 2nd) to obtain LA-Stability with saturating actuators i.e. u k >ρ is admitted for trajectories emanating from S. Furthermore an implicit objective in the synthesis of the compensators gains and determination of S is to maximize (8) 4236

3 the size of the basin of attraction of the closed-loop system. III. PRELIMINARY RESULTS Consider a parameter dependent Lyapunov function (PDLF) V(ξ k α k ) : R 2n Ξ R. The level set associated to V(ξ k α k ) is given by L V ={ξk R 2n ; V(ξ k α k ) 1 α k Ξ} (1) The following definition of contractivity 11 is adapted to consider the parameter uncertainties the sector bounded characterization of ϕ(.) and a degree of performance. Definition 2: Consider a real scalar λ (1. The level set L V is robustly absolutely λ-contractive with respect to the trajectories of system (9) if V λ (ξ k α k ) = V(ξ k+1 α k+1 ) λv(ξ k α k )< (11) ξ k L V ϕ(.) Ω and α k Ξ. In the sequel we consider the PDLF of the form V(ξ k α k )=ξ k Q (α k )ξ k (12) where Q(α k )= N α k(i) Q i Q i =Q i > with Q i R 2n 2n. Thus we have that the level set obtained from (12) and (1) is the intersection of ellipsoidal sets 8 1: L V = E(Q (α k ))=...N E { Q } (13) i where for...n : E(Q i )={ξ k R 2n ; ξ k Q i ξ k 1} (14) Let us also consider a (parameter-dependent) polyhedral set given by: S(ρα k ) = { ξ k R 2n and ϕ(z k ) R p ; (15) Γ 1 (α k )ξ k + Γ 2 (α k )ϕ(z k ) ρ α k Ξ} where by definition: Γ1 (α k ) Γ 2 (α k ) = Σ N α k(i) Γ1i Γ 2i with Γ1i R m 2n Γ 2i R m p. The following lemma describes an inclusion condition which is instrumental to apply the modified sector condition associated to the dead-zone nonlinearity in the derivation of the desired LMI stability and stabilization results 1. Lemma 3: Given the sets E ( Q (α) ) and S(ρα) the inclusion E ( Q (α) ) S(ρα) is obtained for any ϕ(.) verifying the sector condition (3) if there exist U R 2n 2n and a positive diagonal matrix R p p such that: Q i +U +U U L i Ω U Γ 1i(l) 2 Γ 2i(l) (16) ρ 2 (l)...n and l=1...m. Y1 (α) Y 2 (α) = Furthermore by setting 1 For the sake of space we omit in the sequel: i) the indice k in various expressions; and ii) the proofs of the proposed results which can be requested to the corresponding author. K Γ1 (α) F c Γ 2 (α) inequalities (16) imply Ψ(u) S (Ψ(u) Y 1 (α)ξ Y 2 (α)ϕ(z)) (17) for all ξ E ( Q (α) ) and for any diagonal positive definite matrix S R m m. Local Asymptotic (LA-)Stability results will now be derived by guaranteeing the robust absolute λ-contractivity of E ( Q (α) ) i.e. relation (11). In the saturation avoidance case we have Ψ(u)=u sat(u)= and E ci =. Lemma 4: (LA-Stability with saturation avoidance) Let A ci B ci G ci C c D c and F c be given feedback gains that form controller (5). For a given real scalar λ (1 consider the existence of symmetric positive definite matrices Q i R 2n 2n a positive diagonal matrix R p p and a matrix U R 2n 2n satisfying : M ji = Q j A i U G i λ(q i U U ) U L i Ω < 2...N and j = 1...N (18) Q i +U +U U L i Ω U K (l) N i = 2 F c (l) ρ 2 (l)...n and l=1...m. (19) { Then the set S = i E(Q i ) } = E ( Q (α) ) is robustly absolutely λ-contractive and is included in S(ρ)= { ξ R 2n ; ϕ(.) R p ; Kξ+F c ϕ(z) ρ }. Hence the origin of the corresponding closed-loop system (9) is asymptotically stable for any initial condition ξ S and for any ϕ(.) verifying the sector condition (3). Furthermore the control constraint u k ρ is respected for any trajectory emanating from S. Lemma 5: (LA-Stability with saturating actuators) Let A ci B ci G ci E ci C c D c and F c be given feedback gains that form controller (5). For a given real scalar λ (1 consider the existence of symmetric positive definite matrices Q i R 2n 2n of positive diagonal matrices R p p and S R m m and of matrices U R 2n 2n H 1i R m 2n and H 2i R m p satisfying inequalities: M ji...n B i S H 1i H 2i 2S and j = 1...N < (2) Q i +U +U U L i Ω U K (l) H 1i(l) 2 F c (l) H 2i(l) ρ 2 (l)...n and l=1...m. (21) { Then the set S = i E(Q i ) } = E ( Q (α) ) is robustly absolutely λ-contractive. Hence the origin of the corresponding saturating closed-loop system (9) is asymptotically 4237

4 stable for any initial condition ξ S for any ϕ(.) verifying the sector condition (3). IV. MAIN RESULTS Inspired in the work of Scherer and co-workers 17 we define the n-dimensional square real matrices N M X Y W and Z such that: X N U= Z U = Y M W Y In Θ= W It follows that: XY + NW = Y X+ MZ = I n. We also have: In X UΘ= = Θ Z Y T UΘ= X where by construction: T = Y X+W Z. Furthermore using Q11i Q the partitioning Q i = 12i we have: Q 22i Θ Q i Θ = Y Q 11i Y +W Q 12i Y+ Y Q 11i + Y Q 12i W +W Q 22i W W Q 12i Q 11i ˆQ = 11i ˆQ 12i (22) ˆQ 22i A. LMI stabilization conditions Proposition 6: (LA-Stabilization with saturation avoidance) Given a real scalar λ (1 suppose that there exist matrices ( ˆQ 11i ˆQ 12i ˆQ 22i X YT) R n n a positive diagonal matrix R p p and matrices  i ˆB i Ĝ i Ĉ ˆD ˆF satisfying the two LMI conditions (23) and (24). Let matrices N Z and the nonsingular W be such that: I n. NW = I n XY and W Z = T Y X. (25) Then the controller (5) with D c = ˆD C c = ( Ĉ D c CX ) Z B ci =(W ) ( ˆB i Y ) B i D c Fc = ˆF A ci = (W ) (  i W B ci CX Y B i C c Z (26) Y (A i + B i D c C)X)Z G ci =(W ) ( Ĝ i Y (G i + B i F c ) ) and the set S = E(Q (α k ))=...N E ( Q ) i with Qi = Q11i Q 12i that verifies (22) are solutions to Problem Q 22i 1 for any nonlinearity ϕ(z) satisfying relation (3). Proposition 7: (LA-Stabilization with saturating actuators) Given a real scalar λ (1 suppose that there exist matrices ˆQ 11i ˆQ 12i ˆQ 22i X YT positive diagonal matrices S R m m and R p p and matrices  i ˆB i Ĝ i Ê i Ĉ ˆD ˆF (Ĥ 11i Ĥ 12i ) R m n Ĥ 2i R m p satisfying the two LMI conditions (27) and (28). Let N Z and the nonsingular matrix W be such that (25) is verified. Then the controller (5) with the gains given by (26) and by E ci =(W ) ( Ê i S Y ) B i (29) and the set S = E(Q (α))=...n E ( Q ) i with Qi = Q11i Q 12i are solutions to Problem 1 for any nonlinearity ϕ(z) satisfying relation (3). Q 22i B. Synthesis of the dynamic output controllers gains For optimization of the size of the set S = E(Q (α)) we adopt the shape set approach. Thus consider a ball with radius β denoted by E(I 2n β) where the scalar β> is the scaling factor to be maximized subject to the following inclusion: E(I 2n β)= { ξ R 2n ; ξ ξ β } E(Q (α)). By considering µ=1/β the above inclusion is equivalent to: µi2n I 2n...N Q i which by pre and post-multiplication by diag(i 2n Θ ) and its transpose respectively is also equivalent to: µi n Y I n µi n W ˆQ 11i ˆQ 12i...N. (3) ˆQ 22i The invertibility of W can be guaranteed by: W +W > (31) Theorem 8: Given a real scalar λ (1 consider that the following CPPs are feasible. Then the corresponding controller gain matrices obtained from (26) and (29) and the set S = E(Q (α)) solve Problem 1: Synthesis with saturation avoidance: min µ ˆQ 11i ˆQ 12i ˆQ 22i XYWT  i ˆB i Ĝ i Ĉ ˆD ˆF subject to LMIs (23) (24) (3) and (31) Synthesis with saturating actuators: min µ ˆQ 11i ˆQ 12i ˆQ 22i XYWT S  i ˆB i Ĉ ˆDÊ i ˆFĜ i Ĥ 11i Ĥ 12i Ĥ 2i subject to LMIs (27) (28) (3) and (31) V. EXAMPLE (32) (33) Consider the following data for system (2)-(4) from 2:.1.4 A 1 = B 1 = B =.2.7 A 2 = G = G 2 = 1 L 1 = 1.1 L 2 =.9 C= 1 Ω=.55 ρ=1 and ϕ(z)=.275z(1+sin(z)). Table 1 shows the optimal scaling factors and compensators gain matrices obtained from CPPs (32) and (33) for 4238

5 ˆQ 11 j ˆQ 12 j Y A i + ˆB i C Â i Ĝ i ˆQ 22 j A i + B i ˆDC A i X+ B i Ĉ G i +B i ˆF λ ( ˆQ 11i Y Y ) λ ( ˆQ 12i T I ) L i Ω λ ( ˆQ 22i X X ) X L i Ω 2 <...N j = 1...N (23) ˆQ 11i +Y +Y ˆQ 12i + T + I L i Ω ( ˆD (l) C) ˆQ 22i + X + X X L i Ω Ĉ (l) 2 ˆF (l) ρ 2 (l) >...N l=1...m (24) ˆQ 11 j ˆQ 12 j Y A i + ˆB i C Â i Ĝ i Ê i ˆQ 22 j A i + B i ˆDC A i X+ B i Ĉ G i +B i ˆF B i S λ ( ˆQ 11i Y Y ) λ ( ˆQ 12i T I ) L i Ω Ĥ 11i λ ( ˆQ 22i X X ) X L i Ω Ĥ 12i 2 Ĥ 2i 2S <...N j = 1...N (27) ˆQ 11i +Y +Y ˆQ 12i + T + I L i Ω ( ˆD (l) C) Ĥ 11(l) ˆQ 22i + X + X X L i Ω Ĉ (l) Ĥ 12(l) 2 ˆF (l) ρ 2 (l) >...N l=1...m (28) λ=1. Figure 1 shows the corresponding restrictions of sets S to the state space of the plant denoted S r : { { } } S r = E r Q i = x R 2 x ; Q x i 1 2 We see that the size of S r in the saturation avoidance case is more conservative than the size obtained in the saturating actuators case. Thus this last kind of solution can be privileged in practice unless the actuators may be easily damaged due to saturations occurrence. Figure 2 shows the control efforts corresponding to discrete-time simulations with the solution obtained with saturating actuators for λ=1 using two different random sequences for the time-varying parameter vector α k Ξ. The corresponding initial condition ξ = x where x = belongs to the boundary of S r as shown in figure 1. Figure 2 shows that the saturation is kept active during some discrete times for both sequences. However saturations may not occur for other sequences and/or initial conditions in S r. Finally figure 3 shows the conflicting trade-off existing between the performance requirement represented by the coefficient of contractivity λ and the size of the region where the desired performance is guaranteed represented by S r. We see that the greater is the performance requirement the smaller is the region where it is guaranteed. x with saturating act. sat. avoidance λ = x1 Fig. 1. S r obtained from CPPs (32) and (33) VI. CONCLUSION An LMI framework has been proposed to design dynamic output compensators for a class of nonlinear parametervarying discrete-time systems subject to actuators saturations. The proposed results technically differ from 5 mainly due to some tools used to deal with discrete-time and 4239

6 CPP Saturation Avoidance Saturating actuators β A c1 A c B c1 B c E c1 E c G c1 G c C c D c F c TABLE I NUMERICAL RESULTS OBTAINED FOR λ=1 sat(u) x sequence 1 sequence k Fig Control effort - solution with saturating actuator λ = 1. ( ) λ =.9 ( ) λ =.8 ( ) x1 Fig. 3. S r obtained from CPP (33) for different values of λ time-varying parameters: the parameter-varying Lyapunov function the associated non-ellipsoidal contractive level sets and a modification of the technique proposed in E. B. Castelan U. Moreno and E. R. de Pieri. Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations. In IEEE International Symposium on Circuits and Systems pages Greece E. B. Castelan S. Tarbouriech and Isabelle Queinnec. Control design for a class of nonlinear continuous-time systems. Automatica 44(8): J. M. Gomes da Silva Jr. J. Corso and E. B. Castelan. Output feedback stabilization for systems presenting sector-bounded nonlinearities and saturating inputs. In Proc. of the 17th IFAC World Congress pages J. M. Gomes da Silva Jr. D. Limon and T. Alamo. Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints. IEEE Transactions on Automatic Control pages J. M. Gomes da Silva Jr. and S. Tarbouriech. Anti-windup design with guaranteed regions of stability: an lmi-based approach. IEEE Transactions on Automatic Control 5(1): T. Hu and Z. Lin. Composite quadratic Lyapunov functions for constrained control systems. IEEE Transactions on Automatic Control 48(3): T. Hu Z. Lin and B. M. Chen. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica 38: Marc Jungers and E. B. Castelan. Gain-scheduled control design for a class of discrete-time nonlinear systems with saturating actuators. Systems & Control Letters page submitted H. K. Khalil. Nonlinear Systems - Third Edition. Prentice Hall T. Kiyama S. Hara and T Iwasaki. Effectiveness and limitation of circle criterion for lti robust control systems with control input nonlinearities of sector type. Int. J. Robust and Nonlinear Control (15): T. Kiyama and T. Iwasaki. On the use of multi-loop circle for saturating control synthesis. Systems & Control Letters 41: E. F. Mulder M. V. Kothare and M. Morari. Multivariable anti-windup controller synthesis using linear matrix inequalities. Automatica pages Sep C. Pittet S. Tarbouriech and C. Burgat. Stability regions for linear systems with saturating controls via circle and popov criteria. In 36th IEEE Conf. on Decision and Control (CDC 97) pages San Diego USA K. Sawada T. Kiyama and T. Iwasaki. Generalized sector synthesis of output feedback control with anti-windup structure. Systems & Control Letters 58: C. Scherer P. Gahinet and M. Chilali. Multiobjective output-feedback control via LMI optimization. IEEE Trans. on Automat. Contr. 42(7): Q. Zheng and F. Wu. Output feedback control of saturated discretetime linear systems using parameter-dependent lyapunov functions. Syst. & Contr. Letters 57: REFERENCES 1 S. Boyd L. El Ghaoui E. Feron and V. Balakrishnan. Linear matrix inequalities in system and control theory. SIAM Studies in Applied Mathematics E. B. Castelan J. Corso and U. F. Moreno. Stability and stabilization of a class of uncertain nonlinear discrete-time systems with saturating actuators. In 3 rd IFAC Symposium on System Structure and Control Foz do Iguassu - Brazil October