ROBUST CONTROLLER DESIGN WITH HARD INPUT CONSTRAINTS. TIME DOMAIN APPROACH. Received August 2015; revised November 2015

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1 International Journal of Innovative Computing, Information and Control ICIC International c 2016 ISSN Volume 12, Number 1, February 2016 pp ROBUST CONTROLLER DESIGN WITH HARD INPUT CONSTRAINTS. TIME DOMAIN APPROACH Vojtech Veelý 1 and Adrian Ilka 2 1 Intitute of Robotic and Cybernetic Slovak Univerity of Technology Bratilava, Ilkovicova 3, Slovak Republic vojtech.veely@tuba.k 2 Department of Signal and Sytem Chalmer Univerity of Technology Goteborg SE , Sweden ilka@chalmer.e Received Augut 2015; revied November 2015 Abtract. A novel approach to robut controller deign with hard input contraint i preented. The propoed robut controller deign procedure i baed on the robut tability condition developed uing Affine or Parameter-Dependent Quadratic Stability approach. The obtained feaible deign procedure are in the form of BMI or LMI. The obtained deign reult and their propertie are illutrated in imulation example. Keyword: Robut controller, Affine, Parameter-dependent Lyapunov function, Hard input contraint 1. Introduction. All control actuation device are ubject to magnitude and/or rate limit and thi lead to degradation of the performance and even intability of cloed loop control ytem. Hard input contraint belong to the very important tak in the controller deign. A chronological bibliography on aturating actuator can be found in 2,7. Neceary and ufficient condition for controllability of linear ytem ubject to input/tate contraint are given in 5. Robut tabilization of uncertain ytem ubject to input contraint i conidered in 6,14. A piecewie linear control enuring the input contraint i parameterized by algebraic Riccati equation. Youla parametrization of uncertain plant and tabilizing controller which guarantee precribed hard bound on the control ignal can be found in 9. The ue of a two tage IMC anti-windup deign for table plant and input contraint i decribed at 1. The invariant et and an algorithm approach imilar to Soft Variable Structure Control enuring oft input contraint for the model predictive plant control ytem are decribed by 15. In the paper by 10,11 deal with the deign of linear ytem with aturating actuator, where the actuator limitation have to be incorporated into control deign. The emiglobal approach include tabilization, input additive diturbance rejection, and robut tabilization. In the book by 12 method are propoed and algorithm are deigned on the bai of the tate pace approach to overcome the effect of actuator aturation enuring the global tability of cloed loop ytem. Book by 13 i devoted to thee direction: firt, anti-windup trategie, a econd model predictive control with contraint and finally development of a tability and tabilization method for contrained ytem. The paper 16 invetigated the problem of decentralized robut tabilization of a cla of uncertain nonlinear ytem with input aturation. Robut contrained control for MIMO nonlinear ytem can be found in

2 162 V. VESELÝ AND A. ILKA The above hort urvey implie that in the field of robut controller deign with input contraint there are many different approache. The above obervation motivated u to tudy the following reearch problem which ha not been ufficiently olved yet. Deign the robut controller with hard input contraint which will guarantee: tability and robutne propertie of the cloed loop ytem when the uncertain plant parameter Π Ω lie in the given polytopic (convex) uncertainty box; guaranteed cot (performance) for the cloed loop ytem for all Π Ω; Affine or Parameter-Dependent Quadratic Stability; the BMI or LMI deign procedure. In thi paper, we provide, to the author bet knowledge, an alternative novel approach to the robut controller deign with hard input contraint. The propoed uncertainty model, introduced in Section 2, i ued to formulate a robut control problem with hard input contraint. The main reult are preented in Section 3. The deign procedure i baed on the new developed robut tability condition for the hard input contraint controller and numerical example in Section 4 illutrate the effectivene of the propoed approache. Some concluion are made in Section Problem Formulation and Preliminarie. Thi paper i concerned with the cla of uncertain linear ytem that can be decribed a ( ) ( ) p p ẋ = A 0 + A i θ i x + B 0 + B i θ i u (1) y = Cx where x R n, u R m, y R l are the tate, control input and controlled output, repectively; matrice A i, B i, i = 0, 1,..., p are contant with appropriate dimenion; θ = θ 1,..., θ p i a vector of uncertain and poible time varying real parameter atifying θ θ θ Ω, θ θ θ Ω t which belong to the known boundarie. There are two poibilitie for θ parameter change: parameter θ i unknown but time varying, parameter θ i unknown and contant. For the firt cae the modified affine quadratic tability will be adapted to obtain the cloed loop tability condition with input contraint and for the econd cae the modified parameter dependent quadratic tability will be ued. For the cla of uncertain ytem (1) the following problem i tudied in thi paper. Deign a robut tatic output feedback controller with control algorithm u = F (θ 0 )y = F (θ 0 )Cx (2) with hard contraint u j u Mj, j = 1, 2,..., m uch that for the cloed-loop ytem (3) the deigned controller guarantee robut affine or parameter dependent quadratic tability and guaranteed cot with repect to the performance index (4) ẋ = (A(θ) + B(θ)F (θ 0 )C)x = A c x (3) where F (θ 0 ) = F θ 0, θ 0 = diag{θ 0j }, j = 1, 2,..., m, θ 0 θ 0 θ 0, θ0 θ0 θ0 Φ i known time varying parameter which erve for enure the hard input contraint. To ae the performance quality, the following performance index i aociated with ytem (1) ( J = x T Qx + u T Ru + ẋ T Sẋ ) dt = J(t)dt (4) 0 0

3 ROBUST CONTROLLER DESIGN 163 The repective notion of guaranteed cot i given in the next definition. Definition 2.1. Conider ytem (1) and controller (2). If there exit a control law u and a poitive calar J uch that the repective cloed-loop ytem (3) i table and the value of the cloed-loop cot function (4) atifie J J, then J i aid to be the guaranteed cot and u i aid to be the guaranteed cot control law for ytem (1). Recall the well known reult from LQ theory which will be ued below to prove one of the main reult. Lemma Conider the ytem (1) with control algorithm (2). Control algorithm (2) i the guaranteed cot control law for the cloed-loop ytem (3) if and only if there exit a Lyapunov function V (θ) uch that for all θ Ω, θ Ω t and θ 0, θ 0, the following condition hold { } dv (θ) B e (θ) = + J(t) = εx T x ε 0 ε 0 (5) dt Uncertain ytem (1) with control algorithm (2) conforming to Lemma 2.1 i called robut table with guaranteed cot. Note that for concrete tructure of V (θ) if and only if may to be decreaed to if. 3. Main Reult Affine Quadratic Stability. Thi ubection formulate the theoretical approach to the robut tatic output feedback controller deign for uncertain ytem (1) which enure cloed-loop ytem affine quadratic-tability and guaranteed cot (4), for all uncertain plant parameter Π Ω and hard input contraint. Definition The linear time varying ytem (3) i affine quadratically table if there exit p + 2 ymmetric matrice P, P 0, P 1,..., P p uch that p P (θ) = P + P 0 θ 0 + P i θ i > 0 (6) dv (θ) = x (A T T c P (θ) + P (θ)a c + P 0 θ0 + dt for all θ Ω, θ Ω t, (θ 0, θ ) 0 Φ. Then for the cloed loop ytem, Lyapunov function hold Equation (7) can be rewritten a follow ( P θ) dv (θ) dt V (θ) = x T P (θ)x = ẋ T x 0 P (θ) T ( P (θ) P θ) ) p P i θi x 0 (7) ẋ x = P 0 θ0 + p P θ i i. To iolate two matrice P (θ) and A c uing auxiliary matrice N 1, N 2 R n n from (3) we obtain or (2N 1 ẋ + 2N 2 x) T (ẋ A c x) = 0 ẋt x T N1 T + N 1 N1 T A c + N 2 ẋ A T c N 1 + N2 T N2 T A c A T c N 2 x (8) = 0 (9)

4 164 V. VESELÝ AND A. ILKA Subtitute control algorithm (2) to performance (4) J(t) = ẋ T x T S 0 0 Q + C T F T RF Cθ 2 0 Subtituting (8), (9) and (10) to (5) one obtain where ẋ x (10) B e = ẋ T x T W ẋ T x T T 0 (11) W = W 01 + W 02 θ 0 + W 03 θ p (W i1 + W i2 θ 0 )θ i (12) N T 1 + N 1 + S N1 T A 0 + N 2 + P ( W 01 = N2 T A 0 A T 0 N 2 + Q + P θ) 0 N T W 02 = 1 B 0 F C + P 0 N2 T B 0 F C (B 0 F C) T N 2 W 03 = 0 C T F T RF C 0 N1 T A i + P i W i1 = P i A T i N 1 N2 T A i A T i N 2 0 N1 W i2 = T B i F C (B i F C) T N 1 N2 T B i F C (B i F C) T N 2 i = 1, 2,..., N. The firt main reult for deign of robut controller which enure the Affine Quadratic Stability with input contraint are ummarized in the next theorem. Theorem 3.1. Conider the uncertain ytem governed by (1) with robut control algorithm (2) and hard input contraint u j u Mj, j = 1, 2,..., m. Cloed loop ytem (3) i robut affine quadratically table with guaranteed cot if there exit auxiliary matrice N 1, N 2, p + 2 ymmetric matrice P, P 0, P 1,..., P p and output feedback gain matrix F uch that inequality W 0 (13) hold for all θ Ω, θ Ω t and (θ 0, θ ) 0 Φ. Variable θ 0 will guarantee the hard input contraint u j u Mj, j = 1, 2,..., m if it i calculated a { 1 if uj < u Mj, for all j = 1, 2,..., m θ 0cj = u Mj if for ome u j > u Mj, j = 1, 2,..., m (14) max j u j To enure that rate of θ 0 change within θ 0 θ0, θ 0 and to obtain the real value of θ 0 it i recommended to ue a firt order filter with tranfer function θ 0j = θ 0cj ( θ0j ) 1+1. Proof: The proof of ufficient condition are baed on Equation (6), (8), (9) and (10). Becaue matrix W 03 0, matrix W with repect to θ 0, θ i, i = 1, 2,..., p i convex, matrix W i negative definite (emidefinite) if and only if it take negative definite (emidefinite) at the corner of θ 0 and θ. For robut tability analyi (13) reduce to LMI; for robut control ynthei we obtain BMI.

5 ROBUST CONTROLLER DESIGN Parameter Dependent Quadratic Stability. Thi ubection formulate the theoretical approach to the robut tatic output feedback controller deign for ytem (1) which enure cloed loop ytem parameter dependent quadratic tability and guaranteed cot (4), for all uncertain plant parameter Π Ω and hard input contraint. When one ubtitute to Equation (1) maximal and minimal value of θ a polytopic ytem model with N = 2 p vertice i obtained in the form. or {(A 1v, B 1v, C), (A 2v, B 2v, C),..., (A Nv, B Nv, C)}, N = 2 p (15) {A(α), B(α)} = (A iv, B iv )α i, α i = 0 α i = 1, α i 0 where α i 0 1, i = 1, 2,..., N i uncertain contant or time varying parameter. The parameter dependent Lyapunov function i given a follow V (α) = x T P (α)x, P (α) = P 0 θ 0 + P iv α i (16) From (16) and (8) one obtain the time derivative of the Lyapunov function for polytopic ytem a follow dv (α) = ẋ T x 0 P T 0 θ 0 + N P ivα i ẋ dt P 0 θ 0 + N P ivα i P 0 θ0 + N P (17) iv α i x The cloed-loop ytem can be obtained from (15) and (2) ẋ = (A iv + B iv F θ 0 C)x = A ci x i = 1, 2,..., N (18) Uing the ame procedure a for affine quadratic tability (9), (10) and (11) one obtain the cloed loop ytem robut parameter-dependent quadratic tability condition with guaranteed cot in the form where where W v = B ev = ẋ T x T W v ẋt x T T 0 (19) N T 1 + N 1 + S N T 1 A c (α) + N 2 + P (α) φ(α, θ 0 ) φ(α, θ 0 ) = N T 2 A c (α) A c (α) T N 2 + Q + P 0 θ0 + P iv α i + C T F T RF Cθ0 2 Inequality (19) i linear with repect to uncertain parameter α i, and inequality (19) can be plit to N inequalitie a where W i = W iq + W il θ 0 + W ik θ 2 0 0, i = 1, 2,..., N (20) N T W iq = 1 + N 1 + S N1 T A iv + N 2 + P iv φ i 0 N1 W il = T B iv F C + P 0 P 0 (B iv F C) T N 1 N2 T B iv F C (B iv F C) T N 2

6 166 V. VESELÝ AND A. ILKA W ik = 0 C T F T RF C φ i = N T 2 A iv A T ivn 2 + Q + P θ 0 + P iv α i Robut tability condition are ummarized in the following theorem. Theorem 3.2. Conider the uncertain linear ytem (1) or (15). The cloed loop ytem (18) with control algorithm (2) i parameter dependent quadratically table with input contraint u j u Mj, j = 1, 2,..., m if there exit matrice N 1, N 2 R n n, N + 1 ymmetric matrice P 0, P 1v,..., P Nv uch that P 0 θ 0 + N P ivα i 0, N α i = 1, α i 0, θ 0 θ 0 θ 0, θ0 θ0 θ0, ymmetric poitive definite matrice Q, R, S and output feedback gain matrix F uch that W i = W iq + W il θ 0 + W ik θ (21) for i = 1, 2,..., N, θ 0 = θ 0 and θ 0 = θ 0. Variable θ 0 i calculated a in (14). Becaue matrix W ik 0, matrix W i with repect to θ 0 i convex, matrix W i i negative definite (emidefinite) if and only if it take negative definite (emidefinite) at the corner of θ 0 and N vertice of polytopic ytem. For robut tability analyi (21) reduce to LMI; for robut control ynthei we obtain BMI. Proof: The proof of Theorem ufficient condition are baed on the Equation (10), (15), (17) and (18). 4. Example. In order to prove applicability of the propoed robut controller deign procedure with input contraint, two example are preented. The firt example i imulation of randomly generated ytem near to the boundary of tability and the econd one i lightly modified firt example to make the ytem untable. Two control algorithm are calculated for each of the two example: for the firt algorithm the affine quadratic tability approach i ued and for the econd one the parameter dependent quadratic tability. In the both cae the problem i to deign two robut decentralized PI controller which enure the affine (parameter dependent) quadratic tability and guaranteed cot with input contraint. A level of hard input contraint i equal to 200 percent value of nominal input variable. The rate of θ 0 change need to be more than 3. The tate of third order ytem due to PI controller deign need to be increaed to five tate. Example 4.1. Parameter of randomly generated example (1) are given a follow: A 0 = B = with eigenvalue Eig(A 0 ) = 0; 0; ; ; A 1 = B 1 =

7 A 2 = and output matrix C ROBUST CONTROLLER DESIGN C = B 2 = For the parameter: performance Q = qi, q = 0.01, R = ri r, r = 1, S = I, = 0.001; rate of uncertain parameter change θ i = 0, α i = 0; input contraint θ 0.5, 1, θ 0 8, 8 ; Lyapunov matrice contraint P 0.1 I, ro I, ro = 1000; uncertaintie θ i 1, 1, i = 1, 2 the following two robut PI controller are obtained AQS R 11 () = ; R 12 () = PDQS R 11 () = ; R 12 () = For imulation we have ued the nominal plant model and robut controller deigned with AQS. Simulation reult (Figure 1-5) confirm, that Theorem 3.1 hold. Figure 1 and 2 how the imulation reult without input contraint and Figure 3-5 how imulation reult with hard input contraint u 1 4 and u 2 6 which prove that Theorem 3.1 hold and guarantee hard input contraint. Figure 5 how the calculated parameter θ 01 and θ 02 which change the correponding controller gain for guarantee of the hard input contraint. Figure 1. Simulation reult w(t), y(t) without input contraint Figure 2. Controller output u(t) without input contraint

8 168 V. VESELÝ AND A. ILKA Figure 3. Simulation reult w(t), y(t) with input contraint Figure 4. Controller output u(t) with input contraint Figure 5. Calculated parameter θ(t) Example 4.2. To obtain the ytem parameter for the econd example we change the following entrie A 0 (2, 2) = 0.135, A 0 (3, 3) = 0.85 in matrix A 0 (firt example). The obtained eigenvalue of matrix A 0 for econd example are Eig(A 0 ) = 0; 0; ; ; All other ytem parameter are the ame a for the firt example, except for θ 0 rate contraint θ 0 3, 3. Parameter of two deigned robut controller are a follow AQS R 21 () = ; R 22 () = PDQS R 21 () = ; R 22 () = For imulation we have ued the plant model with θ 1 = 1, θ 2 = 1 and the robut controller deigned with PDQS. Simulation reult (Figure 6-10) confirm, that Theorem 3.2 hold. Figure 6 and 7 how the imulation reult without input contraint and Figure 8-10 how imulation reult with input contraint u 1 2 and u 2 2 which prove that Theorem 3.2 hold and guarantee hard input contraint. Figure 10 how the calculated cheduled parameter θ 1 and θ 2.

9 ROBUST CONTROLLER DESIGN 169 Figure 6. Simulation reult w(t), y(t) without input contraint Figure 7. Controller output u(t) without input contraint Figure 8. Simulation reult w(t), y(t) with input contraint Figure 9. Controller output u(t) with input contraint Figure 10. Calculated cheduled parameter θ(t)

10 170 V. VESELÝ AND A. ILKA 5. Concluion. A novel robut controller deign approach with hard input contraint ha been propoed. The obtained reult, illutrated on the two example, how the applicability of the deigned robut controller and it ability to cope with model uncertaintie. Several form of parameter dependent/affine quadratic Lyapunov function are preented and teted by imulation. The propoed robut controller deign approach with parameter dependent/affine Lyapunov function conider quick parameter change to enure the hard input contraint. Simulation reult prove the potential ability of the deigned cloed-loop to withtand alo thee change. The obtained robut controller deign procedure are in the form of BMI and LMI approache. The propoed approach contribute to the deign tool for robut controller which guarantee the affine/parameter dependent quadratic tability, guaranteed performance and hard input contraint. Acknowledgement. The work ha been upported by Grant 1/0475/16 of Slovak Grant Agency. The author alo gratefully acknowledge the helpful comment and uggetion of the reviewer, which have improved the preentation. REFERENCES 1 A. A. Adegbege and W. P. Heath, Stability condition for contrained two-tage internal model control, The 49th IEEE Conference on Deciion and Control, Atlanta, pp , D. S. Berntein and A. V. Michel, A chronological bibliography on aturating actuator, Int. Journal of Robut and Nonlinear Control, pp , M. Chen, P. Shi and C. C. Lim, Robut contrained control for nonlinear ytem baed on diturbance oberver, IEEE Tran. Automatic Control, P. Gahinet, P. Apkarian and M. Chilali, Affine parameter-dependent Lyapunov function and real parametric uncertainty, IEEE Tran. Automatic Control, vol.41, no.3, pp , V. P. M. Heemel and M. K. Camlibel, Controllability of linear ytemw with input contraint, Proc. of the 46th IEEE CD-ROM, pp , D. Henrion, G. Gracia and S. Tarbouriech, Piecewie linear robut control of ytem with input contraint, European Journal of Control, pp , V. Kapila and K. M. Grigariadi, Actuator aturation control, Contr. Eng. Serie, Marcel Dekter, V. M. Kuncevic and M. M. Lycak, Control Sytem Deign Uing Lyapunov Function Approach, Nauka, Mokau, 1977 (in Ruian). 9 W. Reinelt and M. Canale, Robut Control of SISO Sytem Subject to Hard Input Contraint, A. Saberi, Z. Lin and A. R. Teel, Control of linear ytem with aturating actuator, IEEE Tran. Automatic Control, vol.41, no.3, pp , P. Shi, Y. Yin, F. Liu and J. Zhang, Robut control on aturated Markov jump ytem with miing information, Information Science, vol.265, pp , S. Tarboriech, G. Garcia, J. M. G. da Silva Jr. and I. Queinnec, Stability and Stabilization of Linear Sytem with Saturating Actuator, Springer, S. Tarboriech, G. Garcia and A. H. Glattfelder, Advanced Strategie in Control Sytem with Input and Output Contraint, Springer/Verlag Berlin, Heidelberg, V. Veely and A. Ilka, Robut Controller Deign with Input Contraint. Time Domain Approach, Proc. of IFAC ROCOND, Bratilava, V. Veely, D. Roinova and M. Foltin, Robut model predictive control deign with input contraint, ISA Tranaction, vol.49, pp , Y. Ma, N. Zhang and X. Zhong, Decentralized robut control for uncertain nonlinear deciptor largecale compoite ytem with input contraint, International Journal of Innovative Computing, Information and Control, vol.9, no.10, pp , 2013.