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Robust Controller Design with Input Constraints. Time Domain Approach.∗
Proceedings of the 8th IFAC Symposium on Robust Control Design Proceedings of the 8th IFAC Symposium on Robust Control Design Proceedings of the 8th IFAC Symposium on Robust Control Design July 8-11, 2015. Bratislava, Republic Proceedings of the 8th IFACSlovak Symposium on Robust Control Design July 8-11, 2015. Bratislava, Slovak Republic Available online at www.sciencedirect.com July 8-11, 2015. Bratislava, Slovak Republic July 8-11, 2015. Bratislava, Slovak Republic
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Robust Controller Design Robust Controller Design Robust Controller Design Robust Controller Design Constraints. Time Domain Constraints. Time Domain Constraints. Time Domain Constraints. Time Domain
with Input with Input with Input with Input⋆ ⋆⋆ Approach. Approach. Approach. Approach. ⋆
∗ ∗ Vojtech y Vojtech Vesel´ Vesel´ y ∗∗∗ ,, Adrian Adrian Ilka Ilka ∗∗∗ Vojtech Vesel´ y , Adrian Ilka Vojtech Vesel´ y , Adrian Ilka ∗ ∗ Faculty of Electrical Engineering and Information Technology, Slovak ∗ Faculty of Electrical Engineering and Information Technology, Slovak ∗ Faculty ofofElectrical Engineering and Technology, Slovak University Technology in Ilkovicova 812 Faculty ofofElectrical Engineering and Information Information Slovak University Technology in Bratislava, Bratislava, Ilkovicova 3, 3, Technology, 812 19 19 Bratislava, Bratislava, University of Technology in Bratislava, Ilkovicova 3, 812 19 Bratislava, (e-mail: [email protected], [email protected]) University of Technology in Bratislava, Ilkovicova 3, 812 19 Bratislava, (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected], [email protected]) [email protected]) (e-mail: Abstract: A novel approach to robust controller design with hard input constraints is presented. Abstract: A A novel novel approach approach to to robust robust controller controller design design with with hard hard input input constraints constraints is is presented. presented. Abstract: The proposed design procedure is based on the robust stability condition developed using Abstract: A novel approach to robust controller design with hard input constraints is presented. The proposed design procedure is based on the robust stability condition developed using Affine Affine The proposed is on robust condition developed using Affine or Quadratic Stability approach. The feasible design procedures The proposed design design procedure procedure is based based on the the robust stability stability condition developed Affine or Parameter-Dependent Parameter-Dependent Quadratic Stability approach. The obtained obtained feasible designusing procedures or Parameter-Dependent Quadratic Stability approach. The feasible design procedures are in LMI. obtained design and properties are illustrated or The obtained obtained design areParameter-Dependent in the the form form of of BMI BMI or orQuadratic LMI. The TheStability obtainedapproach. design results results and their theirfeasible properties are procedures illustrated are in the form of BMI or LMI. The obtained design results and their properties are illustrated in simulation examples. are in the form of BMI or LMI. The obtained design results and their properties are illustrated in simulation simulation examples. in examples. in simulation examples. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2015, IFAC (International Keywords: Robust Lyapunov Keywords: Robust controller, controller, Affine, Affine, Parameter-Dependent Parameter-Dependent Lyapunov Lyapunov function, function, Hard Hard input input Keywords: Robust controller, Affine, Parameter-Dependent function, Hard input constraints Keywords: constraints Robust controller, Affine, Parameter-Dependent Lyapunov function, Hard input constraints constraints 1. The 1. INTRODUCTION INTRODUCTION The above above short short survey survey implies implies that that in in the the field field of of robust robust 1. INTRODUCTION The above short survey implies that in the field of controller design with input constraints there are many 1. INTRODUCTION The above short survey implies that in the field of robust robust controller design with input constraints there are many controller design with input constraints there are many different approaches. The above observation motivated us All control actuation devices are subject to magnitude controller design with input constraints there are many different approaches. The above observation motivated us All control actuation devices are subject to magnitude different approaches. The above observation motivated us All control actuation devices are subject to magnitude to study the following research problem which has not been and/or rate limits and this leads to degradation of the different approaches. The above observation motivated us All control actuation devices are to subject to magnitude to study the following research problem which has not been and/or rate limits and this leads degradation of the to study the following research problem which has not been and/or rate limits and this leads to degradation sufficiently solved yet. Design the robust controller with of the performance and even instability of closed loop control to study the following research problem which has not been and/or rate limits and instability this leads of to closed degradation of the sufficiently solved yet. Design the robust controller with performance and even loop control sufficiently solved Design the robust performance and even instability of closed closed loop control hard systems. input constraints belong to very imporsolved yet. yet. which Designwill theguarantee: robust controller controller with with performance even instability of hard input input constraints constraints which will guarantee: systems. Hard Hardand input constraints belong to the theloop very control impor- sufficiently hard input constraints which will guarantee: systems. Hard input constraints belong to the very important task in the controller design. A chronological biblioghard input constraints which will guarantee: systems. Hard input constraints belong to the very impor•• stability tant task task in in the the controller controller design. A A chronological chronological bibliogtant stability and and robustness robustness properties properties of of the the closed closed loop loop design. bibliography on saturating actuators can found in Bernstein tant task in the controller design. Abe chronological bibliog•• system stability and robustness properties of the closed loop when the uncertain plant parameters Π ∈ raphy on saturating actuators can be found in Bernstein stability and robustness properties of the closed loop raphy on saturating actuators can be found in Bernstein system when the uncertain plant parameters Π ∈Ω Ω and Michel, (1995) and Kapila and Grigoriadis , (2002). raphy on saturating actuators can beGrigoriadis found in Bernstein system when the uncertain plant parameters Π ∈ lie in the given polytopic (convex) uncertainty box; and Michel, (1995) and Kapila and , (2002). system when uncertain plant parameters ∈Ω Ω and Michel, Michel,and (1995) and Kapila Kapila and for Grigoriadis , (2002). (2002). lie in the giventhe polytopic (convex) uncertaintyΠbox; Necessary sufficient conditions controllability of and (1995) and and Grigoriadis , polytopic (convex) uncertainty box; •• lie guaranteed cost (performance) for the closed loop Necessary and and sufficient sufficient conditions conditions for for controllability controllability of of lie in in the the given given polytopic (convex) uncertainty box; Necessary guaranteed cost (performance) for the closed loop linear systems subject to input/state constraints are given Necessary andsubject sufficient conditions for controllability of •• guaranteed cost (performance) for linear systems systems to input/state input/state constraints are given given system Π Ω; guaranteed cost for the the closed closed loop loop linear subject to constraints are system for for all all Π∈ ∈(performance) Ω; in Heemels and Carmlibel, (2007). Robust stabilization linear systems subject to input/state constraints are given system for all Π ∈ Ω; • affine or Parameter-Dependent Quadratic Stability; in Heemels and Carmlibel, (2007). Robust stabilization system for all Π ∈ Ω; in uncertain Heemels and and Carmlibel, (2007). Robust stabilization • affine or Parameter-Dependent Quadratic Stability; of systems subject to constraints is in Heemels Carmlibel, stabilization •• affine or Quadratic BMI or LMI design procedure. of uncertain uncertain systems subject(2007). to input inputRobust constraints is conconaffine or Parameter-Dependent Parameter-Dependent Quadratic Stability; Stability; of •• the the BMI or LMI design procedure. systems subject to input constraints is considered in Henrion et al, (1998). A piecewise linear of uncertain systems subject to input constraints is conthe BMI or LMI design procedure. sidered in Henrion et al, (1998). A piecewise linear con•this thepaper, BMI or LMI design procedure. sidered in Henrion et al, (1998). A piecewise linear conIn we provide, to the authors’ best knowledge, trol ensuring the input constraints is parameterized by sidered in Henrion et al,constraints (1998). A piecewise linear conthis paper, we provide, to the authors’ best knowledge, trol ensuring ensuring the input input is parameterized parameterized by In trol the constraints is by In paper, provide, to authors’ best an alternative novel approach the robust controller algebraic Riccati equation. Youla parametrization of unIn this this paper, we we provide, to the the to authors’ best knowledge, knowledge, trol ensuring the input constraints is parameterized by an alternative novel approach to the robust controller algebraic Riccati equation. Youla parametrization of ununan alternative novel approach to the robust controller algebraic Riccati equation. Youla parametrization of design with hard input constraints. The proposed uncercertain plant and stabilizing controllers which guarantee an alternative novel approach to the robust controller algebraic Riccati equation. Youla parametrization of un- design with hard input constraints. The proposed uncercertain plant and stabilizing controllers which guarantee certain plant and stabilizing controllers which guarantee design with hard input constraints. The proposed uncertainty model, introduced in Section 2, is used to formulate prescribed hard bounds on the control signal can be found design with hard input constraints. The proposed uncercertain plant and stabilizing controllers which guarantee tainty model, introduced in Section 2, is used to formulate prescribed hard bounds on the control signal can be found prescribed hard bounds on the control signal can be found tainty model, introduced in Section 2, is used to formulate a robust control problem with hard input constraints. in Reinelt an Canale, (2001). The use of a two stage IMC tainty model, introduced in Section 2, is used to formulate prescribed hard bounds on the control signal can be found control problem with hard input constraints. in Reinelt Reinelt an an Canale, Canale, (2001). (2001). The The use use of of aa two two stage stage IMC IMC aa robust in control problem with hard input constraints. The main results are presented Section The design antiwindup design for stable plant input constraints is a robust robust control problem with in hard input3. constraints. in Reinelt an Canale, The and use of a two stage IMC main results are presented in 3. design antiwindup design for (2001). stable plant plant and input constraints is The The main results are presented in Section Sectionrobust 3. The Thestability design antiwindup design for stable and input constraints is procedure is based on the new developed described at Adegbege and Heath, (2011). The invariant The main results are presented in Section 3. The design antiwindup design for stable plant and input constraints is procedure is based on the new developed robust stability described at Adegbege and Heath, (2011). The invariant described at Adegbege and Heath, procedure is based on the new developed robust stability (2011). The invariant condition the input controller and set similar to Soft Variable is based on the new constraints developed robust stability described Adegbege approach and Heath, (2011). condition for for the hard hard input constraints controller and set and and an anatalgorithm algorithm approach similar to The Soft invariant Variable procedure set and an algorithm to Soft Variable condition for the hard input constraints controller and approach similar numerical examples in Section 4 illustrate the effectiveness Structure Control ensuring soft input constraints for the condition for the hard input constraints controller and set and an algorithm approach similar to Soft Variable numerical examples in Section 4 illustrate the effectiveness Structure Control ensuring soft input constraints for the Structure Control ensuring soft input constraints numerical examples in Section 4 illustrate the effectiveness for the of the proposed approaches. model predictive plant control systems is described by numerical examples in Section 4 illustrate the effectiveness Structure Control plant ensuring soft input constraints for the of the proposed approaches. model predictive control systems is described by of the proposed approaches. modely predictive predictive plant control systems is described described by Vesel´ In the by and al, model plant systems is Vesel´ y et et al, al, (2010). (2010). In control the paper paper by Saberi Saberi and at at by al, of the proposed approaches. Vesel´ y et al, (2010). the paper by Saberi In and at al, (1996) the design linear with satuVesel´ y deals et al, with (2010). the of paper bysystems Saberi and al, 2. (1996) deals with the In design of linear systems with at satu(1996) deals with the design of linear systems with satu2. PROBLEM PROBLEM FORMULATION FORMULATION AND AND rating actuators, where the actuator limitations have to be (1996) deals with the design of linear systems with satu2. PROBLEM FORMULATION PRELIMINARIES rating actuators, where the actuator limitations have to be 2. PROBLEM FORMULATION AND AND rating actuators, where the actuator limitations have to be PRELIMINARIES incorporated into control design.The semiglobal approach rating actuators, thedesign.The actuator limitations to be PRELIMINARIES incorporated intowhere control semiglobalhave approach PRELIMINARIES incorporated into control design.The semiglobal approach includes input additive disturbance incorporated into control design.The semiglobal rejection, approach This paper is concerned with the class of uncertain linear includes stabilization, stabilization, input additive disturbance disturbance rejection, includes stabilization, input additive rejection, and robust stabilization. In the book by Tarboriech paper is concerned with the class of uncertain linear includes stabilization, input additive disturbance rejection, and robust robust stabilization. stabilization. In In the the book book by by Tarboriech Tarboriech et et This This is concerned with class systems that be described as This paper paper is can concerned with the the class of of uncertain uncertain linear linear and et al, (2011) methods are proposed and algorithms designed systems that can be described as and robust stabilization. In the book by Tarboriech et systems that can be described as al, (2011) methods are proposed and algorithms designed p p systems that can be described as al, (2011) methods are proposed and algorithms designed p p on the basis of the state space approach to overcome the al, methods and algorithms designed on (2011) the basis basis of the the are stateproposed space approach approach to overcome overcome the p p x p Ai θi )x + (B0 + p Bi θi )u on the of state space to the effects of actuator ensuring the global stability x˙˙ = = (A (A00 + + Ai θi )x + (B0 + Bi θi )u on the basis of thesaturation state space approach to overcome the (1) effects of actuator saturation ensuring the global stability x ˙ = (A + A θ )x + (B + B i i 0 0 i=1 i=1 effects of actuator saturation ensuring the global stability (1) x ˙ = (A + A θ )x + (B + Bii θθii )u )u of closed loop systems. Book by Tarboriech at al. (2007) is i i 0 0 effects of actuator saturation ensuring the global stability i=1 i=1 (1) of closed loop systems. Book by Tarboriech at al. (2007) is (1) i=1 i=1 of closed loop systems. Book by Tarboriech at al. (2007) is y = Cx devoted to these directions: first, anti-windup strategies, i=1 i=1 of closed to loop systems. Book by Tarboriech at al.strategies, (2007) is y = Cx devoted these directions: first, anti-windup y = Cx devoted to these these directions: directions: first, first, anti-windup strategies, adevoted with and n m y =RCx strategies, , y ∈ Rll are the state, control a second secondtomodel model predictive predictive control controlanti-windup with constraints constraints and where u ∈ ∈ R Rm where x x ∈ ∈ Rnnn ,, u second predictive m , y ∈ Rl are the state, control model control with constraints and finally development of a stability and stabilization method aafinally second model predictive control with constraints and m , u ∈ R ∈ state, where x ∈ R input A development of of aa stability stability and and stabilization stabilization method method where R ,, yy respectively; ∈ R Rl are are the thematrices state, control control x ∈controlled R , u ∈ output, input and and controlled output, respectively; matrices Aii ,, B Bii finally development for constrained systems. finally development of a stability and stabilization method input and controlled output, respectively; matrices A i ,, B i = 0, 1, . . . p are constants with appropriate dimensions for constrained systems. input and controlled output, respectively; matrices A Bii i i = 0, 1, . . . p are constants with appropriate dimensions for constrained systems. for constrained systems. i = 0, 1, . . . p are constants with appropriate dimensions ; θ = [θ , ...θ ] is a vector of uncertain and possible ⋆ p ] is 1,[θ. 11. ,. ...θ p are constants with appropriateand dimensions work has been supported by the Slovak Scientific Grant ;i; = θθ 0, = of possible ⋆ The p ] is a has been supported by the Slovak Scientific Grant = [θ1 ,, ...θ ...θ a vector vector satisfying of uncertain uncertain and possible ⋆ The work p Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters θ ∈ �θ θ�] ∈ Ω, work has been supported by the Slovak Scientific Grant ; θ = [θ ] is a vector of uncertain and possible ⋆ The 1 p The work has been supported by the Slovak Scientific Grant Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters satisfying θ ∈ �θ θ�] ∈ Ω, Ω, θ�] ∈ Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters satisfying θ ∈ �θ Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters satisfying θ ∈ �θ θ�] ∈ Ω,
Copyright 2015 198 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015,IFAC IFAC (International Federation of Automatic Control) Copyright ©©2015 IFAC 198 Copyright © 198 Peer review underIFAC responsibility of International Federation of Automatic Copyright © 2015 2015 IFAC 198 Control. 10.1016/j.ifacol.2015.09.457
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˙ ∈ Ωt which belong to the known boundaries. θ˙ ∈ �θ˙ θ� There are two possibilities for θ parameter changes: • parameter θ is unknown but time varying, • parameter θ is unknown and constant. For the first case the modified affine quadratic stability will be adapted to obtain the closed loop stability conditions with input constraints and for the second case the modified parameter dependent quadratic stability will be used. For the class of uncertain systems (1) the following problem is studied in this paper . Problem 1. Design a robust static output feedback controller with control algorithm u = F (θ0 )y = F (θ0 )Cx (2) with hard constraints |uj | ≤ uMj , j = 1, 2, ...m such that for the closed-loop system (3) the designed controller guarantees robust affine or parameter dependent quadratic stability and guaranteed cost with respect to the performance index (4) x˙ = (A(θ) + B(θ)F (θ0 )C)x = Ac x (3) where F (θ0 ) = F θ0 , θ0 = diag{θ0j }, j = 1, 2, ...m, [θ0 ∈ �θ0 θ0 �, θ˙0 ∈ �θ˙0 θ˙0 �] ∈ Φ is known time varying parameter which serve for ensure the hard input constraints. To assess the performance quality, the following performance index is associated with system (1) ∞ ∞ J= J(t)dt (4) (xT Qx + uT Ru + x˙T S x)dt ˙ = 0
0
The respective notion of guaranteed cost is given in the next definition. Definition 1. Consider system (1) and controller (2). If there exist a control law u∗ and a positive scalar J ∗ such that the respective closed-loop system (3) is stable and the value of the closed-loop cost function (4) satisfies J ≤ J ∗ , then J ∗ is said to be the guaranteed cost and u∗ is said to be the guaranteed cost control law for system (1). Recall the well known result from LQ theory which will be used below to prove one of the main results. Lemma 1. (Kuncevic and Lycak , 1977) Consider the system (1) with control algorithm (2). Control algorithm (2) is the guaranteed cost control law for the closed-loop system (3) if and only if there exists a Lyapunov function V (θ) such that the following condition holds dV (θ) + J(t)} ≤ 0 (5) Be (θ) = min{ u dt for all θ ∈ Ω, θ˙ ∈ Ωt and θ0 , θ˙0 Uncertain system (1) with control algorithm (2) conforming to Lemma 1 is called robust stable with guaranteed cost. Note that for concrete structure of V (θ) ”if and only if” may to be decreased to ”if”. 3. MAIN RESULTS 3.1 Main results. Affine Quadratic Stability This subsection formulates the theoretical approach to the robust static output feedback controller design for uncertain system (1) which ensures closed-loop system affine 199
199
quadratic-stability and guaranteed cost (4), for all uncertain plant parameters Π ∈ Ω and hard input constraints. Definition 2. (Gahinet et al., 1996) The linear time varying system (3) is affine quadratically stable if there exists p + 2 symmetric matrices P, P0 , P1 , ...Pp such that p Pi θi > 0 (6) P (θ) = P + P0 θ0 + i=1
p
dV (θ) = xT (ATc P (θ) + P (θ)Ac + P0 θ˙0 + Pi θ˙i )x ≤ 0 dt i=1 (7) for all θ ∈ Ω, θ˙ ∈ Ωt , (θ0 , θ˙0 ) ∈ Φ where for the Lyapunov function of closed loop system holds V (θ) = xT P (θ)x Equation (7) can be rewritten as follows dV (θ) T T 0 P (θ) x˙ = x˙ x (8) ˙ x P (θ) P ( θ) dt ˙ = P0 θ˙0 + p Pi θ˙i P (θ) i=1 To isolate two matrices P (θ) and Ac using auxiliary matrices N1 , N2 ∈ Rn×n from (3) we obtain (2N1 x˙ + 2N2 x)T (x˙ − Ac x) = 0 or
−N1T Ac + N2 N1T + N1 T T −Ac N1 + N2 −N2T Ac − ATc N2
x˙ =0 x (9) Substitute control algorithm (2) to performance (4) T T S 0 x˙ (10) J(t) = x˙ x x 0 Q + C T F T RF Cθ02 x˙ T xT
Substituting (8), (9) and (10) to (5) one obtains T ≤0 (11) Be = x˙ T xT W x˙ T xT where p W = W01 + W02 θ0 + W03 θ02 + (Wi1 + Wi2 θ0 )θi (12) W01 =
i=1 + N1 + S −N1T A0 + N2 + P T ∗ −N2 A0 − AT0 N2 + Q + 0 −N1T B0 F C + P0 W02 = ∗ −N2T B0 F C − (B0 F C)T N2
N1T
W03 =
0 0 0 C T F T RF C
˙ P (θ)
0 −N1T Ai + Pi Wi1 = Pi − ATi N1 −N2T Ai − ATi N2 0 −N1T Bi F C Wi2 = −(Bi F C)T N1 −N2T Bi F C − (Bi F C)T N2 i = 1, 2, ...N The first main results for design of robust controller which ensure the Affine Quadratic Stability with input constraints are summarized in the next theorem Theorem 1. Consider the uncertain systems governed by (1) with robust control algorithm (2) and hard input constraints |uj | ≤ uMj , j = 1, 2, ...m. Closed loop system (3) is robust affine quadratically stable with guaranteed cost if there exist auxiliary matrices N1 , N2 , p + 2 symmetric
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matrices P, P0 , P1 , ...Pp and output feedback gain matrix F such that inequality W ≤0 (13) holds for all θ ∈ Ω, θ˙ ∈ Ωt and (θ0 , θ˙0 ) ∈ Φ. Variable θ0 will guarantee the hard input constraints |uj | ≤ uMj , j = 1, 2, ...m if it is calculated as � 1 if |uj | < uMj , for all j = 1, 2, ...m uMj θ0cj = if for some |uj | > uMj , j = 1, 2, ...m maxj |uj | (14) To ensure that rate of θ0 change within θ˙0 ∈ �θ˙0 , θ˙0 � and to obtain the real value of θ0 it is recommended to use a θ . first order filter with transfer function θ0j = ˙ 0cj −1 (θ0j )
s+1
Proof 1. The proof of sufficient conditions are based on equations (6),(8), (9) and (10). Because matrix W03 ≥ 0, matrix W with respect to θ0 , θi , i = 1, 2, ...p is convex, therefore matrix W is negative definite (semidefinite) if and only if it takes negative definite (semidefinite) at the corners of θ0 and θ. For robust stability analysis (13) reduces to LMI, for robust control synthesis we obtain BMI. 3.2 Main results. Parameter Dependent Quadratic Stability This subsection formulates the theoretical approach to the robust static output feedback controller design for system (1) which ensures closed loop system parameter dependent quadratic stability and guaranteed cost (4), for all uncertain plant parameters Π ∈ Ω and hard input constraints. When one substitute to equation (1) maximal and minimal value of θ a polytopic system model with N = 2p vertices is obtained in the form. {(A1v , B1v , C), (A2v , B2v , C)....(AN v , BN v , C)}, N = 2p (15) or N N � � {A(α), B(α)} = αi = 1, αi ≥ 0 (Aiv , Biv )αi , i=1
i=1
N �
α˙ i = 0
i=1
where αi ∈< 0 1 >, i = 1, 2, ...N is uncertain constant or time varying parameter. The parameter dependent Lyapunov function is given as follows N � V (α) = xT P (α)x, P (α) = P0 θ0 + Piv αi (16) i=1
From (16) and (8) one obtains the time derivative of the Lyapunov function for polytopic systems as follows dV (α) = dt N � 0 P0 θ0 + Piv αi � � x˙ � T T � i=1 x˙ x N N x � � ˙ Piv α˙ i Piv αi P0 θ0 + P0 θ0 + i=1
i=1
(17)
200
The closed-loop system can be obtained from (15) and (2) x˙ = (Aiv + Biv F θ0 C)x = Aci x i = 1, 2, ...N (18) Using the same procedure as for affine quadratic stability (9), (10) and (11) one obtains the closed loop system robust parameter-dependent quadratic stability conditions with guaranteed cost in the form � �T � � Bev = x˙ T xT Wv x˙ T xT ≤0 (19) where � T � N1 + N1 + S −N1T Ac (α) + N2 + P (α) Wv = ∗ ϕ(α, θ0 ) where ϕ(α, θ0 ) = −N2T Ac (α) − Ac (α)T N2 + Q + P0 θ˙0 + N �
Piv α˙ i + C T F T RF Cθ02
i=1
Inequality (19) is linear with respect to uncertain parameter αi , inequality (19) can be split to N inequalities as (20) Wi = Wiq + Wil θ0 + Wik θ02 ≤ 0, i = 1, 2, ...N where � � T N1 + N1 + S −N1T Aiv + N2 + Piv Wiq = ∗ ϕi � � T 0 −N1 Biv F C + P0 Wil = P0 − (Biv F C)T N1 −N2T Biv F C − (Biv F C)T N2 � � 0 0 Wik = 0 C T F T RF C ϕi = −N2T Aiv − ATiv N2 + Q + P θ˙0 +
N �
Piv α˙ i
i=1
Robust stability conditions are summarized in the following theorem. Theorem 2. Consider the uncertain linear system (1) or (15). The closed loop system (18) with control algorithm (2) is parameter dependent quadratically stable with input constraints |uj | ≤ uMj , j = 1, 2, ...m if there exist matrices N1 , N2 ∈ Rn×n , N + 1 symmetric matrices P0 , P1v , ...PN v � �N such that P0 θ0 + N i=1 Piv αi ≥ 0, i=1 αi = 1, αi ≥ 0, ˙ ˙ ˙ θ0 ∈ �θ0 θ0 �, θ0 ∈ �θ0 θ0 �, symmetric positive definite matrices Q, R, S and output feedback gain matrix F such that Wi = Wiq + Wil θ0 + Wik θ02 ≤ 0 (21) for i = 1, 2, ...N , θ0 = θ0 and θ0 = θ 0 . Variable θ0 calculated as in (14). Because matrix Wik ≥ 0, matrix Wi with respect to θ0 is convex, therefore matrix Wi is negative definite (semidefinite) if and only if it takes negative definite (semidefinite) at the corners of θ0 and N vertices of polytopic system. For robust stability analysis (21) reduces to LMI, for robust control synthesis we obtain BMI. Proof 2. The proof of Theorem sufficient conditions are based on the equations (10),(15),(17) and (18). 4. EXAMPLES In order to prove applicability of the proposed robust controller design procedure with input constraints, two examples are presented. The first example is simulation
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of randomly generated system near to the boundary of stability and the second one is slightly modified the first example to making the system unstable. Two control algorithms are calculated for each of the two examples: for the first algorithm the affine quadratic stability approach is used and for the second one the parameter dependent quadratic stability. In the both cases the problem is to design two robust decentralized PI controllers which ensure the affine (parameter dependent) quadratic stability and guaranteed cost with input constraints. A level of hard input constraints is equal to 200 percent value of nominal input variable. The rate of θ0 changes need to be more than 3. The states of third order system due to PI controller design need to be increased to five states. Example 1. Parameters of randomly generated example (1) are given as follows: 0.1 1 −1.1 1 .5 0 0 1 .1 0.4 −0.435 0.2 0 0 .25 −1.85 0 0 B0 = 0 .1 A0 = 0.3 1 0 0 0 0 0 0 0 0 0 0 1 0 0 with eigenvalues Eig(A0 ) = 0; 0; 0.0301; −2.0152; −1.4. 0.02 0.2 −0.2 0 0.2 0 0 0.05 0.03 0.028 0 0.02 0 0 A1 = 0.03 0.075 −0.05 0 0 B1 = 0.025 0.01 0 0 0 0 0 0 0 0 0 0 0 0 00 0.1 0.05 0.07 0.1 0 0.03 A2 = 0.025 0.1 −0.1 0 0 0 0 0 0 and output matrix C
0 0 0 0 0
0 0 0 0 0
0.035 0.03 0.01 0.2 B2 = 0.02 0.05 0 0 0 0
Fig. 1. Simulations results w(t), y(t) without input constraints.
Fig. 2. Controller output u(t) without input constraints.
Fig. 3. Simulations results w(t), y(t) with input constraints.
1 0 00 0 0 0 1 0 0 C = 0 0 0 1 0 0 0 00 1 For the parameters: performance Q = qI, q = 0.01, R = rIr , r = 1, S = sI, s = 0.001; rates of uncertain parameters changes θ˙i = 0, α˙ i = 0; input constraints θ0 ∈ �.5, 1�, θ˙0 ∈ �−8, 8�; Lyapunov matrices constraints P ∈ �0.1 ∗ I, ro ∗ I�, ro = 1000; uncertainties θi ∈ �−1, 1�, i = 1, 2 the following two robust PI controllers are obtained AQS R11 (s) = −6.0907 −
201
Fig. 4. Controller output u(t) with input constraints.
1.2702 0.9607 ; R12 (s) = −1.4507 + s s
PDQS 1.1603 1.3597 ; R12 (s) = −1.7537 + s s For simulation we have used the nominal plant model and robust controller designed with AQS. Simulation results (Fig. 1-5) confirm, that Theorem 1 holds. Fig. 1 and 2 show the simulation results without input constraints and Figs. 3-5 show simulation results with input constraints |u1 | ≤ 4 and |u2 | ≤ 6 which proofs that Theorem 1 holds and guarantees hard input constraints. Fig. 5 shows the calculated parameters θ01 and θ02 . Example 2. To obtain the system parameters for the second example we change the following entries A0 (2, 2) = R11 (s) = −5.0674 −
201
Fig. 5. Calculated parameters θ(t) −0.135, A0(3, 3) = −0.85 in matrix A0 (first example). The obtained eigenvalues of matrix A0 for second example are Eig(A0 ) = 0; 0; 0.3023; −0.8698; ; −1.5175. All other system parameters are the same as for the first example, except for θ0 rates constraints θ˙0 ∈ �−3, 3�. Parameters of
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two designed robust controllers are as follows AQS R21 (s) = −5.1684 − PDQS
0.5782 0.6987 ; R22 (s) = −0.6081 + s s
0.5175 0.6809 ; R22 (s) = −0.7636 + s s For simulation we have used the plant model with θ1 = 1, θ2 = −1 and the robust controller designed with PDQS. Simulation results (Figs. 6-10) confirm, that Theorem 2 holds. Figs. 6 and 7 show the simulation results without input constraints and Figs. 8-10 show simulation results with input constraints |u1 | ≤ 2 and |u2 | ≤ 2 which proofs that Theorem 2 holds and guarantees hard input constraints. Fig. 10 shows the calculated scheduled parameters θ1 and θ2 . R21 (s) = −4.587 −
Fig. 9. Controller output u(t) with input constraints.
Fig. 10. Calculated scheduled parameters θ(t)
Fig. 6. Simulations results w(t), y(t) without input constraints.
and tested by simulations. The proposed robust controller design approach with parameter dependent/affine Lyapunov function consider quick parameter changes to ensure the hard input constraints. Simulation results prove the potential ability of the designed closed-loop to withstand also these changes. The obtained robust controller design procedures are in the form of BMI and LMI approaches. The proposed approach contributes to the design tools for robust controllers which guarantee the affine/parameter dependent quadratic stability, guaranteed performance and hard input constraints. 6. ACKNOWLEDGEMENT The work has been supported by Grant 1/1241/12 and 1/2256/12 of Slovak Grant Agency
Fig. 7. Controller output u(t) without input constraints.
REFERENCES
Fig. 8. Simulations results w(t), y(t) with input constraints. 5. CONCLUSION A novel robust controller design approach with hard input constraints has been proposed. The obtained results, illustrated on the two examples, show the applicability of the designed robust controller and its ability to cope with model uncertainties. Several forms of parameter dependent/affine quadratic Lyapunov functions are presented 202
Adegbege A.A. and Heath W.P. (2011) Stability Conditions for Constrained Two-Stage Internal Model Control In: 49th IEEE Conference on Decision and Control (CDC), Atlanta, 15-17 December 2010, pp. 5251-5256. Bernstein D.S. and Michel A.V. (1995). A Chronological bibliography on saturating actuators. Int. Journal of Robust and Nonlinear Control, Vol, 375-380. Desoer Ch. A. and Vidyasagar M. (2009) Feedback Systems. Input-Output Properties Siam, Philadelphia Gahinet, P., Apkarian, P., and Chilali, M. (1996). Affine parameter-dependent Lyapunov functions and real parametric uncertainty. IEEE Transactions on Automatic Control, 41(3), 436-442. Heemels V.P.M. and Camlibel M.K. (2007). Controllability of linear systemsw with input constraints. In Proceedings of 46th IEEE CD-ROM, 536-541. Henrion D, Gracia G., and Tarbouriech S. (1998) . Piecewise linear Robust Control of Systems with Input Constraints. European Journal of Control, 157-166 Kapila V., Grigariadis K.M. (2002). Actuator saturation control. Contr. Eng. Series, Marcel Dekter.
IFAC ROCOND 2015 July 8-11, 2015. Bratislava, Slovak Republic
Vojtech Veselý et al. / IFAC-PapersOnLine 48-14 (2015) 198–203
Kozakova A. and Vesely V. (2009). Design of robust decentralized controllers using M − ∆ structure robust stability conditions. IJSS, Vol40,5, 497-505 Kuncevic V.M. and Lycak M.M. (1977) Control systems design using Lyapunov function approach Nauka, Moskau, 1977, in russian. Reinelt W. and Canale M. (2001). Robust control of SISO systems subject to hard input constraints. http://www.control.isy.lin.se,Report N LiTHISY-R-2349. Saberi A., Lin Z., and Teel A.R. (1996). Control of linear systems with saturating actuators. IEEE Tans on AC V41, N3 368-375. Skogestad S. and Postlethwaite I. (2005) Multivariable Fedback Control. Analysis and Design John Wiley ans
203
203
sons Tarboriech S., Garcia G., Jo o Manoel Gomes da Silva Jr, and Queinnec I. (2011). Stability and Stabilization of linear systems with Saturating Actuators. Springler, 430page. Tarboriech S., Garcia G.,and Glattfelder A.H. (Eds.) (2007). Advanced Strategies in Control Systems with Input and Output Constraints. Springler/Verlag Berlin Heidelberg 2007, 456page. Vesely V., Rosinova D., and Foltin M. (2010). Robust model predictive control design with input constraints. ISA Trans. 49 , 114-120. Zhou K., Doyle J.C. and Glover K. (1996) Robust and optimal control Prentice Hall, New Jersey
ScienceDirect IFAC-PapersOnLine 48-14 (2015) 198–203
Robust Controller Design Robust Controller Design Robust Controller Design Robust Controller Design Constraints. Time Domain Constraints. Time Domain Constraints. Time Domain Constraints. Time Domain
with Input with Input with Input with Input⋆ ⋆⋆ Approach. Approach. Approach. Approach. ⋆
∗ ∗ Vojtech y Vojtech Vesel´ Vesel´ y ∗∗∗ ,, Adrian Adrian Ilka Ilka ∗∗∗ Vojtech Vesel´ y , Adrian Ilka Vojtech Vesel´ y , Adrian Ilka ∗ ∗ Faculty of Electrical Engineering and Information Technology, Slovak ∗ Faculty of Electrical Engineering and Information Technology, Slovak ∗ Faculty ofofElectrical Engineering and Technology, Slovak University Technology in Ilkovicova 812 Faculty ofofElectrical Engineering and Information Information Slovak University Technology in Bratislava, Bratislava, Ilkovicova 3, 3, Technology, 812 19 19 Bratislava, Bratislava, University of Technology in Bratislava, Ilkovicova 3, 812 19 Bratislava, (e-mail: [email protected], [email protected]) University of Technology in Bratislava, Ilkovicova 3, 812 19 Bratislava, (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected], [email protected]) [email protected]) (e-mail: Abstract: A novel approach to robust controller design with hard input constraints is presented. Abstract: A A novel novel approach approach to to robust robust controller controller design design with with hard hard input input constraints constraints is is presented. presented. Abstract: The proposed design procedure is based on the robust stability condition developed using Abstract: A novel approach to robust controller design with hard input constraints is presented. The proposed design procedure is based on the robust stability condition developed using Affine Affine The proposed is on robust condition developed using Affine or Quadratic Stability approach. The feasible design procedures The proposed design design procedure procedure is based based on the the robust stability stability condition developed Affine or Parameter-Dependent Parameter-Dependent Quadratic Stability approach. The obtained obtained feasible designusing procedures or Parameter-Dependent Quadratic Stability approach. The feasible design procedures are in LMI. obtained design and properties are illustrated or The obtained obtained design areParameter-Dependent in the the form form of of BMI BMI or orQuadratic LMI. The TheStability obtainedapproach. design results results and their theirfeasible properties are procedures illustrated are in the form of BMI or LMI. The obtained design results and their properties are illustrated in simulation examples. are in the form of BMI or LMI. The obtained design results and their properties are illustrated in simulation simulation examples. in examples. in simulation examples. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2015, IFAC (International Keywords: Robust Lyapunov Keywords: Robust controller, controller, Affine, Affine, Parameter-Dependent Parameter-Dependent Lyapunov Lyapunov function, function, Hard Hard input input Keywords: Robust controller, Affine, Parameter-Dependent function, Hard input constraints Keywords: constraints Robust controller, Affine, Parameter-Dependent Lyapunov function, Hard input constraints constraints 1. The 1. INTRODUCTION INTRODUCTION The above above short short survey survey implies implies that that in in the the field field of of robust robust 1. INTRODUCTION The above short survey implies that in the field of controller design with input constraints there are many 1. INTRODUCTION The above short survey implies that in the field of robust robust controller design with input constraints there are many controller design with input constraints there are many different approaches. The above observation motivated us All control actuation devices are subject to magnitude controller design with input constraints there are many different approaches. The above observation motivated us All control actuation devices are subject to magnitude different approaches. The above observation motivated us All control actuation devices are subject to magnitude to study the following research problem which has not been and/or rate limits and this leads to degradation of the different approaches. The above observation motivated us All control actuation devices are to subject to magnitude to study the following research problem which has not been and/or rate limits and this leads degradation of the to study the following research problem which has not been and/or rate limits and this leads to degradation sufficiently solved yet. Design the robust controller with of the performance and even instability of closed loop control to study the following research problem which has not been and/or rate limits and instability this leads of to closed degradation of the sufficiently solved yet. Design the robust controller with performance and even loop control sufficiently solved Design the robust performance and even instability of closed closed loop control hard systems. input constraints belong to very imporsolved yet. yet. which Designwill theguarantee: robust controller controller with with performance even instability of hard input input constraints constraints which will guarantee: systems. Hard Hardand input constraints belong to the theloop very control impor- sufficiently hard input constraints which will guarantee: systems. Hard input constraints belong to the very important task in the controller design. A chronological biblioghard input constraints which will guarantee: systems. Hard input constraints belong to the very impor•• stability tant task task in in the the controller controller design. A A chronological chronological bibliogtant stability and and robustness robustness properties properties of of the the closed closed loop loop design. bibliography on saturating actuators can found in Bernstein tant task in the controller design. Abe chronological bibliog•• system stability and robustness properties of the closed loop when the uncertain plant parameters Π ∈ raphy on saturating actuators can be found in Bernstein stability and robustness properties of the closed loop raphy on saturating actuators can be found in Bernstein system when the uncertain plant parameters Π ∈Ω Ω and Michel, (1995) and Kapila and Grigoriadis , (2002). raphy on saturating actuators can beGrigoriadis found in Bernstein system when the uncertain plant parameters Π ∈ lie in the given polytopic (convex) uncertainty box; and Michel, (1995) and Kapila and , (2002). system when uncertain plant parameters ∈Ω Ω and Michel, Michel,and (1995) and Kapila Kapila and for Grigoriadis , (2002). (2002). lie in the giventhe polytopic (convex) uncertaintyΠbox; Necessary sufficient conditions controllability of and (1995) and and Grigoriadis , polytopic (convex) uncertainty box; •• lie guaranteed cost (performance) for the closed loop Necessary and and sufficient sufficient conditions conditions for for controllability controllability of of lie in in the the given given polytopic (convex) uncertainty box; Necessary guaranteed cost (performance) for the closed loop linear systems subject to input/state constraints are given Necessary andsubject sufficient conditions for controllability of •• guaranteed cost (performance) for linear systems systems to input/state input/state constraints are given given system Π Ω; guaranteed cost for the the closed closed loop loop linear subject to constraints are system for for all all Π∈ ∈(performance) Ω; in Heemels and Carmlibel, (2007). Robust stabilization linear systems subject to input/state constraints are given system for all Π ∈ Ω; • affine or Parameter-Dependent Quadratic Stability; in Heemels and Carmlibel, (2007). Robust stabilization system for all Π ∈ Ω; in uncertain Heemels and and Carmlibel, (2007). Robust stabilization • affine or Parameter-Dependent Quadratic Stability; of systems subject to constraints is in Heemels Carmlibel, stabilization •• affine or Quadratic BMI or LMI design procedure. of uncertain uncertain systems subject(2007). to input inputRobust constraints is conconaffine or Parameter-Dependent Parameter-Dependent Quadratic Stability; Stability; of •• the the BMI or LMI design procedure. systems subject to input constraints is considered in Henrion et al, (1998). A piecewise linear of uncertain systems subject to input constraints is conthe BMI or LMI design procedure. sidered in Henrion et al, (1998). A piecewise linear con•this thepaper, BMI or LMI design procedure. sidered in Henrion et al, (1998). A piecewise linear conIn we provide, to the authors’ best knowledge, trol ensuring the input constraints is parameterized by sidered in Henrion et al,constraints (1998). A piecewise linear conthis paper, we provide, to the authors’ best knowledge, trol ensuring ensuring the input input is parameterized parameterized by In trol the constraints is by In paper, provide, to authors’ best an alternative novel approach the robust controller algebraic Riccati equation. Youla parametrization of unIn this this paper, we we provide, to the the to authors’ best knowledge, knowledge, trol ensuring the input constraints is parameterized by an alternative novel approach to the robust controller algebraic Riccati equation. Youla parametrization of ununan alternative novel approach to the robust controller algebraic Riccati equation. Youla parametrization of design with hard input constraints. The proposed uncercertain plant and stabilizing controllers which guarantee an alternative novel approach to the robust controller algebraic Riccati equation. Youla parametrization of un- design with hard input constraints. The proposed uncercertain plant and stabilizing controllers which guarantee certain plant and stabilizing controllers which guarantee design with hard input constraints. The proposed uncertainty model, introduced in Section 2, is used to formulate prescribed hard bounds on the control signal can be found design with hard input constraints. The proposed uncercertain plant and stabilizing controllers which guarantee tainty model, introduced in Section 2, is used to formulate prescribed hard bounds on the control signal can be found prescribed hard bounds on the control signal can be found tainty model, introduced in Section 2, is used to formulate a robust control problem with hard input constraints. in Reinelt an Canale, (2001). The use of a two stage IMC tainty model, introduced in Section 2, is used to formulate prescribed hard bounds on the control signal can be found control problem with hard input constraints. in Reinelt Reinelt an an Canale, Canale, (2001). (2001). The The use use of of aa two two stage stage IMC IMC aa robust in control problem with hard input constraints. The main results are presented Section The design antiwindup design for stable plant input constraints is a robust robust control problem with in hard input3. constraints. in Reinelt an Canale, The and use of a two stage IMC main results are presented in 3. design antiwindup design for (2001). stable plant plant and input constraints is The The main results are presented in Section Sectionrobust 3. The Thestability design antiwindup design for stable and input constraints is procedure is based on the new developed described at Adegbege and Heath, (2011). The invariant The main results are presented in Section 3. The design antiwindup design for stable plant and input constraints is procedure is based on the new developed robust stability described at Adegbege and Heath, (2011). The invariant described at Adegbege and Heath, procedure is based on the new developed robust stability (2011). The invariant condition the input controller and set similar to Soft Variable is based on the new constraints developed robust stability described Adegbege approach and Heath, (2011). condition for for the hard hard input constraints controller and set and and an anatalgorithm algorithm approach similar to The Soft invariant Variable procedure set and an algorithm to Soft Variable condition for the hard input constraints controller and approach similar numerical examples in Section 4 illustrate the effectiveness Structure Control ensuring soft input constraints for the condition for the hard input constraints controller and set and an algorithm approach similar to Soft Variable numerical examples in Section 4 illustrate the effectiveness Structure Control ensuring soft input constraints for the Structure Control ensuring soft input constraints numerical examples in Section 4 illustrate the effectiveness for the of the proposed approaches. model predictive plant control systems is described by numerical examples in Section 4 illustrate the effectiveness Structure Control plant ensuring soft input constraints for the of the proposed approaches. model predictive control systems is described by of the proposed approaches. modely predictive predictive plant control systems is described described by Vesel´ In the by and al, model plant systems is Vesel´ y et et al, al, (2010). (2010). In control the paper paper by Saberi Saberi and at at by al, of the proposed approaches. Vesel´ y et al, (2010). the paper by Saberi In and at al, (1996) the design linear with satuVesel´ y deals et al, with (2010). the of paper bysystems Saberi and al, 2. (1996) deals with the In design of linear systems with at satu(1996) deals with the design of linear systems with satu2. PROBLEM PROBLEM FORMULATION FORMULATION AND AND rating actuators, where the actuator limitations have to be (1996) deals with the design of linear systems with satu2. PROBLEM FORMULATION PRELIMINARIES rating actuators, where the actuator limitations have to be 2. PROBLEM FORMULATION AND AND rating actuators, where the actuator limitations have to be PRELIMINARIES incorporated into control design.The semiglobal approach rating actuators, thedesign.The actuator limitations to be PRELIMINARIES incorporated intowhere control semiglobalhave approach PRELIMINARIES incorporated into control design.The semiglobal approach includes input additive disturbance incorporated into control design.The semiglobal rejection, approach This paper is concerned with the class of uncertain linear includes stabilization, stabilization, input additive disturbance disturbance rejection, includes stabilization, input additive rejection, and robust stabilization. In the book by Tarboriech paper is concerned with the class of uncertain linear includes stabilization, input additive disturbance rejection, and robust robust stabilization. stabilization. In In the the book book by by Tarboriech Tarboriech et et This This is concerned with class systems that be described as This paper paper is can concerned with the the class of of uncertain uncertain linear linear and et al, (2011) methods are proposed and algorithms designed systems that can be described as and robust stabilization. In the book by Tarboriech et systems that can be described as al, (2011) methods are proposed and algorithms designed p p systems that can be described as al, (2011) methods are proposed and algorithms designed p p on the basis of the state space approach to overcome the al, methods and algorithms designed on (2011) the basis basis of the the are stateproposed space approach approach to overcome overcome the p p x p Ai θi )x + (B0 + p Bi θi )u on the of state space to the effects of actuator ensuring the global stability x˙˙ = = (A (A00 + + Ai θi )x + (B0 + Bi θi )u on the basis of thesaturation state space approach to overcome the (1) effects of actuator saturation ensuring the global stability x ˙ = (A + A θ )x + (B + B i i 0 0 i=1 i=1 effects of actuator saturation ensuring the global stability (1) x ˙ = (A + A θ )x + (B + Bii θθii )u )u of closed loop systems. Book by Tarboriech at al. (2007) is i i 0 0 effects of actuator saturation ensuring the global stability i=1 i=1 (1) of closed loop systems. Book by Tarboriech at al. (2007) is (1) i=1 i=1 of closed loop systems. Book by Tarboriech at al. (2007) is y = Cx devoted to these directions: first, anti-windup strategies, i=1 i=1 of closed to loop systems. Book by Tarboriech at al.strategies, (2007) is y = Cx devoted these directions: first, anti-windup y = Cx devoted to these these directions: directions: first, first, anti-windup strategies, adevoted with and n m y =RCx strategies, , y ∈ Rll are the state, control a second secondtomodel model predictive predictive control controlanti-windup with constraints constraints and where u ∈ ∈ R Rm where x x ∈ ∈ Rnnn ,, u second predictive m , y ∈ Rl are the state, control model control with constraints and finally development of a stability and stabilization method aafinally second model predictive control with constraints and m , u ∈ R ∈ state, where x ∈ R input A development of of aa stability stability and and stabilization stabilization method method where R ,, yy respectively; ∈ R Rl are are the thematrices state, control control x ∈controlled R , u ∈ output, input and and controlled output, respectively; matrices Aii ,, B Bii finally development for constrained systems. finally development of a stability and stabilization method input and controlled output, respectively; matrices A i ,, B i = 0, 1, . . . p are constants with appropriate dimensions for constrained systems. input and controlled output, respectively; matrices A Bii i i = 0, 1, . . . p are constants with appropriate dimensions for constrained systems. for constrained systems. i = 0, 1, . . . p are constants with appropriate dimensions ; θ = [θ , ...θ ] is a vector of uncertain and possible ⋆ p ] is 1,[θ. 11. ,. ...θ p are constants with appropriateand dimensions work has been supported by the Slovak Scientific Grant ;i; = θθ 0, = of possible ⋆ The p ] is a has been supported by the Slovak Scientific Grant = [θ1 ,, ...θ ...θ a vector vector satisfying of uncertain uncertain and possible ⋆ The work p Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters θ ∈ �θ θ�] ∈ Ω, work has been supported by the Slovak Scientific Grant ; θ = [θ ] is a vector of uncertain and possible ⋆ The 1 p The work has been supported by the Slovak Scientific Grant Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters satisfying θ ∈ �θ θ�] ∈ Ω, Ω, θ�] ∈ Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters satisfying θ ∈ �θ Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 time varying real parameters satisfying θ ∈ �θ θ�] ∈ Ω,
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˙ ∈ Ωt which belong to the known boundaries. θ˙ ∈ �θ˙ θ� There are two possibilities for θ parameter changes: • parameter θ is unknown but time varying, • parameter θ is unknown and constant. For the first case the modified affine quadratic stability will be adapted to obtain the closed loop stability conditions with input constraints and for the second case the modified parameter dependent quadratic stability will be used. For the class of uncertain systems (1) the following problem is studied in this paper . Problem 1. Design a robust static output feedback controller with control algorithm u = F (θ0 )y = F (θ0 )Cx (2) with hard constraints |uj | ≤ uMj , j = 1, 2, ...m such that for the closed-loop system (3) the designed controller guarantees robust affine or parameter dependent quadratic stability and guaranteed cost with respect to the performance index (4) x˙ = (A(θ) + B(θ)F (θ0 )C)x = Ac x (3) where F (θ0 ) = F θ0 , θ0 = diag{θ0j }, j = 1, 2, ...m, [θ0 ∈ �θ0 θ0 �, θ˙0 ∈ �θ˙0 θ˙0 �] ∈ Φ is known time varying parameter which serve for ensure the hard input constraints. To assess the performance quality, the following performance index is associated with system (1) ∞ ∞ J= J(t)dt (4) (xT Qx + uT Ru + x˙T S x)dt ˙ = 0
0
The respective notion of guaranteed cost is given in the next definition. Definition 1. Consider system (1) and controller (2). If there exist a control law u∗ and a positive scalar J ∗ such that the respective closed-loop system (3) is stable and the value of the closed-loop cost function (4) satisfies J ≤ J ∗ , then J ∗ is said to be the guaranteed cost and u∗ is said to be the guaranteed cost control law for system (1). Recall the well known result from LQ theory which will be used below to prove one of the main results. Lemma 1. (Kuncevic and Lycak , 1977) Consider the system (1) with control algorithm (2). Control algorithm (2) is the guaranteed cost control law for the closed-loop system (3) if and only if there exists a Lyapunov function V (θ) such that the following condition holds dV (θ) + J(t)} ≤ 0 (5) Be (θ) = min{ u dt for all θ ∈ Ω, θ˙ ∈ Ωt and θ0 , θ˙0 Uncertain system (1) with control algorithm (2) conforming to Lemma 1 is called robust stable with guaranteed cost. Note that for concrete structure of V (θ) ”if and only if” may to be decreased to ”if”. 3. MAIN RESULTS 3.1 Main results. Affine Quadratic Stability This subsection formulates the theoretical approach to the robust static output feedback controller design for uncertain system (1) which ensures closed-loop system affine 199
199
quadratic-stability and guaranteed cost (4), for all uncertain plant parameters Π ∈ Ω and hard input constraints. Definition 2. (Gahinet et al., 1996) The linear time varying system (3) is affine quadratically stable if there exists p + 2 symmetric matrices P, P0 , P1 , ...Pp such that p Pi θi > 0 (6) P (θ) = P + P0 θ0 + i=1
p
dV (θ) = xT (ATc P (θ) + P (θ)Ac + P0 θ˙0 + Pi θ˙i )x ≤ 0 dt i=1 (7) for all θ ∈ Ω, θ˙ ∈ Ωt , (θ0 , θ˙0 ) ∈ Φ where for the Lyapunov function of closed loop system holds V (θ) = xT P (θ)x Equation (7) can be rewritten as follows dV (θ) T T 0 P (θ) x˙ = x˙ x (8) ˙ x P (θ) P ( θ) dt ˙ = P0 θ˙0 + p Pi θ˙i P (θ) i=1 To isolate two matrices P (θ) and Ac using auxiliary matrices N1 , N2 ∈ Rn×n from (3) we obtain (2N1 x˙ + 2N2 x)T (x˙ − Ac x) = 0 or
−N1T Ac + N2 N1T + N1 T T −Ac N1 + N2 −N2T Ac − ATc N2
x˙ =0 x (9) Substitute control algorithm (2) to performance (4) T T S 0 x˙ (10) J(t) = x˙ x x 0 Q + C T F T RF Cθ02 x˙ T xT
Substituting (8), (9) and (10) to (5) one obtains T ≤0 (11) Be = x˙ T xT W x˙ T xT where p W = W01 + W02 θ0 + W03 θ02 + (Wi1 + Wi2 θ0 )θi (12) W01 =
i=1 + N1 + S −N1T A0 + N2 + P T ∗ −N2 A0 − AT0 N2 + Q + 0 −N1T B0 F C + P0 W02 = ∗ −N2T B0 F C − (B0 F C)T N2
N1T
W03 =
0 0 0 C T F T RF C
˙ P (θ)
0 −N1T Ai + Pi Wi1 = Pi − ATi N1 −N2T Ai − ATi N2 0 −N1T Bi F C Wi2 = −(Bi F C)T N1 −N2T Bi F C − (Bi F C)T N2 i = 1, 2, ...N The first main results for design of robust controller which ensure the Affine Quadratic Stability with input constraints are summarized in the next theorem Theorem 1. Consider the uncertain systems governed by (1) with robust control algorithm (2) and hard input constraints |uj | ≤ uMj , j = 1, 2, ...m. Closed loop system (3) is robust affine quadratically stable with guaranteed cost if there exist auxiliary matrices N1 , N2 , p + 2 symmetric
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matrices P, P0 , P1 , ...Pp and output feedback gain matrix F such that inequality W ≤0 (13) holds for all θ ∈ Ω, θ˙ ∈ Ωt and (θ0 , θ˙0 ) ∈ Φ. Variable θ0 will guarantee the hard input constraints |uj | ≤ uMj , j = 1, 2, ...m if it is calculated as � 1 if |uj | < uMj , for all j = 1, 2, ...m uMj θ0cj = if for some |uj | > uMj , j = 1, 2, ...m maxj |uj | (14) To ensure that rate of θ0 change within θ˙0 ∈ �θ˙0 , θ˙0 � and to obtain the real value of θ0 it is recommended to use a θ . first order filter with transfer function θ0j = ˙ 0cj −1 (θ0j )
s+1
Proof 1. The proof of sufficient conditions are based on equations (6),(8), (9) and (10). Because matrix W03 ≥ 0, matrix W with respect to θ0 , θi , i = 1, 2, ...p is convex, therefore matrix W is negative definite (semidefinite) if and only if it takes negative definite (semidefinite) at the corners of θ0 and θ. For robust stability analysis (13) reduces to LMI, for robust control synthesis we obtain BMI. 3.2 Main results. Parameter Dependent Quadratic Stability This subsection formulates the theoretical approach to the robust static output feedback controller design for system (1) which ensures closed loop system parameter dependent quadratic stability and guaranteed cost (4), for all uncertain plant parameters Π ∈ Ω and hard input constraints. When one substitute to equation (1) maximal and minimal value of θ a polytopic system model with N = 2p vertices is obtained in the form. {(A1v , B1v , C), (A2v , B2v , C)....(AN v , BN v , C)}, N = 2p (15) or N N � � {A(α), B(α)} = αi = 1, αi ≥ 0 (Aiv , Biv )αi , i=1
i=1
N �
α˙ i = 0
i=1
where αi ∈< 0 1 >, i = 1, 2, ...N is uncertain constant or time varying parameter. The parameter dependent Lyapunov function is given as follows N � V (α) = xT P (α)x, P (α) = P0 θ0 + Piv αi (16) i=1
From (16) and (8) one obtains the time derivative of the Lyapunov function for polytopic systems as follows dV (α) = dt N � 0 P0 θ0 + Piv αi � � x˙ � T T � i=1 x˙ x N N x � � ˙ Piv α˙ i Piv αi P0 θ0 + P0 θ0 + i=1
i=1
(17)
200
The closed-loop system can be obtained from (15) and (2) x˙ = (Aiv + Biv F θ0 C)x = Aci x i = 1, 2, ...N (18) Using the same procedure as for affine quadratic stability (9), (10) and (11) one obtains the closed loop system robust parameter-dependent quadratic stability conditions with guaranteed cost in the form � �T � � Bev = x˙ T xT Wv x˙ T xT ≤0 (19) where � T � N1 + N1 + S −N1T Ac (α) + N2 + P (α) Wv = ∗ ϕ(α, θ0 ) where ϕ(α, θ0 ) = −N2T Ac (α) − Ac (α)T N2 + Q + P0 θ˙0 + N �
Piv α˙ i + C T F T RF Cθ02
i=1
Inequality (19) is linear with respect to uncertain parameter αi , inequality (19) can be split to N inequalities as (20) Wi = Wiq + Wil θ0 + Wik θ02 ≤ 0, i = 1, 2, ...N where � � T N1 + N1 + S −N1T Aiv + N2 + Piv Wiq = ∗ ϕi � � T 0 −N1 Biv F C + P0 Wil = P0 − (Biv F C)T N1 −N2T Biv F C − (Biv F C)T N2 � � 0 0 Wik = 0 C T F T RF C ϕi = −N2T Aiv − ATiv N2 + Q + P θ˙0 +
N �
Piv α˙ i
i=1
Robust stability conditions are summarized in the following theorem. Theorem 2. Consider the uncertain linear system (1) or (15). The closed loop system (18) with control algorithm (2) is parameter dependent quadratically stable with input constraints |uj | ≤ uMj , j = 1, 2, ...m if there exist matrices N1 , N2 ∈ Rn×n , N + 1 symmetric matrices P0 , P1v , ...PN v � �N such that P0 θ0 + N i=1 Piv αi ≥ 0, i=1 αi = 1, αi ≥ 0, ˙ ˙ ˙ θ0 ∈ �θ0 θ0 �, θ0 ∈ �θ0 θ0 �, symmetric positive definite matrices Q, R, S and output feedback gain matrix F such that Wi = Wiq + Wil θ0 + Wik θ02 ≤ 0 (21) for i = 1, 2, ...N , θ0 = θ0 and θ0 = θ 0 . Variable θ0 calculated as in (14). Because matrix Wik ≥ 0, matrix Wi with respect to θ0 is convex, therefore matrix Wi is negative definite (semidefinite) if and only if it takes negative definite (semidefinite) at the corners of θ0 and N vertices of polytopic system. For robust stability analysis (21) reduces to LMI, for robust control synthesis we obtain BMI. Proof 2. The proof of Theorem sufficient conditions are based on the equations (10),(15),(17) and (18). 4. EXAMPLES In order to prove applicability of the proposed robust controller design procedure with input constraints, two examples are presented. The first example is simulation
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of randomly generated system near to the boundary of stability and the second one is slightly modified the first example to making the system unstable. Two control algorithms are calculated for each of the two examples: for the first algorithm the affine quadratic stability approach is used and for the second one the parameter dependent quadratic stability. In the both cases the problem is to design two robust decentralized PI controllers which ensure the affine (parameter dependent) quadratic stability and guaranteed cost with input constraints. A level of hard input constraints is equal to 200 percent value of nominal input variable. The rate of θ0 changes need to be more than 3. The states of third order system due to PI controller design need to be increased to five states. Example 1. Parameters of randomly generated example (1) are given as follows: 0.1 1 −1.1 1 .5 0 0 1 .1 0.4 −0.435 0.2 0 0 .25 −1.85 0 0 B0 = 0 .1 A0 = 0.3 1 0 0 0 0 0 0 0 0 0 0 1 0 0 with eigenvalues Eig(A0 ) = 0; 0; 0.0301; −2.0152; −1.4. 0.02 0.2 −0.2 0 0.2 0 0 0.05 0.03 0.028 0 0.02 0 0 A1 = 0.03 0.075 −0.05 0 0 B1 = 0.025 0.01 0 0 0 0 0 0 0 0 0 0 0 0 00 0.1 0.05 0.07 0.1 0 0.03 A2 = 0.025 0.1 −0.1 0 0 0 0 0 0 and output matrix C
0 0 0 0 0
0 0 0 0 0
0.035 0.03 0.01 0.2 B2 = 0.02 0.05 0 0 0 0
Fig. 1. Simulations results w(t), y(t) without input constraints.
Fig. 2. Controller output u(t) without input constraints.
Fig. 3. Simulations results w(t), y(t) with input constraints.
1 0 00 0 0 0 1 0 0 C = 0 0 0 1 0 0 0 00 1 For the parameters: performance Q = qI, q = 0.01, R = rIr , r = 1, S = sI, s = 0.001; rates of uncertain parameters changes θ˙i = 0, α˙ i = 0; input constraints θ0 ∈ �.5, 1�, θ˙0 ∈ �−8, 8�; Lyapunov matrices constraints P ∈ �0.1 ∗ I, ro ∗ I�, ro = 1000; uncertainties θi ∈ �−1, 1�, i = 1, 2 the following two robust PI controllers are obtained AQS R11 (s) = −6.0907 −
201
Fig. 4. Controller output u(t) with input constraints.
1.2702 0.9607 ; R12 (s) = −1.4507 + s s
PDQS 1.1603 1.3597 ; R12 (s) = −1.7537 + s s For simulation we have used the nominal plant model and robust controller designed with AQS. Simulation results (Fig. 1-5) confirm, that Theorem 1 holds. Fig. 1 and 2 show the simulation results without input constraints and Figs. 3-5 show simulation results with input constraints |u1 | ≤ 4 and |u2 | ≤ 6 which proofs that Theorem 1 holds and guarantees hard input constraints. Fig. 5 shows the calculated parameters θ01 and θ02 . Example 2. To obtain the system parameters for the second example we change the following entries A0 (2, 2) = R11 (s) = −5.0674 −
201
Fig. 5. Calculated parameters θ(t) −0.135, A0(3, 3) = −0.85 in matrix A0 (first example). The obtained eigenvalues of matrix A0 for second example are Eig(A0 ) = 0; 0; 0.3023; −0.8698; ; −1.5175. All other system parameters are the same as for the first example, except for θ0 rates constraints θ˙0 ∈ �−3, 3�. Parameters of
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two designed robust controllers are as follows AQS R21 (s) = −5.1684 − PDQS
0.5782 0.6987 ; R22 (s) = −0.6081 + s s
0.5175 0.6809 ; R22 (s) = −0.7636 + s s For simulation we have used the plant model with θ1 = 1, θ2 = −1 and the robust controller designed with PDQS. Simulation results (Figs. 6-10) confirm, that Theorem 2 holds. Figs. 6 and 7 show the simulation results without input constraints and Figs. 8-10 show simulation results with input constraints |u1 | ≤ 2 and |u2 | ≤ 2 which proofs that Theorem 2 holds and guarantees hard input constraints. Fig. 10 shows the calculated scheduled parameters θ1 and θ2 . R21 (s) = −4.587 −
Fig. 9. Controller output u(t) with input constraints.
Fig. 10. Calculated scheduled parameters θ(t)
Fig. 6. Simulations results w(t), y(t) without input constraints.
and tested by simulations. The proposed robust controller design approach with parameter dependent/affine Lyapunov function consider quick parameter changes to ensure the hard input constraints. Simulation results prove the potential ability of the designed closed-loop to withstand also these changes. The obtained robust controller design procedures are in the form of BMI and LMI approaches. The proposed approach contributes to the design tools for robust controllers which guarantee the affine/parameter dependent quadratic stability, guaranteed performance and hard input constraints. 6. ACKNOWLEDGEMENT The work has been supported by Grant 1/1241/12 and 1/2256/12 of Slovak Grant Agency
Fig. 7. Controller output u(t) without input constraints.
REFERENCES
Fig. 8. Simulations results w(t), y(t) with input constraints. 5. CONCLUSION A novel robust controller design approach with hard input constraints has been proposed. The obtained results, illustrated on the two examples, show the applicability of the designed robust controller and its ability to cope with model uncertainties. Several forms of parameter dependent/affine quadratic Lyapunov functions are presented 202
Adegbege A.A. and Heath W.P. (2011) Stability Conditions for Constrained Two-Stage Internal Model Control In: 49th IEEE Conference on Decision and Control (CDC), Atlanta, 15-17 December 2010, pp. 5251-5256. Bernstein D.S. and Michel A.V. (1995). A Chronological bibliography on saturating actuators. Int. Journal of Robust and Nonlinear Control, Vol, 375-380. Desoer Ch. A. and Vidyasagar M. (2009) Feedback Systems. Input-Output Properties Siam, Philadelphia Gahinet, P., Apkarian, P., and Chilali, M. (1996). Affine parameter-dependent Lyapunov functions and real parametric uncertainty. IEEE Transactions on Automatic Control, 41(3), 436-442. Heemels V.P.M. and Camlibel M.K. (2007). Controllability of linear systemsw with input constraints. In Proceedings of 46th IEEE CD-ROM, 536-541. Henrion D, Gracia G., and Tarbouriech S. (1998) . Piecewise linear Robust Control of Systems with Input Constraints. European Journal of Control, 157-166 Kapila V., Grigariadis K.M. (2002). Actuator saturation control. Contr. Eng. Series, Marcel Dekter.
IFAC ROCOND 2015 July 8-11, 2015. Bratislava, Slovak Republic
Vojtech Veselý et al. / IFAC-PapersOnLine 48-14 (2015) 198–203
Kozakova A. and Vesely V. (2009). Design of robust decentralized controllers using M − ∆ structure robust stability conditions. IJSS, Vol40,5, 497-505 Kuncevic V.M. and Lycak M.M. (1977) Control systems design using Lyapunov function approach Nauka, Moskau, 1977, in russian. Reinelt W. and Canale M. (2001). Robust control of SISO systems subject to hard input constraints. http://www.control.isy.lin.se,Report N LiTHISY-R-2349. Saberi A., Lin Z., and Teel A.R. (1996). Control of linear systems with saturating actuators. IEEE Tans on AC V41, N3 368-375. Skogestad S. and Postlethwaite I. (2005) Multivariable Fedback Control. Analysis and Design John Wiley ans
203
203
sons Tarboriech S., Garcia G., Jo o Manoel Gomes da Silva Jr, and Queinnec I. (2011). Stability and Stabilization of linear systems with Saturating Actuators. Springler, 430page. Tarboriech S., Garcia G.,and Glattfelder A.H. (Eds.) (2007). Advanced Strategies in Control Systems with Input and Output Constraints. Springler/Verlag Berlin Heidelberg 2007, 456page. Vesely V., Rosinova D., and Foltin M. (2010). Robust model predictive control design with input constraints. ISA Trans. 49 , 114-120. Zhou K., Doyle J.C. and Glover K. (1996) Robust and optimal control Prentice Hall, New Jersey