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Robust Switched Controller Design Robust Switched Controller Design Robust Switched Controller Design Robust Switched Controller Design Nonlinear Continuous Systems Nonlinear Continuous Systems Nonlinear Nonlinear Continuous Continuous Systems Systems

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Vojtech Vesel´ y ∗∗ , Adrian Ilka ∗∗ ∗ , Adrian Ilka ∗ Vojtech Vesel´ y Vojtech Vesel´ y Vojtech Vesel´ y ∗ ,, Adrian Adrian Ilka Ilka ∗ ∗ ∗ Faculty of Electrical Engineering and Information Technology, Slovak ∗ Faculty of Electrical Engineering and Information Technology, Slovak ∗ Faculty ofofElectrical Engineering and Slovak University Technology in Bratislava, Ilkovicova 3, Technology, 812 19 Bratislava, Faculty ofofElectrical Engineering and Information Information Technology, Slovak University Technology in Bratislava, Ilkovicova 3, 812 19 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: (e-mail: [email protected], [email protected], [email protected]) [email protected]) Abstract: A novel approach is presented to robust switched controller design for nonlinear Abstract: approach to switched design nonlinear Abstract: A A novel novel approach is presented to robust robust signal switched controller design for for approach. nonlinear continuous-time systems under is anpresented arbitrary switching usingcontroller the gain scheduling Abstract: A novel approach is presented to robust switched controller design for nonlinear continuous-time systems under an arbitrary switching signal using the gain scheduling approach. continuous-time systems under an arbitrary switching signal using the gain scheduling approach. The proposed design procedure isarbitrary based onswitching the robust multi parameter dependent quadratic continuous-time systems under an signal using the gain scheduling approach. The proposed design procedure based on the robust multi parameter dependent quadratic The proposed design procedure is based the multi parameter dependent quadratic stability condition. The obtained is controller design procedure for nonlinear The proposed design procedure isswitched based on on the robust robust multi parameter dependentcontinuousquadratic stability condition. The obtained switched controller design procedure for nonlinear continuousstability condition. The obtained switched controller design procedure for nonlinear continuoustime systems is in at bilinear matrix form (BMI). The properties of the obtained design are stability condition. The obtained switched controller design procedure for nonlinear continuoustime in matrix time systems systems is in at at bilinear bilinear matrix form form (BMI). (BMI). The The properties properties of of the the obtained obtained design design are are illustrated on is simulation examples. time systems is in at bilinear matrix form (BMI). The properties of the obtained design are illustrated on simulation examples. illustrated on simulation examples. illustrated on simulation examples. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Robust controller, Switched controller, Gain scheduled Controller, Continuous time Keywords: systems, Robust controller, controller, Switched controller, Gain Gain scheduled scheduled Controller, Controller, Continuous Continuous time time Keywords: Robust Nonlinear Lyapunov Switched function controller, Keywords: Robust controller, Nonlinear systems, systems, Lyapunov Switched function controller, Gain scheduled Controller, Continuous time Nonlinear Lyapunov function Nonlinear systems, Lyapunov function 1. INTRODUCTION scheduled parameters many researches have tackled the 1. INTRODUCTION INTRODUCTION scheduled parameters many researches have tackled the 1. scheduled parameters many researches have the design problem of the gain scheduled controllers for linear 1. INTRODUCTION scheduled parameters many researches have tackled tackled the design problem of the gain scheduled controllers for linear design problem of the gain scheduled controllers for linear time varying system (LPV) using linear matrix inequalities Switched systems play an important role in the past design problem of the gain scheduled controllers for linear varying system (LPV) using linear matrix inequalities SwitchedMotivation systems play play an important important rolesystems in the the past past time time varying system (LPV) using matrix inequalities Switched systems an role in and the Lyapunov approach decade. for studying switched time varying system (LPV)function using linear linear matrix (Montagner inequalities Switched systems play an important rolesystems in thestems past (LMI) (LMI) and the Lyapunov function approach (Montagner decade. Motivation for studying switched stems (LMI) and the Lyapunov function approach (Montagner decade. Motivation for studying switched systems stems and Peres , 2004),(Sato and Peaucelle , 2013),(Vesel´ y and from two facts (LMI) and the Lyapunov function approach (Montagner decade. Motivation for studying switched systems stems and Peres ,, 2004),(Sato and Peaucelle ,, 2013),(Vesel´ y and two facts facts and Peres 2004),(Sato and Peaucelle 2013),(Vesel´ y from two Ilka , 2013),(Adegas and Stoustrup , 2011). Reviews of the -from switched systems have numerous application in the conand Peres , 2004),(Sato and Peaucelle , 2013),(Vesel´ y and and from two facts Ilka ,, scheduled 2013),(Adegas and Stoustrup ,, 2011). Reviews of the - switched switched systems have numerous application in the conIlka 2013),(Adegas and Stoustrup 2011). Reviews of the -trol systems have numerous application in the congain controller design can be found in (Rugh of real systems plants, and Ilka , 2013),(Adegas and Stoustrup , 2011). Reviews of the -trol switched have numerous application in the congain scheduled controller design can be found in (Rugh of real plants, and gain scheduled controller design can be found in (Rugh trol real plants, and and Shamma, 2000), (Leith and Leithead, 2000). -trol in of real control, there are dynamical systems that cangain scheduled controller design can be found in (Rugh of real plants, and Shamma, (Leith Leithead, 2000). - in in be realstabilized control, there there arecontinuous dynamicalstatic systems that cancan- and and Shamma, 2000), (Leith and Leithead, 2000). -not real control, are dynamical systems that this paper 2000), a method is and proposed of robust by any output/state and Shamma, 2000), (Leith and Leithead, 2000). switched -not in be realstabilized control, there arecontinuous dynamicalstatic systems that can- In In this paper a method is proposed of robust by any output/state In this paper a method is proposed of robust switched not be stabilized by any continuous static output/state controller design for nonlinear continuous time switched feedback control law, but a stabilizing switching control In this paper a method is proposed of robust switched not be stabilized by any continuous static output/state controller design for nonlinear continuous time switched feedback control law, but a stabilizing switching control controller design for nonlinear continuous time switched feedback control law, but a stabilizing switching control systems using the gain scheduling approach. There are only scheme can be found. controller design for nonlinear continuous time switched feedback control law, but a stabilizing switching control systems using the gain scheduling approach. There are only scheme can be found. systems using the gain scheduling approach. There are scheme can be found. some results in the field of continuous time switched gain Therefore switched system stability and the controller desystems usinginthe gain scheduling approach. There are only only scheme canswitched be found. some results the field of continuous time switched gain Therefore system stability and the controller desome results in the field of continuous time switched gain Therefore switched system stability and the controller descheduled controller design for continuous time systems sign procedure are the most important issues, especially some results in the field of continuous time switched gain Therefore switched system stability and the controller de- scheduled controller design for continuous time systems sign procedure are the most important issues, especially scheduled controller design for continuous time systems sign procedure are the most important issues, especially (Hanifzadegan and Nagamune ,continuous 2013), (Olalla et al. , for nonlinear systems. Stability under arbitrary switching scheduled controller design for time systems sign procedure are the most important issues, especially (Hanifzadegan and Nagamune , 2013), (Olalla et al. , for nonlinear systems. Stability under arbitrary switching (Hanifzadegan 2013), et for nonlinear arbitrary switching et and al. , Nagamune 2012) and ,,stabilization of switched is guaranteed by the Stability existence under of a common (Hanifzadegan and Nagamune 2013), (Olalla (Olalla et al. al. ,, for nonlinear systems. systems. Stability under arbitrary Lyapunov switching 2012),(Horri 2012),(Horri et al. , 2012) and stabilization of switched is guaranteed by the existence of a common Lyapunov 2012),(Horritime et ,, 2012) and of switched is guaranteed the of Lyapunov linear systems (Geromel and function for allby subsystems, (El-Farra et al. , continuous et al. al. 2012) and stabilization stabilization of Colanery switched is guaranteed byswitching the existence existence of aa common common Lyapunov continuous time linear systems (Geromel and Colanery function for all all switching subsystems, (El-Farra et2013), al. ,, 2012),(Horri continuous time linear systems (Geromel and Colanery function for switching subsystems, (El-Farra et al. , 2006),(Allerhand and Shaked , 2011). Representative 2005), (Long and Zhao , 2011), (Sun and Wang , continuous time linear systems (Geromel and Colanery function for all switching subsystems, (El-Farra et al. , , 2006),(Allerhand and Shaked , 2011). Representative 2005), (Long (Long and Zhao Zhao , 2011), (Sun (Sun and Wang Wang 2013), and Shaked 2011). Representative 2005), and and ,,, 2013), two last references Colanery , 2006), (Lunze and Lamnabhi-Lagarrigue , 2009), (Shaker and ,,are2006),(Allerhand 2006),(Allerhand and (Geromel Shaked ,, and 2011). Representative 2005), (Long and Zhao ,, 2011), 2011), (Sun and Wang 2013), are two last references (Geromel and Colanery ,, 2006), (Lunze and Lamnabhi-Lagarrigue , 2009), (Shaker and are two last references (Geromel and Colanery (Lunze and Lamnabhi-Lagarrigue , 2009), (Shaker and (Allerhand and Shaked , 2011). In the first paper the How , 2010), ( Hai Lin and Antsaklis, , 2009), 2009). For switched are two last references (Geromel and Colanery , 2006), 2006), (Lunze and Lamnabhi-Lagarrigue (Shaker and (Allerhand and Shaked , 2011). In the first paper the How , 2010), ( Hai Lin and Antsaklis , 2009). For switched (Allerhand and Shaked , 2011). In the first paper the How , 2010), ( Hai Lin and Antsaklis , 2009). For switched authors introduce the notion of the dwell-time T (minlinear systems, finding the common Lyapunov function is d (Allerhand and Shaked , 2011). In dwell-time the first paper the How , 2010), ( Hai Lin and Antsaklis , 2009). For switched authors introduce the notion of the T (minlinear systems, finding the common Lyapunov function is d authors introduce the notion of the dwell-time T (minlinear systems, finding the common Lyapunov function is imal time interval between switching) into the switched d relatively easy but for nonlinear systems it is difficult. A authors introduce the notion of the dwell-time T (minlinear systems, finding the common Lyapunov function is d imal time interval between switching) into the switched relatively easy but but for for nonlinear nonlinear systems it is is difficult. A imal time interval between into the relatively easy systems it A controller In the stability consurvey of switched design for nonlinear systems time design intervalprocedure. between switching) switching) into analysis the switched switched relatively easy but controller for nonlinear systems it is difficult. difficult. A imal controller design procedure. In the stability analysis survey of switched controller design for nonlinear systems controller design procedure. In the stability analysis consurvey of switched controller design for nonlinear systems dition of switched systems the dwell-time is includedconto can be of found in (El-Farra et al. , 2005), (Long andsystems Zhao , controller design procedure. In the stability analysis consurvey switched controller design for nonlinear of switched systems the dwell-time is included toa can be be found found in Wang (El-Farra et al. al.(Aleksandrov , 2005), 2005), (Long (Long and Zhao , dition Ac T d dition of switched systems the dwell-time is included can in (El-Farra et , and Zhao , the term e (A -closed-loop matrix (4)). In such 2011),(Sun and , 2013), et al. , 2011) c of switched systems the dwell-time is included to to Ac T d can be found in Wang (El-Farra et al.(Aleksandrov , 2005), (Long and Zhao , dition Ac Td (Ac -closed-loop matrix (4)). In such a the term e 2011),(Sun and , 2013), et al. , 2011) A T the term term e c d (A (Adesign -closed-loop matrix (4)). In such such a 2011),(Sun and Wang ,, 2013), (Aleksandrov et al. ,, 2011) way the proposed procedure for (4)). stability analyc -closed-loop and references therein. Due to the switched controller the e matrix In a 2011),(Sun and Wang 2013), (Aleksandrov et al. 2011) c way the proposed design procedure for stability analyand references references therein. Due to totime the nonlinear switched systems controller the proposed design procedure analyand Due the switched controller sis and switching controller design forfor thestability real switching design problemstherein. for continuous in way way the proposed design procedure for stability analyand references therein. Due to the switched controller and switching design for the real switching design problems for continuous continuous time nonlinear systems in sis sis and switching controller design for switching design problems for nonlinear in interval T >controller Td becomes conservative. The ”dwellthis paper we pursue the idea to time use, instead of asystems nonlinear sis and switching controller design for the the real real switching design problems for continuous time nonlinear systems in time time interval T > T becomes conservative. The ”dwellthis paper we pursue the idea to use, instead of a nonlinear d time interval T > T becomes conservative. The this paper we pursue the idea to use, instead of a nonlinear term” for continuous-time systems very complicated d switched plant model a switched gain scheduled plant time interval Tcontinuous-time > Td becomes systems conservative. The ”dwell”dwellthis paperplant we pursue theaidea to use, gain instead of a nonlinear time term” for very complicated switched model switched scheduled plant time term” for continuous-time systems very complicated switched model aa robust switched gain scheduled plant the switched controller design systems procedure. Incomplicated the paper model andplant to design the switched gain scheduled time term” for continuous-time very switched plant model switched gain scheduled plant the switched controller design procedure. In the paper model and andguaranteeing to design design theclosed robust switched gain scheduled the switched controller In model to robust switched gain Shaked ,design 2011) procedure. sufficient conditions are controller loop stability andscheduled guaran- (Allerhand the switchedand controller design procedure. In the the paper paper model andguaranteeing to design the theclosed robust switched gain scheduled (Allerhand and Shaked , 2011) sufficient conditions are controller loop stability and guaran(Allerhand and Shaked , 2011) sufficient conditions are controller guaranteeing closed loop stability and guarangiven for the stability of linear systems with dwell-time teed cost for all operating points of the switched nonlinear and Shakedof, linear 2011) systems sufficientwith conditions are controller guaranteeing closed loop stability andnonlinear guaran- (Allerhand given for the stability dwell-time teed cost for all operating points of the switched given for the stability of linear systems with dwell-time teed cost for all operating points of the switched nonlinear and with polytopic type parameter uncertainty. Lyapunov system. given for the stability of linear systems with dwell-time teed cost for all operating points of the switched nonlinear and with polytopic polytopic typeform parameter uncertainty. Lyapunov system. with type parameter system. functions, in quadratic for eachuncertainty. mode, whichLyapunov are nonGain scheduling control deals with systems subject to and and with polytopic typeform parameter uncertainty. Lyapunov system. functions, in quadratic for each mode, which are Gain scheduling control deals with systems subject to functions, in quadratic form for each mode, which are nonGain scheduling control deals with systems subject to increasing at the switching instants are assigned to noneach parametric variations, which include linear systems with functions, in quadratic form for each mode, which are nonGain scheduling control deals with systems subject to increasing at the switching instants are assigned to each parametric variations, which include linear systems with increasing at the switching instants are assigned to each parametric variations, which include linear systems with mode. During the dwell-time this function varies piecetime varying parameters or nonlinear systems modeled at the switching instants are assigned topieceeach parametric variations, which include linear systems with increasing mode. During the dwell-time this function varies time varying parameters or nonlinear systems modeled mode. During the dwell-time this function varies piecetime varying parameters or nonlinear systems modeled wise linearly in time after switching occurs. The proposed by a family of linear parameter varying systems, (Monmode. During the dwell-time this function varies piecetime varying parameters or nonlinear systems modeled wise linearly in time time after switching occurs. occurs. The proposed proposed by aa family family of linear linear parameter varying systems, (Monlinearly in switching The by of parameter varying systems, method was applied to stabilization via a state feedback tagner and Peres , 2004). Because of time varying(Mongain wise linearly in time after after switching occurs. The proposed by a family of linear parameter varying systems, method was applied to stabilization via aa state feedback tagner and Peres Peres , 2004). 2004). Because of time time varying(Mongain wise method was applied to stabilization via state feedback tagner and , Because of varying gain both for nominal and uncertain cases. Since within the method was applied to stabilization via a state feedback tagner and Peres , 2004). Because of time varying gain The work has been supported by the Slovak Scientific Grant both for nominal and uncertain cases. Since within the both for nominal and uncertain cases. Since within the dwell-time the Lyapunov function varies piecewise linearly The work has been supported by the Slovak Scientific Grant both for nominal and uncertain cases. Since within the Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 work has been supported by the Slovak Scientific Grant The dwell-time the Lyapunov function varies piecewise linearly The work has been supported by the Slovak Scientific Grant dwell-time the Lyapunov function varies piecewise linearly Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 dwell-time the Lyapunov function varies piecewise linearly Agency VEGA, Grant No. 1/1241/12. and 1/2256/12

Agency VEGA, Grant No. 1/1241/12. and 1/2256/12 Copyright IFAC 2015 1079Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright IFAC 2015 1079 Copyright © IFAC 2015 1079 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 1079Control. 10.1016/j.ifacol.2015.09.335

MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Vojtech Veselý et al. / IFAC-PapersOnLine 48-11 (2015) 1068–1073

and the real switching time interval T > Td , the switching controller design procedure become rather conservative. The switched controller design procedure for continuoustime systems proposed in this paper does not use the approach of the ”dwell-time”, therefore there is no such drawback as mentioned in the above references. Vast references on the switched controller design are concentrated on the case where switching can occur immediately (ideal case), for a large number of switched system the realistic case is where the rate of change of the switching signal is finite (non-ideal case). A assumption of a non-ideal case of the switching variable will be used in this paper which gives other opportunities for controller designer. Some results about the stability of a class of uncertain linear varying systems ( transform to switched systems) can be found in (Montagner and Peres , 2003). The remainder of the paper is organized as follows. In Section 1 and 2 we present the class of switched gain scheduled control systems. In Section 3 we address the output feedback PI switched gain scheduled robust controller design procedure for continuous time gain scheduled plant model. Finally, in Section 4, the proposed design procedure is demonstrated on a simple example. Our notation is standard, P ∈ Rm×n denotes the set of real m × n matrices, P > 0(P ≥ 0) ∈ Rn×n is a real symmetric, positive definite (semidefinite) matrix. σ ∈ S indicates the arbitrary switching algorithm and σ + 1 is the first next mode to mode σ.”*” in matrices denotes the respective transposed (conjugate) term to make matrix symmetric. Im is an m × m identity matrix, 0m denotes the zero matrix. 2. PROBLEM STATEMENT AND PRELIMINARIES

1069

Ui , y ∈ Y abovesets we get i , i = 1, 2, ...p. Summarizing p p p x ∈ X = i=1 Xi , u ∈ U = i=1 Ui , y ∈ Y = i=1 Yi . p Aσj (ξ)θj ∈ Rn×n Aσ (ξ, θ) = Aσ0 (ξ) + B σ (ξ, θ) = Bσ0 (ξ) +

j=1 p j=1

(3)

Bσj (ξ)θj ∈ R

Matrices Aσj (ξ), Bσj (ξ), j = 0, 1, 2, ...p belong to the convex set a polytope with K vertices that can formally defined as K Ωσ = {Aσj (ξ), Bσj (ξ) = (Aσij , Bσij )ξi i=1

j = 0, 1, 2, 3, ...p,

K i=1

(4)

ξi = 1, ξi ≥ 0, ξi ∈ Ωξ }

where ξi , i = 1, 2, ...K are constant or possible time varying but unknown parameters, Aσij , Bσij , C are constant matrices of corresponding dimensions, θ ∈ Rp is a vector of known constant or time varying gain scheduled parameter. Assume that both lower and upper bounds are available, that is θ ∈ Ωs = {θ ∈ Rp : θj ∈ θj , θj } (5) θ˙ ∈ Ωt = {θ˙ ∈ Rp : θ˙j ∈ θ˙j , θ˙j } 2.2 Problem formulation For each plant mode consider the uncertain gain scheduled LPV plant model in the form (2),(3) and (4) p p x˙ = (Aσ0 (ξ) + Aσj (ξ)θj )x + (Bσ0 (ξ) + Bσj (ξ)θj )u j=1

2.1 Uncertain LPV plant model for switched systems

n×m

j=1

y = Cx

Consider family of nonlinear switched systems z˙ = fσ (z, v, w) σ ∈ S = {1, 2, ...N } y = h(z)

(1)

where z ∈ Rn is the state, the input v ∈ Rm , the output y ∈ Rl , exogenous input w ∈ Rk which captures parametric dependence of the plant (1) on exogenous input. The arbitrary switching algorithm σ ∈ S is a piecewise constant, right continuous function which specifies at each time the index of the active system, (Muller and Liberzon , 2010). Assume that f(.) is locally Lipschitz for every σ ∈ S. Consider that the number of equilibrium points for each switching modes is equal to p, that is for each mode σ ∈ S the nonlinear system can be replaced by a family of p linearized plant. For more details how to obtain the gain scheduled plant model see excellent surveys (Rugh and Shamma, 2000), (Leith and Leithead, 2000). To receive the model uncertainty of the gain scheduled plant it is necessary to obtain other family of linearized plant models around the p equilibrium points. Finally, one obtains the gain scheduled uncertain plant model in the form x˙ = Aσ (ξ, θ)x + B σ (ξ, θ)u y = Cx

σ∈S

(6) For a robust gain scheduled I part controller design, the states x of (6) need to be extended in such a way that a static output feedback control algorithm can provide proportional (P) and integral (I) parts of the designed controller, for more detail see (Vesel´ y and Rosinov´ a , 2013). Assume that system (6) allows PI controller design with a static output feedback. The feedback control law is considered in the form p u = Fσ (θ)y = (Fσ0 + Fσj θj )Cx (7) j=1

where Fσ (θ) is the static output feedback gain scheduled controller for mode σ.The closed loop system is (8) x˙ = Aσc (ξ, θ, α)x where Aσc (ξ, θ, α) = N N (Aσ (ξ, θ) + B σ (ξ, θ)Fσ (θ)C)ασ = Aσ (ξ, θ)ασ ,

(2)

σ=1

σ=1

T

α = [α1 , α2 , ...αN ],

where x = z − ze , u = v − ve , y = y − y e , (ze , ve , y e ) define the equilibrium family for plant (1). Assume, that for i − th equilibrium point one obtain the sets x ∈ Xi , u ∈

N

σ=1

ασ = 1,

N

α˙ σ = 0

σ=1

αj = 1 when σj is active plant mode, else αj = 0. Assume α ∈ Ωα , α˙ ∈ Ωd .

1080

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To access the system performance, we consider an original weighted scheduled quadratic cost function ∞ J= J(t)dt (9)

lσ (1, 2) = −N1T Aσ (ξ, θ)+N2 +

t=0 T

where J(t) = xT Q(θ)x + u Ru, and p Qj θj , Qj ≥ 0, Q(θ) = Q0 + j=1

This section formulates the theoretical approach to the robust switched gain scheduled controller design with control law (7) which ensure closed loop multi parameter dependent quadratic stability and guaranteed cost for an arbitrary switching algorithm σ ∈ S. Assume that in Theorem1 the Lyapunov function is in the form V (x, ξ, θ, α) = xT P (ξ, θ, α)x (11) where the Lyapunov multi parameter dependent matrix is

i=1

(P0i +

N

(Pσ0i +

σ=1

p

Pσij θj )ασ )ξi (12)

j=1

Time derivative of the Lyapunov function(11) is 0 P (ξ, θ, α) x˙ V˙ (.) = [x˙ T xT ] x P (ξ, θ, α) P˙ (ξ, θ, α) where K N DPσi ασ ξi P˙ (.) = DPσi =

N

σ=1

Using equality

i=1 σ=1 p

Pσij θ˙j +

Pσ0i α˙ σ +

(2N1 x˙ + 2N2 x)T (x˙ −

S(θ) = S0 +

p

Sj θ j +

j=1

Sj =

lσ (1, 1) = N1T + N1

(17)

p p

Skj θk θj +

j=1 k>j

p

Skk θk2

k=1

T

T S0 = Q0 + C Fσ0 RFσo C T T T Qj + C (Fσ0 RFσj + Fσj RFσ0 )C

T Skk = C T Fσk RFσk C The switched model plant (8) can be rewritten to the form p p p Mj (ξ)θj + Mjk (ξ)θj θk + Aσ (ξ, θ) = M0 (ξ) + j=1

j=1 k>j

(18)

p

Mkk (ξ)θ2

k

where

M0 (ξ) = Aσ0 (ξ) + Bσ0 (ξ)Fσ0 C Mj (ξ) = Aσj + (Bσ0 (ξ)Fσj + Bσj (ξ)Fσ0 )C Mjk (ξ) = (Bσj (ξ)Fσk + Bσk (ξ)Fσj )C Mkk (ξ) = Bσk (ξ)Fσk C Due to Theorem1 the closed loop switched gain scheduled system is multi parameter dependent quadratically stable with guaranteed cost for σ ∈ S, ξi , i = 1, 2, ...K if the following inequalities hold (19) Be = [x˙ T xT ]W (ξ, σ, θ)[x˙ T xT ]T ≤ 0 where W (ξ, σ, θ) = {wij (σ, ξ)}2×2 w11 (σ, ξ) = N1T + N1 p K P0i + Pσ0i + w12 (σ, ξ) = Pσij θj ξi i=1

j=1

− N1T Aσ (ξ, θ) + N2 w22 (σ, ξ) = −N2T Aσ (ξ, θ) − Aσ (ξ, θ)T N2 K + DPσi ξi + S(θ)

(14)

i=1

(15)

equation (13) can be rewritten as

Lσ (ξ, θ) = {lσ (i, j)}2×2

DPσi ξi

i=1

T T Sjk = C T (Fσj RFσk + Fσk RFσj )C

(13)

σ=1

N dV (.) T T x˙ = x˙ x Lσ (ξ, θ) x dt σ=1

K

where

Pσij α˙ σ θj

Aσ (ξ, θ)ασ x) = 0

Pσij θj )ξi

J(t) = xT S(θ)x

j=1 σ=1

j=1

N

p N

p j=1

i=1

lσ (2, 2) = −N2T Aσ (ξ, θ) − ATσ (ξ, θ)N2 +

3. MAIN RESULTS

K

(P0i +Pσ0i +

where N1 , N2 ∈ Rn×n are auxiliary matrices. On substituting (7) to (9) one obtains

R>0

. Definition 1. Consider a stable closed loop switched system (8) with N modes. If there is a control algorithm (7) and a positive scalar J ∗ such that the closed loop cost function (9) satisfies J ≤ J ∗ for all θ ∈ Ωs , α ∈ Ωα , then J ∗ is said to be a guaranteed cost and ”u” is said to be a guaranteed cost control algorithm for arbitrary switching algorithm σ ∈ S. Theorem 1. (Kuncevic and Lycak , 1977) Control algorithm (7) is the guaranteed cost control law for the switched closed loop system (8) if and only if there is Lyapunov function V (x, ξ, θ, α) > 0, matrices Q(θ), R and gain matrices Fσk ; k = 0, 1, ...p such that for σ ∈ S the following inequality holds dV (x, ξ, θ, α) Be = + J(t) ≤ −εxT x, ε → 0 (10) dt

P (ξ, θ, α) =

K

(16)

Inequality (19) implies : - for all σ ∈ S the inequality is linear with respect to uncertain parameter ξi , i = 1, 2, ...K, - for all σ ∈ S the inequality is a quadratic function with respect to the gain scheduled parameters θi , i = 1, 2, ...p. For the next development the following theorem is useful. Theorem 2. (Gahinet et al., 1996) Consider a scalar quadratic function of θ ∈ Rp p p p p f (θ) = a0 + aj θj + ajk θj θk + akk θk2 (20) j=1

1081

j=1 k>j

k

MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Vojtech Veselý et al. / IFAC-PapersOnLine 48-11 (2015) 1068–1073

therefore inequality (22) for each σ ∈ S split to K inequalities of type Lσi (θ) < 0 and Wσikk ≥ 0. • Eq. (12) implies that in dependence on the chosen structure of the Lyapunov matrix P (ξ, θ, α) one should obtained different types of stability conditions from quadratic to multi parameter dependent quadratic stabilities. Different types of stability conditions determine the conservatism of the design procedure and the rate of change of corresponding variables.

and assume that if f (θ) is multiconvex, that is δ 2 f (θ) = 2akk ≥ 0, k = 1, 2, ...p δθk2 then f (θ) is negative definite in the hyper rectangle (5) if and only if it takes negative values at the vertices of (5), that is if and only if f (θ) < 0 for all vertices of the set given by (5). Due to (14), (17) and (18) the robust stability conditions of switched system can be rewritten as N

W (ξ, σ, θ) = + +

L(θ, ξ)ασ =

σ=1 p

Wσj (ξ)θj +

j=1 p

k=1

N

4. EXAMPLE (Wσ0 (ξ)+

σ=1 p p

Wσjk (ξ)θj θk + (21)

j=1 k>j

Wσkk θk2 )ασ ≤ 0

σ σ where Wσ0 (ξ) = {w0ij }2×2 , Wσj (ξ) = {wjik }2×2 σ w011 = N1T + N1

σ w012 = −N1T M0 (ξ) + N2 +

K

(P0i + Pσ0i )ξi

i=1

σ T M0 (ξ) − M0T (ξ)N2 +S0 + w022 = −N2 p K + Pσij θ˙j ξi Pσ0i α˙ σ + i=1

j=1

σ σ wj11 = 0; wj12 = −N1T Mj (ξ) +

K

i=1

Pσij ξi

Wσjk (ξ) =

σ=1

0 −N1T Mjk (ξ) T ” ∗ ” −N2 Mjk (ξ) − Mjk (ξ)T N2 + Sjk

0 −N1T Mkk (ξ) ” ∗ ” −N2T Mkk (ξ) − Mkk (ξ)T N2 + Skk The main results on the robust stability condition for the switched gain scheduled control system is given in the next theorem. Theorem 3. Closed loop switched system (8) is robust multi parameter dependent quadratically stable with guaranteed cost if there is a positive definite matrix P (ξ, θ, α) ∈ Rn×n (12), matrices N1 , N2 ∈ Rn×n , positive definite (semidefinite) matrices Q(θ), R and gain scheduled controller matrix Fσ (θ), such that for σ ∈ S Wσkk (ξ) =

(1)

Lσ (ξ, θ) < 0

Consider a simple nonlinear switched system with two modes as σ = 1, x˙ = −asinx + bu (23) σ = 2, x˙ = −acosx + bu, y = x where a ∈ 0.8, 1, when a = 0.8 then b = 1 and a = 1, b = 0.5. One can linearized model (23) in the three working points x0 = {0, π/4, π/2}. For PI gain scheduled controller design the system state space needs to be increased, finally for matrices A(σ, ξ, θ), B(σ, ξ, θ) one obtains 0.117 0 −0.4 0 −0.28 0 θ A(1, ξ, θ) = { 1 0 + + 1 0 0 0 0 θ2 }ξ1 0.14645 0 −0.5 0 −0.35355 0 θ +{ 1 0 + + 1 0 0 0 0 θ2 }ξ2 1 0.5 B(1, ξ, θ) = 0 ξ1 + 0 ξ2 (24) −0.287 0 −0.117 0 0.4 0 A(2, ξ, θ) = { 1 0 + 0 0 θ1 + 0 0 θ2 }ξ1 −0.14645 0 0.5 0 −.35955 0 + +{ 1 0 + θ 1 0 0 θ2 }ξ2 0 0 0.5 1 B(2, ξ, θ) = 0 ξ1 + 0 ξ2 (25) 10 C = 01 For each mode, the robust gain scheduled controller has been designed for the following parameters Q0 = 0.01 ∗ I, Q1 = 0.001 ∗ I, Q2 = 0.005 ∗ I, R = I, max α˙ σ = 100 1s (rate of switching parameters changing) max θ˙j = 1 1s maximal rate of gain scheduled parameter changes. Constraints for the Lyapunov matrix (12) are 0 < P (ξ, θ, α) < 1000 ∗ I. The obtained first gain scheduled controller parameters are:

i=1

σ wj22 = −N2T Mj (ξ) − Mj (ξ)T N2 + Sj + N K Pσij α˙ σ ξi +

1071

(22)

(2) Wσkk ≥ 0, σ ∈ S, θ ∈ Ωs , k = 1, 2, ...p The proof of theorem sufficient conditions immediately follows from eqs. (12)-(21). Notes. • Lσ (θ, ξ) is linear with respect to uncertain parameter K ξi , i = 1, 2, ...K, it holds Lσ (θ, ξ) = i=1 Lσi (θ)ξi ,

σ=1 R1 = −7.8434 −

2.6287 3.852 + {−4.794 − }θ1 + s s

(26) 0.866 }θ2 {0.3278 + s σ=2 4.7241 6.3453 R2 = −13.8691 − + {−8.0619 − }θ1 + s s 1.2511 }θ2 {0.4297 + s (27) The maximal closed loop eigenvalues for the case of θj = 0, j = 1, 2, ...p are σ = 1, maxeig(CLS) = −0.4899 σ = 2, maxeig(CLS) = −0.4888

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MICNON 2015 1072 Vojtech Veselý et al. / IFAC-PapersOnLine 48-11 (2015) 1068–1073 June 24-26, 2015. Saint Petersburg, Russia

Amplitude

Under the same conditions other results have been obtained for the case of max α˙ σ = 300/s and maxθ˙j = 5/s. The second gain scheduled controller parameters are: σ=1 R1 = −7.1769 −

2.4355 2.0309 + {−1.2374 + }θ1 + (28) s s

2.7108 }θ2 {4.9032 − s σ=2 1.3464 1.7337 + {−02912 + }θ1 + (29) R2 = −3.9991 − s s 1.4535 }θ2 {1.1521 − s The maximal closed loop eigenvalues for the case of θj = 0, j = 1, 2, ...p are σ = 1, maxeig(CLS) = −0.2656 σ = 2, maxeig(CLS) = −0.5729 Note that the negative sign means a negative feedback. Simulation results for the two designed gain scheduled switched controller are given in Figures1-5

20

Amplitude Amplitude Amplitude

Amplitude Amplitude

55

60

65

70

0.1 0 −0.1 90

95

100 t(s)

105

110

1

40

60 t(s)

80

100

120

0.5 0

0

0.05

0.1

0.15

t(s)

1

Fig. 3. Development of the switching variable α1 (t)

0.5

θ2(t)

0

θ1(t)

robust gain scheduled controller and its ability to cope with model uncertainties. In the paper several forms of parameter dependent/quadratic Lyapunov functions are proposed. The obtained results are in the form of BMI. The proposed approach contributes to the design tools for switched robust gain scheduled controllers for nonlinear systems.

0.5 1

35

t(s)

0 0

30

0.9 0.8 0.7 50

y(t)

0.5

25 t(s)

w(t)

1

20

Fig. 2. Simulation results w(t), y(t) with the first gain scheduled switched controller-zoomed

Fig. 2

1.5

1.6 1.5 1.4

0

20

40

60 t(s)

80

100

120

Fig. 1. Simulation results w(t), y(t) with the first gain scheduled switched controller

6. ACKNOWLEDGEMENT The work has been supported by Grant 1/1241/12 and 1/2256/12 of Slovak Grant Agency

5. CONCLUSION

REFERENCES

In the paper a novel switched robust gain scheduled controller design procedure has been proposed for switched control of nonlinear systems. The proposed method is based on an uncertain gain scheduled plant, multi parameter dependent Lyapunov function and guaranteed cost. To access the system performance, we consider an original weighted scheduled quadratic cost function which allowed to obtain different performance dependence on the working points, which opens new possibilities for the controller designer. The obtained results, illustrated on examples, show the applicability of the designed switched

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MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Vojtech Veselý et al. / IFAC-PapersOnLine 48-11 (2015) 1068–1073

Fig. 5

Amplitude

1.5

w(t)

1 0.5

y(t)

0 0

20

40

60 t(s)

80

100

120

Amplitude

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40

60 t(s)

80

100

120

0.2 0 −0.2

Amplitude

Amplitude

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Fig. 4. Simulation results w(t), y(t) with second gain scheduled switched controller

15

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25 t(s)

30

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65 t(s)

70

75

105 t(s)

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1

0.5

55

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1.6 1.4 95

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Fig. 5. Simulation results w(t), y(t) with second gain scheduled switched controller

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using multiple Lyapunov function Systems and Control Letters, 54, 12 (2005), 1163-1182. 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. Geromel J. and Colanery P. (2006) Stability and stabilization of continuous time switched linear systems. SIAM Journal on Control and Optimization , 45 (5), (2006), 1915- 1930. 1084

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