Robust controller design method for linear systems with parametric uncertainty using minimum eigenvalue approach

Robust controller design method for linear systems with parametric uncertainty using minimum eigenvalue approach

5th International Conference on Advances in Control and Optimization of Dynamical Systems 5th International Conference on Advances in Control and Opti...

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5th International Conference on Advances in Control and Optimization of Dynamical Systems 5th International Conference on Advances in Control and Optimization of Dynamical Systems 5th International Conference on Advances in Control and February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems Available online at www.sciencedirect.com February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems 5th International Conference on Advances in Control and February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

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IFAC PapersOnLine 51-1 (2018) 610–614 Robust controller design method for linear Robust controller design method for linear Robust controller design method for Robust controller design method for linear linear systems with parametric uncertainty using systems with parametric uncertainty using Robust controller design method for linear systems with parametric uncertainty using systems with parametric uncertainty using minimum eigenvalue approach minimum eigenvalue approach systems with parametric uncertainty minimum eigenvalue approach minimum∗ eigenvalue approach ∗using ∗ Mminimum Venkatesh Sourav Patra ∗ Goshaidas Ray ∗ eigenvalue approach M Venkatesh ∗∗ Sourav Patra ∗ Goshaidas Ray ∗

M Venkatesh ∗ Sourav Patra ∗ Goshaidas Ray ∗ M Venkatesh Sourav Patra Goshaidas Ray ∗ of Electrical ∗ ∗ Department M Venkatesh Sourav Patra ∗Engineering, Goshaidas Ray ∗ of Electrical Engineering, ∗ Department ofTechnology Electrical Kharagpur, Engineering,India Indian Institute of ∗ Department Department of Electrical Kharagpur, Engineering, Indian Institute Indian Institute of of Technology Technology Kharagpur, India India (e-mails: [email protected], ∗[email protected], Indian Institute ofofTechnology Kharagpur, Department Electrical Engineering,India (e-mails: [email protected], [email protected], (e-mails: [email protected], [email protected], [email protected]). (e-mails: [email protected], [email protected], Indian Institute of Technology Kharagpur, India [email protected]). [email protected]). [email protected]). (e-mails: [email protected], [email protected], Abstract: In this paper, a static [email protected]). state feedback robust controller design method for linear Abstract: In In this this paper, paper, aa static static state state feedback feedback robust robust controller controller design design method method for for linear linear Abstract: systems withInparametric uncertainty is presented. closed-loop analysis for is carried Abstract: this paper, a static state feedback The robust controllerstability design method linear systems with parametric uncertainty is presented. The closed-loop stability analysis is carried systems withsense parametric uncertainty presented. The closed-loop stability analysis is carried out in the of Lyapunov and is controller design parameters with constraints are systems withIn parametric uncertainty isthe presented. The closed-loop stability analysis is carried Abstract: thisof a static feedback robust controller design method for linear out in in the the sense ofpaper, Lyapunov andstate the controller design parameters with constraints are out sense Lyapunov and the controller design parameters with constraints are formulated in linear matrix inequality (LMI) framework. A less conservative robust stability out in the sense of matrix Lyapunov and isthe controller design parameters with constraints are systems with parametric uncertainty presented. The closed-loop stability analysis is carried formulated in linear inequality (LMI) framework. A less conservative robust stability formulated in linear inequality A less conservative stability result is obtained by matrix introducing a new(LMI) set offramework. sufficient conditions developed robust which contains formulated in linear inequality (LMI) A parameters less conservative stability out sense of matrix Lyapunov and the controller design with robust constraints are resultinis isthe obtained by introducing new set offramework. sufficient conditions developed which contains contains result obtained by introducing aa new set of sufficient conditions developed which the information of minimum eigenvalue of the perturbation matrices. The control effort is also result is obtained by introducing a new(LMI) set ofperturbation sufficient conditions developed which contains formulated in linear matrix inequality framework. A less conservative robust stability the information of minimum eigenvalue of the matrices. The control effort is also the information of minimum eigenvalue of theinperturbation TheThe control effort design is also reduced by imposing additional constraints the designedmatrices. framework. proposed the information of by minimum eigenvalue thein matrices. TheThe control effort is also result is by obtained introducing a newofset ofperturbation sufficient conditions developed which contains reduced by imposing additional constraints the designed framework. proposed design reduced imposing additional constraints in the designed framework. Thethe proposed design technique is illustrated by considering several numerical examples that show superiority of reduced byis imposing additional constraints innumerical the designed framework. The proposed design the information of minimum eigenvalue of the perturbation matrices. The control effort is also technique illustrated by considering several examples that show the superiority of technique is illustrated by considering several numerical examples that show the superiority of the proposed technique over the existing methods. technique is imposing illustrated by considering several numerical examples that show the superiority of reduced by additional constraints in the designed framework. The proposed design the proposed technique over the existing methods. the proposed technique over the existing methods. the proposed technique by over the existing methods. technique is illustrated considering several numerical examples that show superiority of © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. the All rights reserved. Keywords: Linear system; Lyapunov function; Static state feedback controller; Linear matrix the proposed technique over the existing methods. Keywords: system; Lyapunov Keywords: Linear Linear system; Lyapunov function; function; Static Static state state feedback feedback controller; controller; Linear Linear matrix matrix inequality; Minimum Eigenvalues. Keywords: Linear system; Lyapunov function; Static state feedback controller; Linear matrix inequality; Minimum Eigenvalues. inequality; Minimum Eigenvalues. inequality; Linear Minimum Eigenvalues. Keywords: system; Lyapunov function; Static state feedback controller; Linear matrix 1. INTRODUCTION ditions are established in LMI framework with a priori inequality; Minimum Eigenvalues. 1. INTRODUCTION INTRODUCTION ditions are are established established in in LMI LMI framework framework with with aa priori priori 1. ditions choice ofare theestablished Lyapunov in matrix (2004)). In a(Chilali 1. INTRODUCTION ditions LMI (Lien framework with priori choice of the Lyapunov matrix (Lien (2004)). In (Chilali choice of the aLyapunov matrix (Lien (2004)). Inwith (Chilali et al. (1999)), mixed H2matrix /H∞ synthesis problemIn pole Uncertainties and1.parameter perturbations widely exist ditions ofare theestablished (Lien (2004)). INTRODUCTION LMI framework withwith a(Chilali priori et al. al. (1999)), (1999)), aLyapunov mixed H H2in /H problem pole ∞ synthesis Uncertainties and and parameter parameter perturbations widely widely exist exist choice et a mixed /H synthesis problem with pole placement constraint in a specified region has been solved 2 ∞ Uncertainties perturbations in most of the practical systems and have widely adverseexist ef- choice et al. (1999)), aLyapunov mixed in H2amatrix /H problem pole of the (Lien (2004)). Inwith (Chilali placement constraint specified region has been solved ∞ synthesis Uncertainties and parameter perturbations in most of the practical systems and have adverse efplacement constraint in a specified region has been solved in LMI framework. In (Siljak (1989)), a design approach in most of thethe practical systems and have adverse ef- et fects on both performance and stability of control placement constraint in a specified region has been solved al. (1999)), a mixed H /H synthesis problem with pole in LMI framework. In (Siljak (1989)), a design approach 2 ∞ in most the practical systems and have widely adverse ef- is Uncertainties and parameter perturbations exist fects on of both the performance and stability of control control LMI framework. In (Siljak (1989)),robust a design approach introduced for solving the quadratic stabilization fects on both the performance and stability of systems. The design of robust controller in the presence of in in LMI framework. In (Siljak (1989)), a design approach placement constraint in a specified region has been solved is introduced for solving the quadratic robust stabilization fects on both the performance and stability of control in most of the practical systems and have adverse efsystems. The The design design of of robust robust controller controller in in the the presence presence of of problem is introduced for solving the quadratic robustetstabilization of nonlinear systems. In (Kheloufi al. (2013), systems. parametric uncertainty is a crucial requirement in many is introduced for solving the quadratic robust stabilization LMI framework. Insystems. (Siljak (1989)), a design approach problem of nonlinear In (Kheloufi et al. (2013), systems. design of robust controller in the presence of in fects on The both the performance andrequirement stability ofin parametric uncertainty is aa crucial crucial requirement incontrol many problem of nonlinear systems. In (Kheloufi et al. (2013), Wang and Jiang (2014) and Kheloufi et al. (2015)), an parametric uncertainty is many feedback control problems. Even though the uncertainties problem of nonlinear systems. In (Kheloufi et al. (2013), is introduced for solving the quadratic robust stabilization Wang and Jiang (2014) and Kheloufi et al. (2015)), an parametric uncertainty is aEven crucial requirement in many systems. The design of robust controller in the presence of feedback control problems. though the uncertainties Wang and Jiang (2014) and Kheloufi et al. (2015)), an observer based controller design method for linear systems feedback control problems. Even though the uncertainties and perturbations are frequently encountered in practical Wang and Jiang (2014) and Kheloufi et al. (2015)), an problem of nonlinear systems. In (Kheloufi et al. (2013), observer based controller design method for linear systems feedback control problems. Even though the uncertainties parametric uncertainty is a crucial requirement in many and perturbations are frequently encountered in practical observer based controller design method for linear systems in the presence of norm-bounded uncertainty has been and perturbations are frequently encountered in practical systems, obtaining are an frequently exact mathematical model is quite observer based controller design method for linear systems Wang and Jiang (2014) and Kheloufi et al. (2015)), an in the presence of norm-bounded uncertainty has been and perturbations encountered in practical feedback control problems. Even though the uncertainties systems, obtaining obtaining an an exact exact mathematical mathematical model model is is quite quite presented the presence norm-bounded been in LMIof using uncertainty the Young’s has relation. systems, difficult (Gu et al. an (2005), (1996)). This is isdue to in in the presence offramework norm-bounded uncertainty has been observer based controller design method for linear systems presented in LMI framework using the Young’s relation. systems, obtaining exactDoyle mathematical model quite and perturbations are frequently encountered in practical difficult (Gu et al. (2005), Doyle (1996)). This is due to presented in LMI using the Young’s relation. (Kheloufi et al.framework (2016)), robust observer based H∞ difficult (Gu et al. noises, (2005), data Doyleerrors, (1996)). This issystems, due to In the environmental aging of is presented in LMI using uncertainty the Young’s relation. in presence offramework norm-bounded has been In (Kheloufi et al. (2016)), robust observer based H difficult (Gu et al. noises, (2005), Doyle (1996)). This to control systems, obtaining an exactdata mathematical model isdue quite the environmental environmental data errors, aging of systems, systems, In the (Kheloufi et al. (2016)), robust observer based H∞ for a class of Lipschitz nonlinear discrete time the noises, errors, aging of uncertain or slowly varying parameters, etc. Thus, in last In (Kheloufi et al. (2016)), robust observer based H∞ presented in LMI framework using the Young’s relation. control for a class of Lipschitz nonlinear discrete time ∞ the environmental noises, data errors, aging of issystems, difficult (Gu et al. (2005), Doyle (1996)). This due to uncertain or slowly varying parameters, etc. Thus, in last control for a class of Lipschitz nonlinear discrete time systems with parameter uncertainties has been designed. uncertain or slowly varying parameters, etc. been Thus,devoted in last control few decades, a decent amount of efforts has for a class of Lipschitz nonlinear discrete time In (Kheloufi et al. (2016)), robust observer based H systems with parameter uncertainties has been designed. ∞ uncertain or slowly varying parameters, etc. Thus, in last the environmental noises, data errors, aging of systems, few decades, decades, aa decent decent amount amount of of efforts efforts has has been been devoted devoted In parameter uncertainties been designed. (Kim with (2016)), delay uncertainties forhas linear systems usfew to the robustastability and stabilization problem of linear systems systems with uncertainties has been designed. control for a parameter class of uncertainties Lipschitz nonlinear discrete time In (Kim (2016)), delay for linear systems usfew decades, decent amount of efforts has been devoted uncertain or slowly varying parameters, etc. Thus, in last to the robust stability and stabilization problem of linear In (Kim (2016)), delay uncertainties for linear systems usLyapunov-Krasovskii functional for along with improved to the robust stability and stabilizationand problem ofend, linear ing systems with stability parameter to thisof In (Kim (2016)), delay uncertainties linear systems uswith parameter uncertainties has been designed. ing functional along with to the robust anduncertainty; stabilization problem linearaa systems few decades, decent amount of effortsand has systems withaparameter parameter uncertainty; and tobeen thisdevoted end, ing Lyapunov-Krasovskii Lyapunov-Krasovskii functional along with improved improved inequalities has been developed. systems with uncertainty; to this end, a integral plenty of design methods have already been established ing Lyapunov-Krasovskii functional along with improved In (Kim (2016)), delay uncertainties for linear systems usintegral inequalities has been developed. systems with parameter uncertainty; and to this end, a to the robust stability and have stabilization of linear integral inequalities has been developed. plenty of design design methods have already problem been established established plenty of methods already been and applied to themethods various have practical problems. integral inequalities hasstate been developed. ing Lyapunov-Krasovskii functional along with design improved In this paper, a static feedback controller for plenty ofwith design already been established systems parameter uncertainty; and to this end, a and applied to the various practical problems. In paper, aa static feedback controller design and applied to the various practical problems. In this thisuncertain paper, static state feedback controller design for for inequalities hasstate been developed. linear systems is proposed using the Lyapunov and applied to the various practical problems. plenty ofanalysis design methods have already been has established Robust of uncertain linear systems become integral In thisuncertain paper, a static state feedback controller design for linear systems is proposed using the Lyapunov Robust analysis of uncertain linear systems has become linear uncertain systems is proposed using the Lyapunov stability approach by introducing a new set of sufficient Robust analysis of various uncertain linear systems has become and applied to the practical problems. a research focus in areas of control systems. Using In systems is proposed usingset the Lyapunov thisuncertain paper, a static feedback controller for stability approach by introducing aa new sufficient Robust analysis of many uncertain systems has become research focus in in many areaslinear of control control systems. Using linear stability approach by state introducing new set of of design sufficient conditions which contains the minimum eigenvalue inforaa research focus many areas of systems. Using feedback control, the robust stabilisation problem for unstability approach by introducing a new set of sufficient linear uncertain systems is proposed using the Lyapunov conditions which contains the minimum eigenvalue inforafeedback research focus in many areas of control systems. Using Robust analysis of uncertain linear systems has become control, the robust stabilisation problem for un- conditions which contains the minimum mation of the perturbation matrices. Toeigenvalue the best ofinforour feedback control, the robust stabilisation problem for uncertain linear systems for both continuousand discreteconditions which contains the minimum eigenvalue inforapproach by introducing a new set of sufficient mation of the perturbation matrices. To the best of feedback control, the robust stabilisation problem for un- stability a research focus in many areas of control systems. Using certain linear systems for both continuousand discretemation of the perturbation matrices. To the best of our our knowledge, this result is new compared to the existing certain linear systems for both continuousand discretetime plant models has been solved (Khargonekar et al. mation of the perturbation matrices. Toeigenvalue the bestexisting ofinforour conditions which contains the minimum knowledge, this result is new compared to the certain linear systems for both continuousand discretefeedback control, the robust stabilisation problem for untime plant models has been solved (Khargonekar et al. knowledge, this result is new compared to the existing literature and the effectiveness of the method is demontime plant models has been solved (Khargonekar et al. mation (1990)). In models (Crusiushas andbeen Trofino (1999) and (Kuo et thisthe result is new compared to the existing of the perturbation matrices. To the best of our literature and effectiveness of the method is demontime plant solved (Khargonekar al. knowledge, certain linear systems for both continuousand discrete(1990)). In (Crusius and Trofino (1999) and (Kuo literature and the effectiveness of theThe method is effort demonstrated through numerical examples. control is (1990)). Instatic (Crusius andfeedback Trofino controller (1999) and (Kuo et al. (2004)), a output is designed to literature and the effectiveness of the method is demonknowledge, this result is new compared to the existing strated through numerical examples. The control effort is (1990)). Instatic (Crusius andfeedback Trofino (1999) and time plant models has been solved (Khargonekar et al. (2004)), output controller is (Kuo designed to strated through numerical examples. The control effort is reduced by imposing some additional constraints in the (2004)), aauncertain static output feedback controller is designed to stabilize plants, where a set of necessary and strated through numerical examples. The control effort is literature and the effectiveness of the method is demonreduced by imposing some additional constraints in the (2004)), aIn static output feedback controller is (Kuo designed to design (1990)). (Crusius and Trofino (1999) and et al. stabilize uncertain plants, where a set of necessary and reduced by imposing some additional constraints in the framework. The LMI formulation of the design stabilize uncertain plants, where aforset of necessary and sufficient uncertain conditionsplants, is established stability analysisand of strated reduced by imposing some additional constraints in the through numerical examples. The control effort is design framework. The LMI formulation of the design stabilize where a set of necessary (2004)), a static output feedback controller is designed to sufficient conditions conditions is is established established for for stability stability analysis analysis of of problem design framework. The LMI formulation of advantages the design leads to considerable computational sufficient such systems. On theisother hand, different approaches are design framework. The LMI formulation of advantages the design by imposing some additional constraints in the problem leads to considerable computational sufficient conditions established forset stability analysisand of reduced stabilize uncertain plants, where a of necessary such systems. On the other hand, different approaches are problemetleads to considerable computational advantages al. (1994)). The proposed design technique is such systems. On the other hand, different approaches are (Boyd available in theOn control literature, which some of the problem leads to considerable computational advantages framework. TheThe LMI formulation oftechnique the design (Boyd et al. (1994)). proposed design is such systems. theisother hand,among different approaches are sufficient conditions established for stability analysis of design available in the control control literature, among which some of the the (Boyd et al. (1994)). The proposed designexamples technique is illustrated by considering several numerical that available in the literature, among which some of methods deal with output feedback stabilisation problem (Boyd et al. (1994)). The proposed design technique is problem leads to considerable computational advantages illustrated by considering several numerical examples that available in theOn control literature, among which some of the such systems. the other hand, different approaches are methods deal with output feedback stabilisation problem illustrated by considering several numerical examples that show the superiority of the proposed method over the methods deal Matrix with output feedback stabilisation problem with Bilinear Inequality (BMI) constraints (Lens illustrated by considering several numerical examples that (Boyd et al. (1994)). The proposed design technique is show the superiority of the proposed method over the methods deal with output feedback stabilisation problem available in the control literature, among which some of the with Bilinear Bilinear Matrix Matrix Inequality Inequality (BMI) (BMI) constraints constraints (Lens (Lens existing show thetechniques superiority(Lien of the proposed methodand overJiang the (2004) and Wang with (2009), Toker Matrix and Ozbay (1995)) leading to a serious show thetechniques superiority of the proposed method overJiang the by considering several numerical examples that existing (Lien (2004) and Wang and with Bilinear Inequality (BMI) constraints (Lens illustrated methods deal with output feedback stabilisation problem (2009), Toker and Ozbay (1995)) leading to a serious existing techniques (Lien (2004) and Wang and Jiang (2009), Toker difficulty. and Ozbay (1995)) leading to a serious computational In some works, the stability con- (2014)). existing (2004) and Wang thetechniques superiority(Lien of the proposed methodand overJiang the (2014)). (2009), Toker Matrix and Ozbay (1995)) leading to a serious with Bilinear Inequality (BMI) constraints (Lens computational difficulty. In some some works, the stability stability con- show (2014)). computational difficulty. In works, the con(2014)). existing techniques (Lien (2004) and Wang and Jiang computational In some works, the stability con(2009), Toker difficulty. and Ozbay (1995)) leading to a serious Copyright © 2018 difficulty. IFAC computational In some works, the stability con-642 (2014)). 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 642Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 642Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 642 10.1016/j.ifacol.2018.05.102 Copyright © 2018 IFAC 642

5th International Conference on Advances in Control and Optimization of Dynamical Systems M Venkatesh et al. / IFAC PapersOnLine 51-1 (2018) 610–614 February 18-22, 2018. Hyderabad, India

2. NOTATIONS Rn Rn×m X>0 AT λmin (.) . I (∗)

n-dimensional Euclidean space. Set of n×m real matrices. Positive-definite symmetric matrix. Transpose of matrix A. Minimum eigenvalue of a matrix. Euclidean norm of a vector. Identity matrix of appropriate dimension. Represents symmetric elements in a symmetric matrix.

3. THE PROBLEM STATEMENT AND SOME PRELIMINARY RESULTS Let a continuous-time uncertain linear system be considered as: x(t) ˙ = (A + ∆A)x(t) + Bu(t), y(t) = (C + ∆C)x(t) + Du(t),

x(0);

(1)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the input vector, y(t) ∈ Rp is the output vector and x(0) is the initial condition. The pairs (A, B) and (A, C) are assumed to be stabilizable and detectable, respectively; the perturbed matrices are represented as √ √ ∆A = α1 ( √γMA )( √γNA ) (2) ∆C = α2 ( γMC )( γNC ) where 0 ≤ α1 ≤ 1 and 0 ≤ α2 ≤ 1; the known constant matrices MA , NA , MC and NC are with appropriate dimensions having MA and MC full rank; γ is a positive scalar known as perturbation factor. For the uncertain system (1), a control law u(t) = −Kx(t)

(3)

has to be designed such that the closed loop system

is stable.

x(t) ˙ = (A + ∆A − BK)x(t)

(4)

In the present work, the stability analysis is performed using a Lyapunov candidate function as V (x(t)) = xT (t)P x(t)

(5)

where P > 0. Taking derivative with respect to time and using (1), we get

where ρ > 0. In LMI framework, assuming V (x(t)) = xT (t)x(t) i.e. P = I, the system (1) is stabilizable by the control law (3) for a given positive scalar γ, if there exist positive scalars 1 , ρ and a matrix K ∈ Rm×n such that the following LMI is feasible:  T  √ A + A − K T B T − BK γMA   < 0. (8) +1 γNAT NA + ρI (∗) −1 I Also in (Wang and Jiang (2014)), the quadratic stabilization problem of uncertain linear systems is solved. The system (1) is stabilizable by the control law (3) for a given positive scalar γ, if there exist positive scalar 1 , a symmetric positive definite matrix Z ∈ Rn×n and a matrix ˆ ∈ Rn×m such that the following LMI is feasible: K   √ ˆ T − BK ˆT ZAT + AZ − KB γZNAT   < 0, (9) +1 γMA MAT (∗) −1 I ˆ T Z −1 . where the controller gain is obtained as K = K It is worth pointing out that the proposed result established in the following section will be compared with the conditions (8) and (9) which are respectively derived in (Lien (2004)) and (Wang and Jiang (2014)). 4. MAIN RESULTS In this section, a new controller design method will be presented and the effectiveness of the proposed method will be compared with the existing results. The robust stability result is obtained by introducing a new set of sufficient conditions which contains the minimum eigenvalue information of the perturbation matrices. Theorem 1. System (1) is asymptotically stabilizable by the control law given in (3), if for a given positive scalar γ there exist positive scalars 1 , kkh , kz , a symmetric positive ˆ ∈ Rn×m such definite matrix Z ∈ Rn×n and a matrix K that the following optimization problem is solvable. Minimize (kkh + kz ) subject to  ˆ T − BK ˆT ZAT + AZ − KB T  +1 γλmin (MA MA )I (∗) ZNAT MAT

˙ V˙ (x(t))=x˙ T (t)P x(t) + xT (t)P x(t) =xT (t)(A + ∆A − BK)T P x(t) + xT (t)P (A + ∆A − BK)x(t)   =xT (t) (A + ∆A − BK)T P + P (A + ∆A − BK) x(t). (6)    Σ

Then V˙ (x(t)) < 0 ∀ x(t) = 0 if Σ < 0. In (Lien (2004)), an asymptotic stability result for linear uncertain system (1) with uncertainty ∆A, as given in (2), is developed using the Lyapunov function (5). The system (1) is asymptotically stable if V˙ (x(t)) ≤ −ρ||x(t)||2

(7) 643

611





γZNAT

−1 I



 < 0,

+ MA NA Z − 1 λmin (MA MAT )I <0,    ˆT −kkh I K Z I > 0. < 0, ˆ I kz I K −I

(10)

ˆ T Z −1 . The controller gain matrix K = K

Proof. The linear uncertain system (1) is asymptotically stable using the control law (3) with the Lyapunov candidate function (5), if V˙ (x(t)) < 0 ∀ x(t) = 0, i.e., Σ < 0. Pre- and post-multiplying Σ with Z = P −1 and defining ˆ = ZK T , we have K

5th International Conference on Advances in Control and 612 M Venkatesh et al. / IFAC PapersOnLine 51-1 (2018) 610–614 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

   ˆ T −KB ˆ T. ˜ = ZAT +AZ+α1 γZNAT MAT +α1 γMA NA Z−B K T Σ ˆ T − BK ˆ T ( γλmin (N T NA ))Z + AZ − KB ZA A  (11)  T  < 0,  γλ (M M )I + T T 1 min A A Now if ∃ 1 > 0 such that ZNA MA + MA NA Z < (∗) −1 I 1 λmin (MA MAT )I, then ZNAT MAT + MA NA Z − 1 λmin (MA MAT )I <0,     ˆT α1 γZNAT MAT + α1 γMA NA Z <1 γλmin (MA MAT )I −kkh I K Z I > 0. < 0, γ ˆ I kz I K −I + ZNAT NA Z (12) 1 (15) for 0 ≤ α1 ≤ 1 and γ > 0. Since (12) holds and if ˆ T Z −1 . Then the controller gain K = K ˆ T − KB ˆ T + 1 γλmin (MA M T )I ZAT + AZ − B K A Proof: It follows the same line of proof of Theorem 1. γ T + ZNA NA Z < 0, (13) 1 Table 1 gives a comparitive study between the proposed ˙ then V (x(t)) < 0 and hence, the stability is guaranteed. method and the existing ones in terms of the no of decision Using Schur complement lemma, (13) is equivalent to the variables. The proposed one involves more no of variables, constraint given in (10). In order to restrict the size of the however, yields less conservative results. gain K, two positive scalars kkh and kz are introduced and Table 1 No of decision variables minimized by assigning the following inequalities: Method Nc (Lien (2004)) LMI (8) nm +2 T ˆ −1 ˆ Z < kz I. K K < kkh I, These two constraints are equivalent to the following LMIs: 

 ˆT −kkh I K < 0, ˆ K −I This completes the proof.



Z I I kz I



> 0.

The conservatism can further be reduced by using the information of the minimum eigenvalues of the perturbation matrices MA and NA which is given in the following corollary. Corollary 1. System (1) is asymptotically stabilizable by the control law given in (3), if for a given positive scalar γ there exist positive scalars 1 , kkh , kz , a symmetric positive ˆ ∈ Rn×m such definite matrix Z ∈ Rn×n and a matrix K that the following optimization problem is solved. Minimize (kkh + kz ) subject to

K

Several numerical examples in this section illustrate the effectiveness of the proposed stability criteria which shows the superiority over the existing techniques. Example 1: Let us consider a system state-space matrices as considered in (Lien (2004)): A=



 1 1 1 0 −2 1 , 1 −2 −5

B=

C = [1 0 1],



 1 0 0 1 , 0 0

D = 0,

(16)

with ∆A=30%|A|. MA and NA are considered as:   −0.3421 −0.1639 −0.4568 MA = −0.3997 −0.5998 0.1557 , −1.2257 0.2413 0.0767   −0.2644 −0.6058 −1.1585 NA = 0.0522 −0.5940 0.2987 . −0.4775 0.0102 0.1036 For the linear uncertain system (1), with parameters in (16), the maximum allowable parameter uncertainty γ is obtained from Theorem 1 and Corollary 1 which are depicted in Table 2. From the table, it is clear that the proposed technique gives higher perturbation bound with less control effort compared to the existing methods.

Table 2 Comparison with the existing methods LMI (8) (Lien (2004)) LMI (9)(Wang and Jiang (2014)) LMI (10) 3.32 4.57 4.97       7250.56 0.98 4.04 21.39 −5.95 36.74 37.75 −89.22 24.41 0.98 7248.59 1.96 1.39 4.20 4.01 −61.43 181.18 −19.06

644

0.5n2 + 0.5n + nm + 1 0.5n2 + 0.5n + nm + 3

5. NUMERICAL EXAMPLES

(14)

Remark 1: In (Lien (2004)), a robust stability analysis has been carried out using the Lyapunov function approach and the posed BMI constraint is handled with the equality constraint P = I. However in (Wang and Jiang (2014)), the BMI constraint is handled using the change of variables technique and the conservatism is reduced using the Young’s relation. In this paper, the stability result is obtained using the change of variables technique and introducing a new set of sufficient conditions which contains the minimum eigenvalue information of the perturbation matrices.

Method γmax

(Wang and Jiang (2014)) LMI (9) Theorem 1



LMI (15) 5.11  53.59 −139.73 27.35 −100.18 302.82 −27.55

5th International Conference on Advances in Control and Optimization of Dynamical Systems M Venkatesh et al. / IFAC PapersOnLine 51-1 (2018) 610–614 February 18-22, 2018. Hyderabad, India

Example 2:

less conservative result than the other existing techniques.

We now consider a spring-mass-damper system which is described by the following second-order ordinary differential equation (Gu et al. (2005)):

Table 4 Comparison with the existing methods Method γmax

2

m

d x(t) dx(t) +c + kx(t) = f (t) dt dt

K

LMI (8) (Lien (2004))

LMI (9) (Wang and Jiang (2014))

LMI (10)

LMI (15)

0.40

4.08

4.22

8.77

[0.01 110.9]

[107.4 22.6]

[384.3 154.0]

[545.5 283.1]

Example 3:

where m is the mass; c is the damping constant; k is the spring stiffness; x(t) is the displacement and f(t) is the external force. By defining x1 and x2 as the displacement variable and its first derivative (velocity), we get the satespace equations

Let us consider an LTI system with state-space matrices as follows: 

−8  6 A= 45 −30

x˙ 1 = x2 , x˙ 2 =

1 m [−kx1

y = x1 .

− cx2 + f ],

A=

0 −1

 1 , −0.2

C = [1 0],

 −0.1 0.995 , −0.995 −0.1



C = [ 0.7071 0 ] ,

−2.4456  −2.2798 MA =  −3.8847 −3.6700

0.2828 −2.0060 −1.2043 2.3325

1.5319 1.5637 −2.0010 0.1259

−2.8022  0.1912 NA =  −2.5671 0.2281

−2.1813 1.4782 1.1785 0.7296

−2.7072 −2.8293 0.6429 0.2031



D = 0.



D = [0 0],

(17)

with ∆A=35%|A|; we have MA and NA as

  0 B= , 1

B=

 36 26  , −43  −52

 10 4  , 0  −1

C = [1 0 0 0],

Converting into a block diagonalized form as 

19 37 −36 10

4  −4 B= −1 −0.5

With the physical parameters as given in the Table 3 (Hussein (2015)), the above equations can be written in the state-space form as 

−20 −12 −12 34



Table 3 Physical parameters of Mass-spring-damper system (Hussein (2015)) m Mass 1.0 Kg k Spring constant 1.0N/m c damping constant 0.2Ns/m f input force 1.0N

A=

613

 0 , 1.4213

 −0.7650 0.4789  , −0.1857  0.4088

 −4.4496 0.8763  . 0.6478  −0.6249

For the maximum allowable parameter uncertainty γ, for the linear uncertain system has been shown in Table 5. From the table, it is clear that the proposed stability criteria gives considerably less conservative result than the existing methods.

D = 0,

Table 5 Comparison with the existing methods

with ∆A=25%|A|, we have MA and NA as follows:

MA =



NA =



−0.3700 −0.3700 −0.3700 0.3345



−0.3345 , 0.3345

Method

LMI (8) (Lien (2004))

LMI (9) (Wang and Jiang (2014))

LMI (10)

LMI (15)

γmax

2.03

6.51

21.8

49.3

6. CONCLUSIONS



−0.3700 . −0.3345

The maximum allowable parameter uncertainty γ, for the linear spring-mass-damper uncertain system is obtained by applying the proposed stability criteria given in Theorem 1 and Corollary 1 which is depicted in Table 4. It can be seen that the proposed stability criteria gives considerably 645

In this paper, a static state feedback controller design for continuous-time linear uncertain system with a higher perturbation bound is presented. The controller gain matrix is computed through the posed LMI framework. A less conservative stability result is obtained by introducing a new set of sufficient conditions that involve the minimum eigenvalue information of the known perturbation matrices. Also, the control effort is reduced by imposing some additional constraints in the designed framework. The proposed design technique is illustrated

5th International Conference on Advances in Control and 614 M Venkatesh et al. / IFAC PapersOnLine 51-1 (2018) 610–614 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

by considering three numerical examples. The results show the effectiveness of the proposed method compared to the existing techniques. REFERENCES Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. SIAM. Chilali, M., Gahinet, P., and Apkarian, P. (1999). Robust pole placement in lmi regions. IEEE transactions on Automatic Control, 44(12), 2257–2270. Crusius, C.A. and Trofino, A. (1999). Sufficient lmi conditions for output feedback control problems. IEEE Transactions on Automatic Control, 44(5), 1053–1057. Doyle, J. (1996). Robust and optimal control. In Decision and Control, 1996., Proceedings of the 35th IEEE Conference on, volume 2, 1595–1598. IEEE. Gu, D.W., Petkov, P., and Konstantinov, M.M. (2005). R Robust control design with MATLAB. Springer Science & Business Media. Hussein, M.T. (2015). Modeling mechanical and electrical un- certain systems using functions of robust control R matlab toolbox. International Journal of Advanced Computer Science & Applications, 1(6), 79–84. Khargonekar, P.P., Petersen, I.R., and Zhou, K. (1990). Robust stabilization of uncertain linear systems: quadratic stabilizability and h/sup infinity/control theory. IEEE Transactions on Automatic Control, 35(3), 356–361. Kheloufi, H., Bedouhene, F., Zemouche, A., and Alessandri, A. (2015). Observer-based stabilisation of linear systems with parameter uncertainties by using enhanced lmi conditions. International Journal of Control, 88(6), 1189–1200. Kheloufi, H., Zemouche, A., Bedouhene, F., and Boutayeb, M. (2013). On lmi conditions to design observer-

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