Robust output-control for uncertain linear systems: Homogeneous differentiator-based observer approach

Transcription

1 Robust output-control for uncertain linear systems: Homogeneous differentiator-based observer approach Hector Ríos, Manuel Mera, Denis Efimov, Andrey Polyakov To cite this version: Hector Ríos, Manuel Mera, Denis Efimov, Andrey Polyakov Robust output-control for uncertain linear systems: Homogeneous differentiator-based observer approach International Journal of Robust and Nonlinear Control, Wiley, 27, 27 ), pp <2/rnc3643> <hal-45383> HAL Id: hal Submitted on 2 Dec 26 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not The documents may come from teaching and research institutions in France or abroad, or from public or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés

2 25; : 9 Robust Output-Control for Uncertain Linear Systems: Homogeneous Differentiator-based Observer Approach H Ríos, M Mera 2, D Efimov 3 and A Polyakov 3 Electrical & Computer Engineering Department, University of California, Santa Barbara, CA , USA 2 UPIBI-Instituto Politécnico Nacional, Av Acueducto de Guadalupe S/N, 734, México, DF, México 3 Non-A Inria, Parc Scientifique de la Haute Borne, 4 avenue Halley, 5965 Villeneuve d Ascq, France; CRIStAL UMR CNRS 989), Ecole Centrale de Lille, BP 48, Cité Scientifique, 5965 Villeneuve-d Ascq, France; and Department of Control Systems and Informatics, Saint Petersburg State University of Information Technologies Mechanics and Optics ITMO), Kronverkskiy av 49, Saint Petersburg, 97, Russia SUMMARY This paper deals with the design of a robust control for linear systems with external disturbances using a homogeneous differentiator-based observer based on a Implicit Lyapunov Function approach Sufficient conditions for stability of the closed-loop system in the presence of external disturbances are obtained and represented by Linear Matrix Inequalities The parameter tuning for both controller and observer is formulated as a semi-definite programming problem with Linear Matrix Inequalities constraints Simulation results illustrate the feasibility of the proposed approach and some improvements with respect to the classic linear observer approach Received KEY WORDS: Uncertain linear systems; Closed-loop stability; Homogeneous differentiator State of the Art and Motivation INTRODUCTION In the last thirty years, the problems of system stability analysis and designing a controller for a given system that guarantees the closed-loop stability and, at the same time, ensures given performance requirements in presence of uncertainties, have been widely studied see [, 2, 3, 4]) In this sense, the robust control is a branch of modern control theory whose aim is to achieve robust performance and/or stability in the presence of bounded modeling errors and/or uncertainties The classic control design, based on the frequency domain, was reasonable robust This was the beginning of the robust control theory Likely the most important results of the robust control theory are H and µ analysis see, eg the books [5, 4, 6, 7]) These methods minimize the sensitivity of a system over its frequency spectrum guaranteeing that the system will have a sufficiently small deviation from expected trajectories when disturbances affect the system Another approach to robust controller design is the so-called Sliding- Mode Control technique see, eg [8, 9, ]) This approach has attracted the attention because it possesses important properties such as: insensitivity more than robustness) with respect to so-called Correspondence to: H Ríos, Electrical & Computer Engineering, University of California, Santa Barbara, Harold Frank Hall 32, Santa Barbara, CA , USA hector riosbarajas@eceucsbedu This work was supported in part by the Government of Russian Federation Grant 74-U), and the Ministry of Education and Science of Russian Federation Project 4Z533) [Version: 2/3/27 v2]

3 2 H RÍOS ET AL matched uncertainties/disturbances acting in the same channel as the control; and finite/fixed time convergence see, eg [, 2, 3, 4]) Another approach is the Attractive Ellipsoid Method AEM) [5, 6] The AEM, originally introduced in [7] and later formalized for linear systems in [8, 9], and for nonlinear systems in [2]; is convenient for the analysis of systems with unmatched and non-vanishing perturbations and uncertainties) An additional advantage of this method is that the design parameters can be obtained directly from a linear optimization problem subject to a set of Linear Matrix Inequalities LMIs), which in general is desirable from the computational point of view Because the AEM is a Lyapunov based approach, it is natural to use the Implicit Lyapunov Function ILF) [2] to obtain additional properties for the closed-loop system, such as finite-time convergence to the ellipsoidal set [22] This property is crucial considering that many applications require control inputs that guarantee convergence of the closed-loop system trajectories to a set of values in a given amount of time When dealing with the limited information problem the finite-time property is also important for the observer design [23] It is important to mention that most of the robust output-control approaches are based on linear observers, and therefore the type of convergence is not faster than exponential In this sense, the ILF originally presented in [24], and later revisited in [25, 2], can be used to design nonlinear observers based on a Lyapunov function defined implicitly by an algebraic equation and providing faster rate of convergence than the exponential one, ie finite-time [26] On the other hand, the stability analysis of the error dynamics then does not require the explicit solution of this equation, instead the stability conditions can be derived directly using the implicit function theorem Therefore, part of the motivation of this paper is founded on the necessity of regulating dynamical systems even in the presence of external disturbances, where the state estimation time is crucial and a faster convergence is required 2 Main Contribution Motivated by the features of the ILF approach and the fact that in general there does not exist a constructive way to design robust output-control based on nonlinear observers, this paper contributes to the design of a constructive robust output-control for linear systems with external disturbances that uses a homogeneous differentiator-based observer This approach provides the following features: Sufficient conditions for the stability of the closed-loop system in the presence of bounded external disturbances are given based on the ILF approach 2 Such conditions are expressed in terms of matrix inequalities 3 The parameter tuning for both controller and observer is formulated as the semi-definite programming problem with LMIs constraints 4 Simulation results show some improvements with respect to the classic linear observer approach Structure of the Paper: The problem statement is given in Section 2 while some preliminaries are presented in Section 3 The robust control design, the ILF homogeneous differentiator observer and the conditions for the stability of the closed-loop system are described in Section 4 Some simulation results are depicted in Section 5 and concluding remarks in Section 6 Finally, the finitetime convergence proof of the ILF homogeneous differentiator, and the proof for the closed-loop stability are given in the Appendix 2 PROBLEM STATEMENT Consider the following class of linear dynamical systems with external disturbances, ie ẋ = Ax + Bu + Dwt), ) y = Cx, 2)

4 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS 3 where x R n, u R p, y R m, and w R q are the system state, control input, measurable output and the unknown external disturbance vector, respectively The known matrices A, B, C, and D have suitable dimensions It is assumed that all considered inputs allow the existence and extension of solutions to the whole semi-axis t The goal is to build a robust control, based on the state estimation given by a robust state observer, that can maintain closed-loop stability in spite of certain class of external disturbances 3 PRELIMINARIES Some preliminaries and results of homogeneity, finite-time stability and Implicit Lyapunov Function see, eg [27, 28, 29]) are introduced in this section Let q denote the Euclidian norm of a vector q, and, r a sequence of integers,, r Let Ω be an open subset of R n with Ω Let us consider that for all x Ω the solutions of a differential equation, ie xt, x ), are defined for all t 3 HOMOGENEITY In the homogeneity framework [27], for any r i >, i =, n and λ >, define the dilation matrix Λ r λ) = diagλ ri ), i =, n, and the vector of weights r = r,, r n ) T Definition The function g : R n R is called r-homogeneous r i >, i =, n), if for any x R n the relation gλ r λ)x) = λ d gx) holds for some d R and all λ > Respectively, the vector field f : R n R n is called r-homogeneous r i >, i =, n), if for any x R n the relation fλ r λ)x) = λ d Λ r λ)fx) holds for some d min r i and all λ > In both cases, the constant i n d is called the degree of homogeneity A dynamical system ẋ = fx), x) = x, 3) is called r-homogeneous of degree d if this property is satisfied for the vector field f in the sense of Definition 32 STABILITY Definition 2 The origin of system 3) is said to be : Stable if for any ɛ > there is δɛ) such that for any x Ω the solutions are defined for all t and, if x δɛ), then xt, x ) ɛ, for all t ; AS if it is Stable and for any ɛ > there exists T ɛ, κ) such that for any x Ω, if x κ, then xt, x ) ɛ, for all t T ɛ, κ); FTS if it is AS and for any x Ω there exists T x < + such that xt, x ) =, for all t T x If Ω = R n, then x = is said to be globally Stable GS), AS GAS), or FTS GFTS), respectively see, for more details [3]) Now, the following result, given by [28], represents the main application of homogeneity to finite-time stability and finite-time stabilization Theorem Let f : R n R n be a continuous r-homogeneous vector field with a negative degree If the origin of the system 3) is locally AS then it is GFTS 33 IMPLICIT LYAPUNOV FUNCTION The following theorems provide the background for asymptotic and finite-time stability analysis, respectively, of 3) using the ILF Approach [29] The acronyms AS and FTS correspond to Asymptotically Stable and Finite-Time Stable, respectively The function T x ) = inf{t x : xt, x ) = t T x } is called the uniform settling time of the system 3)

5 4 H RÍOS ET AL Theorem 2 If there exists a continuous function G : R + R n R, V, x) GV, x), satisfying the following conditions: ) G is continuously differentiable outside the origin for all positive V R + and for all x R n \{}; 2) for any x R n \{} there exists V R + such that GV, x) = ; 3) let Φ = {V, x) R + R n \{} : GV, x) = }, then, lim x V = +, lim V x =, lim x V = +, for all V, x) Φ; 4) the inequality GV,x) V < holds for all V R + and for all x R n \{}; 5) GV,x) x fx) < holds for all V, x) Φ; then the origin of 3) is GAS Theorem 3 If there exists a continuous function G : R + R n R that satisfies the conditions -4 of Theorem 2, and there exist c > and < µ <, such that GV, x) fx) cv x µ GV, x), V holds for all V, x) Φ, then the origin of 3) is GFTS and T x ) V µ cµ where GV, x ) = is the settling time function, 4 ROBUST CONTROL DESIGN 4 CLOSED-LOOP SYSTEM DYNAMICS For simplicity, let us start with the case m =, ie y R Thus, the following assumptions are introduced Assumption The pair A, C) is observable Assumption 2 The unknown external input wt) is bounded by a known positive constant w +, ie wt) w +, for all t Let Assumptions and 2 be satisfied Then, the observer for system )-2) takes the following structure: ˆx = P A P ˆx + P ā y + Bu P R[y C ˆx], 4) where ˆx R n is the estimation of x, the term R : R R n represents a nonlinear injection and it will be designed further, and the nonsingular matrix P R n n satisfies the following form where P AP = A + ā CP, CP = ), 5) A =, ā = a,, a n ) T, and a i, i =, n are the coefficients of the characteristic polynomial of matrix A Let us define the estimation error as e := x ˆx Hence, the estimation error dynamics is given by ė = A P A P P ā C ) x + P A P e + Dw + P R[Ce] 6) From 5) it follows that A = P A P + P ā C Thus, 6) is rewritten as follows ė = P A P e + Dw + P R[Ce] 7) Now, let us introduce the following assumption that ensures the possibility for designing a state feedback control u Note that Assumption implies that the matrix P always exists

6 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS 5 Assumption 3 The pair A, B) is controllable Thus, consider that the control u is designed as a feedback control based on the state estimation, ie u = K uˆx, where K u R p n is a matrix feedback gain which will be designed further Hence, substituting u = K uˆx = K u x e) in ), it follows that ẋ = A K x BK u e + Dwt), 8) where A K = A + BK u Thus, the complete closed-loop system dynamics can be written as follows ) ) ) ) ) ẋ AK BK = u x D ė n n P + w + n A P e D P 9) R[Ce] Note that the complete closed-loop system dynamics 9) is a cascade system since the dynamics of e is totally independent of x Thus, its analysis can be decomposed on two separate steps: Optimization of performance ellipsoid size with respect to w) of 7) by selecting the form and shape of the nonlinear injection R 2 Optimization of performance ellipsoid size with respect to w) of 8) by a choice of K u In the following the designs of the nonlinear injection term R and the matrix feedback gain K u are presented, respectively 42 NONLINEAR INJECTION DESIGN Let us consider the estimation error dynamics 7) and the linear transformation = P e Thus, taking into account that Assumption holds, the corresponding dynamics is given as follows = A + Dw + R[CP ], ) e y = CP =, ) where D := P D In this work the nonlinear term is designed as follows R[ ] = Λ r V h))k = Λ r V h))kh T, 2) where k := k,, k n ) T R n is a vector of gains, h :=,,, ) T R n, Λ r ) is a dilation matrix with a vector of weights r := µ, 2µ,, nµ) T, where µ, and V : R n R + is a positive definite function implicitly defined by GV, ) =, for any R n \{}, and GV, ) := T Λ r V )P Λ r V ), 3) with a positive definite matrix P = P T R n n and r := + n )µ, + n 2)µ,, ) T R n Note that GV h), h) = implies that V h)) 2+2n )µ = 2 ht P h Therefore, in an expanded way, the nonlinear injection 2) may be written as follows µ 2+2µn )) k p +µn 2) +µn ) 2µ 2+2µn )) k 2 p +µn 3) +µn ) R[ ] = nµ 2+2µn )) k n p µ +µn ) where p = h T P h and s α := s α signs), for any s R and α >, 4)

7 6 H RÍOS ET AL The dynamics ) with the nonlinear injection 2) and w = describes the error dynamics of the homogeneous differentiator given in [26], ie iµ = 2 + k +µn 2) +µn ), 2 = 3 + k 2 +µn 3) +µn ), n = k n µ +µn ), where k 2+2µn )) i = k i p, i =, n Remark For the case µ =, 2) takes the form of the HOSM differentiator [3] For this case, a similar observer for linear systems with external disturbances has been presented in [32], where ˆx is a linear combination of an estimation given by a Luenberger observer and the compensation of its estimation error provided by the HOSM differentiator [3] Remark 2 The nonlinear injection term R[ ] may be implemented using the implicit 2) or the explicit 4) form, in the first case V can be calculated by means of the algorithm presented in [29] which is based on the bisection numerical method The following theorem describes the finite-time convergence of the error dynamics ) to a certain neighborhood of the origin Theorem 4 Let the nonlinear injection 2) be applied to the observer 4) Define ϕ := D w + If the following matrix inequalities P δdiagh)p diagh), 5) P A + A T P ) + P kh T + hk T P + HP + P H + βp P P γϕ I, 6) n k ) T I n Λ rλ) Λr λ)p Λ r λ) ) I n Λ rλ) k β 2 λ 2µ h T, λ [, λ ], λ = δ /2+2n )µ), 7) P h P H + HP γϕ + β) /2 P >, γϕ + β <, 8) are feasible for some positive scalars β and γ, a positive definite matrix P = P T R n n, a matrix H = diag{r i } n i=, and a vector k Rn, then the estimation error et) converges in a finite time to a neighborhood of the origin, ie et) with the following settling time function where V ) such that GV ), ) =, and µ, ) Proof See the Appendix λ /2 min P T P P ), t T + )), 9) T + ) V µ ) µ γϕ β), 2) Remark 3 The main difference with respect to the HOSM observer [32] is that in this work an auxiliary Luenberger observer is not required, and moreover, the proposed observer gains are designed in a constructive way using LMIs Remark 4 Since most of the HOSM observers, for instance [32], do not have a Lyapunov function to prove the convergence, it is more complicated to prove the closed-loop stability when they are implemented for robust output-control design

8 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS 7 42 Numerical Aspects The set of matrix inequalities 5)-8), presented in Theorem for the homogeneous differentiator design, is bilinear with respect to matrices k and P and nonlinear with respect to λ, γ and β The solution to this problem is in general complex but it is possible to apply some algorithms see, eg [33] and [26]), to find a numerical solution for the system of nonlinear inequalities For some fixed β R +, γ R + and µ, δ, ), if the following system of LMIs P δh P H, h T P h =, P A + A T P + Xh T + hx T + H P + P H + β P ) P P γϕ I, 2) n P H + H P γϕ + β) /2 P >, γϕ + β <, is feasible for a positive definite matrix P = P T R n n and X R n, then the variables P = P and k = P X satisfy the set of matrix inequalities 5), 6) and 8) given in Theorem For the inequality 7) it is possible to find first a feasible solution P, X), for the set of LMIs 2), and then evaluate numerically the inequality 7) This inequality can be easily checked on a grid with a sufficiently small step size for λ = λ j N, for j =,, 2,, N; where N is a sufficiently large number 422 Optimization Problem Considering the implicitly defined function V in 3), such that GV, ) =, it is clear that the region to which the error converge is related to the size of matrix P In fact, the maximization of P is equivalent to the minimization of the ellipsoidal set T Λ r V )P Λ r V ) for V = In order to reduce the size of this region the following optimization problem is proposed trp ) max P,X,β,γ subject to 2) 22) If ˆP and are the solution of the above optimization problem, then the minimizing gain is given by ˆk = ˆX ˆP 43 MATRIX FEEDBACK GAIN DESIGN Let us consider the closed-loop dynamics 8) Thus, based on [5], the following result is established Theorem 5 Let the linear control u = K uˆx be applied to the system ) Let Assumption 2, 3 and the statements of the Theorem 4 hold If the following matrix inequality P x A K + A T K P x + 2α x Px Px BK u Px D α x P T n q, 23) α x Q w is feasible for α x R +, and some positive definite matrices Q w = Q T w R q q and Px = Px T { R n n, P T := P T P P R n n with P that satisfies 5), then the ellipsoid EP x ) := x R n : x T Px x < } is exponentially attractive for the closed-loop system 8) Proof See the Appendix 43 Numerical Aspects In order to apply the result given by Theorem 5 it is needed to solve the nonlinear matrix inequality 23) with respect to the variables Px and K u for a positive scalar α x The following proposition provides one simple scheme that can be used for practical selection of the design parameters

9 8 H RÍOS ET AL Proposition Let α x R +, P defined in 5) If the following LMIs ) 2Px + R x I n, 24) I n P T AP x + BY u + AP x + BY u ) T + 2α x P x BY u D α x R x n q, 25) α x Q w hold for some symmetric positive definite matrices Q w R q q, Px R n n, R x R n n, P T R n n, and the matrix Y u R p n, then the matrix inequality 23) holds, and the matrix feedback gain is chosen as K u = Y u Px Proof Post and pre-multiplying 23) by diagp x, P x, I q q ) it follows that A KP x + P x A T K + 2α xp x BKP x D α x P x P T P x n q α x Q w 26) Applying the inequality 2X T Y X T Λ X + Y T ΛY for any positive definite matrix Λ = Λ T to the term P xp T P x one obtains P x P T P x 2P x + P T Let us assume that 2P x + P T R x Then, the LMI 24) is just the last inequality written using the Schur s complement Finally, defining Y u = K u P x, and taking into account that P x P T P x R x, the LMI 25) is provided from 26) Hence, if 24) and 25) hold, then 23) is satisfied 432 Optimization Problem 2 Similarly as in section 422, it is possible to minimize the ellipsoidal EP x ) by minimizing the size of matrix P x One simple way to achive this is to minimize the trace of P x trp x ) subject to 24), 25) min 27) P x,y u,r x,α x If ˆP x and Ŷu are the solution of 27), the optimal gain can be obtained as ˆK u = Ŷu ˆP x Remark 5 Both of the optimization problems presented above are bilinear with respect to the scalar variables In order to solve them it is necessary to use a Bilinear Matrix Inequality solver, eg PENBMI, or to use an iterative method, eg Algorithm 3 in [5] 44 CLOSED-LOOP STABILITY Based on the previous results, the following theorem is provided for the complete closed-loop stability Theorem 6 Let Assumption, 2 and 3 hold Let observer 4) designed according to Theorem 4, and the linear control u = K uˆx designed according to Theorem 5, be applied to the system ) Then the trajectories of the closed-loop system satisfy et) β )µt + V ) µ ) +n )µ µ, t < T + )), 28) λ /2 min P T P )λ min P )) µ+n )µ) 2+2n )µ xt) ɛ x, t, 29)

10 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS 9 with V ) µ = λ min P T P )λ max P )) µ 2 e) µ, ɛ x >, and et) λ /2 min P T ), t T + )), 3) xt), as t λ /2 min Px ) 3) Proof Based on the statements given by Theorems 4 and 5, it is obtained that the trajectories of the closedloop system 9) satisfy 3) and 3) From the proof of Theorem 4, it is possible to show that V β )V µ, V >, 32) λ /2+2µn )) max P ) /+µn )) V λ /2 maxp ), 33) where β = γϕ + β and µ, ) Therefore, is always bounded, and thus e is also bounded for all t From 9), since A K is Hurwitz, it follows that t xt) = e AKt x) + e AKt τ) Dwτ) BK u eτ))dτ, k x e λxt x) + k ) x D w + + BK u sup eτ) k x x) + k x λ x λ x τ t D w + + BK u sup eτ) τ t ) = ɛ x, where e A Kt k x e λxt, for some k x > and λ x > Thus, since e is bounded for all t, the state x is also bounded, ie xt) ɛ x for all t Hence, the closed-loop system is forward complete [34], and thus the trajectories of the closedloop system do not have a growth faster than exponential one The rest of the result is deduced from the comparison principle [3] applied to 32) and using 33) to provide the corresponding bounds for e 45 MULTIPLE-OUTPUT MULTIPLE-INPUT CASE 45 Nonlinear Injection Design The observer 4) can be applied for the case m, p 2 under the following assumption Assumption 4 The output of the system )-2) is such that the observability index σ j for each output y j, j =, m, satisfies σ + + σ m = n Then, the observer takes the following structure: ˆx = P A P ˆx + P A y + Bu P R[y C ˆx], 34) where the nonsingular matrix P R n n satisfies the following canonical form where A = diag PAP = A + A CP, 35) σ σ,, σ m σ m R n n,

11 H RÍOS ET AL A = diag a a σ σ, a σ+ a σ+σ 2,, σ2 a σ++σ m a σ++σ m R n m, CP = diag T σ, T σ 2,, T σ m σ m Rm n, with a i, i =, n the coefficients of the characteristic polynomial of matrix A, and the nonlinear injection as R[y C ˆx] = R [e y ] R m [e ym ] = Λ r V σ e y h ))k e y Λ r m V σm e ym h m ))k m e ym where k j := k j,,, k j,σj ) T R σj are the vectors of gains, Λ rj ) are the dilation matrices with vectors of weights r j := µ j, 2µ j,, σ j µ j ) T, where µ j, and V σj : R σj R + are positive definite functions implicitly defined by G j V σj, σj ) =, for any σ :=,, σ ) R σ \{},, σm := σ++σ m,, σ++σ m ) R σm \{}, j =, m, and G j V σj, σj ) := T σ j Λ rj V σj )P σj Λ rj V σj ) σj, j =, m, with positive definite matrices P σj = P T σj R σj σj, and r j := + σ j )µ j, + σ j 2)µ j,, ) T R σj, j =, m Remark 6 Due to Assumption 4, the observer 34) takes a diagonal block form for each output error e yj = y j c j ˆx, j =, m, where c j is the j th row of the matrix C Then, applying the results given by Theorem 4 for each output error e yj, it is possible to form the positive definite matrix P = diagp σ,, P σm ) R n n, where each P σj = P T σj R σj σj, j =, m, is the solution of the set of LMIs 2) for each e yj, j =, m Based on Theorem 4, the observation results for the multiple-output case can be deduced as an extension of the results obtained for the single output case applied in a block form for every output Therefore, the estimation error e will converge in a finite time to a neighborhood of the origin, ie et) λ /2 min PT P P), t T + )) 452 Matrix Feedback Gain Design The results given by Theorem 5 can be directly applied for the case m, p 2 defining the matrix P T in 23) as P T := P T P P R n n Based on Theorem 5, if the matrix inequality 23) is feasible for the new P T, then the ellipsoid EP x ) = { x R n : x T Px x < } will be exponentially attractive for the closed-loop system Thus, the matrix feedback gain is taken as K u = Y u Px, with Px R n n and Y u R p n the solutions of the set of LMIs given by Proposition with P T = P T P P 453 Closed-Loop Stability Based on the previous results, the following theorem is provided for the complete closed-loop stability for the multiple-output multiple-input case Theorem 7 Let Assumption 2, 3 and 4 hold Let observer 34) designed according to Theorem 4 in a block form, and the linear control u = K uˆx designed according to Theorem 5, be applied to the

12 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS system ) for the case m, p 2 Then the trajectories of the closed-loop system satisfy et) β )µt + V ) µ ) +n )µ µ, t < T + )), 36) λ /2 min PT P)λ min P )) µ+n )µ) 2+2n )µ xt) ɛ x, t, 37) with V ) µ = λ min P T P)λ max P )) µ 2 e) µ, ɛ x >, and Proof The proof follows the same steps as the proof of Theorem 6 et) λ /2 min P T ), t T + )), 38) xt), as t λ /2 min P x ) 39) 5 SIMULATION RESULTS 5 SPACECRAFT Consider the following linear simplified model of a spacecraft [5] ẋ = x ) y = x, u wt), where x and x 2 are the rotation angle and angular velocity of the spacecraft, respectively; x 3 is the generalized coordinate and x 4 is its velocity The initial conditions are x = 8,, 8, ) T and the perturbation term wt) = ) ) sint) + sint) , Q sint) + sint) + 5 w = It is possible to check that Assumptions 2 and 4 are satisfied, with ϕ = 59 and σ = σ 2 = 2, respectively The transformation matrix P takes the following form P = 426 For this example, the two nonlinear injections R and R 2 are designed in the same way Then, selecting β = 5, γ = 425, δ =, µ = 6, and fixing ϕ = 59, the ellipsoidal matrix and the gains, obtained from the solution of the LMI problem stated in Theorem 4 for the observer 4), are the following P σ = P σ2 = ) 6323, k = k 2 = ) T

13 2 H RÍOS ET AL For the matrix feedback gain design, let us fix α x =, then the solutions of the LMIs 24)-25), given by Proposition, are the following P x = , R x = , K u = ) In order to have a reference for the improvements with respect to the classic linear design control schemes, the following Luenberger observer-based feedback control is implemented u = K ˆx, ˆx = Aˆx + Bu + Ly C ˆx) The control gain matrix K and the observer gain matrix L are calculated by means of the AEM see Appendix 6) The obtained results are the following K = ) , L = The simulations have been done using the explicit Euler method with a sampling time equal to seconds while the solutions for the LMIs were found using SeDuMi solver among YALMIP in MATLAB The results are depicted in the Figures -4 From Figures -4 one may conclude that if the control based on homogeneous differentiator is applied, then the trajectories of the system converge to a smaller region than the trajectories with the control based on AEM design Moreover, for the control with homogeneous differentiator the trajectories of the system do not present an overshoot, and a better precision for the state estimation is achieved than with Luenberger observer Note that in the Figures 2 and 4 the presence of oscillations can be detected in the state and in the control signa This effect is caused by the homogeneous differentiator structure itself and it might be seen as a trade-off with the faster convergence rate It is noteworthy that it is possible to tune the parameters of the homogeneous differentiator, specifically µ, to reduce this effect with the cost of having a slower convergence rate 6 CONCLUSIONS This work deals with the design of a robust control for linear systems with external disturbances based on a homogeneous differentiator observer Sufficient conditions for stability of the closedloop system in the presence of external disturbances are obtained and represented by LMIs The parameter tuning for both controller and observer is formulated as the semi-definite programming problem with LMIs constraints Simulation results illustrate the feasibility of the proposed approach and some improvements with respect to the classic linear observer approach The following auxiliary result is required APPENDIX

14 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS 3 5 x ˆx x 2 ˆx Time [sec] 5 x 3 ˆx x 4 ˆx Time [sec] Figure Real State vs Estimated State In this figure the state estimation given by the homogeneous differentiator observer is illustrated The fast rate of converge can be seen in these graphs Control based Linear Obs Control based Homogeneous Diff Obs xt) Time [sec] Figure 2 Trajectories of the Real System The trajectories of the system when the control based homogeneous differentiator observer is used converge to a smaller region than the trajectories for the control based Luenberger observer and there does not appear overshoot

15 4 H RÍOS ET AL Control based Linear Obs Control based Homogeneous Diff Obs et) Time [sec] Figure 3 State Estimation Error It is evident that the homogeneous differentiator observerprovides a better precision for the state estimation than the Luenberger observer based on AEM Control based Linear Obs Control based Homogeneous Diff Obs ut) Time [sec] Figure 4 Control Signal The signal control based on the Luenberger observer spends more energy during the transient than the homogeneous differentiator observer After this transient, the corresponding control signals are practically the same for both control approaches Lemma Let P R n n, P = P T >, the vectors R n, h =,,, ) T R n, and the scalar δ, ) Consider the function GV, ) defined as in 3) If GV, ) = and the inequality 5) is fulfilled, then V h) V ) with R the first component of the vector δ /2+2n )µ), 4) Proof Let us introduce the notation V h := V h) and V := V ) Denote the elements of matrix P as p ij It is clear that for = the inequality 4) holds trivially Then, let us consider the case when Note that GV h, h) = implies that 2 ht P h = V 2+2n )µ h Thus, GV, ) =,

16 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS 5 for >, can be rewritten in the following form V +n )µ h p 2 n T Λ r V )P Λ r V ) V +n )µ h p 2 n = 4) ) The case when < can be analyzed analogously Denote p = p2 p p,, n p R n and p 22 p 2n P = R n ) n ) p n2 p nn Hence, for >, 4) can be rewritten as follows where i = i V i)µ 42) is rewritten as 2 n T /V +n )µ ) p T 2 p P = n V V h ) 2+2n )µ, 42) h, i = 2, n Equivalently, in terms of = 2,, n ) T R n, V V h ) 2+2n )µ = T P + p T + T p + Since P >, the right hand side of the last equality has a minimal value Thus, it can be shown that this value will be attained for = p, ie P V V h ) 2+2n )µ p T P p Finally, the inequality 5) implies that p T obtained and the theorem is proved P p δ, with δ, ) Therefore, 4) is Proof of Theorem 4: The error dynamics given in ) is rewritten as follows ie = A + Λ r V h))kh T ) + d w + d, where d w = Dw, and d = Λ r V h ) Λ rv ) ) kh T To prove the convergence of to zero, let us compute the time derivative of the function V by means of the implicit function theorem, ie Then, it follows that [ ] G G V = V > G = V T Λ r V ) P H + HP ) Λ r V ), V and G = 2T Λ r V )P Λ r V ) A + Λ r V h))kh ) ) T + d w + d

17 6 H RÍOS ET AL Considering the homogeneity properties of the dilation matrix, ie one obtains G = 2V µ Λ r V Λ r V )A [Λ r V )] = V µ A, h))kh T = V µ [Λ r V )] kh T Λ r V ), [ T Λ r V ) P A + P kh ) T Λ r V ) +V µ Let us introduce the extended vector ξ, ie ξ := Λ rv ) Λ r V )d Λ r V )d w Adding and subtracting the following terms V µ V µ ] T Λ r V )P Λ r V )d w + V µ Λ r V )P Λ r V )d 43) T Λ r V ) HP + P H) Λ r V ), β d T Λ r V )P Λ r V )d, V µ γϕ dt wλ 2 rv )d w, where ϕ := Λ r V )d w ϕ := D w + for V > Then, from 43), it is given that G = V µ ξ T Ωξ T Λ r V ) HP + P H) Λ r V ) ) +γϕ ϕ2 + β V 2µ d T Λ r V )P Λ r V )d, where P A + A T P + P kh T + hk T P + HP + P H V µ P V µ P Ω = V µ P β V 2µ P 44) V µ P V 2µ γϕ In Applying Schur s complement to 44) and using the fact that ϕ ϕ, one obtains LMI 6) Thus, the term G is upper bounded for V > as follows with G V µ [ T Λ r V ) HP + P H) Λ r V ) + γϕ + βs, V, V h ) ], 45) S, V, V h ) = β 2 V 2µ d T Λ r V )P Λ r V )d Taking into account that 2 ht P µ h = V h) 2+2n )µ, it follows that S, V, V h ) = kt Λ r V h ) Λ rv ) ) Λ r V )P Λ r V ) Λ r V h ) Λ r) ) k, β 2 h T P hv h /V ) 2 2n )µ and since V +n )µ h Λ r V h ) = Λ rv h ), denoting λ = V h /V, it is given that S, V, V h ) = kt I n Λ r λ)) Λ r λ)p Λ r λ) I n Λ r λ)) k λ 2µ β 2 h T P h Now, from Lemma, it follows that λ [, λ ], with λ = δ /2+2n )µ) Then, inequality 7) implies that S, V, V h ), for any λ [, λ ] Hence, 45) can be upper bounded for all V >

18 ROBUST OUTPUT-CONTROL FOR UNCERTAIN LINEAR SYSTEMS 7 as G µ V T Λ r V ) HP + P H) Λ r V ) + γϕ + β ) 46) For the time derivative of V, it is given that V T Λ r V = + ) HP + P H) Λ r V ) + γϕ + β T Λ r V ) HP + P H) Λ r V ) γϕ + β T Λ r V ) HP + P H) Λ r V ) V µ ) V µ, V > Then, if GV, ) = and β 2 := γϕ + β is such that LMI 8) is satisfied, it follows that for all V > ) β V 2 + T Λ r V )P Λ r V ) T Λ r V ) HP + P H) Λ r V V µ β ) V µ 47) ) Therefore, since T Λ r V )P Λ r V ) = it follows that λ min P ) 2 T P Thus, the estimation error e converges in a finite time to a neighborhood of the origin given by 9) Finally, the computation of the settling time function 2) is straightforward from 47) Hence, the theorem is proven Proof of Theorem 5: Under Assumption 3 the matrix K u always exists, then due to Assumption 2 and the statements of Theorem 4 it follows that w T Q w w, e T P T e, 48) for the nonsingular matrix P defined in 5), and the positive definite matrix P = P T > defined in Theorem 4 such that P T = P T P P, and Q w = Q T w > such that w + = λ min Q w )) /2 Let us consider the quadratic function V x x) = x T Px x, with P x = Px T > Its derivative along the trajectories of the system 8) is V x = x e w = x e w T P x A K + A T K P x + 2α x Px Px BK u Px D n n n q q q T P x A K + A T K P x 2α x x T P x x + α x e T P T e + α x w T Q w w, + α x Px Px BK u Px D α x P T n q α x Q w x e w x e w for some positive scalar α x Taking into account the assumptions given in 48), it follows that V x 2α x x T Px x ) + x T e P x A K + A T K P x + α x Px Px BK u Px D α x P T n q w α x Q w x e w Therefore, if the matrix inequality 23) holds, then the time derivative of V x is negative definite outside the ellipsoid x T Px x, which implies by [5] that EP x ) is the attractive ellipsoid of the closed-loop system 8) 6 AEM Linear Case Let us consider, based on [5], the linear version of the AEM to compare the results and the implementability of the approach presented in this paper The linear AEM formulation consists

19 8 H RÍOS ET AL in using an observer based feedback u = K ˆx, where K R m n is a constant gain and ˆx is the state estimation whose dynamics is given by the classic Luenberger observer, ie ˆx = Aˆx + BK ˆx + Ly C ˆx), with the matrix gain L R n m Define the estimation error as e = x ˆx Now, let us select a quadratic storage function V c = ˆx T P cˆx + e T P e e, with P c = P T c > and P e = P T e > The time derivative of the function V c can be written as V c = 2ˆx T P c [A + BK)ˆx + LCe] + 2e T P e [A LC)e + Dw] = ˆx T e P ca + BK) + A + BK) T P c P c LC P e A LC) + A LC) T P e P e D ˆx e ω ω Then, adding and subtracting α c x T P c x = α c ˆx + e) T P c ˆx + e) and α c ω T Qω, it is given that where z c = ˆx T, e T, w T ) T, and W = V c = zc T W z c α c x T Pc x + α c ω T Qω, P ca k + A T k P c + α c P c P c LC + α c P c P e A l + A T l P e + α c P c P e D, 49) α c Q where A k = A + BK and A l = A LC Thus, taking into account that ω T Qω, if W then an upper bound for the derivative of V c is given by V c α c x T P c x) To facilitate the implementation it is possible to approximate the nonlinear inequality W by applying the quadratic transformation W = T W T T with T = diagp e Pc, I n, I n ) Then, let us define Y := KPc and Y 2 := P e L, and inroduce R and R 2 such that ) P e AP c + BY + Y T B T + Pc A T + α c Pc Pe R <, < P c R 2 Applying Λ inequality and Schur s complement it follows that R 2Pc I n I n AP c ) + BY + APc + BY ) T + α c Pc, 5) ) R2 I n 5) I n P c Therefore, W is upper estimated by W 2, ie W W 2, where W 2 = R Y 2 C + α c P e P e A Y 2 C + P e A Y 2 C) T + α c R 2 P e D 52) α c Q Hence, it is clear that if W 2, then W is true, and the ellipsoid characterized by the matrix P c is an attractive ellipsoid for the system ) Now, to minimize the size of the ellipsoid, the function trpc ) is selected as the objective, and the following optimization problem is proposed trp c ) min P c,p ey,y 2,R,R 2 subject to 5), 5), 52)

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