Robust Performance Analysis of Affine Single Parameter-dependent Systems with. of polynomially parameter-dependent Lyapunov matrices.

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1 Preprints of the 9th World Congress he International Federation of Automatic Control Robust Performance Analysis of Affine Single Parameter-dependent Systems with Polynomially Parameter-dependent Lyapunov Matrices Mahdieh Sadat Sadabadi and Alireza Karimi Automatic Control Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland ( s: and Abstract: his paper addresses the problem of H and H 2 performance analysis of continuoustime affine single parameter-dependent systems with polynomially parameter-dependent Lyapunov matrices First, some necessary and sufficient conditions in terms of linear matrix inequalities (LMIs) are provided using (D, G) scaling approach hen, the results are used to deal with the problem of fixed-order H and H 2 controller design via bilinear matrix inequalities (BMIs) in a nonconservative way Simulation results demonstrate the effectiveness of the proposed approach INRODUCION Robust performance analysis of uncertain linear time invariant (LI) systems using linear matrix inequalities (LMIs) is a challenging issue in the robust control theory he starting point for dealing with this problem is based on the concept of quadratic stability which considerably suffers from the conservatism imposed by the use of fixed Lyapunov matrix (eg Palhares et al [997) o improve the results and reduce the conservatism, special structures for the Lyapunov matrices have been considered, such as affine linear parameter-dependent (Feron et al [996, Peaucelle et al [2, de Oliveira et al [24a,b, Ebihara and Hagiwara [26) and more recently, (homogeneous) polynomially parameter-dependent Lyapunov matrices (Chesi et al [25a,b, Oliveira and Peres [25, 27, Ebihara et al [29, Zhang et al [2) In Ebihara and Hagiwara [26 and Ebihara et al [29, necessary and sufficient conditions in terms of LMIs for robust Hurwitz stability analysis of affine single parameter-dependent matrices with polynomially parameter-dependent Lyapunov matrices have been proposed hen, their results have been extended to deal with the affine multiple parameter-dependent systems and some sufficient conditions for the existence of the linear parameter-dependent Lyapunov matrices have been developed he proposed results of Ebihara et al [29 are applicable to H performance analysis problem of systems with an affine single parameter-dependent state matrix A and parameter-independent matrices (B,C) In this paper, the problem of H and H 2 performance analysis of continuous-time LI affine single parameterdependent systems is formulated as a set of LMIs by means of polynomially parameter-dependent Lyapunov matrices he proposed approach is based on (D, G) scaling his research work is financially supported by the Swiss National Science Foundation under Grant No (Meinsma and Fu [997, Ebihara [28, Zhang et al [2) which can convert inequality conditions depending on an uncertain parameter, being hence a feasibility problem of infinite dimension, to finite-dimensional LMIs he obtained results can be used to provide some necessary and sufficient conditions in terms of Bilinear Matrix Inequalities (BMIs) for fixed-order controller design of affine single parameter-dependent systems o the best of our knowledge, this is the first time that such necessary and sufficient conditions for fixed-order control synthesis have been developed he contributions of this work with respect to Ebihara et al [29 are the extension of the results to H performance analysis in the case of the affine single parameter-dependent uncertainty in all matrices A, B, andc; H 2 performance analysis; fixed-order controller design: a BMI-based approach he organization of the paper is as follows: Section 2 presents some preliminaries and the problem statement Sections 3 and 4 are devoted to the main results he problem of fixed-order controller synthesis is presented in Section 5 Simulation examples are given in Section 6 he paper ends with concluding remarks in Section 7 he notation used in this paper is standard In particular, I n and n are the n n identity matrix and the zero vector of dimension n, respectively he symbols trace(a),, and A(i : j, :) denote the trace of matrix A, the Kronecker product, and the extraction of the i th through the j th row of matrix A, respectively he symbol He{A} is a notation for A+A For the simplicity of presentation, symbol indicates the symmetric blocks 2 PRELIMINARIES Consider a continuous-time LI affine single parameterdependent system as follows: Copyright 24 IFAC 64

2 ẋ = A(θ)x + B(θ)u () y = C(θ)x x R n, u R m,andy R p he real matrices A, B, andc are of appropriate dimensions It is assumed that the triplet (A, B, C) belongs to the following uncertainty domain: A(θ) =A + θa B(θ) =B + θb (2) C(θ) =C + θc θ [,, without loss of generality he assumptions about the system are as follows: Matrix A(θ) is robustly stable, its eigenvalues lie in the open left-half plane for all θ [, he system is strictly proper which is a reasonable assumption since all physical systems are strictly proper Moreover, it is a necessary assumption to have a bounded H 2 norm he transfer matrix from the input u to the output y is given by: H(s, θ) =C(θ)(sI A(θ)) B(θ) (3) Lemmas and 2 provide some necessary and sufficient conditions for the satisfaction of H and H 2 norm bounds of transfer matrix H(s, θ) given in (3) Lemma (Bounded Real Lemma for continuous-time systems) he inequality H(s, θ) < γ holds for all θ [, if and only if there exists a symmetric matrix P (θ) > such that (Boyd et al [994): A(θ) P (θ)+p(θ)a(θ) +γ C(θ) C(θ) +γ P (θ)b(θ)b(θ) (4) P (θ) < Lemma 2 he inequality H(s, θ) 2 2 < υ holds for all θ [, if and only if there exist symmetric matrices P (θ) > andq(θ) > such that (Boyd et al [994): A(θ) P (θ)+p(θ)a(θ)+p(θ)b(θ)b(θ) P (θ) < (5) P (θ) C(θ) > C(θ) Q(θ) (6) trace(q(θ)) υ< It should be noted that P (θ) andq(θ) in Lemma and Lemma 2 do not have special structures Lemma 3 ((D, G) Scaling) Let Φ R n(k+) n(k+) hen, the following matrix inequality (Zhang et al [2): (θ [k I n ) Φ(θ [k I n ) < (7) θ [k = [ θθ 2 θ k,holdsforallθ [, if and only if there exist a positive-definite matrix D and a real skew-symmetric matrix G R nk nk R nk nk such that: Φ+ n k (D, G) < (8) n k (D, G) Rn(k+) n(k+) is defined as follows: n k (D, G) = [ Īn k Ĩn k [ D G G D [Īn k Ĩn k ; k (9) Īk n =[I k k I n Ĩk n =[ () k I k I n and n (D, G) = he inequality in (8) is an LMI with respect to the matrices Φ, D, andg 3 MAIN RESULS In this section, some necessary and sufficient conditions in terms of LMIs for the existence of a linearly parameterdependent Lyapunov matrix P (θ) andq(θ) given by: P (θ) =P + θp () Q(θ) =Q + θq (2) which satisfy (4), (5), and (6), are provided he main idea behind of these conditions is (D, G) scaling approach which can be used to convert the positivity (negativity) of a polynomial matrix in the following form into the feasibility problem of some LMIs independent on θ N J(θ) = θ k J k, θ [, (3) k= J k (k =,,, N) are Hermitian matrices 3 H Performance Analysis by Means of Linearly In this part, the problem of H performance analysis via linearly parameter-dependent Lyapunov matrices is proposed and the results are given in the following theorem heorem For an affine single parameter-dependent system, the inequality given in (4) holds with P (θ) in () if and only if there exist symmetric matrices P and P, D>, and a real skew-symmetric matrix G of appropriate dimensions such that: P + P > (4) P P > and W + n 2 (D, G) < (5) Y γi m+p { } P W =He P [ A A (6) B Y = P B P + B P B P (7) C C and n 2 (D, G) is given in (9) with k =2 he inequality (5) is an LMI with respect to γ and the matrices P, P, D, andg Proof: First, the inequalities in (4) imply that Lyapunov matrix P (θ) in () is positive definite hen, the inequality given in (4) can be rewritten as follows: I n Φ I n < (8) θ 2 I n θ 2 I n Φ =He { } P P [ A A + γ [ P B P B + P B + γ C C [ C C [ P B P B + P B (9) 642

3 Based on (D, G) scaling, (8) is equivalent to the following inequality: Φ + n 2 (D, G) < (2) he last term in the inequality (9) contains the product of unknown Lyapunov matrices; therefore, the inequality (2) is not an LMI Schur Complement lemma (A Albert [969) can be applied on (9) to convert it to LMI in (5) 32 H 2 Performance Analysis by Means of Linearly he results of H 2 performance analysis with linearly parameter-dependent Lyapunov matrices are presented in heorem 2 heorem 2 For an affine single parameter-dependent system, the inequalities in (5) and (6) hold with P (θ) and Q(θ) given in () and (2), respectively if and only if there exist symmetric matrices P, P, Q and Q,apositive definite matrix D, and a real skew-symmetric matrix G of appropriate dimensions such that: W + n 2 (D, G) < (2) Y ( : m, :) I m [ P + P C + C > (22) C + C Q + Q [ P P C C > (23) C C Q Q trace(q + Q ) <υ (24) trace(q Q ) <υ W and Y are defined in (6) and (7), respectively he above inequalities are LMIs with respect to υ and the matrices P, P, Q, Q, D, andg Proof: he inequality given in (5) can be rewritten as follows: I n θ 2 I n Φ 2 I n θ 2 I n < (25) { P } Φ 2 =He P [ A A [ P B [ P B (26) + P B + P B P B + P B By applying (D, G) scaling approach and then the Schur Complement lemma on (25), the inequality given in (2) is obtained Inequalities in (6) are linear with respect to θ herefore, they hold for all θ [, if and only if they are satisfied just for θ = and θ = In this way, the other inequalities in (22)-(24) result In the sequel, the problem of H and H 2 analysis based on polynomially parameter-dependent Lyapunov matrices is proposed 4 EXENSION O POLYNOMIALLY PARAMEER-DEPENDEN LYAPUNOV MARICES It has been shown in Bliman [24 that the inequalities in (4)-(6) depending continuously on the scalar parameter θ have polynomial type solutions with respect to θ of the following form: N P N (θ) = θ i P i (27) Q N (θ) = i= N θ i Q i (28) i= with sufficiently high degree N herefore, in this section, we are interested in the extension of the previous results to polynomially parameter-dependent Lyapunov matrix P N (θ) andq N (θ) he results are summerized in the following subsections As the degree of the polynomials P (θ) and Q(θ) increases, the results converge to the optimal ones Finally, necessary and sufficient conditions for the robust performance analysis of affine single parameterdependent systems are derived 4 H Performance Analysis by Means of Polynomially heorem 3 For an affine single parameter-dependent system, the inequality in (4) holds with P N (θ) given in (27) if and only if there exist symmetric matrices P i for i =,,,N, positive definite matrices D and L, and real skew-symmetric matrices G and K such that: WN + n N+ (D, G) < (29) Y N γi m+p 2P P P j P j P P j+ 2 P j P + n j (L, K) < (3) 2j P j P j+ P 2j 2P 2j N 2 if N is even j = (3) N+ 2 if N is odd P P W N =He P [ A A (32) N Y N = P B C P B + P B C + P 2 B P N B + P N B P N B (33) and n N+ (D, G) isdefinedin(9)withk = N + Note that if N is an odd number, 2j = N +; therefore, P 2j = is considered Proof: he inequality given in (3) directly expresses the positivity of matrix P N (θ) based on (D, G) scaling 643

4 approach in two cases N is either even or odd he remaining of the poof is similar to that of heorem 42 H 2 Performance Analysis by Means of Polynomially heorem 4 For an affine single parameter-dependent system, the inequalities in (5) and (6) hold with P N (θ) and Q N (θ) given in (27) and (28), respectively if and only if there exist symmetric matrices P i and Q i for i =,,,N, positive definite matrices D, L, andh and skew-symmetric matrices G, K, and such that: WN + n N+ (D, G) < (34) Y N ( : m, :) I m 2Z Z Z j Z j Z Z j+ 2 Z j Z + n+p j (L, K) < 2j Z j Z j+ Z 2j 2Z 2j (35) 2(q υ) q q j q j 2 q q j+ q j q 2j q j q j+ q 2j 2q 2j + j(h, ) < (36) j, W N,andY N are given in (3), (32), and (33), respectively Pi C i for i =, C i Q i Z i = (37) Pi for i =2,,N Q i q i = trace(q i ); i =,,,N (38) Note that Z 2j =andq 2j =ifn is an odd number Proof: he inequalities given in (5) and (6) can be rewritten in the form of (3) as follows: I n θ N+ I n Φ 2N I n θ N+ I n < (39) (Z + Z θ + + Z N θ N ) < (4) q υ + q θ + + q N θ N < (4) Φ 2N = W N + YN ( : m, :)Y N ( : m, :) (42) Z i and q i for i =,,N are defined in (37) and (38) hen, by applying (D, G) scaling approach to the inequalities in (39)-(4), the linear matrix inequalities in (34)-(36) are obtained 5 FIXED-ORDER H AND H 2 CONROLLER DESIGN Results of heorem 3 and 4 can be utilized to design fixedorder dynamic output feedback controllers for affine single parameter-dependent systems represented by the following state space realization: ẋ g (t) =A g (θ)x g (t)+b g (θ)u(t)+b w (θ)w(t) z(t) =C z (θ)x g (t)+d zu (θ)u(t) (43) y(t) =C g x g (t) the matrices A g (θ), B g (θ), B w (θ), C z (θ), and D zu (θ) belong to the following uncertainty area: A g (θ) =A g + θa g B g (θ) =B g + θb g B w (θ) =B w + θb w (44) C z (θ) =C z + θc z D zu (θ) =D zu + θd zu θ [, he signals x g R n, u R ni, w R r, y R no,andz R s are the state, the control input, the exogenous input, the measured output, and the controlled output, respectively he objective is to design a stabilizing dynamic outputfeedback controller satisfying some H and H 2 norm bounds on the transfer function between w and z, H zw (s, θ) he fixed-order controller K c (s) ispresented as follows: ẋ c (t) =A c x c (t)+b c y(t) (45) u(t) =C c x c (t)+d c y(t) A c R m m and B c, C c,andd c are of appropriate dimensions hen, the closed-loop system H zw (s, θ) has the following state space realization: ẋ(t) =A(θ)x(t) +B(θ)w(t) (46) z(t) =C(θ)x(t) x(t) =[x g (t) x c (t) and Ag (θ)+b A(θ) = g (θ)d c C g B g (θ)c c B c C g A c Bw (θ) (47) B(θ) = C(θ) =[C z (θ)+d zu (θ)d c C g D zu (θ)c c he closed-loop matrices A(θ), B(θ), and C(θ) belongto (2); Ag + B A = g D c C g B g C c ; B c C g A c Ag + B A = g D c C g B g C c ; (48) Bw Bw B = ; B = ; C =[C z + D zu D c C g D zu C c ; C =[C z + D zu D c C g D zu C c Remark: If the matrix C g belongs to (2) and B g is fixed, the same results are obtained; however, the case of uncertainty in both matrices (B g,c g ) has not been considered in this contribution Inequalities given in (29) and (3) are necessary and sufficient conditions for fixed-order H controller design In this case, H performance constraints are expressed by a set of BMIs by means of polynomially parameter-dependent Lyapunov matrices 644

5 Similar to the H control problem, the problem of fixedorder H 2 dynamic output-feedback design of the affine single parameter-dependent systems described by (43) can be reformulated as an optimization problem subject to a set of BMI and LMI constraints in (34)-(36) o solve optimisation problems involving BMI constraints, several local and global approaches have been developed in the literature (eg Safonov et al [994, Kocvara and Stingl [26, Kanev et al [24, Dinh et al [22) hese methods can be conveniently employed to solve the BMI problem of fixed-order controller design proposed in this paper 6 SIMULAION EXAMPLES In this section, three illustrative examples are provided It should be noted that LMI problems in the examples are solved using YALMIP (Löfberg [24) as the interface and SDP3 (oh et al [999) as the solver Example : In this example, the problem of H analysis of the following stable affine single parameter-dependent system is considered [ A = B = C = [ ; B = ; A = [ ; C = [ [ (49) he state matrices A and A are borrowed from Ebihara and Hagiwara [26 hey have shown that there does not exist a linearly parameter-dependent Lyapunov matrix for the stability analysis of A(θ) =A +θa for all θ [, By applying a fine griding on θ, the worst case H norm of the system can be computed, H(s, θ) wc = 5336 he results of Sections 3 and 4 are applied to determine the upper bound of H norm of the system (A(θ),B(θ),C(θ)) he LMI conditions in (4) and (5) become infeasible by linearly parameter-dependent Lyapunov matrices However, the second-degree polynomially parameter-dependent Lyapunov matrices reach to the worst case H norm Example 2: As the second example, consider the following system borrowed from de Oliveira et al [24a: A = A = (5) B = B = ; C = C = he worst case H 2 norm of the system is H(s, θ) 2 wc = 492 (a fine girding has been used) able shows the upper bound of H 2 norm of the system (A(θ),B,C) and the dual system (A (θ),c,b ) with polynomially parameter-dependent Lyapunov matrices of order N In able, the symbol indicates that the LMIs in (34)-(36) able Upper bound of H(s, θ) 2 in Example 2 with polynomially parameter-dependent Lyapunov matrices of order N N H(s, θ) H (s, θ) able 2 Upper bound of H zw (s, θ) Example 3 Method γ K c Crusius et al, [ Shaked, [ Dong et al, [ Sadabadi et al, [ Proposed method with N = 662 [ Proposed method with N = [72e3 2234e3 are infeasible with the polynomially parameter-dependent Lyapunov matrices P N (θ) andq N (θ) of the related order N By increasing the order of polynomially parameterdependent matrices P N (θ) andq N (θ) in (27) and (28), better results are obtained and finally the worst case H 2 norm can be reached with P N (θ) and Q N (θ) of order N =4 Example 3: Consider the continuous-time polytopic system with two vertices in Dong and Yang [23 he system can be easily converted to an affine single parameterdependent system in (43);, [ [ A g(θ) = θ B g = + θ ; B w = 3 C g = [ ; C z = [ D zu = [ (5) θ [, he objective here is to design a static output feedback H controller with polynomially parameter-dependent Lyapunov matrices o this end, an optimization problem, which is the minimization of γ subject to the LMI and BMI constraints in (4) and (5), should be solved he BMI constraints are solved by using PENBMI, version 2, (Kocvara and Stingl [26) with initial controller K c = [ he problem is solved after 3 iterations (CPU time = 9823sec) Resulting static output feedback is given in able 2 he results are compared with the LMI-based methods in Crusius and rofino [999, Shaked [23, Dong and Yang [23, Sadabadi and Karimi [23 It can be observed from able 2 that the proposed BMI-based method in this paper with polynomially parameter-dependent Lyapunov matrices of order two leads to the best results among the others in 645

6 7 CONCLUSION In this paper, the necessary and sufficient LMI conditions for H and H 2 performance analysis of continuous-time affine single parameter-dependent systems with polynomially parameter-dependent Lyapunov matrices have been proposed he fundamental idea of the proposed approach is based on the use of (D, G) scaling which can convert inequality conditions depending on an uncertain parameter to a set of parameter-independent LMIs he results have been employed to develop the necessary and sufficient conditions for fixed-order H and H 2 dynamic output feedback controller design for affine single parameterdependent systems in terms of BMIs he extension of the results to the robust performance analysis/controller synthesis of affine multi parameter-dependent systems by linearly parameter-dependent Lyapunov matrices leads to only sufficient conditions REFERENCES A Albert Conditions for positive and nonnegative definiteness in terms of pseudoinverses SIAM Journal on Applied 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