Filter Design for Feedback-loop Trade-off of L 1 Adaptive Controller: A Linear Matrix Inequality Approach

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1 AIAA Guidance, Navigation and Control Conference and Exhibit August 2008, Honolulu, Hawaii AIAA Filter Design for Feedback-loop Trade-off of L 1 Adaptive Controller: A Linear Matrix Inequality Approach Dapeng Li, Naira Hovakimyan University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Chengyu Cao University of Connecticut, Storrs, CT, 06269, USA Kevin A. Wise The Boeing Company, Integrated Defense Systems, St. Louis, MO 63166, USA This paper presents a convex optimization method for the feedback-loop tradeoff of L 1 adaptive controller. Both problems of performance improvement and time-delay margin maximization are shown to be cast into Linear Matrix Inequality (LMI) type conditions. First, each of these conditions is studied separately towards a distinct objective, and next two similar LMI algorithms are proposed for optimization of one of the objectives with a prespecified constraint on the other. I. Introduction References 1,2 introduced a new paradigm for design of adaptive control systems, known as L 1 adaptive controller, that leads to uniform performance bounds and guaranteed time-delay margin. 3 Reference 4 further addressed the problem of filter optimization for L 1 adaptive controller. The objective was to determine a systematic design methodology for design of the underlying filter that would improve the performance dependent upon the L 1 gain of the cascaded system, 1,2 meantime maximizing the time-delay margin. 3 A multi-objective optimization algorithm, realized by minimizing a weighted combination of the time-delay margin and the L 1 -gain of the cascaded system, 1,2 was therefore proposed. A MATLAB optimization solver was employed, but no converging results were obtained. The so-called optimal results came from the first few iterations of function optimization. The choice of the weighting factor for trade-off between the performance and robustness was not straightforward. This drawback motivated the approach of this paper. In this paper, we develop a Linear Matrix Inequality (LMI) approach for L 1 -gain and timedelay margin optimization. 3 The L 1 -gain (or peak-to-peak gain) calculation for discrete-time linear This material is based upon work supported by the United States Air Force under Contract No.FA and NASA under Contracts NNX08AB97A and NNX08AC81A. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force and NASA. Corresponding author dpli@vt.edu. Graduate Research Assistant, Student Member AIAA, Department of Mechanical Science and Engineering; li63@illinois.edu (Corresponding Author). Professor, Associate Fellow AIAA, Department of Mechanical Science and Engineering; nhovakim@illinois.edu Research Assistant Professor, Member AIAA,Department of Mechanical Engineering; ccao@engr.uconn.edu Copyright 2008 by Dapeng SeniorLi. Technical Published by Fellow, the American Associate Institute Fellow of Aeronautics AIAA, and kevin.a.wise@boeing.com Astronautics, Inc., with permission.

2 systems has been studied extensively since early 1980 s (see Ref. 5 and references therein). In Ref. 6 discretization is used to address the problem of L 1 -gain computation for continuous-time systems. Reference 7 derives an upper bound for the L 1 -gain by solving a LMI feasibility problem. This LMI approach is further developed in this paper to solve the L 1 -gain minimization problem for performance improvement. On the other hand, delay-dependent as well as delay-independent stability criteria for time-delay LTI systems have been developed in vast number of references such as Refs, 8 10 which can also be cast into LMI-type conditions. These LMIs allow for computation of a lower bound on the time-delay margin. Reference 8 proves that the delay-time dependent stability criteria are equivalent to small-gain type argument, and therefore can be easily transformed into LMI-type conditions. In this paper, we follow the LMI conditions from Ref. 10 for time-delay margin maximization of the L 1 adaptive controller. The approach in this paper unfolds in two steps: first two LMI feasibility problems are formulated for verification of the upper bound on the L 1 -gain of the cascaded systems and the lower bound of the time-delay margin of the corresponding LTI system respectively. Note that following standard arguments in LMI literature, 11 these problems can be cast into generalized eigenvalue problems (GEVP). We formulate two GEVPs respectively. The solution to these GEVPs yields the parameters for the filter design. We also introduce a special structure for the underlying matrices that takes care of the equality constraint related to the unity DC gain of the filter. Next, the constrained optimization result is presented, which allows for optimization of one of the objectives while retaining a prespecified bound on the other. The paper is organized as follows. In Section II, an overview of L 1 adaptive controller is given, followed with a statement of the problem formulation. Section III focuses on unconstrained L 1 - gain upper bound computation and its minimization via LMIs. In Section IV, similar procedure is presented for time-delay margin maximization. In Section V, the constrained optimization result is presented, which allows for optimization of one of the objectives while retaining a prespecified bound on the other. The numerical example in Section VI illustrates the results, while Section VII concludes the paper. II.A. Overview of L 1 Adaptive Control II. Introduction Consider the following single-input single-output system dynamics: ẋ(t) = A m x(t) + b(u(t) θ x(t)), x(0) = x 0 y(t) = c x(t), (1) where x R n is the system state vector (measurable), u R is the control signal, b,c R n are known constant vectors, A m is a given n n Hurwitz matrix, y R is the regulated output, and the unknown parameter θ belongs to a given compact convex set θ Θ. Let ẋ m (t) = A m x m (t) + bk g r(t), x m (0) = x 0, y m (t) = c x m (t) (2) be the ideal reference system, where x m R n, k g is the feedforward gain achieving asymptotic tracking of step commands: 1 k g = lim s 0 c H(s) = 1 c H(0), (3) and H(s) = (si A m ) 1 b. (4)

3 For the linearly parameterized system in (1), we consider the following state predictor ˆx(t) = A mˆx(t) + b(u(t) ˆθ (t)x(t)), ˆx(0) = x 0 ŷ(t) = c ˆx(t), (5) in which the adaptive law for ˆθ(t) is given by ˆθ(t) = ΓProj(ˆθ(t),x(t) x (t)pb), ˆθ(0) = ˆθ0, (6) where x(t) = ˆx(t) x(t) is the prediction error, Γ R n n = Γ c I n n is the matrix of adaptation gains, P = P is the solution of the algebraic equation A mp + PA m = Q, Q > 0, while Proj denotes the projection operator, 12 ensuring boundedness of the parametric estimates by definition. Letting r(t) = ˆθ (t)x(t), (7) we consider the following filtered adaptive controller: u(s) = C(s) ( r(s) + k g r(s) ), (8) where u(s), r(s),r(s) are the Laplace transformations of u(t), r(t),r(t), respectively, while C(s) is a stable and strictly proper system with unity DC gain C(0) = 1. Letting θ max = max θ Ω n θ i, where θ i is the i th element of θ, the complete L 1 adaptive controller consists of (5), (6), (8) subject to the following upper bound i=1 λ H(s)(1 C(s)) L1 θ max < 1, (9) where H(s)(1 C(s)) L1 is the L 1 -gain of the stable transfer function H(s)(1 C(s)). 1 We define a linear time-invariant reference system using the non-adaptive version of (8) ( ) x ref (s) = H(s) k g C(s)r(s) + (C(s) 1)θ x ref (s) ( ) u ref (s) = C(s) k g r(s) + θ x ref (s) (10) (11) along with the design system y ref (s) = c x ref (s) (12) y des (s) = c G(s)r(s) = C(s)k g c H(s)r(s), (13) ( ) u des (s) = k g C(s) 1 + C(s)θ H(s) r(s). (14) Notice that y des (t) does not depend upon the uncertain parameters and can be used to define the control specifications. The main result from Ref. 1 claims that subject to (9) one has x x ref L γ 1 / Γ c, u u ref L γ 2 / Γ c, (15)

4 where γ 1 = H 2 (s) L1 θmax /λ min (P),H 2 (s) = I + (I Ḡ(s)θ ) 1 (Ḡ(s)θ + (C(s) 1)I),γ 2 = C(s) 1 c o Ho(s)c o L1 θmax /λ min (P) + C(s)θ K L1 γ 1, θ max = max θ Ω n i=1 4θ2 i and further y ref y des L u ref u des L λ 1 λ c L1 G(s) L1 r L, (16) λ 1 λ C(s)θ K L1 G(s) L1 r L. (17) We notice that when C(s) = 1, u ref (t) reduces to the following ideal controller u ideal (t) = k g r(t) + θ x ref (t), (18) and (10) reduces to (2) by canceling the uncertainties exactly. We note that the control law u ref (t) is not implementable, since its definition involves the unknown parameters. However, the L 1 adaptive controller ensures that x(t) and u(t) follow the state x ref (t) and the control signal u ref (t) of this reference system both in transient and steady-state, if the adaptation rate is selected sufficiently large. Reference 1 further provides design guidelines for selection of C(s) to ensure that the output y ref (t) of the reference system in (10) can satisfy desired control specifications. Further we recall the following lemma from Ref. 1 Lemma 1 Let C(s) = the following is true: ω s + ω, ω > 0. For any strictly proper asymptotically stable system H(s) lim (C(s) 1)H(s) L ω 1 = 0. The time-delay margin τ of L 1 adaptive controller has been introduced in Ref. 3 In the presence of fast adaptation it is lower bounded as: where P(H o (s)) is the phase margin of the open-loop system τ T (H o (s)) = P(H o (s))/ω c, (19) H o (s) = C(s)(1 + θ H(s))/(1 C(s)), H(s) = (si Am bθ ) 1 b, (20) and ω c is the cross-over frequency of H o (s). The structure of H o (s) is shown in Fig. 1, where r b is the input signal and ξ(t) is the state. Figure 1. H o(s). The L 1 adaptive control architecture and its complete design and analysis framework is developed in Refs. 1 3

5 II.B. Problem Statement From the relationships in (9), (15) and (16), (17 ) it is straightforward to notice that, in addition to increasing the rate of adaptation Γ c, one needs to select C(s) to minimize H(s)(1 C(s)) L1 for performance improvement, which can be achieved by increasing the bandwidth of C(s), as suggested by Lemma 1. It further follows from (20) that increasing the bandwidth of C(s) will reduce the time-delay margin to zero. For the purpose of identifying the optimal trade-off in between the two objectives, we consider the following two constrained optimization problems separately: Problem 1: min C(s) (1 C(s))H(s) L 1 s.t. C(0) = 1 and τ η, (21) where η > 0 is a pre-specified lower bound on the time-delay margin. Problem 2: max T (H o(s)) C(s) s.t. C(0) = 1 and H(s)(1 C(s)) L1 γ, (22) where γ is a pre-specified upper bound on the L 1 -gain of (1 C(s))H(s). We will use the following dynamical system for the state-space realization of C(s): C(s) : ẋ f = A f x f + B f u f, y f = C f x f, (23) where x f R l is the state of the filter, y f R is the output, and u f R is the input, A f, B f and C f are matrices of appropriate dimensions. The proposed algorithms seek to determine the optimal values for these matrices in the context of the above formulated problems. Although these three matrices can be treated equally, we fix B f in the later discussion to reduce the dimension of the parameter space. III. L 1 -Gain Optimization In this section, we investigate constraint-free L 1 -gain minimization of the cascaded system (1 C(s))H(s) by resorting to the -norm, which serves as an upper bound of the L 1 -gain of the given LTI system. 7 Consider the following LTI system: G : ẋ = Ax + Bv, (24) y = Cx, where x R n is the state, y R q is the system output, and v R is the exogenous input which is bounded. The next theorem provides a non-conservative upper-bound for the L 1 gain of the above system. 7,13 Theorem 1 If there exits a positive definite matrix P α R n n solving the following LMI: AP α + P α A + αp α + 1 α BB 0, P α > 0, (25)

6 for some α > 0, then where denotes the Euclidean norm. G L1 CP α C, (26) Remark 1 Since the upper bound is a function of α, one can consider the least conservative bound dependent upon α. The least upper bound of the L 1 gain over all possible α is known as -norm, 14 and is defined as G = inf α CP αc. Let the state-space representation of the cascaded system H(s)(1 C(s)), for which we will minimize the L 1 gain, be given by: ] ] A ξ = m bc f b ξ + u, (27) 0 A f B f ] y = I 0 ξ. Note that the state space representation of H(s)(1 C(s)) bears the same form of (24). Therefore, direct application of Theorem 1 leads to the following result: Theorem 2 If there exist matrices Z G R 1 n,y G R n n, and positive definite matrices P G1 R n n and P G2 R l l, solving the following LMI s ] ] Ω bz G b Φ(P G1,P G2,Z G,Y G ) ZG b Y G + YG + αp ] G 2 B f 0, (28) α b B f P G1 γi, (29) for some α > 0, γ > 0 and B f R n 1, where Ω = A m P G1 + P G1 A m + αp G 1, then H(s)(1 C(s)) L1 γ. The corresponding filter is given by: A f = Y G P 1 G 2 and C f = Z G P 1 G 2. ] P Proof. First, let P α = G1 0, Z G C f P G2 and Y G A f P G2. It is straightforward to see 0 P G2 that the inequality in (28) is equivalent to ] ] A m P G1 + P G1 A m + αp G 1 bc f P G2 b P G2 Cf b P G2 A f + A f P G 2 + αp ] G2 B f 0. (30) α Considering Schur s complement, the above inequality can be reduced to A m bc f ] b B f P G1 0 0 A f 0 P G2 ] P +α P G2 α ] ] ] P + G1 0 A m bc f 0 P G2 0 A f ] ] b b 0. (31) B f B f

7 ] A Applying Theorem 1, by substitution of m bc f b with A, B with 0 A f B ] f I 0, we have ] ] ] H(s)(1 C(s)) L1 P G1 0 I I 0 0 P G2 0 = P G 1. ], and C with Thus, P G1 γi implies H(s)(1 C(s)) L1 γ. This completes the proof. Next, we are interested in finding the minimal upper bound of the L 1 gain. Note that γ is a generalized eigenvalue for the matrix pair (P G1, I). Let γ H(s)(1 C(s)) L1 be the upper bound to be minimized. Straightforward application of standard LMI arguments yields the following generalized eigenvalue problem (GEVP): 11 min γ, Y G,Z G,P G1 >0,P G2 >0 s.t. (28) holds and P G1 γi. (32) The optimal filter C(s) is then realized via (A f = Y G PG 1 2,B f,c f = Z G PG 1 2 ). Note that we use γ to distinguish from the fixed constant γ in (29). IV. Time-Delay Margin Optimization Similarly, we develop LMI tools for constraint-free time-delay margin optimization. Consider the following LTI system with time-delay δ > 0 ẋ(t) = Ax(t) + A d x(t δ), (33) where x(t) R n is the state of the system, A and A d R n n are matrices of appropriate dimensions. Theorem 3 The system (33) is asymptotically stable for any 0 δ η, if there exist P > 0, P 1 > 0, and P 2 > 0 satisfying H ηpa ηpa d ηap ηp 1 0 0, (34) ηa d P 0 ηp 2 where H = P(A + A d ) + (A + A d )P + ηa d (P 1 + P 2 )A d. The proof can be found in Ref. 10 Remark 2 Since the system (33) remains stable for all δ less than η, the actual time-delay margin of the system (33) is lower-bounded by η. To apply Theorem 3 to the time-delay margin maximization problem, we first prove the following result. Corollary 1 If the following LMI P(A + A d ) + (A + A d )P ηpa ηpa d ηap ηp 0 ηa d P ηp has a positive definite solution for P > 0, then the condition in (34) holds. 0 (35)

8 Proof. Let P = P 1 = P 2. Then by Schur s complement the relationship in (35) leads to (34). Next we consider the time-delay margin maximization problem for L 1 adaptive controller. Let the state-space realization for H o (s) be given by Ẋ(t) = A f + B f C f B f C f 0 0 A f B f θ 0 0 A m + bθ X(t) B f C f B f C f 0 bc f bc f 0 X(t δ), (36) where X(t) R n+2l is the state of H o (s). The following theorem is an application of Corollary 1 to the LTI system in (36). Theorem 4 If for some fixed matrix B f and prespecified lower bound η > 0, there exist matrices Y T R n n, Z T R 1 n, and positive definite matrices P T1 R l l and P T2 R n n satisfying Λ + Λ + + ηλ η Ψ(P T1,P T2,Y T,Z T ) ηλ ηp T 0 0, (37) η ηp where Λ = = P T = Y T + B f Z T B f Z T 0 0 Y T B f θ P T2 0 0 A m P T2 + bθ P T B f Z T B f Z T 0 bz T bz T 0 P T P T1 0, 0 0 P T2, then the system (36) is asymptotically stable for any 0 δ η. Since the time-delay margin is the maximum time-delay for which the system is not losing its stability, the actual time-delay margin of the system (36) should be greater than η. We then have the following corollary. Corollary 2 The time-delay margin τ of the closed-loop system (1) with L 1 adaptive controller, defined via (5), (6,) (7) and (8), is lower bounded by η if there exist matrices Y T R n n, Z T R 1 n and positive definite matrices P T1 R l l and P T2 R n n satisfying the inequality in (37). The corresponding filter C(s) is then realized via (Y T P 1 T 1,B f,z T P 1 T 1 ). Proof. From (19), we have τ T (H o (s)). Further, Corollary 2 yields T (H o (s)) η. Therefore τ η. Notice that η in (37) can be viewed as a generalized eigenvalue. We can formulate the following GEVP for time-delay margin maximization, where we use η to denote the optimization objective: s.t. Ψ(PT1,P T2,Y T,Z T ) max Z T,Y T,P T1 >0,P T2 >0 η, Λ + Λ + + ηλ η ηλ ηp T 0 η ηp T The optimal filter C(s) is then given by C(s) = (Y T P 1 T 1,B f,z T P 1 T 1 ). 0. (38) (39)

9 V. Constrained Optimization for L 1 gain and Time-delay Margin In this section, we address the constrained optimization problems in (21) and (22) via LMI formulations. For constrained L 1 gain optimization, the GEVP optimization problem in (32) is addressed, with an additional LMI condition as provided by Theorem 4 for the time-delay margin. Similarly, for the constrained time-delay margin optimization, the GEVP optimization problem in (39) is used, together with the LMI condition in Theorem 2 for the L 1 gain. We first prove the following theorem to handle the non-convex constraint C(0) = 1 in the LMI optimization problem. Theorem 5 Let the state-space realization of C(s) be given by (A f,b f,c f ) with 0 B f = Let Y AP and Z CP, where P R l l is a symmetric positive definite matrix. If there exists a P such that Y and Z have the following structure: Y =..... ] 0 0 1, Z = ρ z 1 z l 1, (40) ρ y 1 y l 1 where ρ, y i and z i (i = 1,2,...,l 1) are arbitrary constants, then C(0) = 1. Proof. It s straightforward to see that C(0) = C f A 1 f B f = Z Y 1 B f = 1. We now address the constrained optimization problem (21). Theorem 6 Given η as a desired lower bound for the time-delay margin of the closed loop L 1 adaptive control system, if there exist matrices P C > 0, P G1 > 0, P T2 > 0, Y and Z solving the following GEVP min γ, (41) P T2,P G1,P C,Y,Z s.t. P C γi, Φ(P G1,P C,Z,Y ) 0, (42) and Ψ(P C,P T2,Z,Y ) 0, (43) where Y, Z and B f comply with the structure in (40), then problem (21) is solved by choosing C(s) = (A f = Y P 1 C,B f,c f = ZP 1 C ).

10 Proof. Let γ be the upper bound of the L 1 gain to be minimized and η be a given lower bound on the time-delay margin to be satisfied. Consider the L 1 gain optimization problem (32) and the LMI condition in (37). Let P C = P G2 = P T1, Y = Y G = Y T, and Z = Z G = Z T, where Y, Z, and B f comply with the structure in (40). By substituting P C, Z and Y into (32) and (37), we get (42) and (43), respectively. It follows from Theorem 5 that the choice of B f, Y, and Z complying with the structure in (40) ensures that C(0) = 1. Then the optimization problem (32), together with the constraint (37), yields the filter C(s) via the realization of (A f = Y P 1 C,B f,c f = ZP 1 C ). For the time-delay margin optimization problem in (22), we have the following result, where the proof is omitted since the idea is similar to Theorem 6. Theorem 7 Given γ as a desired upper bound of the L 1 gain, if there exist P C > 0, P T2 > 0 P G2 > 0, Y and Z solving the following LMI GEVP subject to max η (44) P T2,P G1,P C,Y,Z P C γi, Φ(P G1,P C,Z,Y ) 0, (45) and Ψ(P C,P T2,Z,Y ) 0, (46) where Y, Z and B f comply with the structure in (40), then the problem in (22) is solved by choosing C(s) = (A f = Y P 1 C,B f,c f = ZP 1 C ). Remark 3 We note that existence of a solution to the above stated LMIs and GEVPs is not guaranteed in general case. VI. Numerical Example A numerical example is given for the L 1 gain optimization problem in (21). Consider a scalar plant with a m = 2, b = 1, and θ = 1.7 and choose a second order filter C(s) : A f R 2 2, ] C f R 2 1 and B f = 0 1. Set α = 3.2 and let 0.6 be the desired time-delay margin. Then ] 0 1 ] A f and C f are determined by applying Theorem 6. Let Y = and Z = ρ z. We ρ y have the following Generalized Eigenvalue Problem: min γ, ρ,y,z,p T2 >0,P G1 >0,P C >0 s.t. P G1 γi 2 2, (47)

11 Y + Y + B f Z + Z Bf Z Bf B f Z 0 Y + Z Bf Z Bf + B f Z Y + Y B f Z Z B f Z b 0 bz 2P G2 A m + 2P G2 θ b 0 Y + B f Z B f Z 0 P C 0 Y P G2 θ B f P G2 A m + P G2 θ b B f Z B f Z 0 0 bz bz Z B f Z b Y 0 0 Z B f Z b B f θ P G2 P G2 A m + P G2 θ b P C , (48) 0 P T /3P C /3P C /3P T2 A m P C + P C A m + αp C Z b b B f bz Y ] + Y + αp G2 P C P T P G2 ] b B f α ] 0, (49) > 0. (50) The above generalized eigenvalue problem is solved using the function gevp of the MATLAB LMI Toolbox, yielding the following filter parameters: ] ] A f =,C f = , and the transfer function C(s) = s s s The minimum upper bound of the L 1 gain is It can be verified that C(0) = 1. VII. Conclusion In this paper a convex optimization approach for optimal filter design of L 1 adaptive controller is proposed. The LMI conditions for minimizing the L 1 gain and maximizing the time-delay margin are first considered separately. Next, a constrained LMI algorithm is stated that allows for optimizing one while retaining the desired bound for the other. A numerical example is given to illustrate the proposed algorithm.

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