Synthesis of Static Output Feedback SPR Systems via LQR Weighting Matrix Design

Transcription

1 49th IEEE Conference on Decision and Control December 15-17, 21 Hilton Atlanta Hotel, Atlanta, GA, USA Synthesis of Static Output Feedback SPR Systems via LQR Weighting Matrix Design Jen-te Yu, Ming-Li Chiang and Li-Chen Fu Abstract In this paper we propose an approach using linear quadratic regulator (LQR) weighting matrices to synthesize strictly positive real (SPR) systems by static output feedback. The systems being considered are linear time-invariant (LTI). We first recall full state feedback LQR design. The two weighting matrices for state and control input respectively in the performance index are then used as two free parameters to design the SPR controller. By connecting strictly positive realness with full state feedback LQR through the algebraic Riccati equation associated with the latter and imposing wellposed condition in terms of positive definiteness on the weighting matrices, we show that the proposed formula for weighting matrices in this paper can render the resulting closed loop system SPR. The stabilizing static output feedback gain which is designed to make the closed loop system SPR becomes readily available once the two LQR weighting matrices are determined. Moreover, from the derived explicit form of control gain, we can achieve SPR synthesis even when system matrices are partially known. We provide in the end a numerical example to validate the approach. I. INTRODUCTION It is well-known that a strictly positive real (SPR) system is passive, asymptotically stable, and minimum phase. Various definitions and equivalent conditions of strict positive realness can be found in [1. SPR system possesses high degree of robustness against uncertainty. Typical applications of SPR systems include stability analysis for adaptive controls and inter-connected dynamic systems. In this paper we focus on the synthesis of strictly positive real systems using static output feedback which is derived from the weighting matrices in linear quadratic regulator (LQR). The systems considered here are linear and time-invariant (LTI). Some earlier works have given in-depth discussions from different aspects about this topic and provided some interesting and important results, e.g., [2 and [3. In [4, they attacked the problem using singular optimal control approach. A coordinate transformation used to reduce the order of the positive real lemma equations is introduced. With the proposed transformation and the accompanied properties, the solution of SPR synthesis problem is presented in a form of linear matrix inequality (LMI). In [5, the authors parameterized all static output feedback controllers for SPR synthesis problem using Parrott s Theorem and LMI. This work was supported by National Science Council under the grant NSC E MY3. Jen-te Yu and M. L. Chiang are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan. jenteyu@hotmail.com, d919215@ntu.edu.tw L. C. Fu is with the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan. lichen@ntu.edu.tw Moreover, an important result is shown in that paper: For a minimum phase system, if no constant output feedback exists that renders the system SPR, then no dynamic output feedback exists that makes the system SPR. Based on the results of [4 and [5, [6 presents a simple formula for SPR synthesis problem and shows that if the control gain is sufficiently large, then the system can be made SPR. A dynamic output feedback stabilization controller is proposed in [7 to relax the restriction of SPR synthesis problem from being minimum phase to be weakly minimum phase. In [9, a dynamic output feedback compensator plus a variable structure design is proposed for minimum phase systems to cope with polytopic uncertainties. However, the assumptions for unknown plants are still very restrictive. A mixed H and positive real synthesis problem is considered in [1. With the assumptions of system symmetry, the authors propose a static output feedback gain to guarantee a mixed H and positive real performance. The results in [5 and [6 offered some insight in the construction of a component matrix (denoted as P in this paper) and provided some historical review as well. While some material of current paper may parallel that in the above papers, we intend to revisit the same problem but from a different perspective, and the objective is to further clarify this problem and make it more transparent. We provide an explicit formulation for the SPR synthesis problem based on properly designed weighting matrices in LOR control. Using the explicit formulations of weighting matrices Q and R in performance index, we can design an optimal LQR controller which simultaneously renders the system SPR. Furthermore, we extend the result to the case when system matrix A is not completely known. It is shown that if we have some information about the system s zeros and a relation about CAB, then we can still design the output feedback gain to make the system SPR. This paper is organized as follows. First, we give the problem statement which concerns how to make a linear time-invariant system SPR using static output feedback. We provide two coupled matrix equations to characterize the SPR requirement, which is quite standard in the controls literature. Then we discuss the solution of the coupled matrix equations which is similar to existing results in literatures. Following that we briefly recall and provide the main result from full state feedback LQR theory, namely the optimal state feedback gain and the associated Riccati equation. We then show the connection of the LQR solution to the SPR property. The result is a well-known solution with a special structure tailored to satisfy one of the two coupled SPR /1/$ IEEE 499

2 equations, which is at least positive semi-definite. We will prove that this solution is in fact positive definite, satisfying the SPR requirement. After that we then show that the wellposedness condition that must be imposed on the two weighting matrices for the state and input respectively in the LQR performance index, which is nothing but positive definiteness (possibly semi-definiteness) will naturally lead us to the static output feedback gain, which is required to render the closed loop system SPR. A partially unknown system case is also discussed. Then, we provide a numerical example to validate the approach. Following that we conclude the paper in the end. II. PROBLEM STATEMENT Consider a controllable and observable LTI system ẋ = Ax+Bu y = Cx, (1) where x R n, u, y R m, A R n n, B R n m and C R m n. We want to find a static output feedback gain F such that the closed loop system with u = Fy is SPR. From the well-known results of SPR conditions (e.g., [1, [5), the problem considered in this paper is equivalent to the following: design a static output feedback gain F such that there exists a symmetric positive definite matrix P that satisfies the positive real lemma equations: A T FP +PA F < (2) PB = C T, (3) where A F = A BFC. For system (1), necessary conditions of existence of such P can be found in Theorem 1 in [5 or Theorem 2 in [6. Based on which we make the following assumptions to guarantee the solution existence of our problem. (A1) System (1) is minimum phase. (A2) CB = (CB) T >. III. SOLUTIONS OF POSITIVE REAL LEMMA EQUATIONS Before designing the static output feedback gain, we discuss the characteristics of positive real lemma equations (2) and (3). First we consider what class of P will satisfy (3). From CB = (CB) T > and (3), it can be shown that ([5) P = C T (CB) C +N T P N, (4) where P is a symmetric positive definite matrix and N will be defined later. Note that the dimensions of P and P are not the same. From [6, P can be constructed through the following Lyapunov equation where N and M satisfy (NAM) T P +P (NAM) = Q, (5) NB =, CM =, and NM = I. (6) Q is any symmetric positive definite matrix and for simplicity, one can choose Q = Q T = I. Obviously P of (4) is at least positive semi-definite. Given below we provide a procedure to construct N and M which is slightly different from that in [4. Let us perform singular value decomposition (SVD) on B and C respectively. From (A2) we know that both B and C are full rank, and based on which the unitary and singular matrices of B and C are partitioned accordingly. B can be represented as where C can be represented as where Let Perform SVD on B = XYZ T = X 1 Y 1 Z T (7) X = [X 1,X 2 [ Y1 Y =. C = USV T = US 1 V 1 T, (8) ˆN ˆM S = [S 1, V = [V 1,V 2. ˆN = X T 2 ˆM = V 2. ˆN ˆM = X T 2 V 2 = EGH T (9) We may now define N and M as follows N = G /2 E T ˆN = G /2 E T X T 2 M = ˆMHG /2 = V 2 HG /2 (1) One can verify easily that NB =,CM =, andnm = I. From the above we can get CB = US 1 V T 1 X 1Y 1 Z T (11) S 1 UT (CB)ZY 1 = V T 1 X 1 > (12) As P is at least positive semi-definite, we show that it is impossible to find such a vector v that yields v T C T (CB) Cv = and v T N T P Nv = simultaneously, which is equivalent to proving the positive definiteness of P. Lemma 1: P of (4) is positive definite for any symmetric positive definite P. proof: Suppose there exists a vector v such that v T N T P Nv =. (13) Obviously Nv = must hold as P >. Therefore we get X T 2 v = v = a i X 1i, (14) 4991

3 where X 1i denotes any column vectors in X 1 and a i is a constant scalar. Since (CB) >, the only possibility for v T C T (CB) 1Cv = to hold is Cv =, i.e. V1 T v =, which further implies a i V1 T X 1i = and obviously contradicts the fact that V1 TX 1 >. On the other hand, suppose that there exists a vector v such that v T C T (CB) Cv = (15) Obviously Cv = must hold, i.e., V1 T v =. If this v also yields v T N T P Nv =, namely Nv =, i.e. X2 Tv =, which will further imply v = b i X 1i, where b i is a constant scalar, then from which we get b i V1 T X 1i =. This again contradicts the fact that V1 TX 1 >. Based on the above argument we conclude that P is positive definite. Now we have the structure of P = P T > for (3) and based on which we will design the feedback gain for synthesis of SPR systems. Remark 1: A useful coordinate transformation utilizing N and M is proposed in [4 and verification of the SPR equations (2), (3) is reduced to that of a lower dimensional subsystem. Here we also provide an alternative construction for N and M and show that (4) is always positive definite for any symmetric positive definite P. In the following, we will design the output feedback gain satisfies (2) based on P in the form of (4). IV. MAIN RESULTS In this section, we will show how to design the static output feedback u = Fy such that the closed-loop system (1) is SPR based on selection of the weighting matrices in the performance index of LQR control. We clarify the relationship between LQR and SPR synthesis that was implicit in existing literatures, and present an algorithm to design the feedback gain that simultaneously achieves SPR synthesis and LQR control. A. Full State Feedback LQR and Output Feedback SPR Synthesis The connection of SPR with optimal control has been discussed in-depth, for example, in [4, but from a singular optimal control perspective. We now recall the main result from full state feedback LQR theory - the optimal feedback gain and its associated Riccati equation to design our static feedback gain. J = (x T Qx+u T Ru)dt (16) u = Kx (17) K = R B T P (18) = A T P +PA PBR B T P +Q (19) where matrix Q is the weighting for the state and matrix R is the weighting for the input respectively in the performance index. An equivalent form of (19) is as follows = (A BK) T P +P(A BK)+K T RK +Q. (2) As SPR condition requires that PB = C T, or equivalently B T P = C, which when combined with the LQR result yields the following u = Kx = R B T Px = R Cx = R y. (21) Obviously the above expression suggests that the static output feedback gain F can be chosen as the inverse of the input weighting matrix, namely F = R. (22) In order to forcep to satisfy both the Riccati equation (19) and the positive real lemma equation (3) simultaneously, (19) should be re-written as A T P +PA C T R C +Q = (23) Substitution of the (special-structured) matrix P from (4) into the (23) yields the following A T [C T (CB) C +N T P N +[C T (CB) C +N T P NA C T R C +Q = (24) From a different perspective, one can choose the state weighting matrix Q according to (24) in the LQR design as follows Q = A T [C T (CB) C +N T P N [C T (CB) C +N T P NA+C T R C (25) Since the inverse of the control weighting matrix R appears in Q, these two weighting matrices are no longer independent, which is reasonable in our approach as P must satisfy one extra requirement, namely, the positive real lemma equations. B. Synthesis of SPR Systems via LQR Weighting Matrix Design We now show how one can find the static output feedback gain F via LQR such that (2) and (3) are satisfied. Recall that in the full state feedback LQR design, for well-posedness reason, the weighting matrix Q for the state x must be at least positive semi-definite, and the weighting matrix R for the input u must be positive definite respectively in the performance index. Thus we have to ensure that Q = Q T, (26) R = R T >, (27) K T RK +Q >. (28) Note that from (18), (28) is equivalent to and thus from (2) C T R C +Q >, A T FP +PA F <. (29) 4992

4 Consider the following transformation T = [M B(CB), (3) whose inverse exists and is also used in [6 and [4, that makes the above requirement imposed on the weighting matrices more transparent. Since Q = Q T implies T T QT and vice versa, the condition Q can be transformed to [ M T QM M T QB(CB) (CB) B T QM (CB) B T QB(CB). (31) By Schur complement, (31) is equivalent to the following LMIs : M T QM = Q > (32) (CB) B T QB(CB) (33) (CB) B T QB(CB) [(CB) B T QMQ [MT QB(CB). (34) Substitute (25) into (33), then we have R (CB) [CAB +(CAB) T (CB). (35) Combing (25) with (34), we can conclude that if we choose R = (CB) { [(CAB)+(CAB) T +R Q } (CB) where +ǫi >, (36) R Q = [ CAM +(NAB) T P Q [CAM +(NAB) T P T, (37) is symmetric positive semidefinite andǫi is used to guarantee the positive definiteness of R in which ǫ is a positive constant, then (34) will hold. Since the matrix in (31) is symmetric, the condition of (31) is equivalent to conditions (32), (33) and (34) [8. Thus, Q will be positive semidefinite if R satisfies (36). Note that R satisfying (36) will automatically imply (35). It is clear that from (29) and (4), the positive real lemma equations (2) and (3) will be satisfied with the proposed design. Hence by the design of R and Q in (36) and (25), positive definiteness of these weighting matrices are guaranteed and the closed-loop system with u = R Cx = Fy is SPR. We summarize the design process of SPR synthesis based on LQR weighting matrix design as follows: Step. Given system (1), verify conditions (A1) and (A2). If yes, go to Step 1. Step 1. Generate N and M s.t. (6) is satisfied (e.g., by (1)). NB =, CM =, and NM = I. Step 2. Specify Q = Q T > and solve P in (5), (NAM) T P +P (NAM) = Q. Step 3. Construct R = R T > and Q = Q T by (36) and (25) with R = (CB) { [(CAB)+(CAB) T R Q +R Q }(CB) +ǫi >, = [ CAM +(NAB) T P Q [CAM +(NAB) T P T, Q = A T [C T (CB) C +N T P N [C T (CB) C +N T P NA +C T R C, where ǫ is chosen s.t. R is positive definite. Step 4. Choose u = Fy = R y. Remark 2: The resultant gain above is similar to that in [6. However, here we provide an explicit solution formulation via LQR weighting matrix design rather than implicit forms represented by LMIs. In an earlier work [11, it is proved that optimal output feedback LQR is equivalent to constrained optimal full state feedback LQR, where F = KC +, (the superscript + stands for generalized inverse) and the constraint is expressed as KV 2 =. For the case of SPR,. Obviously the constraint KV 2 = is satisfied automatically. One can easily verify that F = R = KC +. We therefore conclude that the output feedback gain by this design method not only renders the system SPR but also is optimal in the LQR sense. In other words, using the explicit formulations ofqanr, we can design an optimal LQR controller which simultaneously renders the system SPR. K = R B T P = R C = R US 1 V T 1 C. System Matrix A Unknown Now we consider a minimum phase system {A,B,C} with matrix A unknown. Assume that B and C are known and satisfy CB = (CB) T >. In addition, assume we have the information of the system as follows: (A3) A < a, and a is a known constant. (A4) Re(z i ) > α >, where z i is a zero of the system and α is a known constant, (A5) [CAB+(CAB) T +βi >, β is a known positive constant. Here denotes the the induced matrix Euclidean norm. Before we design the control gain F = R, it is clear that R 1 = (CB) { [(CAB)+(CAB) T +βi +R Q } (CB) is a symmetric positive semidefinite matrix. (38) Lemma 2 [12: Suppose that X = X T > and Y = Y T. Then X > Y iff λ(yx ) < 1, where λ( ) stands for the spectral radius. By Lemma 2, we can conclude that R 1 < r 1 I iff R 1 < r 1, since R 1 is symmetric positive semidefinite. 4993

5 Now we start the design of the output feedback gain. Since eigenvalues of NAM are the zeros of the system (see [4, Remark 3.2), we can obtain the stability margin of NAM from the zeros of the system matrices, and thus e (NAM)t ce αt. Then, from (5), we can obtain P = e (NAM)Tt Q e (NAM)t c2 2α q = p, where q = Q. From above analysis and since B, C are known, we can find a positive constant r that depends on B,C,Q, a, and α such that R Q r. Let (CB) = (cb), if we choose the gain R to be symmetric positive definite and R r = (cb) 2 (2β +r ) r 1 >, (39) then from Lemma 2, R > ri > r 1 I > R 1 and thus the closed-loop system will be SPR. From above discussion, we know that given a minimum phase system that satisfies assumptions (A1)-(A5) with unknown matrix A and known B and C, we can design the static output feedback gain by (39) to construct a SPR system. V. NUMERICAL EXAMPLES Consider system (1) with B = A = , CT =, This is a minimum phase system with its zeros located at and It is easy to verify that the relative degree one condition CB = (CB) T > is satisfied. Following the design procedures as presented above and choose Q = I, we can construct N, M and P as N = [ 1 1 P =, M = [ , Choose ǫ = 1 and we obtain the static output feedback gain F as [ R = F = Moreover, R = [ , and Q = The resultant symmetric positive definite matrix P for positive real lemma equations is P = One can check easily that the closed-loop system is SPR as the coupled matrix equations (2) and (3) are satisfied. It is not difficult to verify that P satisfies the LQR Riccati equation (19) as well, provided the two weighting matrices Q and R are used. Continued with this example, now we consider the case when A is unknown in subsection IV. C. From (A3), (A4) and (A5), suppose that a = 2, β = 1 and α =.3. By computation we estimater = 5. By (39), the gain is chosen as r = (1) 2 (2 1+5) = 7, and we can design the output feedback gain F 2 = ri. Since P determined by (4) is not directly effected by A, we only need to check (2) for strict positive realness. In this case we have A T F 2 P +PA F2 = <. Hence the control gain makes the closed-loop system SPR. VI. CONCLUSIONS In this paper we present a new perspective on the static output feedback based SPR controller synthesis problem and further clarified its connection to the full state feedback LQR problem. Specifically, for the latter to be well-posed, it is natural that one imposes positive semi-definiteness condition on the state weighting matrix, and positive definiteness condition on the input weighting matrix respectively in the performance index, which when combined with a transformation, automatically leads one to the solution of the static output feedback gain that renders the closed loop system SPR. Thus the proposed weighting matrix design simultaneously satisfies optimal control and SPR synthesis for the system. Moreover, the derived formulation provides a solution to SPR synthesis problem for the case when system matrix A is partially known. The numerical examples illustrated the construction steps of the control design and validated the design concept. 4994

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