روش LMI به تجزیه و تحلیل سیستم مثبت خطی و سنتز

This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron–Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This new result leads to an alternative proof of the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the H∞H∞ performance of continuous-time positive systems. In addition, we further derive a new LMI for the H∞H∞ performance analysis where the variable corresponding to the Lyapunov matrix is allowed to be non-symmetric. We readily extend these results to discrete-time positive systems and derive new LMIs for the H∞H∞ performance analysis and synthesis. We finally illustrate their effectiveness by numerical examples on robust state-feedback H∞H∞ controller synthesis for discrete-time positive systems affected by parametric uncertainties.

This paper is concerned with the analysis and synthesis of linear time-invariant (LTI) positive systems. A linear system is said to be positive (or more accurately, internally positive) if its state and output are both nonnegative for any nonnegative initial state and nonnegative input. Because of this strong property, there are remarkable, and very peculiar results that are valid only for positive systems. Among them, the existence of a diagonal Lyapunov matrix that characterizes stability is well known [1] and [2]. Recently, Shorten et al. showed that the peculiar “diagonal stability result” can be proved by means of the duality theory in convex optimization. They further obtained new results on the stability of switched positive systems [3], [4], [5] and [6]. Along this line, Tanaka and Langbort proved that the KYP-type linear matrix inequality (LMI) characterizing the H∞H∞ performance of positive systems admits a diagonal Lyapunov matrix [7]. These recent results indicate that the duality theory is a powerful tool for positive system analysis. Along the same line, in this paper, we develop duality-based arguments for positive system analysis. Our novel contribution can be summarized as follows: 1. We provide a duality-based concise proof of the Perron–Frobenius theorem [1] and [2]. In addition to the existence of the Frobenius eigenvalue, we show the existence of the nonnegative eigenvector by duality. 2. Again by a duality-based argument, we clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries. This condition leads to an alternative proof of the result in [7]. The analysis is partly motivated from the observation that the L2L2 and L1L1 induced norm analysis of positive systems can be transformed into the stability analysis of appropriately constructed positive systems [8] and [9]. 3. We derive new LMI conditions for the stability and H∞H∞ performance analysis of continuous-time positive systems, where the common positive definiteness constraint on the Lyapunov matrix PP as in P≻0P≻0 can be relaxed to P+PT≻0P+PT≻0. This implies that PP is not necessarily required to be symmetric. 4. We extend the above results to discrete-time positive systems and derive new LMIs for the H∞H∞ performance analysis and synthesis, some of which are reported also in [10]. We illustrate the effectiveness of these new LMIs by numerical examples on structured robust state-feedback H∞H∞ controller synthesis for discrete-time positive systems affected by parametric uncertainties. Even though we provide LMI-based formulations in this paper, it is known that linear-programming-based formulation is possible in the case where the plant is SISO, see, ex.,[8] and [11]. Note that a conference version of this paper was presented in [12]. In the current paper we include new LMI results for discrete-time positive systems. In particular, we show that a given discrete-time positive system can be converted into a continuous-time positive system preserving the stability and the H∞H∞ norm. This enables us to derive new LMIs for the H∞H∞ performance analysis and synthesis of discrete-time positive systems. We use the following notations in this paper. First, we denote by View the MathML sourceS++n(S+n) the set of positive (semi)definite matrices of size nn. For a symmetric matrix X∈Rn×nX∈Rn×n, we also write View the MathML sourceX≻0(X⪰0) to denote that XX is positive (semi)definite. Similarly, we write View the MathML sourceX≺0(X⪯0) to denote that XX is negative (semi)definite. In addition, we denote by View the MathML sourceD++n the set of diagonal, and positive definite matrices of size nn. For A∈Rn×nA∈Rn×n, we define View the MathML sourceHe{A}=A+AT. The notation λ(A)λ(A) stands for the set of the eigenvalues of AA. A matrix A∈Rn×nA∈Rn×n is said to be Hurwitz stable if View the MathML sourcemaxλ∈λ(A)Reλ<0, and is said to be Schur stable if maxλ∈λ(A)|λ|<1maxλ∈λ(A)|λ|<1. For two given matrices AA and BB of the same size, we write A>BA>B (A≥BA≥B) if Aij>BijAij>Bij(Aij≥Bij)(Aij≥Bij) holds for all (i,j)(i,j), where View the MathML sourceAij(Bij) stands for the (i,j)(i,j)-entry of A(B)A(B). We also define View the MathML sourceR++n×m≔{A∈Rn×m,A>0},R+n×m≔{A∈Rn×m,A≥0}. Turn MathJax on Finally, for a given A∈Rn×nA∈Rn×n, we define by D(A)∈RnD(A)∈Rn the vector composed of the diagonal entries, i.e., View the MathML sourceD(A)≔[A11⋯Ann]T.

In this paper, we showed that several remarkable and peculiar results for positive system analysis can be proved concisely by duality-based arguments. We clarified a necessary and sufficient condition under which a Hurwitz (Schur) stable Metzler (nonnegative) matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This result leads us to an alternative and concise proof for the fact that the KYP-type LMI characterizing the H∞H∞ performance of positive systems admits diagonal Lyapunov matrices as well. On the other hand, we also showed that the Lyapunov matrix in the Lyapunov inequalities and the KYP-type LMIs can be relaxed to non-symmetric in the case of positive systems. Moreover, for the H∞H∞ performance of discrete-time positive systems, we derived new LMIs that are structurally different from the KYP-type LMI. We illustrated the effectiveness of these new LMIs by numerical examples on structurally-constrained robust state-feedback H∞H∞ controller synthesis.