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A Lyapunov variable-free KYP lemma for SISO continuous systems

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H. G. Hoang1 , H. D. Tuan1 , and Pierre Apkarian2

Abstract This paper proposes a novel frequency-selective Kalman-Yakubovich-Popov (FS-KYP) lemma for analysis of single-input single-output (SISO) continuous systems. In contrast to existing approaches, the proposed method only uses a minimal number of variables due to the absence of Lyapunov variables in semidefinite programming (SDP) formulation. The SDP formulation is extended to polytopic uncertain systems also without any additional variable and yields an efficient method for computation of the H∞ gain for polytopic systems. The viability of the proposed method are demonstrated through several numerical examples.

I. I NTRODUCTION The celebrated Kalman-Yakubovich-Popov (KYP) lemma, also known by other equivalent forms such as positive real lemma and bounded real lemmas (see e.g. [1]–[5]), is one of the most important theoretical result in modern control. The lemma and its variations allow a computationally intractable semi-infinite program (SIP) in frequency domain to be characterized by a computationally tractable SDP. The most popular form of KYP lemma for SISO systems (see e.g. [3]) states that given a Hermitian indefinite matrix Θ ∈ C 2×2 , the semi-infinite programming (SIP) condition  H   F (jω) F (jω)   Θ ≥0 ∀ω∈R 1 1

(1)

for a n−order transfer function F (eω ) is equivalent to a SDP involving its state-space realization (A, B, C, D) and a Lyapunov variable. However, SIP conditions on a relevant frequency interval are

also desirable in many practical applications. Thus, generalizations of KYP lemma referred as FS-KYP ∗

The work is supported by the Australian Research Council under grant ARC Discovery Project 0556174.

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School of Electrical Engineering and Telecommunication, the University of New South Wales, Sydney, NSW 2052,

AUSTRALIA; Email: [email protected], [email protected] 2

ONERA-CERT, 2 av. Edouard Belin, 31055 Toulouse, France; Email: [email protected]

DRAFT

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lemmas has been developed [6], [7]. Their common drawback is that they invariably involve Lyapunov variables, whose dimension increases dramatically with respect to dimension of the system. For example, a transfer function of order n = 100 requires a Lyapunov variable of dimension 100 × 100, which is equivalent to 5050 scalar variables. Consequently, the resulting SDP is of very large dimension, thereby is difficult to solve using current general-purpose SDP solvers such as [8]. On the other hand, practical systems are often affected by multiple sources of disturbance and noise. Hence, a robust FS-KYP lemma that is capable of handling such uncertainties is highly desirable. Though there is no universal approach to address general uncertainty, there exist ones that deal with particular classes of uncertain systems. For instance, robust analysis techniques for systems affected by polytopic uncertainties have been studied in [9]–[11]. However, these techniques also experience similar drawbacks of the original KYP lemma: even more Lyapunov variables of dimension n × n are involved in the SDP formulation. The main objective of this paper is to develop a new FS-KYP lemma that are free from Lyapunov variables. The attractive feature of the proposed method is that the resulting LMIs are of low dimension, thus enabling the analysis of very large scale systems to be achieved on a standard personal computer. In conjunction with stability analysis results in robust control [12], [13] it offer a new powerful analysis for single input single output systems. The rest of the paper is structured as follows. Section 2 describes mathematical tools that will be used throughout the paper. Section 3 presents the reduced order SDP formulation for conventional FS-KYP lemma. In section 4, we develop a novel FS-KYP lemma for robust analysis of polytopic systems and a bisection method for computing H∞ gains of such systems. Finally, numerical examples are given in Section 5 and concluding remarks are presented in Section 6. The following notation is used in the paper. Vectors and matrices will be represented by italicized bold lower case and upper case letters, respectively. The superscript conjugation) whereas the superscript

“H”

“T”

denotes the transpose (without

denotes Hermitian transpose. Symbols R and C are used to

denote real and complex spaces. Real part and imaginary part of a complex number w are denoted by