An LMI Approach for Stability Analysis and Output-Feedback Stabilization of Discrete-Time Lur’e Systems Using Zames-Falb Multipliers

Abstract

This paper investigates the problem of stability analysis and output-feedback stabilization of discrete-time Lur'e systems where the nonlinearity is odd and slope bounded. Using the linear matrix inequality (LMI) conditions from the literature to handle the 1 norm and positive realness constraints, an iterative algorithm based on LMIs is constructed to assess stability through the existence of a Zames-Falb multiplier of any given order based on independent positive definite matrices for the 1 norm and positive realness. More important, the method can also deal with output-feedback stabilization. Numerical examples illustrate the performance of the proposed approach when compared with other methods.

Control design of uncertain discrete‐time Lur'e systems with sector and slope bounded nonlinearities

“…Example Consider the six systems investigated in Experiment 1 of Reference 30. The objective is to obtain the maximum value of $$$ \Omega $$$ considering sector bounded nonlinearities and the maximum value of $$$ \Omega =\Lambda $$$ for slope bounded nonlinearities.…”

“…The objective is to obtain the maximum value of $$$ \Omega $$$ considering sector bounded nonlinearities and the maximum value of $$$ \Omega =\Lambda $$$ for slope bounded nonlinearities. In the former case, Algorithm 1 is compared with BPOV20, and in the latter with BOVP22 30 . Both state‐ and output‐feedback control laws are investigated.…”

“…In the former case, Algorithm 1 is compared with BPOV20, and in the latter with BOVP22. 30 Both stateand output-feedback control laws are investigated. All methods were tested with it max = 50 and the results are shown in Table 4.…”

“…Values of Ω Σ and Λ Σ of the output-feedback stabilization problem of Example 2 considering the design conditions from KB17, 18 BPOV20,19 and Algorithm 1 (sector bounded) and Algorithm 1 (sector and slope bounded) Maximum values of Ω (sector bounded) and Ω = Λ (slope bounded) obtained by Algorithm 1, BOVP22,30 and BPOV2019 for state-and output-feedback stabilization in Example 3 (it max = 50 in all cases)…”

Identification of Self-Excited Systems Using Discrete-Time, Time-Delayed Lur'e Models

“…The stability of Lur'e models is a classical problem, expressed by the Aizerman conjecture for sectorbounded nonlinearities [17], [18], [19]. Although the Aizerman conjecture is false, the stability of Lur'e models has been widely studied in both continuous time [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and discrete time [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45].…”