Stability Analysis Of Nonlinear Systems With A Specific Structure

Lur’e control system is a typical nonlinear control system. Scholars study on it fromthe stability problem of aircraft autopilot. If we put the nonlinear part of the system toisolate, the system will be converted to the form of closed loop control system. In thispaper, the specific structure of the nonlinear system is just Lur’e type system. Thenonlinearities are constrained in a finite or infinite sector area. It has a very extensivepractical engineering background in many fields. Therefore, the absolute stability of Lur’esystem occupies a very important position in analysis and design of nonlinear controlsystem.This paper studied the absolute stability of the Lur’e type nonlinear control system.Based on the Lyapunov stability theorem and the knowledge of matrix theory, we use theLMI toolbox in MATLAB, respectively to study the absolute stability of the Lur’esystems and Lur’e systems with time delays. Including the following aspects:Firstly, the problem of absolute stability of Lur’e systems with sector and sloperestricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria areestablished by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The slope-restricted condition plays an important role in improvingthe condition of absolute stability. The criteria strengthen some existing results.Simulations are given to illustrate the effective of the results.Secondly, absolute stability analysis for time-delay Lur’e systems with both sectorand slope restricted nonlinearities is concerned with. A new delay-dependent stabilitycriterion is given by using a Lur’e-Postnikov functional. The additional slope restrictionson nonlinearities play important roles in improving the absolute stability conditions.Moreover, a frequency-domain interpretation for the criterion is presented as well.Numerical examples demonstrate the effectiveness of the proposed criterion.Finally, the absolute stability of Lur’e systems with sector and slope restrictednonlinearities and multiple time-varying delays is considered in this paper. Using Jenseninequality and extended Jensen inequality, some delay-dependent sufficient conditions for the absolute stability of Lur’e system are derived. The maximum upper bound of theallowable delay is obtained by solving a convex optimization problem. A numericalexample shows that the result obtained in this paper is less conservative than previousones.