Abstract
Graphical methods are a key tool to analyze Lur’e systems with time delay. In this letter we revisit clockwise properties of the Nyquist plot and extend results in the literature to critically stable systems and time-delayed systems. It is known that rational transfer functions with no resonant poles and no zeros satisfy the Kalman conjecture. We show that the same class of transfer functions in series with a time delay also satisfies the Kalman conjecture. Furthermore the same class of transfer functions in series with an integrator and delay (which may be zero) satisfies a suitably relaxed form of the Kalman conjecture. Useful results are also obtained where the delay is constant but unknown. Results in this letter can be used as benchmarks to test sufficient stability conditions for the Lur’e problem with time-delay systems.
Highlights
The Lur’e problem [1] consists of analysing the stability of the feedback interconnection between an LTI system and any nonlinearity within a class of systems
The Lur’e system is shown in Fig. 1, which is the negative feedback interconnection between an LTI plant and a nonlinearity that belongs to a class of nonlinearities, and where the injected signal f can be treated as a disturbance or a signal that generates the initial condition of the plant [7]
This paper develops a class of time-delayed stable transfer functions and a class of time-delayed critically stable transfer functions that satisfy the Kalman conjecture
Summary
The Lur’e problem [1] consists of analysing the stability of the feedback interconnection between an LTI system and any nonlinearity within a class of systems. IQC stability conditions for Lur’e systems with critically stable plants have been explored in [7],. Gn that allow us to ensure that G satisfies either the Kalman conjecture or a suitably relaxed version of the Kalman conjecture in the case where G is critically stable We propose such plants as benchmarks to judge the performance of sufficient conditions developed by other stability methods. This paper is to show that this set of plants together with an integrator and/or time delay satisfies the Kalman conjecture (or relaxed version). There are several conventional methods to design compensators for linear time-delayed plants such as the Smith predictor [29] and the Dahlin controller [30] These conventional methods cannot guarantee stability and robustness with respect to actuator saturation. In [32] time-delayed plants were introduced; here we provide results for a wider class of plants
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