Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
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48 citations
TL;DR: The absolute exponential stability and stabilization for a class of switched nonlinear systems whose system matrices are Metzler is designed by the state-feedback and average dwell time switching strategy.
48 citations
TL;DR: A state- and time-dependent coefficient is proposed based on the derived inversion error, which eliminates the need for parameter tuning and ensures the convergence of the sliding surface to the boundary layer without compactness assumptions.
Abstract: A sliding mode controller (SMC) is proposed for a class of systems comprising a hysteresis operator preceding a linear system with an all-pole transfer function. The hysteresis operator is modeled with uncertain piecewise linear characteristics, and a nominal inverse operator is included to mitigate the hysteresis effect. A classical SMC design typically uses a constant coefficient in the switching component, which is tuned via trial-and-error. In this paper, a state- and time-dependent coefficient is proposed based on the derived inversion error, which eliminates the need for parameter tuning and ensures the convergence of the sliding surface to the boundary layer without compactness assumptions. In addition, singular perturbation is used to analyze the system behavior within the sliding-surface boundary layer for the case of a constant coefficient in the classical SMC design. In particular, analytical insight is gained on the frequency-scaling behavior of the tracking error under a periodic reference. Simulation and experimental results based on a piezoelectric actuator-based nanopositioner are presented to illustrate the design and analysis, where the hysteresis nonlinearity is represented by a Prandtal-Ishlinskii operator.
48 citations
TL;DR: In this article, the lower Lyapunov number is defined as the spectral spectrum of the spectrum of a function with bounded variation, and a Fredholm determinant is shown to be the determinant of this spectrum.
Abstract: We call the number ξ the lower Lyapunov number. We will study Spec^) , the spectrum of P \\BV> the restriction of P to the subspace BV of functions with bounded variation. The generating function of P is determined by the orbits of the division points of the partition, and the orbits are characterized by a finite dimensional matrix Φ(z) which is defined by a renewal equation (§ 3). Hence, we can show that D(z)=det(I— Φ(#))> which we call a Fredholm determinant, is the determinant of /— #P=ΣίΓ-o zP in the following sense: Theorem A. Let λ G C and assume that \\\\\\>e~. Then λ belongs to Sρec(F) if and only if z—\\~ is a zero of D(z):
48 citations
TL;DR: The proposed piecewise-linear model based regulator has been successfully applied to the chaos synchronization of general nonlinear Chua's circuits and is guaranteed to be stable and the output tracking is achieved asymptotically.
Abstract: This paper considers the output tracking problem of general piecewise discrete-time linear systems via error feedback scheme. A number of sufficient conditions are obtained based on piecewise-quadratic Lyapunov functions in the framework of output regulation theory. The resulting closed-loop system is guaranteed to be stable and the output tracking is achieved asymptotically. Moreover, the proposed piecewise-linear model based regulator has been successfully applied to the chaos synchronization of general nonlinear Chua's circuits. Simulation results are also given to illustrate the performance and advantages of the proposed approach.
48 citations