S.P. Banks

Multi-periodic repetitive control system: a Lyapunov stability analysis for MIMO systems

A multi-input–multi-output (MIMO) repetitive control problem of tracking and disturbance rejection is considered when both reference and disturbance signals are finite linear combinations of periodic but not necessarily sinusoidal signals. Lyapunov stability analyses under a positive real condition (and a natural relaxation) and exponential stability under a strict positive real condition are provided together with bounds on the induced L 2 and RMS gains of the closed loop system. It is shown that similar Lyapunov stability results apply when the plant is a positive real state-delay system. Extension of the analyses to a class of non-linear systems is discussed and indicates a good degree of robustness in the design.

A generalization of lyapunov's equation to nonlinear systems

Abstract In this paper a generalization of Lyapunovs equation for the stability of linear dynamical systems to globally asymptotically stable nonlinear systems is presented by embedding the system in a linear infinite-dimensional one on a tensor space by using Carleman linearization. This linear representation allows the definition and solution of a Lyapunov equation as in the usual linear case. The converse result is also discussed using the fact that globally asymptotically stable non linear systems are essentially linear.

OPTIMAL CONTROL OF NONHOMOGENEOUS CHAOTIC SYSTEMS

Abstract An optimal control approach for controlling chaotic dynamics is developed. This is possible by using an approximation technique that solves the nonlinear optimal control problem by reducing it to a sequence of linear time-varying systems. The theory is implemented to direct chaotic trajectories of the Duffing equation to a period- n orbit.

Multi-periodic nonlinear repetitive control: Feedback stability analysis

In this paper the stability of multi-periodic repetitive control problem, where two or more periods exist in the reference and disturbance signals, is studied. A Lyapunov analysis is used to prove L 2 m (0,∞) ∩L ∞ m (0,∞) stability for a class of passive nonlinear systems subject to a class of nonlinear perturbations. A proof of exponential stability under a strictly positive real condition is provided.

MIMO Multi-Periodic Repetitive Control Systems: A Lyapunov Analysis

Abstract In this paper a new type of repetitive control problem where two or more periods exist in reference and disturbance signals is considered. A Lyapunov stability analysis under a positive real condition is provided and a new method of designing compensators in a negative feedback loop to create positive realness is outlined.

Global Stabilization and Switching Manifolds

Abstract The global stabilization of nonlinear systems is considered by reducing the problem to a lower dimensional switching manifold which is made globally attracting. The method generalizes the standard Lyapunov approach.

Nonlinear Control Design in the Frequency Domain

Abstract A first step is made towards a complete generalization of the classical linear frequency domain theory of feedback control. First, the theory of partiad fraction expansions is extended to multi-dimensional complex rational functions. As may be expected, the theory is now much more complicated and requires the use of ideal theory and notions from algebraic geometry. It turns out that, as in the linear case, the coefficients of the expansion (which are now polynomials) are obtained by ‘removing the given singularity’ and evaluating on the singular variety, i.e. evaluating the rationed function modulo the singularity in the coordinate ring of the singularity. Multi-dimensional residue theory is based on the use of homology groups of the space (in fact, the compactified version, S2n) minus the singular variety T. We shall show that the inversion of the n-dimensional Laplace transform can be performed by finding a homology basis of Hq(S2n\T), and a dual basis of Hr-1(T ⋃ {∞}), where r + q = n. This will reduce the computation in many cases to a simple application of the n-dimensional version of Cauchy’s theorem. The use of the theory in feedback control design is given with a particular study of a simple second-order bilinear system. We shah define an implicit closed-loop transfer function (which is nonseparable) and then apply norm inequalities in the time domain to complete the stability analysis.

FAULT DETECTION FILTER FOR NONLINEAR SYSTEMS USING LINEAR APPROXIMATIONS

Abstract In this paper, the design of a fault detection filter for nonlinear systems is presented. The nonlinear system is represented as a sequence of linear time-varying approximations and at each linear approximation the design of an optimal stochastic fault detection filter for linear time-varying systems is applied. The final residual is primarily affected by a target fault and minimally by nuisance faults.

Stabilization of nonlinear systems using the associated angular system

Abstract The stabilizability of general nonlinear systems is studied by considering the associated angular system, which leads to a simple stabilizing controller in many cases. Bilinear systems are studied in detail and general nonlinear systems are reduced to infinite dimensional bilinear form by using Lie theory.

Exponential Stability of Large-Scale Nonlinear Systems with Multiple Time Delays

Abstract The exponential stability of large-scale nonlinear parabolic systems with time-varying delays is studied in this paper using a new generalization of Gronwalrs inequality and Lyapunov theory. The systems are interconnected by bounded operators, although this can be generalized. All the time-delays will be assumed to have bounded time derivatives for simplicity here.