Instability of non-linear time-varying systems: combined conditions on the derivatives

An (L₂-)stability criterion is derived for non-linear time-varying (NLTV) systems with a separable gain (k(t).f(.)) in the feedback-path, where f(.) is monotone, using a simple R-L multiplier (s + β)/(s + α)(α > β) and involving the first and second derivatives of k(t). The criterion seems to be very much relaxed for systems with a slowly time-varying gain k(t).     

Stability conditions for a class of time-varying discrete-time systems

This paper deals with the stability conditions for a class of dynamical discrete-time systems, called ‘P-invariants’ and ‘not P-invariants’. Stability conditions are given in terms of polyhedral convex cones and are obtained by using some extensions on M-matrices.     

Indirect adaptive control for a class of non-linear systems with a time-varying structure

In this paper we present an indirect adaptive control method for a class of uncertain non-linear systems with a timevarying structure. We view the non-linear systems as composed of a finite number of 'pieces', which are interpolated by functions that depend on a possibly exogenous scheduling variable. We assume that each piece is in strict feedback form, and show that the indirect adaptive method yields stability of all signals in the closed-loop, as well as convergence of the state vector to a residual set around the equilibrium, whose size can be set by the choice of several design constants. The class of systems considered here is a generalization of the class of strict feedback systems traditionally considered in the backstepping literature. Finally, we apply the indirect adaptive method to the problem of regulation of aircraft wing rock with a time-varying angle of attack.     

Robust output feedback stabilization of a class of time-varying non-linear systems

This paper deals with the problem of robust output feedback stabilization of a class of time-varying non-linear systems. This class of systems involves two kinds of time-varying uncertainties: those norm-bounded and those bounded by a smooth non-linear function of the output. Under the assumption that the zero dynamics of the system are uniformly asymptotically stable and some additional mild conditions, we show via a Lyapunov function approach that the uncertain system can be robustly stabilized by a time-varying non-linear output feedback controller. The order of this controller turns out to be one less than the relative degree of the uncertain system. A systematic design procedure is given for constructing the controller. Illustrative examples are given. Note that the results generalize several previous results on robust output feedback stabilization.     

The decentralized stabilization and control of a class of unknown non-linear time-varying systems

The following problem is considered in this paper. Suppose a system S consists of a set of arbitrary interconnected subsystems S_i, i = 1, 2, …, Ω; is it possible to stabilize and satisfactorily control the whole system S by using only local controllers about the individual subsystems without a knowledge of the manner of the actual interconnections of the whole system? Sufficient conditions are obtained for such a result to hold true; in particular it is shown that a system S consisting of a number of subsystems S_i connected in an arbitrary way between themselves with finite gains: $\text{S}{}_{\mathrm{i}}\text{:}\text{x}\text{?}{}_{\mathrm{i}}\text{=}\text{A}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)}\text{x}{}_{\mathrm{i}}\text{+}\text{b}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)u}{}_{\mathrm{i}}\text{, y}{}_{\mathrm{i}}\text{=}\text{c}\text{′}{}_{\mathrm{i}}\text{(}\text{x}{}_{\mathrm{i}}\text{, t)}\text{x}{}_{\mathrm{i}}$ where A_i and b_i have a particular structure, may be satisfactorily controlled by applying only local controllers C_i about the individual subsystems: C_i: u_i = K′_i(?)x_i where K_i is a constant gain matrix with the scalar ? appearing as a parameter, provided ? is large enough.     

Design of observers for non-linear time-varying systems

Novel stable observers are proposed for discrete, time-varying systems having time-varying, sector-bounded non-linearities. First, the form of the non-linearity is assumed to be known for all time and a stable non-linear filter is presented. Then assuming that the form of the non-linearity is not known but only a sector bound is given, we propose a linear filter and an upper bound on the sector constant to obtain a stable observer. The convergence conditions for the proposed observers are given directly in terms of the design parameters and are, therefore, very easy to check.     

Controller assessment for a class of non-linear systems

The use of autoregressive moving average (ARMA) models to assess the control loop performance for processes that are adequately described by the superposition of a linear dynamic model and linear stochastic or deterministic disturbance model is well known. In this paper, classes of non-linear dynamic/stochastic systems for which a similar result can be obtained are established for single-input single-output discrete system. For these systems, lower mean-square error bounds on performance, can be estimated from the closed-loop routine operating data by using non-linear autoregressive moving average with exogenous inputs (NARMAX) models. It is necessary to know the process time delay. The fitting of these models is greatly facilitated by using efficient algorithms, such as Orthogonal Least Squares or other fast orthogonal search algorithms. These models can also be used to assess the predictive importance of non-linearities over multiple-time horizons.     

Canonical transformation for a class of time-varying multivariable systems

This paper concerns the transformation of time-varying multivariable systems into canonical structures. The study employs differential matrix operators. It leads to a systematic and straightforward technique for developing canonical forms for time-varying multivariable systems that are uniformly observable and «lexicography-fixed’. It shows that canonical forms derived by other authors are special cases of the canonical form of this paper. The derived canonical forms are not unique. However, their structures are controlled by the designer     

一类时变时滞系统的稳定性分析

针对时变时滞系统稳定性问题, 在考虑非线性扰动的情况下, 为了降低时变时滞系统稳定性判据的保守性, 以改进的Jensen 不等式, Wirtinger型双重积分不等式以及优化凸组合技术为基础, 构造增广的Lyapunov-Krasovskii 泛函, 得到了新的时滞相关稳定性判据. 最后, 通过数值仿真对比可知, 该稳定性判据具有较小的保守性和良好的鲁棒性.     

Balancing for a class of non-linear singularly perturbed systems

In this work, a balancing method is investigated for a class of non-linear singularly perturbed systems. The main result presented here shows that the well-known 'two-stages' strategy used with singular perturbations in control theory can be extended to compute a balancing form of non-linear singularly perturbed systems. So, an approximate balancing form is derived from the balancing forms of the slow and fast subsystems both computed separately. This two-stage method avoids the difficult task of solving high dimensional and ill-conditioned Hamilton-Jacobi equations due to the presence of the small singular perturbation parameter.     

Quantitative fault estimation for a class of non-linear systems

This paper develops a methodology for actuator fault diagnostics and quantitative estimation of fault signals in a class of non-linear systems. The class of non-linear systems considered is one in which the non-linearity is an incremental quadratic type of non-linearity that includes Lipschitz, positive real and sector non-linear functions. The methodology provides for both state estimation and fault vector estimation. An LMI procedure can be utilized for explicit computation of the observer gains. The procedure developed can also be specialized to linear time invariant systems. Compared to previous results for LTI systems, the procedure developed herein does not require the LTI system to be minimum phase. The use of the developed methodology is demonstrated through two illustrative examples of real world physical applications.     

Non-linearity measures for a class of SISO non-linear systems

A new measure of non-linearity, that is captured by two linear systems, is proposed to quantify the size of the non-linearity in the input/output behaviour of SISO non-linear systems. As such, the controller design for the non-linear system is simplified to the design of a controller for these two linear systems. It is also shown that when Hammerstein and Wiener models are involved, the controller design for these two structures are equivalent if the linear dynamic term is stable. Two examples, a numerical example and a non-linear system characterized by a time-delay and a variable gain are used to demonstrate these concepts.     

A parametric frequency response method for non-linear time-varying systems

A new parametric frequency response algorithm is introduced to investigate linear and non-linear dynamic systems with time-varying parameters. In the new algorithm the time-varying parameters are regarded as additional inputs of the systems and the non-linear generalised frequency response functions for multi-input-single-output systems are then employed to obtain Zadeh's system functions from a differential equation representation. The parametric frequency response method reveals how the time-varying parameters affect the behaviour of the systems through a time-varying term. The new method can be applied to both linear and non-linear time-varying systems.     

Robust observability for a class of time-varying discrete-time uncertain systems

This paper introduces a concept of robust observability for a class of uncertain discrete-time systems. This notion is an extension of the standard notion of observability for discrete-time, linear time-varying systems. The uncertainty and noise are modelled deterministically via a sum quadratic constraint. A necessary and sufficient condition for robust observability is presented in terms of existence of a suitable solution to a Riccati difference equation. The set of possible initial states given noisy measurements over a finite number of sampling periods is shown to be an ellipsoid.     

A synthesis of Lyapunov functions for non-linear time-varying control systems

The formulation of a class of Lyapunov functions for time-varying nonlinear control systems which occur in the stability analysis of control systems is considered. A new approach is presented, which is an extension of a generation technique for time-invariant non-linear systems. This approach, which uses by analogy the classical theory of Hamilton, permits stability and transient information to be obtained from system equations with time-varying damping as well as time-varying gain. A second and third-order example is used to illustrate the application of this new result.     

Popov-like stability criterion for a class of time-varying discrete systems

A Popov-like stability criterion is derived for use in the analysis of a class of time-varying discrete control systems. The class of systems consists of a time-varying gain followed by a causal time-invariant discrete convolution operator. The stability criterion provides a simple graphical means of obtaining stability information which complements the existing information for this class of systems.     

Closed-loop balancing for a class of non-linear singularly perturbed systems

The aim of this article is to investigate the closed-loop balancing reduction method for a class of non-linear singularly perturbed systems. We show that the well-known two-stage strategy involved commonly within the singular perturbation theory can be used to derive an approximate closed-loop balancing. The proposed two-stage method avoids the difficult task of solving high dimensional and ill conditioned non-linear Hamilton–Jacobi equation due to the presence of the small perturbation parameter.