LTI approximation of nonlinear systems via signal distribution theory

L₂ and L₁ optimal linear time-invariant (LTI) approximation of discrete-time nonlinear systems, such as nonlinear finite impulse response (NFIR) systems, is studied via a signal distribution theory motivated approach. The use of a signal distribution theoretic framework facilitates the formulation and analysis of many system modelling problems, including system identification problems. Specifically, a very explicit solution to the L₂ (least squares) LTI approximation problem for NFIR systems is obtained in this manner. Furthermore, the L₁ (least absolute deviations) LTI approximation problem for NFIR systems is essentially reduced to a linear programming problem. Active LTI modelling emphasizes model quality based on the intended use of the models in linear controller design. Robust stability and LTI approximation concepts are studied here in a nonlinear systems context. Numerical examples are given illustrating the performance of the least squares (LS) method and the least absolute deviations (LAD) method with LTI models against nonlinear unmodelled dynamics.     

Linear approximations of nonlinear FIR systems for separable input processes

Nonlinear systems can be approximated by linear time-invariant (LTI) models in many ways. Here, LTI models that are optimal approximations in the mean-square error sense are analyzed. A necessary and sufficient condition on the input signal for the optimal LTI approximation of an arbitrary nonlinear finite impulse response (NFIR) system to be a linear finite impulse response (FIR) model is presented. This condition says that the input should be separable of a certain order, i.e., that certain conditional expectations should be linear. For the special case of Gaussian input signals, this condition is closely related to a generalized version of Bussgang's classic theorem about static nonlinearities. It is shown that this generalized theorem can be used for structure identification and for the identification of generalized Wiener-Hammerstein systems.     

On linear models for nonlinear systems

Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Fréchet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Fréchet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an ?_∞ signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.     

Switching between stabilizing controllers

This paper deals with the problem of switching between several linear time-invariant (LTI) controllers—all of them capable of stabilizing a specific LTI process—in such a way that the stability of the closed-loop system is guaranteed for any switching sequence. We show that it is possible to find realizations for any given family of controller transfer matrices so that the closed-loop system remains stable, no matter how we switch among the controller. The motivation for this problem is the control of complex systems where conflicting requirements make a single LTI controller unsuitable.     

A unified framework for the study of anti-windup designs

We present a unified framework for the study of linear time-invariant (LTI) systems subject to control input nonlinearities. The framework is based on the following two-step design paradigm: ‘design the linear controller ignoring control input nonlinearities and then add anti-windup bumpless transfer (AWBT) compensation to minimize the adverse effects of any control input nonlinearities on closed loop performance’. The resulting AWBT compensation is applicable to multivariable controllers of arbitrary structure and order. All known LTI anti-windup and/or bumpless transfer compensation schemes are shown to be special cases of this framework. This unification of existing schemes for AWBT compensation under a general framework is the main result of the paper.     

Least-squares LTI approximation of nonlinear systems and quasistationarity analysis

Least-squares linear time-invariant (LTI) approximation of discrete-time nonlinear systems is studied in a generalized harmonic analysis setting extending an earlier result based on quasistationary signals. The least-squares optimal LTI model is such that the crosscorrelation between the input and the LTI model output equals the crosscorrelation between the input and the output of the nonlinear system. New results for limits of sample averages of signals are derived via Riemann-Stieltjes integration theory. These results are applied to crosscorrelation and quasistationarity analysis of input-output signals for several important classes of nonlinear systems, including stable finite memory, Wiener and Hammerstein systems. This analysis demonstrates that the assumptions used in the least-squares LTI approximation setup are fairly mild. Finally, an illustrative example is provided.     

Properties of linear switching time-varying discrete-time systems with applications

We study two discrete-time, linear switching time-varying (LSTV) structures, each of which consists of a periodic switch connected to several linear time-invariant (LTI) systems. Such structures can be used to represent any linear periodically time-varying (LPTV) systems. We give basic properties associated with the LSTV structures in terms of their LTI building blocks, and then apply the results to solve a general approximation problem: How to optimally approximate an LPTV system with period p by an LPTV system with period ? The optimality is measured using norms. The study is extended to general multirate periodic systems.     

Quantitative design of a class of nonlinear systems with parameter uncertainty

This paper considers the case in which a linear time-invariant (LTI) but uncertain plant suffers from nonlinearities y=n(x) which can be expressed as y=K_n+η(x), |η(x)|≤M, with K, a possibly uncertain scalar. This covers a large and very important class of nonlinearities encountered in practice such as friction, backlash, dead zone and quantization. Quantitative design techniques are presented for this class for the satisfaction of specifications. Special attention is paid to the avoidance of limit cycles using describing function theory, although the design method is also amenable of application using other stability criteria such as the circle criteria. Numerical examples are developed illustrating the design procedure.     

基于LMI的小时滞饱和系统稳定域估计方法

基于Pade近似变换,将小时滞饱和系统的稳定域估计转化为估计奇异摄动饱和系统的稳定域问题.证明了此奇异摄动饱和系统的稳定域具有可解耦性,并在此基础上建立LMI优化模型并提出小时滞饱和系统稳定域估计的降阶方法.算例仿真验证了方法的正确性和有效性.     

Multivariable pi-sharing theory and its application on the Lur'eproblem

The pi-sharing theory developed by Lawrence and Johnson (1986) is extended to handle square multivariable continuous-time systems, and for the most essential part of finding usable pi-coefficients, LMI formulations are utilized such that for any finite-dimensional LTI systems, time-invariant pi-coefficients can be obtained easily. Furthermore, pi-coefficients are derived for sector-bounded nonlinearities, and the extended pi-sharing theory is applied to the multivariable Lur'e problem. With this approach, not only can a Lur'e system with known sector bounds on nonlinearities be checked for stability, but also the maximal bounds ensuring stability can be found under some conditions     

Stability and Stabilization of Two‐Dimensional LTI Switched Systems with Potentially Unstable Focus

This paper investigates stability and stabilization of two‐dimensional switched linear time‐invariant (LTI) systems with potentially unstable focus. For the case that the origin is a single common focus of all subsystems, we first give continuous positive definite functions related only to the elements of subsystems' state matrices. Then, based on the continuous positive definite functions obtained, this paper proposes several sufficient conditions of stability/asymptotic stability/instability of the kind of switched LTI systems. By means of the stability results proposed, global asymptotic stabilizing controls (GASC), global asymptotic stabilizing switching paths (GASSP) and corresponding algorithms are designed for two‐dimensional switched LTI systems with focus. Finally, two illustrative examples and numerical simulations demonstrate the effectiveness of the new stability and stabilization results obtained in this paper.     

一种估计奇异摄动饱和系统稳定域的方法

针对奇异摄动饱和系统, 提出了一种估计其稳定域的降阶方法. 结合饱和函数的特殊性质, 证明了此类系统的稳定域可分解为伴随系统的不变集与一个足够大球体的笛卡尔积. 将原系统稳定域估计问题转化为低阶伴随系统稳定域的估计问题, 利用线性矩阵不等式(Linear matrix inequality, LMI)优化方法估计伴随系统的稳定域以减少保守性. 本方法不仅可以克服奇异摄动饱和系统的奇异性, 还可以一定程度克服系统的``维数灾'等问题.     

Minimum Rate Coding for LTI Systems Over Noiseless Channels

This paper studies rate requirements for state estimation in linear time-invariant (LTI) systems where the controller and the plant are connected via a noiseless channel with limited capacity. Using information theoretic arguments, we obtain first for scalar systems, and subsequently for multidimensional systems, lower bounds on the data rates required for state estimation under three different stability criteria, namely monotonic boundedness of entropy, asymptotic stability of distortion, and support size stability. Further, the minimum data rate achievable by any source-encoder is computed under each of these criteria, and the best rate achievable with quantization is shown to be in agreement with the information-theoretic bounds in some specific cases (such as if the system coefficient is an integer or if the criterion is an asymptotic one). Existence of optimal variable-length and fixed-length quantizers are studied and optimal quantizers are constructed under each of these criteria. One observation is that, the uniform quantizer is, in addition to being simple, efficient in linear control systems     

On the stability of fuzzy systems

Studies the global asymptotic stability of a class of fuzzy systems. It demonstrates the equivalence of stability properties of fuzzy systems and linear time invariant (LTI) switching systems. A necessary and sufficient condition for the stability of such systems are given, and it is shown that under the sufficient condition, a common Lyapunov function exists for the LTI subsystems. A particular case when the system matrices can be simultaneously transformed to normal matrices is shown to correspond to the existence of a common quadratic Lyapunov function. A constructive procedure to check the possibility of simultaneous transformation to normal matrices is provided     

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.     

An LMI approach to design robust fault detection filter for uncertain LTI systems

In this paper, the robust fault detection filter design problem for uncertain linear time-invariant (LTI) systems with both unknown inputs and modelling errors is studied. The basic idea of our study is to use an optimal residual generator (assuming no modelling errors) as the reference residual model of the robust fault detection filter design for uncertain LTI systems with modelling errors and, based on it, to formulate the robust fault detection filter design as an H_∞ model-matching problem. By using some recent results of H_∞ optimization, a solution of the optimization problem is then presented via a linear matrix inequality (LMI) formulation. The main results include the development of an optimal reference residual model, the formulation of robust fault detection filter design problem, the derivation of a sufficient condition for the existence of a robust fault detection filter and a construction of it based on the LMI solution parameters, the determination of adaptive threshold for fault detection. An illustrative design example is employed to demonstrate the effectiveness of the proposed approach.     

A general theory of linear time-invariant adaptive feedforward systems with harmonic regressors

Establishes necessary and sufficient conditions for an adaptive system with a harmonic regressor (i.e., a regressor comprised exclusively of sinusoidal signals) to admit an exact linear time-invariant (LTI) representation. These conditions are important because a large number of adaptive systems used in practice have sinusoidal regressors, and the stability and performance of such systems having LTI representations can be completely analyzed by well-known methods. The theory is extended to applications where the LTI conditions do not hold, in which case the harmonic adaptive system can be written as the parallel connection of a purely LTI subsystem and a linear time-varying (LTV) subsystem. An explicit upper bound is established on the induced two-norm of the LTV block, which allows systematic treatment using emerging robust control methods applicable to LTI systems with norm-bounded LTV perturbations.     

Adaptive feedback linearization control of chaotic systems via recurrent high-order neural networks

In the realm of nonlinear control, feedback linearization via differential geometric techniques has been a concept of paramount importance. However, the applicability of this approach is quite limited, in the sense that a detailed knowledge of the system nonlinearities is required. In practice, most physical chaotic systems have inherent unknown nonlinearities, making real-time control of such chaotic systems still a very challenging area of research. In this paper, we propose using the recurrent high-order neural network for both identifying and controlling unknown chaotic systems, in which the feedback linearization technique is used in an adaptive manner. The global uniform boundedness of parameter estimation errors and the asymptotic stability of tracking errors are proved by the Lyapunov stability theory and the LaSalle-Yoshizawa theorem. In a systematic way, this method enables stabilization of chaotic motion to either a steady state or a desired trajectory. The effectiveness of the proposed adaptive control method is illustrated with computer simulations of a complex chaotic system.     

Stochastic Stability and H∞ Disturbance Attenuation of A Class of Nonlinear Stochastic Delayed Systems

In this paper, we consider stochastic stability analysis and H_∞ disturbance attenuation for a class of discrete-time nonlinear stochastic retarded systems whose nonlinearities are described by statistical means. First a sufficient condition on mean square stability, which is independent of delay, is proposed using the stochastic Lyapunov functional approach. Then, H_∞ disturbance attenuation property of the system is analyzed. It is shown that the H_∞ disturbance attenuation of such nonlinear systems can be reduced to a linear matrix inequality problem.     

Squared and absolute errors in optimal approximation of nonlinear systems

Optimal squared error and absolute error-based approximation problems for static polynomial models of nonlinear, discrete-time, systems are studied in detail. These problems have many similarities with other linear-in-the-parameters approximation problems, such as with optimal approximation problems for linear time-invariant models of linear and nonlinear systems. Nonprobabilistic signal analysis is used.Close connections between the studied approximation problems and certain classical topics in approximation theory, such as optimal L₂(−1,1) and L₁(−1,1) approximation, are established by analysing conditions under which sample averages of static nonlinear functions of the input converge to appropriate Riemann integrals of the static functions. These results should play a significant role in the analysis of corresponding system identification and model validation problems. Furthermore, these results demonstrate that optimal modelling based on the absolute error can offer advantages over squared error-based modelling. Especially, modelling problems in which some signals possess heavy tails can benefit from absolute value-based signal and error analysis.