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.     

Routh approximations for reducing order of linear, time-invariant systems

A new method of approximating the transfer function of a high-order linear system by one of lower order is proposed. Called the "Routh approximation method" because it is based on an expansion that uses the Routh table of the original transfer function, the method has a number of useful properties: if the original transfer function is stable, then all approximants are stable; the sequence of approximants converge monotonically to the original in terms of "impulse response" energy; the approximants are partial Padé approximants in the sense that the firstkcoefficients of the power series expansions of thekth-order approximant and of the original are equal; the poles and zeros of the approximants move toward the poles and zeros of the original as the order of the approximation is increased. A numerical example is given for the calculation of the Routh approximants of a fourth-order transfer function and for illustration of some of the properties.     

Worst case system identification in : Optimal algorithms and error bounds

This paper analyses worst case identification of linear shift invariant systems with an error criterion. The assumed apriori information on the plant are bounds on the system gain and decay rate of the impulse response of the unknown system and the experimental data is assumed to be a finite set of corrupted impulse or step response samples of the system.     

Robust admissibility of time-varying singular systems with commensurate time delays

This paper addresses the problem of admissibility of time-varying singular systems with commensurate time delays. By introducing a new method to study the convergence of the fast sub-system, an admissibility condition is obtained in terms of linear matrix inequalities, which guarantees the considered time-varying singular delay system to be regular, impulse free and stable. Furthermore, a robust admissibility condition is proposed for the case when the time-varying system matrices admit the usual constant-plus-norm-bounded structure. It is theoretically established that the proposed robust admissibility condition is less conservative than the existing one in the literature. One of the important features of the results is that the inferred admissibility condition coincides with the usual stability condition for state-space delay systems. Moreover, the results are also applicable to the multiple rational delay case.     

On the existence of a positive realization

The positive realization problem for linear systems is to find conditions, for a given transfer function with nonnegative impulse response, to have a realization such that the resulting system is a positive system. Recently, it has been shown that, under a mild assumption on the long-term behaviour of the impulse response, this problem is related to the maximum modulus poles only. In this paper necessary and sufficient conditions for positive realizability of discrete-time systems are given. They show that also nondominant poles play a role in the most general case. Positive realizability conditions for the continuous-time case are also given.     

Recursive identification of linear and non-linear systems

The paper develops recursive techniques for off-line identification of linear and nonlinear systems. It is shown that if the system is linear and time invariant, impulse response characterization of the system coupled with an orthogonal series approximation can be utilized for the purpose stated above. The techniques of adaptive Kalman filtering are shown to be applicable, which besides permitting recursive evaluation of the coefficients, lead to a number of important advantages. In the second part of the study, the proposed method is extended for recursive identification of a class of non-linear systems which can be represented as a cascade combination of a linear dynamical system and a non-linear zero memory system. The method of Volterra series representation of such systems is utilized. Results are illustrated through numerical examples in each case.     

Comments on the computation of interval Routh approximants

In Bandyopadhyay et al. (1994, 1997), the Routh approximation method was extended to derive reduced-order interval models for linear interval systems. In this paper, the authors show that: 1) interval Routh approximants to a high-order interval transfer function depend on the implementation of interval Routh expansion and inversion algorithms; 2) interval Routh expansion algorithms cannot guarantee the success in generating a full interval Routh array; 3) some interval Routh approximants may not be robustly stable even if the original interval system is robustly stable; and 4) an interval Routh approximant is in general not useful for robust controller design because its dynamic uncertainties (in terms of robust frequency responses) do not cover those of the original interval system     

A Constructive Approach to Approximate Linear Periodic Systems

In this note, the problem on approximating a linear periodic system with period N by a periodic model with period M(0infin norms. Under this framework, both finite impulse response (FIR) and infinite impulse response (IIR) periodic systems can be efficiently approximated by using a constructive linear matrix inequality approach. Finally, some numerical examples are given to illustrate the effectiveness of the proposed approach     

An eigenstructure assignment approach for constrained linear continuous-time singular systems

This paper establishes necessary and sufficient algebraic conditions for positive invariance of convex polyhedra with respect to some linear continuous-time singular systems. They can be considered as an extension for linear singular systems of the classical positive invariance relations for regular linear systems. For a stabilizable and impulse controllable singular system with constrained inputs, a stabilizing state feedback control guaranteeing the closed-loop positive invariance of some polyhedral sets determined from the feedback matrix is studied. An analysis of the closed-loop positive invariance relations is thus presented in terms of eigenstructure and stability properties. An eigenstructure assignment technique is proposed depending on the number of stable finite poles of the singular system.     

Synthesis of MIMO controllers for extending the stability range of periodic solutions in forced nonlinear systems

This paper deals with a central issue in bifurcations and chaos control applications, i.e., the stabilization of periodic motions in sinusoidally forced nonlinear systems. Specifically, the problem of designing multi-input-multi-output (MIMO) finite-dimensional linear time-invariant controllers maximizing the amplitude of the sinusoidal input for which the corresponding periodic solutions are guaranteed to be stable, is considered. By exploiting linearization techniques and the multi-variable circle criterion, a synthesis algorithm is developed to determine the controller which maximizes a suitable lower bound of the amplitude of the input. The algorithm requires the solution of a sequence of linear matrix inequalities (LMIs) of increasing size. The Brusselator oscillator is employed as a case study to show that the synthesized controllers, though optimizing a lower bound, provide quite satisfactory control performance.     

Design of pole placing controllers for stable and unstable systems with pure time delay

Based on improved low order dead time approximants or starting from an overall approximation of linear time invariant systems with pure time delay, pole placing controller design is applied. This procedure yields excellent results for stable and unstable systems. The main advantage of this approach is the use of well-known state feedback control methods which can be used to shape the closed-loop response in a desired manner. As long as the approximations are of sufficient quality, the closed loop transients of model and original loop coincide within small error limits.     

Legendre orthogonal polynomials approximation of system gramians and its application to balanced truncation

The paper presents a procedure for computation of system gramians by using Legendre orthogonal polynomials approximation of the system state impulse and output responses. The proposed approach is trajectory based and relies on the system state and output trajectories snapshots selected either by experiment or computer simulation. It is defined in deterministic settings as opposed to similar approaches defined in stochastic settings by using the data covariance matrix. The advantage of using orthogonal series approximation for the gramians is to avoid solving the usual Lyapunov equations. The proposed method can be equally well applied to linear time-invariant as well as time-varying systems, and even to unstable systems, since the gramians approximation is performed on a finite interval of time. When the observation interval contains the whole energy of the system state impulse and output responses, the proposed method gives similar results as the gramians computed by solving Lyapunov equations. Several experiments are performed showing the good approximation properties of the presented method.     

具线性脉冲的微分系统的稳定性

研究了具有线性脉冲作用的常微分系统平衡态的稳定性问题.应用常数变易法和Lyapunov方法,证明了在文中假设的条件下,总存在线性脉冲控制使一般非自治常微分系统的平衡态是一致渐近稳定.将自治微分系统作为特殊情形,在更一般的条件下得到了相同结论.最后给出了自治微分系统和非自治微分系统的两个例子以说明所得结论.     

基于脉冲响应的输出误差模型的辨识

基于系统脉冲响应参数, 利用相关分析方法, 提出了一种辨识输出误差模型参数的方法. 该方法是利用有限脉冲响应模型逼近输出误差模型, 通过依次递增脉冲响应参数的数目N来提高逼近精度. 理论分析表明, 只要N足够大, 模型的辨识精度可以满足实际要求. 提出的辨识方法可以在假设阶次N =1的条件下, 依次递增计算N较大时的脉冲响应参数和目标函数值, 从而根据脉冲响应确定系统的参数. 仿真试验说明提出的方法估计输出误差模型的参数是有效的.     

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.     

Delay-dependent stabilization of singular Markovian jump systems with state delay

This paper deals with the delay-dependent stabilization problem for singular systems with Markovian jump parameters and time delays. A delay-dependent condition is established for the considered system to be regular, impulse free and stochastically stable. Based on the condition, a design algorithm of the desired state feedback controller which guarantees the resultant closed-loop system to be regular, impulse free and stochastically stable is proposed in terms of a set of strict linear matrix inequalities （LMIs）. Numerical examples show the effectiveness of the proposed methods.     

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.     

基于未建模动态补偿的一类MIMO非线性系统的自适应广义预测解耦切换控制

针对一类不确定的多输入多输出(MIMO)离散时间零动态不稳定非线性系统, 提出了一种基于未建模动态补偿的非线性广义预测解耦切换控制方法. 该控制方法要求系统的未建模动态满足线性增长条件, 放宽了未建模动态全局有界的限制. 建立了所提的自适应控制方法的稳定性和收敛性分析. 而且, 在设计广义预测解耦控制器时, 把“一一映射”与ANFIS的训练相结合来估计系统的未建模动态, 保证了ANFIS的万能逼近特性. 最后, 仿真结果验 证了所提方法的优越性.     

Hamiltonian discretization of boundary control systems

A fundamental problem in the simulation and control of complex physical systems containing distributed-parameter components concerns finite-dimensional approximation. Numerical methods for partial differential equations (PDEs) usually assume the boundary conditions to be given, while more often than not the interaction of the distributed-parameter components with the other components takes place precisely via the boundary. On the other hand, finite-dimensional approximation methods for infinite-dimensional input-output systems (e.g., in semi-group format) are not easily relatable to numerical techniques for solving PDEs, and are mainly confined to linear PDEs. In this paper we take a new view on this problem by proposing a method for spatial discretization of boundary control systems based on a particular type of mixed finite elements, resulting in a finite-dimensional input-output system. The approach is based on formulating the distributed-parameter component as an infinite-dimensional port-Hamiltonian system, and exploiting the geometric structure of this representation for the choice of appropriate mixed finite elements. The spatially discretized system is again a port-Hamiltonian system, which can be treated as an approximating lumped-parameter physical system of the same type. In the current paper this program is carried out for the case of an ideal transmission line described by the telegrapher's equations, and for the two-dimensional wave equation.     

广义时变脉冲系统的时域稳定

研究了状态依赖广义时变脉冲系统的时域稳定问题.基于微分矩阵不等式 (Differential matrix inequalities, DMI) 和S-procedure理论, 给出了两类状态依赖广义时变脉冲系统时域稳定的充分条件.接下来, 根据给出的充分条件设计了状态反馈控制器, 使得闭环系统时域稳定.最后, 给出数值算例来验证结论的有效性.