Hinfinity fuzzy control design for nonlinear singularly perturbed systems with pole placement constraints: an LMI approach.

This paper considers the problem of designing an H infinity fuzzy controller with pole placement constraints for a class of nonlinear singularly perturbed systems. Based on a linear matrix inequality (LMI) approach, we develop an H infinity fuzzy controller that guarantees 1) the L2-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, and 2) the closed-loop poles of each local system to be within a pre-specified LMI stability region. In order to alleviate the ill-conditioned LMIs resulting from the interaction of slow and fast dynamic modes, solutions to the problem are given in terms of linear matrix inequalities which are independent of the singular perturbation, epsilon. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard singularly perturbed non-linear systems. A numerical example is provided to illustrate the design developed in this paper.     

Exponential stability of singularly perturbed systems with time delay and uncertainties

Recently, there are several studies on singularly perturbed systems with time delay and uncertainties. However, the existing results show only the limited results on either time delay or uncertainties. For time delay, small time delay or time delay only on the slow state are considered while, for uncertainties, the uncertain linear singularly perturbed system or the uncertain nonlinear singularly perturbed system with the conservative condition on uncertainties are considered. In this paper, the exponential stability of singularly perturbed systems with both time delay and uncertainties is investigated in the general form.     

Intelligent robust control for uncertain nonlinear time-varying systems and its application to robotic systems.

This paper addresses the problem of designing adaptive fuzzy-based (or neural network-based) robust controls for a large class of uncertain nonlinear time-varying systems. This class of systems can be perturbed by plant uncertainties, unmodeled perturbations, and external disturbances. Nonlinear H(infinity) control technique incorporated with adaptive control technique and VSC technique is employed to construct the intelligent robust stabilization controller such that an H(infinity) control is achieved. The problem of the robust tracking control design for uncertain robotic systems is employed to demonstrate the effectiveness of the developed robust stabilization control scheme. Therefore, an intelligent robust tracking controller for uncertain robotic systems in the presence of high-degree uncertainties can easily be implemented. Its solution requires only to solve a linear algebraic matrix inequality and a satisfactorily transient and asymptotical tracking performance is guaranteed. A simulation example is made to confirm the performance of the developed control algorithms.     

ROBUST CONTROL FOR A CLASS OF UNCERTAIN STATE‐DELAYED SINGULARLY PERTURBED SYSTEMS

This paper considers the problem of robust control for a class of uncertain state‐delayed singularly perturbed systems with norm‐bounded nonlinear uncertainties. The system under consideration involves state time‐delay and norm‐bounded nonlinear uncertainties in the slow state variable. It is shown that the state feedback gain matrices can be determined to guarantee the stability of the closed‐loop system for all ∞ ∞ (0, ∞00) and independently of the time‐delay. Based on this key result and some standard Riccati inequality approaches for robust control of singularly perturbed systems, a constructive design procedure is developed. We present an illustrative example to demonstrate the applicability of the proposed design approach.     

Asymptotic H∞ control of singularly perturbedsystems with parametric uncertainties

This paper deals with the problem of control of singularly perturbed linear continuous-time systems. The authors' attention is focused on the design of a composite linear controller based on the slow and fast problems such that both stability and a prescribed H_∞ performance for the full-order system are achieved. The asymptotic behavior of the composite controller is studied, which is independent of the singular perturbation ϵ when ϵ is sufficiently small. Furthermore, the problem of robust control for the above system with parameter uncertainty is also investigated     

Robust stability of non-standard nonlinear singularly perturbed discrete systems with uncertainties

In the past several decades, the singularly perturbed discrete systems have received much attention for the stability analysis and controller design. Recently, there are some results about the nonlinear singularly perturbed discrete systems. Compared with the existing result, we consider the robust stability of the uncertain nonlinear singularly perturbed discrete systems with the less conservative assumption via the Lyapunov function method. Moreover, the previous results of the singularly perturbed discrete system are only applied to the system, which is composed of the slow part and the fast part, separately. However, we consider the non-standard nonlinear singularly perturbed discrete system in which the slow part and the fast part coexist, that is, a general case of the nonlinear singularly perturbed discrete systems. Then, by using the lower-order subsystems from two standard systems, we present the robust stability of the non-standard nonlinear singularly perturbed discrete system with uncertainties.     

Contraction-based stabilisation of nonlinear singularly perturbed systems and application to high gain feedback

Recent development of contraction theory-based analysis has opened the door for inspecting differential behaviour of singularly perturbed systems. In this paper, a contraction theory-based framework is proposed for stabilisation of singularly perturbed systems. The primary objective is to design a feedback controller to achieve bounded tracking error for both standard and non-standard singularly perturbed systems. This framework provides relaxation over traditional quadratic Lyapunov-based method as there is no need to satisfy interconnection conditions during controller design algorithm. Moreover, the stability bound does not depend on smallness of singularly perturbed parameter and robust to additive bounded uncertainties. Combined with high gain scaling, the proposed technique is shown to assure contraction of approximate feedback linearisable systems. These findings extend the class of nonlinear systems which can be made contracting.     

Robust sampled-data H/sub /spl infin// control of linear singularly perturbed systems

State-feedback H/sub /spl infin// control problem for linear singularly perturbed systems with norm-bounded uncertainties is studied. The fast variables are sampled with fast rates, while for the slow variables both cases of slow and of fast sampling are considered. The recent "input delay" approach to sampled-data control is applied, where the closed-loop system is represented as a continuous one with time-varying input delay. Linear matrix inequalities (LMIs) for solution of H/sub /spl infin// control problem are derived via input-output approach to stability and L/sub 2/-gain analysis of time-delay systems. A numerical example illustrates the efficiency of the method.     

control for fast sampling discrete-time singularly perturbed systems

This paper is concerned with the H_∞ control problem via state feedback for fast sampling discrete-time singularly perturbed systems. A new H_∞ controller design method is given in terms of solutions to linear matrix inequalities (LMIs), which eliminates the regularity restrictions attached to the Riccati-based solution. A method for evaluating the upper bound of singular perturbation parameter with meeting a prescribed H_∞ performance bound requirement is also given. Furthermore, the results are extended to robust controller design for fast sampling discrete-time singularly perturbed systems with polytopic uncertainties. Numerical examples are given to illustrate the validity of the proposed methods.     

奇异摄动系统的鲁棒控制与仿真研究

研究参数不确定奇异摄动系统的状态反馈镇定问题,得到基于线性矩阵不等式（LMI）的系统状态反馈可镇定的充分条件,给出状态反馈控制器的设计算法。算例验证了该方法的可行性与有效性。而且,这种方法同时适用于标准奇异摄动系统和非标准奇异摄动系统的控制。最后通过例子进行了详细的仿真研究。     

On the model-based networked control for singularly perturbed systems with nonlinear uncertainties

In this paper, the model-based networked control is addressed for a class of singularly perturbed control systems with nonlinear uncertainties. An approximate linear slow and fast control system of the plant, which can be obtained by omitting the nonlinear uncertainties, are used as a model to estimate the state behavior of the plant between transmission times. The stability of model-based networked control systems is investigated under the assumption that the controller/actuator is updated with the sensor information at constant time intervals. It is shown that there exists the allowable upper bound of the singular perturbation parameter such that the model-based networked control system is globally exponentially stable.     

Exponential tracking of feedback linearizable non-linear control systems with uncertainties

This paper studies the robust output tracking problem of feedback linearizable non-linear control systems with uncertainties. Utilizing the input-output feedback linearization technique and the Lyapunov method for non-linear state feedback synthesis, a robust globally exponential output tracking controller design methodology for a broad class of non-linear control systems with uncertainties is developed. The underlying theoretical approaches are the differential geometry approach and the composite Lyapunov approach. One utilizes the parametrized coordinate transformation to transform the original non-linear system with uncertainties into a singularly perturbed model with uncertainties and the composite Lyapunov approach is then applied for output tracking. To demonstrate the practical applicability, the paper has investigated a pendulum control system.     

Design of robust PD-type control laws for robotic manipulators with parametric uncertainties

In this article, design of a simple robust control law that achieves desired positions and orientations for robotic manipulators with parametric uncertainties is studied. A discontinuous control law is proposed, which consists of a high-gain linear proportional plus derivative (PD) term and additional terms that compensate for the effect of gravitation. The stability of the robotic system under the proposed control law is proved by LaSalle's stability theorem. Furthermore, by the theory of singularly perturbed systems, it is shown that if the proportional and derivative gain matrices are diagonal with large positive elements then the system is decoupled into a set of first-order linear systems. Simulation results are presented to illustrate the application of the proposed control law to a two-link robotic manipulator.     

Singularly perturbed implicit control law for linear time‐varying delay MIMO systems

This paper proposes a control scheme for the problem of stabilizing partly unknown multiple‐input multiple‐output linear time‐varying retarded systems. The control scheme is composed by a singularly perturbed controller and a reference model. We assume the knowledge of a number of structural characteristics of the system as the boundedness and the knowledge of the bounds for the unknown parameters (and their derivatives) that define the system matrices, as well as the structure of these matrices. The results presented here are a generalization of previous results on linear time‐varying Single‐Input Single‐Output (SISO) and multiple‐input multiple‐output systems without delays and linear time‐varying retarded SISO systems. The closed‐loop system is a linear singularly perturbed retarded system with uniform asymptotic stability behavior. The uniform asymptotic stability of the singularly perturbed retarded system is guaranteed. We show how to design a control law such that the system dynamics for each output is given by a Hurwitz polynomial with constant coefficients. Copyright © 2015 John Wiley & Sons, Ltd.     

基于迭代线性矩阵不等式的奇异摄动系统同时镇定

研究了采用一个线性状态反馈控制器镇定多个线性奇异摄动系统的问题.同时镇定条 件可以表达为一组矩阵不等式条件,所得条件与摄动参数无关,从而有效地回避了病态问题.采 用基于快慢分解的两步法可以得到镇定控制器增益和相应的Lyapunov函数.而在每一步需要利 用迭代线性矩阵不等式技术求解相应的双线性矩阵不等式,相关定理研究了算法的收敛性.本文 所得结论可同时适用于标准与非标准奇异摄动系统.文末给出了相应的仿真算例.     

一类奇异摄动时滞不确定系统鲁棒稳定性

为了解决复杂工业过程控制系统由于工程设计时模型描述的不完全性造成的不稳定，针对一类常见的奇异摄动时滞不确定线性系统，研究了其鲁棒稳定性问题。通过对系统各参数日。范数有界摄动分析，获得了系统时滞有关稳定的充分条件。给出了保证系统稳定的时滞τ及奇异摄动参数ε的上界，在工程设计中为复杂工业过程控制系统稳定边界的判别提供了简单有效的手段。应用实例表明了奇异摄动时滞不确定系统鲁棒稳定性充分条件使用的方便性。     

H∞control of interconnected nonlinear sampled-data systems with parametric uncertainty

This paper studies the problem of robust control design for a class of interconnected uncertain systems under sampled measurements. The class of system under consideration is described by a state space model containing unknown cone bounded nonlinear interaction and time-varying norm-bounded parameter uncertainties in both state and output equations. Our attention is focused on the design of linear dynamic output feedback controllers using sampled measurements. We address the problem of robust H_∞ control in which both robust stability and a prescribed H_∞ performance are required to be achieved irrespective of the uncertainties and nonlinearities. The H_∞ performance measure involves both continuous-time and discrete-time signals. It has been shown that the above problems can be recast into H_∞ syntheses for related N decoupled linear sampled-data systems without parameter uncertainties and unknown nonlinearities, which can be solved in terms of Riccati differential equations with finite discrete jumps. A numerical example is given to show the potential of the proposed technique.     

Robust H/sub /spl infin// control for uncertain discrete-time-delay fuzzy systems via output feedback controllers

This work investigates the problem of robust output feedback H/sub /spl infin// control for a class of uncertain discrete-time fuzzy systems with time delays. The state-space Takagi-Sugeno fuzzy model with time delays and norm-bounded parameter uncertainties is adopted. The purpose is the design of a full-order fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop system and guarantees an H/sub /spl infin// norm bound constraint on disturbance attenuation for all admissible uncertainties. In terms of linear matrix inequalities (LMIs), a sufficient condition for the solvability of this problem is presented. Explicit expressions of a desired output feedback controller are proposed when the given LMIs are feasible. The effectiveness and the applicability of the proposed design approach are demonstrated by applying this to the problem of robust H/sub /spl infin// control for a class of uncertain nonlinear discrete delay systems.     

Robust sampled-data H∞ control of uncertain singularly perturbed systems using time-dependent Lyapunov functionals

In this paper, the problem of robust sampled-data H_∞ control of linear uncertain singularly perturbed systems is investigated. The parametric uncertainties are assumed to be time-varying and norm-bounded. Two types of controller design are considered: (1) with a fast sampling in the fast state and a slow one in the slow state, and (2) with a fast sampling in both states. For each type, a time-dependent Lyapunov functional associated with the sampling pattern is introduced to analyse the exponential stability and L₂-gain performance of the closed-loop system. Linear matrix inequalities based solutions of the robust sampled-data H_∞ control problem are derived. The new results are proved theoretically to be less conservative than the existing results. An illustrative example is given which substantiates the usefulness of the proposed method.     

Order Reduction Is Invalid for Singularly Perturbed Control Problems with a Vector Fast Variable

The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed optimal control problem under consideration. In this paper we show, however, that the set of singularly perturbed optimal control systems for which the order reduction approach is invalid is dense (in the L ^∞ norm) in the class of systems which we consider. This is established under the assumption that the fast variable in the singularly perturbed system is not a scalar. Date received: June 8, 2001. Date revised: December 30, 2001.