Robust exponential stability of time-varying linear systems

This paper investigates the robustness of time-varying linear systems under a large class of complex time-varying perturbations. Previous results⁸ which were restricted to bounded linear perturbations of output feedback type are generalized to unbounded and nonlinear perturbations of multi-output feedback type. We establish a lower bound for the stability radius of these systems and show how it may be possible to improve the bound using time-varying scalar transformations of the state, input and output variables. The results are applied to derive Gershgorin type stability criteria for time-varying linear systems.     

Input-output stability of sampled-data linear time-varying systems

In this paper we consider the input-output stability of feedback systems consisting of a continuous-time, linear, time-varying plant with a discrete time, linear time-varying controller. These results generalize some of the results of Chen and Francis (1991, 1995) where the plant is restricted to be time-invariant     

Design of robust-optimal output feedback controllers for linear uncertain systems using LMI-based approach and genetic algorithm

This paper considers the robust-optimal design problems of output feedback controllers for linear systems with both time-varying elemental (structured) and norm-bounded (unstructured) parameter uncertainties. Two new sufficient conditions are proposed in terms of linear-matrix-inequalities (LMIs) for ensuring that the linear output feedback systems with both time-varying elemental and norm-bounded parameter uncertainties are asymptotically stable, where the mixed quadratically-coupled parameter uncertainties are directly considered in the problem formulation. A numerical example is given to show that the presented sufficient conditions are less conservative than existing ones reported recently. Then, by integrating the hybrid Taguchi-genetic algorithm (HTGA) and the proposed LMI-based sufficient conditions, a new integrative approach is presented to find the output feedback controllers of the linear systems with both time-varying elemental and norm-bounded parameter uncertainties such that the control objective of minimizing a quadratic integral performance criterion subject to the stability robustness constraint is achieved. A design example of the robust-optimal output feedback controller for the AFTI/F-16 aircraft control system with the time-varying elemental parameter uncertainties is given to demonstrate the applicability of the proposed new integrative approach.     

Stability of nonlinear time-varying feedback systems

In this paper, the stability of nonlinear time-varying feedback systems is studied using a “passive operator” technique. The feedback system is assumed to consist of a linear time-invariant operator G(s) in the forward path and a nonlinear time-varying gain function f(·)K(t) in the feedback path. The stability condition indicates that the bound on the time derivative [dK(t)/(dt)] depends both on the nonlinearity f(·) and the multiplier Z(s) chosen to make G(s)Z(s) positive real. It is also shown that the main result in this paper can be specialized to yield many of the results obtained so far for nonlinear time-invariant systems and linear time-varying systems.     

On the stability of multiloop feedback systems

Some extensions on recent work in passive stability theory by Desoer and the author are presented. The stability of feedback systems is considered where the forward-loop and return-loop subsystems each have been subdivided into several systems operating in parallel. Results are obtained in the areas of Lyapunov stability, L2input L2output stability, and bounded-input bounded-state stability. Before applying these results to a specific multiloop feedback system, a pseudoenergy analysis must be done on each sub sub-system. Two specific types of systems are so analyzed. The first is a general linear first-order time-varying system. The second is a linear time-varying infinite-dimensional system; the analysis of this system takes the form of a Kalman-Yacubovich-type lemma. By using these two new systems, along with several others that have been analyzed previously, stability theorems for many specific multiloop feedback systems can be proven. One such example is given.     

Delay-dependent robust stabilization of uncertain systems withmultiple state delays

This paper deals with the problem of robust stabilization for uncertain systems with multiple state delays. The parameter uncertainties are time-varying and unknown but are norm-bounded, and the delays are time-varying. A new method for achieving robust stabilization is presented for a class of uncertain time-delay systems via linear memoryless state feedback control. The results depend on the size of the delays and are given in terms of several linear matrix inequalities     

Arbitrary eigenvalue assignments for linear time-varying multivariable control systems

The problem of eigenvalue assignments for a class of linear time-varying multi-variable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is ‘lexicography-fixedly controllable’ can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.     

一类参数不确定脉冲系统的保成本控制研究

根据脉冲系统的稳定性结果,得到了线性时变脉冲系统一致渐近稳定的充分条件．在此基础上研究了一类参数不确定系统的保成本状态反馈控制问题,根据稳定性理论与鲁棒控制原理得到了此类控制器的一个存在性条件．进一步通过有关参数不确定性的结论证明了该条件等价于一组线性矩阵不等式的可解性问题．最后给出了一个具体示例．     

Designing stabilizing controllers for uncertain systems using theRiccati equation approach

A useful technique for determining a linear feedback control law stabilizes an uncertain system is the Riccati-equation approach of I.R. Petersen and C.V. Hollot (1986). They consider systems with time-varying uncertainty in the system matrix and obtain the constant feedback gains for the linear stabilizing controller in terms of the solutions of a Riccati equation. The technique is extended to include problems with time-varying uncertainty in the input connection matrix. Several examples are included to demonstrate the efficacy of this result     

On feedback stabilization of time-varying discrete linear systems

Results are given for stabilizing time-varying discrete linear systems by means of a feedback control stemming from a receding-horizon concept and a minimum quadratic cost with a fixed terminal constraint. The results parallel those recently obtained for continuous-time systems [8] and extend a well-known method of Kleinman for stabilizing discrete fixed linear systems [7].     

Stabilizability of linear time-varying and uncertain linear systems

Feedback stabilization of linear time-varying and uncertain linear systems is considered. It is proved that given a stabilizing dynamic linear state-feedback controller, one can always construct a stabilizing nondynamic linear state-feedback controller. A similar result is shown for uncertain linear systems. A linear time-varying system can be stabilized by dynamic output feedback if and only if it admits a coprime factorization     

Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay

The stabilization of exponentially unstable linear systems with time-varying input delay is considered in this paper. We extend the truncated predictor feedback (TPF) design method, which was recently developed for systems with all poles on the closed left-half plane, to be applicable to exponentially unstable linear systems. Assuming that the time-varying delay is known and bounded, the design approach of a time-varying state feedback controller is developed based on the solution of a parametric Lyapunov equation. An explicit condition is derived for which the stability of the closed-loop system with the proposed controller is guaranteed. It is shown that, for the stability of the closed-loop system, the maximum allowable time-delay in the input is inversely proportional to the sum of the unstable poles in the plant. The effectiveness of the proposed method is demonstrated through numerical examples.     

The output feedback control for uncertain nonholonomic systems

This paper considers the problems of almost asymptotic stabilization and global asymptotic regulation （GAR） by output feedback for a class of uncertain nonholonomic systems. By combining the nonsmooth change of coordinates and output feedback domination design together, we construct a simple linear time-varying output feedback controller, which can universally stabilize a whole family of uncertain nonholonomic systems. The simulation demonstrates the effectiveness of the proposed controller.     

The output feedback control for uncertain nonholonomic systems

This paper considers the problems of almost asymptotic stabilization and global asymptotic regulation (GAR) by output feedback for a class of uncertain nonholonomic systems. By combining the nonsmooth change of coordinates and output feedback domination design together, we construct a simple linear time-varying output feedback controller, which can universally stabilize a whole family of uncertain nonholonomic systems. The simulation demonstrates the effectiveness of the proposed controller.     

Exponential stability and solution bounds for systems with bounded nonlinearities

Estimation of Lyapunov exponents of systems with bounded nonlinearities plays an essential part in their robust control. Known results in this field are based on the Gronwall inequality yielding relatively conservative bounds for Lyapunov exponents. In this note, we obtained more accurate upper bounds for the general Lyapunov exponent for systems consisting of a known linear time-varying part and an unknown nonlinear component with a bounded Lipschitz constant at zero. Consequently, a sufficient condition for exponential stability of this system is formulated. The systems are indicated for which the obtained bound is precise, i.e., cannot be improved without additional information on the nonlinear term. In the presence of a persisting perturbation, an upper bound for the solution norm is derived and expressed in the norm of the solution of the corresponding linear system. Using the obtained results, a condition for exponential stability of a linear time-varying control system with a nonlinear feedback is derived. Numerical results are obtained for a second-order time-varying system and for the Lienard equation; in the latter case they are favorably compared with stability conditions previously obtained using the Lyapunov function method.     

Robust control through piecewise Lyapunov functions for discrete time-varying uncertain systems

This paper is concerned with the robust stabilization by state feedback of a linear discrete-time system with time-varying uncertain parameters. An optimization problem involving a set of linear matrix inequalities and scaling parameters provides both the robust feedback gain and the piecewise Lyapunov function used to ensure the closed-loop stability. In the case of linear time-varying systems involving the convex combination of two matrices, only two scaling parameters constrained into the interval [0,?1] are needed, allowing a simple numerical solution as illustrated by means of examples.     

Time-varying Lyapunov functions for linear time-varying systems

Adopting special time-varying Lyapunov-function candidates, it is shown that the Popov criterion ensures large-scale asymptotic stability for linear time-varying feedback systems with relaxed conditions on the time-varying gain fc(t). The main result is that li(t)/k(t) need not be bounded for all finite t and need be bounded only when ibecomes arbitrarily large. Again, in the derivations we make use of the Lefschetz version of the Kalman-Yakubovich lemma.     

State feedback stabilization of linear impulsive systems

We address the fundamental problem of state feedback stabilization for a class of linear impulsive systems featuring arbitrarily-spaced impulse times and possibly singular state transition matrices. Specifically, we show that a strong reachability property enables a state feedback law to be constructed that yields a uniformly exponentially stable closed-loop system. The approach adopts a receding horizon strategy involving a weighted reachability gramian in a manner reminiscent of well-known results for time-varying linear systems for both continuous and discrete-time cases.     

A gradient flow approach to on-line robust pole assignment for synthesizing output feedback control systems

Pole assignment is a basic design method for synthesis of feedback control systems. In this paper, a gradient flow approach is presented for robust pole assignment in synthesizing output feedback control systems. The proposed approach is shown to be capable of synthesizing linear output feedback control systems via on-line robust pole assignment. Convergence of the gradient flow can be guaranteed. Moreover, with appropriate design parameters the gradient flow converges exponentially to an optimal solution to the robust pole assignment problem and the closed-loop control system based on the gradient flow is globally exponentially stable. These desired properties make it possible to apply the proposed approach to slowly time-varying linear control systems. Simulation results are shown to demonstrate the effectiveness and advantages of the proposed approach.     

Memoryless stabilization of uncertain linear systems includingtime-varying state delays

The problem of stabilizing a class of uncertain time-delay systems via memoryless linear feedback is examined. The systems under consideration are linear systems with time-varying state delays. They also contain uncertain parameters (possibly time-varying) whose values are known only to within a prescribed compact bounding set. The main contribution given is to enlarge the class of time-delay systems for which one can construct a stabilizing memoryless linear feedback controller. Within this framework, a novel notion of robust memoryless stabilizability is first introduced via the method of Lyapunov functionals. Then a sufficient condition for the stabilizability is proposed. It is shown that solvability of a parameterized Riccati equation can be used to determine whether the time-delay system satisfies the sufficient condition. If there exists a positive definite symmetric solution satisfying the Riccati equation, a suitable memoryless linear feedback law can be derived