Lyapunov stability robustness measures for multivariable,continuous, time-invariant,linear systems

In this paper general robustness measure bounds are introduced for any multivariable, continuous, time-invariant, linear systems. Bounds are obtained for allowable non-linear time-varying perturbations such that the resulting system remains stable. Bounds are also derived for linear perturbations. The robustness measures and the related theorems are applied to optimal LQ state feedback, direct output feedback, and to generalized dynamic output feedback designs.     

Stability robustness bounds for discrete-time linear regulators

Stability robustness analysis and design for linear multivariable discrete-time systems with bounded uncertainties are discussed. Robust stability of the full-state feedback linear quadratic (LQ) regulator in the presence of perturbations (modelling errors) of the system matrices is investigated. These results are based on a recently developed bound on elemental (structured) time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system. Lyapunov theory and singular value decomposition techniques are employed in deriving these bounds. Extensions of these results to linear stochastic systems with the Kalman filter as the stale estimator (LQG regulators) and to reduced-order dynamic compensator feedback are described. A state feedback control design method is presented for LQ regulators, using a quantitative measure called the Stability Robustness Index. Simple examples illustrate these new results.     

Output feedback stabilisation for uncertain nonlinear time-delay systems subject to input constraints

Robust stabilisation of a class of imperfectly known systems with time-varying time-delays via output feedback is investigated. The systems addressed are composed of a nonlinear nominal system influenced by nonlinear perturbations which may be time-, state-, delayed state- and/or input-dependent. The output of the system is modelled by a nonlinear function, which may depend on the delayed states, and inputs, together with a feed-through term. Using bounding information on the perturbations, in terms of specified growth conditions, classes of unconstrained and constrained output feedback controllers are designed in order to guarantee a prescribed stability property for the closed-loop systems, provided appropriate stability criteria hold. Two stability criteria are given: one in terms of a linear matrix inequality (LMI) and the other is algebraic in nature, obtained using a Ger?gorin Theorem.     

一类线性时变系统的输出反馈控制

基于线性时不变系统能控能观标准型变换及非线性系统高增益观测器方法,本文研究了一类线性时变系统 的输出反馈控制问题. 通过引入时变的状态变量坐标变换,分别设计了线性时变系统的状态反馈控制器、状态观测器以及基于 状态观测器的输出反馈控制器. 进一步地,本文分别证明了观测器动态误差是渐近收敛于零的,而状态反馈控制器以及输出反馈控制器可以 保证闭环系统的渐近稳定性.     

Robust stability for linear discrete-time systems with structured perturbations

The robust stability problems of linear discrete-time systems with structured perturbations are considered. The necessary and sufficient condition of interval stability is given and the maximal bound of robust stability is presented for systems with a non-negative or non-positive nominal system matrix. Further, it is shown that the given necessary and sufficient condition and the given maximal robust bound are still valid for time-varying perturbations and nonlinear perturbations.     

On the design of stabilizing controllers for singularly perturbedsystems

The robustness of output feedback control designs for singularly perturbed systems based on the reduced systems is discussed. A frequency-domain approach is presented to determine the conditions for the stability of linear time-invariant systems subject to neglected high-frequency dynamics. In contrast with the qualitative analyses for robust stability that have appeared in the literature, this approach gives an explicit, computable bound on unmodeled dynamics which does not destabilize the systems. Stability conditions for various feedback control schemes are presented. It is noted that the approach, extended to systems with mixed singular-regular perturbations, can be used to derive the robust stability condition     

Quantitative measures of robustness for systems including delayedperturbations

Asymptotically stable linear systems subject to delayed time-varying and nonlinear perturbations are considered. Razumikhin-type theorems are used to obtain easy-to-compute bounds on the perturbations so that the systems remain stable. Results indicate that if delayed perturbations are included, then the bound is reduced as compared to the one for nondelayed perturbations. However, in certain cases previously obtained bounds for the nondelayed perturbations guarantee stability even when delayed perturbations are in effect     

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.     

具有时变状态滞后2-D离散系统的稳定与镇定(英文)

针对一类由局部状态空间(LSS)Fornasini-Marchesini(FM)第二模型描述的,具有时变状态滞后的2-D离散系统,其中时变滞后项的上、下界均为正实数,研究了其稳定性和控制综合问题。首先,利用Lyapunov-Krasovski泛函方法,提出了系统的稳定性准则。再根据这一准则,分别设计状态反馈和动态输出反馈控制器保证系统的稳定性。状态反馈控制律和输出反馈矩阵可由线性矩阵不等式(LMI)求得。最后,通过数值算例说明所得结果的有效性。     

一种推广的组合非线性输出反馈控制

针对多变量饱和线性系统的时变参考输入跟踪问题,研究了一种组合非线性输出反馈控制器的设计方法.基于原始的组合非线性反馈理论,构造了全阶和降阶输出反馈控制器.控制器由线性输出反馈项和非线性反馈项组成,使得闭环系统在包含于吸引域的不变集内渐近稳定.除了能够跟踪时变参考输入外,系统还具有良好的动态性能.仿真结果说明了所开发控制器的有效性.     

Robust stability criteria for dynamical systems including delayedperturbations

Considers the problem of robust stability of uncertain time-delay dynamical systems. A new robust stability criteria for linear dynamical systems subject to delayed time-varying and nonlinear perturbations is derived. The results obtained in this note are less conservative than the ones reported so far in the literature. Some analytical methods are employed to investigate the bound on the perturbations so that the systems are stable. A numerical example is given to demonstrate the utilization of the authors' results     

Robust decentralized stabilization of a class of linear discrete-time systems with non-linear interactions

This paper deals with robust stabilization of a class of linear discrete-time systems under non-linear perturbations via output feedback. A bound on the non-linear perturbations is maximized in the design. It is shown that degree of freedoms by the introduction of instrumental variables employed in this paper lead to much flexibility in obtaining both a robust output feedback controller and a maximal allowable bound of the non-linear perturbations. An improved method involving linear matrix inequalities are suggested to solve the matrix inequalities characterizing a solution of the robust stabilization problem. Consequently, the proposed method can yield a much less conservative result than that of earlier methods. Of major interest is an extension to a class of interconnected systems composed of linear subsystems with non-linear interactions. A robust decentralized controller is presented such that the closed-loop systems are maximally tolerant to interconnected non-linear couplings. Numerical examples illustrate the validity of the proposed approach.     

Robustness measure bounds for optimal model matching controldesigns

The theorems introduced by R.V. Patel et al. (1977) for multivariable linear quadratic state feedback design in the presence of perturbations are extended to optimal model matching control system designs. A robustness measure bound is introduced for optimal model matching control systems. Bounds are obtained for allowable nonlinear time-varying perturbations such that the resulting closed-loop system remain stable. Bounds on a special, but important class of perturbations, in which the perturbation is linear, are also derived     

Dynamic Output Feedback Control of Switched Linear Systems

This paper is devoted to stability analysis and control design of switched linear systems in both continuous and discrete-time domains. A particular class of matrix inequalities, the so-called Lyapunov--Metzler inequalities, provides conditions for open-loop stability analysis and closed-loop switching control using state and output feedback. Switched linear systems are analyzed in a general framework by introducing a quadratic in the state cost determined from a series of impulse perturbations. Lower bounds on the cost associated with the optimal switching control strategy are derived from the determination of a feasible solution to the Hamilton--Jacobi--Bellman inequality. An upper bound on the optimal cost associated with a closed-loop stabilizing switching strategy is provided as well. The solution of the output feedback problem is based on the construction of a full-order linear switched filter whose state variable is used by the mechanism for the determination of the switching rule. Throughout, the theoretical results are illustrated by means of academic examples. A realistic practical application related to the optimal control of semiactive suspensions in road vehicles is reported.     

Linear parameter-varying discrete time-delay systems: Stability and l 2-gain controllers

In this paper, we investigate a class of linear parameter-varying discrete time-delay (LPVDTD) systems where the state-space matrices depend on time-varying parameters and the delay is unknown but bounded. We treat both notions of quadratic stability based on a single quadratic Lyapunov function and affine quadratic stability using parameter-dependent Lyapunov functions. In both cases, we develop LMI-based results of stability testing for time-delay as well as delayless discrete-time systems. Then, we design state-feedback controllers which guarantee quadratic stability and an induced l ₂-norm bound. For the case of dynamic output feedback control, we use a parameter-independent quadratic Lyapunov-Krasovskii function to develop LMI-based solvability conditions which are evaluated at the extreme points of the admissible parameter set. Throughout the paper, complementary results for linear parameter-varying discrete (LPVD) systems without delay are presented.     

Design of performance robustness for uncertain linear systems withstate and control delays

The linear systems considered in this paper are subject to uncertain perturbations of norm-bounded time-varying parameters and multiple time delays in system state and control. The time delays are uncertain, independent of each other, and allowed to be time-varying. The integral quadratic cost criterion is employed to measure system performance. Using solutions of Lyapunov and Riccati equations, a linear state feedback control law is proposed to stabilize the perturbed system and to guarantee an upper bound of system performance, which is applicable to arbitrary time delays     

Memoryless output feedback nullification and canonical forms,for time varying systems

We study the possibility of nullifying time-varying systems with memoryless output feedback. The systems we examine are discrete-time linear single-input single-output finite-dimensional time-varying systems. For generic completely controllable and completely observable discrete-time systems, we show that any state at any time can be steered to the origin within finite time. An algorithm for nullification and an upper bound for nullification time, depending only on the system's dimension, are provided. The algorithm is described using a representation of the system in time-varying controller canonical form. We verify that every completely controllable system has such a representation.     

Robust finite-time stabilisation of uncertain linear systems

This article deals with the problem of finite-time stability and stabilisation of uncertain linear systems. For linear time-varying systems subject to norm-bounded uncertainties some conditions for finite-time stability are provided. These conditions are expressed in terms of differential linear matrix inequalities. Then the problem of controller design is tackled, both for the state feedback and for the output feedback case; in both cases the controller can be found solving a suitable set of LMIs. A typical engineering case-study is included, to illustrate the applicability of the devised conditions.     

The necessity of the small-gain theorem for time-varying andnonlinear systems

It is shown that an appropriately modified version of the small-gain theorem which uses the `asymptotic gain' of a fading-memory system leads to a necessary and sufficient condition for the stable invertibility of certain feedback operators. This modified small-gain theorem is then used to obtain conditions for robust finite-energy input-output stability of: (1) nonlinear plants subject to nonlinear time-varying perturbations; and (2) linear time-varying plants subject to linear time-varying perturbations. These conditions are shown to be necessary as well as sufficient     

On robustness and stabilization of linear systems with delayednonlinear perturbations

In this paper, by employing a Razumikhin-type theorem, a robust stability criterion for a class of linear systems subject to delayed time-varying nonlinear perturbations is given. Then, stabilization of a class of linear systems subject to mismatched delayed time-varying nonlinear perturbations by linear controller is considered. For such systems, conditions which ensure closed-loop asymptotic stability are given. Numerical examples are given to illustrate the results