On Lyapunov stability robustness bounds for linear discrete-time systems with structured time-varying uncertainty

In this paper we analyse the stability robustness of linear discrete-time systems which are described by a state-space model but are perturbed with structured time-varying uncertainty. We present new Lyapunov stability robustness bounds in which the freedom of the matrix Q is utilized more effectively than that used by Kolla et al. (1989) to obtain a larger bound of tolerable time-varying uncertainty, and the similarity transformation is employed more directly and usefully than that proposed by Kolla and Farison (1990) to reduce conservation. Further, the relationship between the matrix Q and the similarity transformation matrix M is given. Improvements are illustrated by an application of our proposed method to a macroeconomic system     

New robust stability bounds of linear discrete-time systems with time-varying uncertainties

This paper presents new uncertain parameter variation bounds for linear discrete-time systems to preserve asymptotic stability. The Lyapunov method is utilized to treat both structured and unstructured uncertainties, and the results are optimized with respect to a parameter in the inequality used. When applied to examples considered by previous authors, our results give less conservative bounds.     

Local stabilization for linear discrete-time systems with bounded controls and norm-bounded time-varying uncertainty

The problem of the local stabilization of linear discrete-time systems subject to bounded controls and suffering from uncertainty of the norm-bounded time-varying type is addressed. From the solution of a certain discrete Riccati equation, a control gain and a set of safe initial conditions are obtained. The asymptotic stability of the saturated closed-loop system is then locally guaranteed for all admissible uncertainties. It is also shown how the control problem can be translated into L.M.I. conditions. The connections between local stability results and disturbance rejection problem are investigated. In the presence of control saturation, it is thus shown that it is possible to reject a certain class of perturbations. Finally, a discretized model of the inverted pendulum allows us to illustrate the results. © 1998 John Wiley & Sons, Ltd.     

Improved bounds for linear discrete-time systems with structuredperturbations

A time-domain analysis of the stability robustness of linear discrete-time systems subject to time-varying structured perturbations is considered. The Lyapunov stability theory is used to obtain bounds on the perturbation such that the systems remain stable. It is shown that these bounds are less conservative than the existing ones. This is illustrated via two numerical examples     

Robust H∞ control for linear discrete-time systems with norm-bounded time-varying uncertainty

The problem of robust H_∞ analysis and synthesis for linear discrete-time systems with norm-bounded time-varying uncertainty is studied in this paper. It will be shown that this problem is equivalent to the problem of H_∞ analysis and synthesis of an auxiliary system. The necessary and sufficient conditions for the equivalency are proved. Thus the original problem can be solved by existing H_∞ control methods.     

A discrete-time iterative learning algorithm for linear time-varying systems

In this paper, an iterative learning algorithm (ILC) is presented for a MIMO linear time-varying system. We consider the convergence of the algorithm. A necessary and sufficient condition for the existence of convergent algorithm is stated. Then, we prove that the same condition is sufficient for the robustness of the proposed learning algorithm against state disturbance, output measurement noise, and reinitialization error. Finally, a simulation example is given to illustrate the results.     

Robust stabilization of linear discrete-time systems with time-varying input delay

This paper presents a sufficient condition to stabilize linear discrete-time systems with time-varying input delay and model uncertainties. The key idea consists of applying Artstein’s reduction method and the scaled bounded real lemma; the robust stabilization problem is cast as a set of linear matrix inequalities on a transformed delay-free system. Hence, a novel convex characterization of the problem, an alternative to bilinear matrix inequalities in prior literature, is provided. Predictor-based controllers and robust state-feedback stabilization are particular cases of the proposal. Furthermore, numerical examples show that this proposal may improve tolerance against delay variations when compared to other methods available in literature.     

Dissipative control of linear discrete-time systems with dissipative uncertainty

This paper focuses on the dissipative control of uncertain linear discrete-time systems. The uncertainty under consideration is characterized by a dissipative system, which contains commonly used uncertainty structures, such as normbounded and positive real uncertainties, as special cases. We consider the design of a feedback controller which can achieve asymptotic stability and strict quadratic dissipativeness for all admissible uncertainties. Both the linear static state feedback and the dynamic output feedback controllers are considered. It is shown that the robust dissipative control problem can be solved in terms of a scaled quadratic dissipative control problem without uncertainty. Linear matrix inequality (LMI) based methods for designing robust controllers are derived. The result of this paper unifies existing results on discrete-time H and positive real control and it provides a more flexible and less conservative control design as it al ows for a bet er trade-off between phase and gain performances.     

Robust static output feedback control for linear discrete-time systems with time-varying uncertainties

This paper is concerned with the problem of designing robust static output feedback controllers for linear discrete-time systems with time-varying polytopic uncertainties. Sufficient conditions for robust static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities, and the results are extended to H₂ and H_∞ static output feedback controller designs. Numerical examples are given to illustrate the effectiveness of the proposed design methods.     

Model-matching and decoupling for continuous- and discrete-time linear time-varying systems

Solutions to the exact model-matching and block-decoupling problems for both continuous- and discrete-time linear time-varying systems are presented. The parametrisation of the whole class of proper solutions is given. For the decoupling, the minimal delay problem is also considered in a time-varying setting. The approach is algebraic and based on the Smith–MacMillan form at infinity of a transfer matrix of a time-varying system which has been recently introduced in systems theory. This avoids the difficulties related to the inversion of the transfer matrices with entries in non-commutative fields over which the determinants (of Dieudonné or Ore type) are much more complicated. The solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice. Examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown.     

Periodically time-varying memory state-feedback controller synthesis for discrete-time linear systems

In this paper, we deal with discrete-time linear periodic/time-invariant systems with polytopic-type uncertainties and propose a new linear matrix inequality (LMI)-based method for robust state-feedback controller synthesis. In stark contrast with existing approaches that are confined to memoryless static controller synthesis, we explore dynamical controller synthesis and reveal a particular periodically time-varying memory state-feedback controller (PTVMSFC) structure that allows LMI-based synthesis. In the context of robust controller synthesis, we prove rigorously that the proposed design method encompasses the well-known extended-LMI-based static controller synthesis methods as particular cases. Through numerical experiments, we demonstrate that the suggested design method is indeed effective in achieving less conservative results, under both periodic and time-invariant settings. We finally derive a viable test to verify that the designed robust PTVMSFC is “exact” in the sense that it attains the best achievable robust performance. This exactness verification test works fine in practice, and we will show via a numerical example that exact robust control is indeed attained by designing PTVMSFCs, even for such a problem where the standard memoryless static state-feedback fails.     

Robust stabilization of linear systems with norm-bounded time-varying uncertainty

In this paper, robust stabilization of a class of linear systems with norm-bounded time-varying uncertainties is considered. It is shown that for this class of uncertain systems quadratic stabilizability via linear control is equivalent to the existence of a positive definite symmetric matrix solution to a (parameter-dependent) Riccati equation. Also, a construction for the stabilizing feedback law is given in terms of the solution to the Riccati equation.     

A synthesis theory for linear time-varying feedback systems with plant uncertainty

Given a feedback system containing a linear, time-varying (LTV) plant with significant plant uncertainty, it is required that the system response to command and disturbance inputs satisfy specified tolerances over the range of plant uncertainty. The synthesis procedure guarantees the latter satisfied, providing that they are of the following form. Leth(t',tau)be the system response att'= t - taudue to a command inputdelta(t - tau), andh_{tau}(s)=int liminf{0}limsup{infty}h(t',tau)e^-{st'}dt'is the Laplace transform ofh(t',tau). There is given a setM_{tau}(omega)={m_{tau}(omega)} , omega in[0, infty), with the requirement that|h_{tau}(jomega)| in M_{tau}(omega), over the range of plant uncertainty. The disturbance response tolerances are of the same form, in response to a disturbance inputdelta (t- tau). The acceptable response setM_{tau}(omega)can depend on τ. The design emerges with a fixed pair of LTV compensation networks and can be considered applicable to time-domain response tolerances, to the extent that a set of bounds on a time function can be translated into an equivalent set on its frequency response. The design procedure utilizes only time-invariant frequency response concepts and is conceptually easy to follow and implement. At any fixed τ, the time-varying system is converted into an equivalent time-invariant one with plant uncertainty, for which an exact solution is available, with "frozen" time-invariant compensation. Schauder's fixed-point theorem is used to prove the equivalence of the two systems. The ensemble over τ of the time-invariant compensation gives the final required LTV compensation. It is proven that the design is stable and nonresonant for all bounded inputs.     

Optimal control of linear systems with parameter uncertainty

A class of infinite-horizon regulator problems is formulated for families of time-invariant linear systems with parameter uncertainty. Under a regularity assumption, the optimal linear, state-feedback control is shown to exist and is defined via a positive-definite solution of a family of Riccati-type, algebraic equations. The solvability of these equations is equivalent to the stabilizability of the family of linear systems by a constant, linear feedback.     

Reliable control for linear systems with time-varying delays and parameter uncertainties

In this paper, reliable control for linear systems with time-varying delays and parameter uncertainties is considered. By constructing newly augmented Lyapunov–Krasovskii functionals and utilizing some mathematical techniques such as Leibnitz's rule, Schur's complement, reciprocally convex combination, and so on, a reliable controller design method for linear systems with time-varying delays and parameter uncertainties will be suggested in Theorem 1. Based on the result of Theorem 1, a non-reliable stabilization criterion will be presented in Corollary 1. Theorem 1 and Corollary 1 are derived within the framework of linear matrix inequalities(LMIs) which can be easily solved by utilizing various optimization algorithms. Two numerical examples are included to show the effectiveness and necessity of the proposed results.     

On perturbations of linear controllable infinite-dimensional time-varying discrete-time systems

It is shown that for a broad class of linear (possibly time-varying and infinite-dimensional) discrete-time systems x_k+1 = A_kx_k + B_ku_k the property of being uniformly equicontrollable is preserved under small perturbations of system parameters. The problem of controllability of asymptotically time-invariant systems is also studied.     

LMI-based analysis of robust adaptive control for linear systems with time-varying uncertainty

Passification-based adaptive control, also known as simple adaptive control, is studied with respect to its robustness to time-varying uncertainties. Results are formulated in terms of LMIs and are therefore testable in polynomial time using semi-definite programming solvers. The main result shows that the adaptive strategy allows, without measurement nor estimation of the uncertain parameters to guarantee asymptotic stability for a wide rage of these parameters. To achieve this result, the stability property is relaxed: convergence is proved to a small neighborhood of the origin, and attractive domain. It is also demonstrated that this attractor can be made as small as required the only limitation being implementation constraints.     

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach

This paper addresses the problem of establishing robust asymptotical stability of discrete-time linear systems polynomially affected by time-varying uncertainty confined into a polytope. A linear matrix inequality (LMI) condition for establishing robust asymptotical stability is proposed by introducing a novel approach for establishing the existence of a common homogeneous polynomial Lyapunov function (HPLF). This approach consists, firstly, of introducing a Gram matrix built with respect to the state and parametrized by an arbitrary vector function of the uncertainty, and secondly, of requiring that a transformation of the introduced Gram matrix is a sum of squares (SOS) of matrix polynomials. The approach, hence, is referred to as a Gram-SOS approach. It is shown that the proposed LMI condition is sufficient for any degree of the HPLF candidate, that includes quadratic robust stability as a special case, and that is also necessary for a sufficiently large degree of the HPLF candidate. Numerical examples also show that the proposed LMI condition can outperform alternative ones in terms of conservatism and computational burden.