Robust stability and stabilization methods for a class of nonlinear discrete-time delay systems (original) (raw)

Robust Stabilization of Discrete-Time Systems with Time-Varying Delay: An LMI Approach

Mathematical Problems in Engineering, 2008

Sufficient linear matrix inequality LMI conditions to verify the robust stability and to design robust state feedback gains for the class of linear discrete-time systems with time-varying delay and polytopic uncertainties are presented. The conditions are obtained through parameter-dependent Lyapunov-Krasovskii functionals and use some extra variables, which yield less conservative LMI conditions. Both problems, robust stability analysis and robust synthesis, are formulated as convex problems where all system matrices can be affected by uncertainty. Some numerical examples are presented to illustrate the advantages of the proposed LMI conditions.

A convex approach for robust state feedback control of discrete-time systems with state delay

2004

In this paper, uncertain discrete-time systems with state delay are investigated. The uncertainty is supposed to belong to a known convex polytope. Linear matrix inequality conditions are given for the robust stability of the system, encompassing quadratic stability based results. Then, convex conditions assuring the existence of a robust state feedback gain are derived, assuring the delay independent quadratic stability of the closed-loop system (thus allowing to deal with time-varying uncertain systems) or, in the time-invariant case, guaranteeing the robust stability irrespective of the value of the delay. Moreover, the feedback control law can also include a term depending on the delayed state which, if the value of the delay is known, can be used to improve the control design. Numerical examples illustrate the effectiveness of the proposed techniques.

Robust Stability and Constrained Stabilization of Discrete-Time Delay Systems

IFAC Proceedings Volumes, 2012

This paper revisits the problem of robust stability and stabilization of uncertain time-delay systems. We focus on the class of non-negative discrete-time delay systems and show that it is asymptotically stable if and only if an associated non-negative system without delay is asymptotically stable. This fact allows one to establish strong result on robust stability and stability radius for this class of systems. An alternative representation of delay systems is also constructed whereby its system matrix is in block companion from. Under the assumption of non-negativity for delay systems, this alternative form represents a conventional non-negative system and similar strong robust stability results are derived. Finally, we consider the problem of constrained stabilization and provide a new LMI feasibility solution for it. This makes it possible to stabilize a general discrete-time delay system such that the closed-loop system admits non-negative structure with desirable properties.

Output Feedback Stabilization for a Discrete-Time System With a Time-Varying Delay

IEEE Transactions on Automatic Control, 2008

This study employs the free-weighting matrix approach to investigate the output feedback control of a linear discrete-time system with an interval time-varying delay. First, the delay-dependent stability is analyzed using a new method of estimating the upper bound on the difference of a Lyapunov function without ignoring any terms; and based on the results, a design criterion for a static output feedback (SOF) controller is derived. Since the conditions thus obtained for the existence of admissible controllers are not expressed strictly in terms of linear matrix inequalities, a modified cone complementarity linearization algorithm is employed to solve the nonconvex feasibility SOF control problem. Furthermore, the problem of designing a dynamic output feedback controller is formulated as one of designing an SOF controller. Numerical examples demonstrate the effectiveness of the method and its advantage over existing methods.

New stabilisation schemes for discrete delay systems with uncertain non-linear perturbations

IET Control Theory & Applications, 2010

New H 1 controller design schemes are provided for a class of discrete-time systems with uncertain non-linear perturbations. The class includes both systems with time-varying delays and systems without delay. One design scheme is generated by state-feedback and the other scheme is based on proportionalsummation -difference (PSD) feedback. An appropriate Lyapunov-Krasovskii functional (LKF) is constructed and a new parametrised characterisation is established in terms of feasibility-testing of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the theoretical developments.

Stability of discrete-time systems with time-varying delays

Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148), 2001

This paper is concerned with the stability analysis of discrete linear systems with time-varying delays. The novelty of the paper comes from the consideration of a new inequality which is less conservative than the celebrated Jensen inequality employed in the context of discretetime delay systems. This inequality is a discrete-time counterpart of the Wirtinger-based integral inequality that was recently employed for the improved analysis of continuous-tine systems with delays. However, differently from the continuous-time case, the proof of the new inequality is not based on the Wirtinger inequality. The method is also combined with an efficient representation of the improved reciprocally convex combination inequality in order to reduce the conservatism induced by the LMIs optimization setup. The effectiveness of the proposed result is illustrated by some classical examples from the literature.

Linear Uncertain Discrete Time Delay System : A Survey

2017

Linear Matrix Inequalities (LMIs) and LMI techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design. Three factors make LMI techniques appealing. a) A variety of design specifications and constraints can be expressed as LMIs. b) Once formulated in terms of LMIs, a problem can be solved exactly by efficient convex optimization algorithms (the “LMI solvers”). c) While most problems with multiple constraints or objectives lack analytical solutions in terms of matrix equations, they often remain tractable in the LMI framework. This makes LMIbased design a valuable alternative to classical “analytical” methods. II. LINEAR MATRIX INEQUALITIES

New stability conditions for class of nonlinear discrete-time systems with time-varying delay

2021

The problem of stability analysis for a class of nonlinear discrete time systems with time varying delay is studied in this work. Such systems are modeled by delayed difference equations. Subsequently, this system is transformed into an arrow form matrix representation. Using M-matrix properties, novel sufficient stability conditions are determined. It is shown how to use our method to design a state feedback controller that stabilizes a discrete time Lure system with time varying delay and sector bounded nonlinearity. The originalities of our findings are shown in their explicit representation, using system’s parameters, as well as in their easiness to be employed. The obtained results demonstrate also that checking stability of a nonlinear discrete time systems with time varying delay can be reduced to an M-matrix test. Several examples are provided to show the effectiveness of the introduced technique.

New Stability Conditions for a Class of Nonlinear Discrete-Time Systems with Time-Varying Delay

Mathematics 2020, 8, 1531; , 2020

In this paper, the stability problem of discrete time delay systems is investigated. The class of systems under consideration is represented by delayed difference equations and models nonlinear discrete time systems with time varying delay. It is transformed into an arrow from matrix representation which allows the use of aggregation techniques and M-matrix properties to determine novel sufficient stability conditions. The originalities of our findings are shown in their explicit representation, using system’s parameters, as well as in their easiness to be employed. The obtained results demonstrate also that checking stability of nonlinear discrete time systems with time varying delay can be reduced to an M-matrix test. Next, it is shown how to use our method in designing a state feedback controller that stabilizes a discrete time Lure system with time varying delay and sector bounded nonlinearity. Finally, several examples are provided to show the effectiveness of the introduced technique.