Stability Of Discrete-time Linear Systems Research Articles

Stabilization of linear discrete-time systems over resource-constrained networks under dynamical multiple description coding scheme

In this paper, the stabilization problem is investigated for a class of linear discrete-time systems in a resource-constrained network environment where only the data with finite bits is permitted to transmit. The data transmissions from the sensors to the controller are subject to random packet dropouts. To improve the reliability of the data transmission, a novel dynamical multiple description coding scheme is developed with guaranteed convergence of the decoding error. By recurring to the matrix inequality theory and the stochastic analysis technique, sufficient conditions are established to ensure the stochastic stability of the closed-loop system. Furthermore, a parametric relationship is discovered that characterizes the influences from the packet-dropout probabilities and the encoder parameters on the stability of the closed-loop system. Finally, a numerical example is provided to illustrate the validity of the proposed control scheme.

Self-Triggered Stabilization of Discrete-Time Linear Systems With Quantized State Measurements

We study the self-triggered stabilization of discrete-time linear systems with quantized state measurements. In the networked control system we consider, sensors may be spatially distributed and be connected to a self-triggering mechanism through finite data-rate channels. Each sensor independently encodes its measurements and sends them to the self-triggering mechanism. The self-triggering mechanism integrates quantized measurement data and then computes sampling times. Assuming that the closed-loop system is stable in the absence of quantization and self-triggered sampling, we propose a joint design method of an encoding scheme and a self-triggering mechanism for stabilization. To deal with data inaccuracy due to quantization, the proposed self-triggering mechanism uses not only quantized data but also an upper bound of quantization errors, which is shared with a decoder.

Stabilization of discrete-time linear systems with infinite distributed input delays

This paper investigates the stabilization problem of discrete-time linear systems with infinite distributed input delays. A novel framework is adopted to analyze the stability of the concerned systems. Under this framework, two truncated predictor feedback controllers are developed for two classes of discrete-time linear systems with infinite distributed input delays via the low gain method respectively. It is shown that under the designed controllers, those two classes systems are globally exponentially stable. To the best of our knowledge, this is the first time that the stabilization problem of discrete-time linear systems with infinite distributed input delays is considered. Two simulation examples are provided to illustrate the effectiveness of the proposed controllers.

A New Asymptotic Stability Criterion for Linear Discrete-Time Systems

In this brief, a new criterion to determine whether the roots of a polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p(z^{-1})$ </tex-math></inline-formula> with real coefficients are strictly inside the unit circle is presented. It is based on representing the polynomial in a form corresponding to the characteristic polynomial of a closed-loop systems in which all-pass transfer functions are included in both direct chain and feedback loop. The novel and simple criterion for assessing the asymptotic stability of linear discrete-time systems is proved applying the small-gain theorem in the discrete-time domain.

Mean square stability of discrete-time linear systems with random impulsive disturbances

Mean square stability of discrete-time linear systems with random impulsive disturbances

Stability analysis of delayed discrete-time systems based on a delay-square-dependent Lyapunov functional

Recent research has shown that one can obtain a less conservative stability criterion for a continuous-time linear system with a time-varying delay by introducing a Lyapunov-Krasovskii functional with a polynomial matrix on the time-varying delay. This paper aims at analysing the stability of discrete-time linear systems with time-varying delays by introducing a delay-square-dependent Lyapunov functional. A novel convex method is presented to formulate a less conservative stability criterion, which is demonstrated through numerical simulation. Moreover, it is also shown that, if the polynomial inequality method is employed, the resultant stability criterion is inapplicable due to its extremely high numerical complexity.

New Computation Method of Minimum Dwell Time for the Schur Stability of Discrete-time Linear Systems

We are interested in Schur stability of discrete-time switched linear systems in this paper. For this purpose, we have obtained an upper bound of the solution of discrete-time switched linear systems depending on Schur stability parameter. Also, we have obtained dwell and average dwell time which satisfies the Schur stability of the system. The method given in this paper is based on Schur stability parameter.

Bounded controls for discrete-time linear systems subject to input time delay

This paper investigates the global stabilization of discrete-time linear systems with input time delay by bounded controls. Based on some special canonical forms containing time delays both in its input and state, two special discrete-time linear systems---multiple integrators and oscillators are first considered. The global stabilizing controllers are respectively established, and moreover, explicit conditions are established to guarantee the stability of the closed-loop systems. Subsequently, a concise design method is proposed for globally stabilizing general discrete-time linear system by combining the design methods for multiple integrators and oscillators. The designed controller is in the explicit form with explicit stability conditions being given, and thus is easier to use than the existing results. Finally, numerical simulations illustrate the effectiveness of the proposed approaches.

Some novel necessary and sufficient conditions of exponential stability for discrete-time systems with multiple delays: A Lyapunov matrix approach

This paper aims at establishing necessary and sufficient conditions of exponential stability for linear discrete-time systems with multiple delays. Firstly, we introduce a new concept—Lyapunov matrix, and investigate its properties, existence and uniqueness by: (i) characterizing the solution of a boundary value problem of matrix difference equations; and (ii) constructing complete type Lyapunov–Krasovskii functionals with pre-specified forward difference. Secondly, a new constructive analysis methodology, named Lyapunov matrix approach, is proposed to establish necessary and sufficient exponential stability conditions for discrete-time systems with multiple delays. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results. It is worth emphasizing that, from a view of computation, the Lyapunov matrix approach proposed here is concerned with three key steps: (i) solve a systems of linear equations; (ii) check whether a constant matrix is of full-column-rank, and (iii) judge whether a constant matrix is positive definite. All of these can be easily realized by using the tool software—MATLAB.

On Second-Moment Stability of Discrete-Time Linear Systems With General Stochastic Dynamics

This article provides a new unified framework for second-moment stability of discrete-time linear systems with stochastic dynamics. Relations of notions of second-moment stability are studied for systems with general stochastic dynamics, and associated Lyapunov inequalities are derived. The system dynamics may depend on any type of stochastic process in our framework. Our results for the unified framework can immediately lead us to more specific and tractable stability conditions when the underlying stochastic process is restricted to a more definite one. Usefulness of the developed framework is demonstrated through three selected applications.

The Multi-step Transformation Control and Exponentially Stability for Linear Discrete-time Systems with Additional Control Delay

Background: Few results about the controller of linear discrete-time system with control delay are reported. Methods: By means of the multi-step transformation with memory, the linear discrete-time systems with additional control delay can be transformed to the equivalent linear discrete-time systems without control delay, and the dimension of the transformed system is not increased. By designing the optimal controller of the finite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained. At the same time, by designing the optimal controller of the infinite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained as well. Results: The corresponding finite horizon optimal controller has proved to render the closed-loop system exponentially stable. And the corresponding infinite horizon optimal controller has proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally, two examples are used to verify the theoretical results of this paper. Conclusion: The controller design and the exponentially stability of discrete-time linear system with additional state delay will be investigated in the future.

New Results on Stability of Linear Discrete-Time Systems With Time-Varying Delay

This paper is concerned with the stability problem for linear discrete-time systems with a time-varying delay. Some relations between two general free-matrix-based summation inequalities are discussed. A novel Lyapunov-Krasovskii functional (LKF) is proposed by modifying the single- and double-summation LKF terms. As a result, a new stability condition is obtained by employing the general free-matrix-based summation inequalities reported recently. Numerical examples are given to show that the obtained stability condition is more relaxed than some of existing results.

Robust $$\mathcal{H}_\infty$$ Observer-based Stabilization of Linear Discrete-time Systems with Parameter Uncertainties

This paper addresses the problem of observer-based stabilization of discrete-time linear systems in presence of parameter uncertainties and l2-bounded disturbances. We propose a new variant of the classical two-step LMI approach. In the first step, we use a slack variable technique to solve the optimization problem resulting from stabilization by a static state feedback. In the second step, a part of the slack variable obtained is incorporated in the $$\mathcal{H}_\infty$$ observer-based stabilization problem, to calculate simultaneously the Lyapunov matrix and the observer-based controller gains. The effectiveness of the proposed design methodology is demonstrated using the dynamic model of a quadruple-tank flow process system. Additional numerical illustrations are presented to show the superiority of the proposed Modified Two-Step Method (MTSM) from a LMI feasibility point of view.

Static Output Feedback Stabilization of Discrete-Time Linear Systems With Stochastic Dynamics Determined by an Independent Identically Distributed Process

This letter discusses an extension of our earlier results about state feedback stabilization toward static output feedback stabilization for discrete-time linear systems with stochastic dynamics determined by an independent and identically distributed (i.i.d.) process. A completely linearized necessary and sufficient inequality condition for static output feedback synthesis is known to be difficult to obtain even in the case with deterministic systems. Hence, this letter derives with the so-called S-variable approach a bilinear matrix inequality condition that can be effectively solved through exploiting the result of state feedback synthesis. An illustrative numerical example is also given for such synthesis.

Global stabilisation of discrete‐time linear systems using event‐triggered bounded controls

This study deals with the problem of global stabilisation of discrete-time linear systems using event-triggered bounded controls. The system dynamics is decomposed into single input subsystems, which explicitly display various dynamic characteristics. Event-triggered bounded control laws, both of the state feedback type and the output feedback type, are constructed based on the dynamics of the subsystems. Event-triggering strategies are designed for updating the control input of the two control laws, respectively. Global asymptotic stabilisation is shown to be achieved under these control laws. Numerical simulation is given to verify the effectiveness of the theoretical results.

Stabilization of Discrete-Time Linear Systems With an Unknown Time-Varying Delay by Switched Low-Gain Feedback

In this note, we investigate the stabilization of discrete-time linear systems with an unknown time-varying delay. Existing literature on the stabilization of time-delay systems require some knowledge, such as an upper bound, of the delay. In this paper, by proposing a switched low-gain feedback control law, we are able to achieve asymptotic stabilization of the system without any knowledge of the delay. In addition, the proposed switched low-gain feedback law also significantly improves the closed-loop system performance in terms of overshoot and convergence speed in comparison with the traditional fixed gain design. Simulation study illustrates the effectiveness of our theoretical results.

Semi-global stabilization of discrete-time linear systems with unsymmetrical saturation

This paper deals with the stabilization problem for discrete-time linear system with unsymmetric saturation. To handle the structural properties of the unsymmetric saturation nonlinearity, the description of the transformation carries from the symmetric case to the unsymmetric case is employed. By using the parametric Lyapunov equation-based low gain feedback design approach, linear matrix inequality formulations for the state feedback and output feedback control design are given. The effectiveness of the proposed method is confirmed by numerical examples.

Stabilization of discrete-time linear systems by delay independent truncated predictor feedback

For a discrete-time linear system with input delay, the predictor feedback law is the product of a feedback gain matrix with the predicted state at a future time instant ahead of the current time instant by the amount of the delay, which is the sum of the zero input solution and the zero state solution of the system. The zero state solution is a finite summation that involves past input, requiring considerable memory in the digital implementation of the predictor feedback law. The truncated predictor feedback, which results from discarding the finite summation part of the predictor feedback law, reduces implementation complexity. The delay independent truncated predictor feedback law further discards the delay dependent transition matrix in the truncated predictor feedback law and is thus robust to unknown delays. It is known that such a delay independent truncated predictor feedback law stabilizes a discrete-time linear system with all its poles at z = 1 or inside the unit circle no matter how large the delay is. In this paper, we first construct an example to show that the delay independent truncated predictor feedback law cannot compensate too large a delay if the open loop system has poles on the unit circle at z ≠ 1. Then, a delay bound is provided for the stabilizability of a general linear system by the delay independent truncated predictor feedback.

Finite-time stability and stabilisation with polyhedral domains for linear systems

ABSTRACT This paper deals with finite-time stability, boundedness and stabilisation of discrete-time linear systems with regards to polyhedral domains. The key observation made in this paper is that finite-time stability properties can be analysed in terms of reachable sets. We first establish necessary and sufficient conditions for finite-time stability and boundedness of nominal systems, expressible as linear programming problems. We then present sufficient conditions that may be used to compute feedback control to render such systems finite-time stable/bounded. Finally we generalise the above results to the case of discrete-time linear systems with polytopic uncertainty subject to a bounded disturbance.

LMI-based stability tests for LPV and switched discrete-time linear systems through redundant equations

Abstract This paper revisits the problem of testing the robust stability of discrete-time linear systems with arbitrarily fast time-varying parameters, which includes the important class of switched discrete-time linear systems with arbitrary switching. By exploring the redundant description of the state space equations, sufficient linear matrix inequality (LMI) conditions of distinct complexities assuring robust stability are given in terms of the number k—1 of redundant equations. As k ≥ 1 increases, the conditions become also asymptotically necessary. Numerical examples borrowed from the literature illustrate that the proposed conditions can be computationally less demanding than other convergent relaxations.