Stability Of Discrete-time Linear Systems Research Articles

Mean Square Stabilization of Linear Discrete-Time Systems Over Power-Constrained Fading Channels

This paper considers the mean square stabilization problem of discrete-time linear time-invariant systems over a power constrained fading channel. Fundamental limitations on the mean square stabilizability are obtained via information theoretic arguments. For scalar systems and two-dimensional systems, necessary and sufficient conditions for the mean square stabilizability are provided. Moreover, an adaptive time division multiple access (TDMA) communication scheme is designed for high-dimensional systems, which achieves a larger stabilizability region than the conventional TDMA communication scheme, and is proved to be optimal under certain situations.

Quantized stabilization of wireless networked control systems with packet losses

This paper considers stabilization of discrete-time linear systems, where wireless networks exist for transmitting the sensor and controller information. Based on Markov jump systems, we show that the coarsest quantizer that stabilizes the WNCS is logarithmic in the sense of mean square quadratic stability and the stabilization of this system can be transformed into the robust stabilization of an equivalent uncertain system. Moreover, a method of optimal quantizer/controller design in terms of linear matrix inequality is presented. Finally, a numerical example is provided to illustrate the effectiveness of the developed theoretical results.

Stabilization of Discrete-Time Linear Systems With Quantization and Noise Input

In this paper, our main concern is the problem of stabilization of discrete-time linear systems with both quantization and noise input. Stabilizability by means of quantizers that perform adaptations called “zooming” is analyzed. The noise input can be separated into an additive white noise part and a deterministic constant input disturbance. The analysis of mean-square stability of the system with noise input is essentially the asymptotic stability of the system disturbed by a constant input. It is shown that the system with the constant input disturbance is asymptotically stabilized by the quantized feedback control policy if the system without noise input can be stabilized by a linear state feedback law. Both the state quantization and the input quantization are studied.

New Results for Finite-Time Stability of Discrete-Time Linear Systems with Interval Time-Varying Delay

The problem of finite-time stability for linear discrete time systems with state time-varying delay is considered in this paper. Two finite sum inequalities for estimating weighted norms of delayed states are proposed in order to obtain less conservative stability criteria. By using Lyapunov-Krasovskii-like functional with power function, two sufficient conditions of finite-time stability are proposed and expressed in the form of linear matrix inequalities (LMIs), which are dependent on the minimum and maximum delay bounds. The numerical example is presented to illustrate the applicability of the developed results. It was shown that the obtained results are less conservative than some existing ones in the literature.

Stability Analysis of Discrete-Time Linear Systems with Unbounded Stochastic Delays

This paper investigates the stability of discrete-time linear systems with stochastic delays. We assume that delays are modeled as random variables, which take values in integers with a certain probability. For the scalar case, we provide an analytical bound on the probability to guarantee the stability of linear systems. In the vector case, we derive a linear matrix inequality condition to compute the probability for ensuring the stability of closed-loop systems. As a special case, we also determine the step size of gradient algorithms with stochastic delays in the unconstrained quadratic programming to guarantee convergence to the optimal solution. Numerical examples are provided to show the effectiveness of the proposed analysis techniques.

Stabilization of discrete-time linear systems with quantization and arbitrary packet losses

In this paper, we mainly focus on the stabilization problem of discrete-time linear systems with both quantized input feedback and arbitrary packet losses. The coarsest quantization strategy is analyzed in detail to guarantee the asymptotic stability of the system. If the coarsest quantizer is logarithmic, the necessary and sufficient conditions to asymptotically stabilize such system are converted into some algebraic Riccati equations and then into some LMIs. Then the infimum of the quantization density for the logarithmic quantizer over all packet loss dependent Lyapunov functions is obtained according to these LMIs. In addition, we also prove that the sector bound approach for the logarithmic quantizer is still valid for the system with arbitrary packet losses. The problem of asymptotic stability can be converted into a robust control problem with sector bound uncertainties. The robust stability of the uncertain system is formulated into some LMIs. Finally, an example is given to show the effectiveness of the results obtained in this paper.

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.

Quantized feedback stabilization of discrete-time linear system with Markovian jump packet losses

In this paper, we mainly focus on the problem of quantized feedback stabilization of a discrete-time linear system with Markovian jump packet losses. Based on the packet loss dependent Lyapunov functions, the coarsest quantizer is analyzed in detail to guarantee the mean square quadratic stabilization of the system. If the system adopts the logarithmic quantizer, the necessary and sufficient conditions, which make the system mean square quadratically stabilized by a linear state feedback controller, can be converted into an algebraic Riccati equation and then into an LMI. Then based on this LMI, we obtain the infimum quantization density of the logarithmic quantizer over all dependent Lyapunov functions. Finally, a numerical example is presented to illustrate the effectiveness of the results obtained in this paper.

Finite-time stability analysis and stabilization for linear discrete-time system with time-varying delay

The problem of finite-time stability for linear discrete-time systems with time-varying delay is studied in this paper. In order to deal with the time delay, the original system is firstly transformed into two interconnected subsystems. By constructing a delay-dependent Lyapunov–Krasovskii functional and using a two-term approximation of the time-varying delay, sufficient conditions of finite-time stability are derived and expressed in terms of linear matrix inequalities (LMIs). The derived stability conditions can be applied into analyzing the finite-time stability and deriving the maximally tolerable delay. Compared with the existing results on finite-time stability, the derived stability conditions are less conservative. In addition, for the stabilization problem, we design the state-feedback controller. Finally, numerical examples are used to illustrate the effectiveness of the proposed method.

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.

New criterion for finite-time stability of linear discrete-time systems with time-varying delay

This paper is concerned with the problem of finite-time stability analysis of linear discrete-time systems with time-varying delay. The time-varying delay has lower and upper bounds. By choosing a novel Lyapunov–Krasovskii-like functional, a new sufficient condition is derived to guarantee that the state of the system with time-varying delay does not exceed a given threshold during a fixed time interval. Then, the corresponding corollary is developed for the case of constant time delay. Numerical examples are provided to demonstrate the effectiveness and merits of the proposed method.

Delay-Partitioning Approach to Stability of Linear Discrete-Time Systems with Interval-Like Time-Varying Delay

This paper is concerned with the problem of global asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By utilizing the concept of delay partitioning, a new linear-matrix-inequality-(LMI-) based criterion for the global asymptotic stability of such systems is proposed. The proposed criterion does not involve any free weighting matrices but depends on both the size of delay and partition size. The developed approach is extended to address the problem of global asymptotic stability of state-delayed discrete-time systems with norm-bounded uncertainties. The proposed results are compared with several existing results.

Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems

Abstract The positive asymptotically stable continuous-time linear systems are approximated by corresponding asymptotically stable discrete-time linear systems. Two methods of the approximation are presented and the comparison of the methods is addressed. The considerations are illustrated by three numerical examples and an example of positive electrical circuit.

Stabilization of Discrete-Time Systems With Multiple Actuator Delays and Saturations

This paper studies the problems of global and semi-global stabilization of discrete-time linear systems with multiple input saturations and arbitrarily large bounded delays. By developing the methodology of truncated predictor feedback (TPF), state feedback control laws using only the current states of the systems are constructed to solve these problems. The feedback gains are dependent on the delay information of the open-loop system and thus are referred to as delay-dependent feedback. A method for determining the exact condition such that the resulting closed-loop system is asymptotically stable is also presented. Moreover, if the delays in the system are time-varying or even unknown, a modified delay-independent TPF is also established to solve the concerned problems. Numerical example has been worked out to illustrate the effectiveness of the proposed approaches.

On exponential stability for linear discrete-time systems in Banach spaces

In this paper we investigate four concepts of exponential stability for difference equations in Banach spaces. Characterizations of these concepts are given. They can be considered as variants for the discrete-time case of the classical results due to Barbashin [6] and Datko [5]. An illustrative example clarifies the relations between these concepts.

Stability Analysis of Linear Discrete-Time Systems with Interval Delay: A Delay-Partitioning Approach

This paper considers the problem of asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By using a delay partitioning-based Lyapunov functional, a new criterion for the asymptotic stability of such systems is proposed in terms of linear matrix inequalities (LMIs). The proposed stability condition depends on both the size of delay and partition size. The presented approach is compared with previously reported approaches.

Parametric Lyapunov Equation Approach to Stabilization of Discrete-Time Systems With Input Delay and Saturation

This paper studies the problem of stabilization of discrete-time linear systems with input delay and saturation nonlinearity. By exploring some further intricate properties of the recently developed parametric Lyapunov equation-based low-gain feedback design approach, solutions are proposed to solve the problems by both state feedback and output feedback. This new approach is not only simpler than the existing method that is based on the eigenstructure assignment technique, but also provides explicit conditions on the low-gain parameter to guarantee the stability of the closed-loop system. Moreover, it is possible by adjusting the low-gain parameter online to achieve global results when the system is subject to both input saturation and time-delay. Also, the delay in the input is allowed to be time-varying in some cases.

Stabilization of linear systems over networks with bounded packet loss and its use in model predictive control

This paper considers stabilization of discrete-time linear systems, where network exists for transmitting the sensor and controller information, and arbitrary bounded packet loss occurs in the sensor–controller link and the controller–actuator link. The stabilization of this system is transformed into the robust stabilization of a set of systems. The stability result for this system is specially applied on model predictive control (MPC) that explicitly considers the satisfaction of input and state constraints. Two synthesis approaches of MPC are presented, one parameterizing the infinite horizon control moves into a single state feedback law, the other into a free control move followed by the single state feedback law. Two simulation examples are given to illustrate the effectiveness of the proposed techniques.

New Stability Criteria for Linear Discrete-Time Systems With Interval-Like Time-Varying Delays

This technical note is concerned with stability of linear discrete-time systems with interval-like time-varying delays. Two new stability criteria are derived by constructing a new Lyapunov functional and utilizing two novel techniques for estimating the forward difference of the Lyapunov functional. The relationship between these stability criteria and some existing ones is established. Through the relationship, it is clear that these stability criteria are less conservative than some existing ones, which is confirmed by a numerical example.

Stochastic Receding Horizon Control: Stability Results

We present new results on stochastic stability of linear discrete-time systems with additive uncertainty under several receding horizon control policies. We guarantee bounded variance of the closed-loop system under each policy via appending a stability constraint to the existing optimization routine of the corresponding policy.