Stabilization Of Uncertain Dynamical Systems Research Articles

The stabilization of uncertain dynamic systems involving the generalized Riemann-Liouville fractional derivative via linear state feedback control

The main objective of this paper is to investigate the stabilization problem of uncertain fractional-order dynamic systems (UFDSs) that involve the generalized Riemann-Liouville fractional derivative using a linear feedback controller. We establish several sufficient conditions for achieving Mittag-Leffler stabilization and asymptotic stabilization of UFDSs. The primary methodology employed in our study is Lyapunov's direct method (LDM), along with other inequality techniques, which have demonstrated significant efficacy in studying the stability theory of nonlinear dynamic systems. To demonstrate the effectiveness of the proposed approach, examples and corresponding simulations are provided.

Model-Based Event-Triggered Control for Systems With Quantization and Time-Varying Network Delays

This paper combines two important control techniques for reducing communication traffic in control networks, namely, model-based networked control systems (MB-NCS) and event-triggered control. The resulting framework is used for stabilization of uncertain dynamical systems and is extended to systems subject to quantization and time-varying network delays. The use of a model of the plant in the controller node not only generalizes the zero-order-hold (ZOH) implementation in traditional event-triggered control schemes but it also provides stability thresholds that are robust to model uncertainties. The effects of quantized measurements are especially important in the selection of stabilizing thresholds. We are able to design error events based on the quantized variables that yield asymptotic stability compared to similar results in event-triggered control that consider nonquantized measurements which, in general, are not possible to use in digital computations. With respect to MB-NCS, the stability conditions presented here do not need explicit knowledge of the plant parameters as in previous work but are given only in terms of the parameters of the nominal model and some bounds in the model uncertainties. We consider the joint adverse effects of quantization and time delays and emphasize the expected tradeoff between the selection of quantization parameters and the admissible network induced delays.

Bounds on eigenvalues of real and complex interval matrices

We present a cheap and tight formula for bounding real and imaginary parts of eigenvalues of real or complex interval matrices. It outperforms the classical formulae not only for the complex case but also for the real case. In particular, it generalizes and improves the results by Rohn (1998) [5] and Hertz (2009) [19]. The main idea behind is to reduce the problem to enclosing eigenvalues of symmetric interval matrices, for which diverse methods can be utilized.The result helps in analysing stability of uncertain dynamical systems since the formula gives sufficient conditions for testing Schur and Hurwitz stability of interval matrices. It may also serve as a starting point for some iteration methods.

Stability and reliable data reconstruction of uncertain dynamic systems over finite capacity channels

In this paper we present an encoder, decoder and a stabilizing controller for reliable data reconstruction and robust stability of uncertain dynamic systems controlled over Additive White Gaussian Noise (AWGN) channels. The uncertainty in the dynamic system is described by a relative entropy constraint. Such an uncertainty description is a natural stochastic generalization of the sum quadratic uncertainty description. This paper complements the results of Farhadi and Charalambous (2008) by showing that the necessary condition presented there can be tight. This is shown by designing an encoder, decoder and a stabilizing controller.

An Improved Delay-Dependent Criterion for Asymptotic Stability of Uncertain Dynamic Systems with Time-Varying Delays

In this paper, the problem of stability analysis for uncertain dynamic systems with time-varying delays is considered. The parametric uncertainties are assumed to be bounded in magnitude. Based on the Lyapunov stability theory, a new delay-dependent stability criterion for the system is established in terms of linear matrix inequalities, which can be solved easily by various efficient convex optimization algorithms. Two numerical examples are illustrated to show the effectiveness of proposed method.

Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays

In this paper, the problem of delay-range-dependent stabilization criterion for uncertain dynamic systems with time-varying delays is considered. The time-varying delays considered is assumed to be belong to a given interval in which lower bound of delay is not restricted to zero. By constructing a suitable augmented Lyapunov’s functional and utilizing free weight matrices, the criterion for stabilization is established in terms of linear matrix inequalities. Three numerical examples are given to show the effectiveness of proposed method.

Exponential stability of uncertain dynamic systems including state delay

In this letter, the exponential stability of uncertain time-delay systems is investigated. Based on the Lyapunov method, a novel stability criterion has been derived in terms of matrix inequalities which can be easily solved using efficient convex optimization algorithms. Two numerical examples are included to show the effectiveness of the proposed method.

Some new results on the asymptotic stability of uncertain dynamical systems with time-varying delay

Several sufficient conditions for the robust stability of uncertain dynamic systems with time-varying delays are presented. The results obtained can be applicable to handle either delay-dependent or-independent tests. The included uncertainties can also be either linear or nonlinear. The Lyapunov method is employed in the discussion.

Robust output feedback controllers for dynamical systems including delayed perturbations

The problem of output feedback stabilization of uncertain dynamical systems including time-varying delay is discussed. Here we assume that the uncertainties satisfy the so-called matching conditions, and that the time-varying delay is any bounded and continuous non-negative function. For such a class of uncertain time-delay systems, we show that under the assumption that the transfer function matrix of the nominal system is strictly feedback positive real, there always exist a class of stabilizing output feedback controllers. We first propose a linear output feedback controller which can guarantee always uniform ultimate boundedness of closed-loop dynamical systems. Furthermore, we modify this linear controller by adding a nonlinear part to obtain a nonlinear output feedback controller which can guarantee always uniform asymptotic stability of closed-loop dynamical systems. As the proposed output feedback controllers are independent of the delay, the results obtained in this paper are applicable to dynamical systems without exact knowledge of the delays, i.e. systems with perturbed delays. Finally, a numerical example is given to demonstrate the validity of the results.

High-order sliding modes in control systems

The article considers the properties of a new type of solutions that arise in discontinuous dynamic systems. A specific feature of these solutions is the tangency of the phase velocity vectors to the manifold of the right-hand side discontinuities, instead of the transversal intersection of the manifold typically observed for ordinary sliding modes. The solutions identified in this case are high-order sliding modes, and the order of the mode is determined by the smoothness of tangency of the sliding manifold. Second-order sliding modes are considered in detail. Examples of systems with such modes are given; application of the theory to stabilization of uncertain dynamic systems is described. It is shown that the sensitivity of high-order sliding modes to small variations in the right-hand side of the discontinuous system is an order of magnitude higher than for ordinary sliding modes.

Memoryless stabilization of uncertain dynamic systems with time-varying delayed states and controls

We consider the feedback stabilization of uncertain dynamic systems with time-varying delays in both states and controls. We extend the so-called Riccati-equation approach to include those systems, and we give an example.

Sliding Mode Control in Banach Space

The regularization principle is considered to describe sliding motion in Banach space. The equivalent control method is proved in application to infinite-dimensional dynamic systems of parabolic type. According to the method unlike that of (Orlov and Utkin, 1987) the sliding motion can be described by the sliding mode equation in the subspace but not in the original space. The decoupling synthesis procedure based on the method is proposed. The concept of the ”unit control” is introduced in Banach space. Using the decoupling synthesis procedure it is shown that the unit control can be designed for stabilization of uncertain dynamic systems in Banach space. The unit control solving the same stabilization problem in Hilbert space is found in the explicit form.

Robust stability of a class of uncertain nonlinear dynamical systems with time-varying delay

We consider the problem of robust stability of a class of uncertain nonlinear dynamical systems with time-varying delay. Based on the assumption that the nominal system (i.e. the system in the absence of uncertainty and delay) is stable, we derive some sufficient conditions on robust stability of uncertain nonlinear dynamical systems with time-varying delay. Some analytical methods and the Bellman-Gronwall inequality are used to investigate such sufficient conditions. The notable features of the results obtained in this paper are their simplicity in testing the stability of uncertain dynamical systems with delay and their clarity in giving an insight into system analysis. Our results are also applicable to perturbed time-delay dynamical systems without exact knowledge of the delay. In addition, a numerical example is given to demonstrate the validity of our results.

Robust stabilization for a class of uncertain dynamical systems with time delay

In this paper, a new and simple approach whereby we derive several sufficient conditions on robust stabilizability for a class of uncertain dynamical systems with time delay is presented. Some analytical methods and the Bellman-Gronwall inequality are employed to investigate these sufficient conditions. The notable features of the results obtained are their simplicity in testing the stability of uncertain dynamical systems with time delay and their clarity in giving insight into system analysis. Finally, several numerical examples are given to demonstrate the utilization of the results.

Output feedback control of uncertain coupled systems

The stabilization of uncertain dynamical systems on ℝ n that admit a decomposition into two coupled subsystems of dimension nc and nr respectively, is studied. We refer to the nc -dimensional sybsystem as the reduced-order system and to the nr -dimensional subsystem as the residual system: the overall n-dimensional system is termed the full system—the prototype for which is a mechanical model arising in the study of the active control of structures. Each subsystem is modelled by a differential equation with a linear nominal part and a nonlinear perturbation of a specified class. The output available for feedback purposes is an ℝ nc -valued linear combination of the state components of the full system and is subject to bounded measurement noise (with known bound). An output feedback strategy is described and sufficient conditions are obtained for the existence of a (calculable) global uniform compact attractor (containing the state origin) for controlled systems of this class.

A modified variable structure control algorithm for stabilization of uncertain dynamical systems

AbstractA modified VS feedback is suggested for robust stabilization of continuous‐time dynamical systems in the presence of parametric and external time‐varying disturbances satisfying the ‘matching conditions’. The main feature of the proposed algorithm is that it contains additional switching ‘integral’ terms which track the unknown disturbances and make it possible to achieve the typical VSS robustness by using switching terms in the control law with arbitrarily small amplitude. This result can be achieved only if the derivatives of the disturbances are bounded.

Robust output feedback stabilization of uncertain dynamic systems with bounded controllers

AbstractWe propose a new class of infinitely many bounded output feedback controllers for uncertain dynamic systems with bounded uncertainties. No statistical information about the uncertainties is assumed. A variable structure systems approach is employed in the synthesis of the proposed output feedback controllers. The role of the system zeros in the output feedback stabilization using the variable structure approach is discussed. We show that the proposed controllers guarantee the practical stability of the closed‐loop system and give estimates of the regions of practical stability.

A variable structure control with simple adaptation laws for upper bounds on the norm of the uncertainties

In deterministic design of feedback controllers for uncertain dynamical systems, upper bounds on the norm of the uncertainties are an important clue to guarantee the stability of the closed-loop system. Sometimes, however these upper bounds may not be easily obtained because of the complexity of structure of the uncertainties. Simple adaptation laws for upper bounds on the norm of the uncertainties are proposed, and these adaptive upper bounds are used to design a variable-structure control system that guarantees the asymptotic stability of uncertain dynamical systems. >

Global stabilization of nonlinear systems with a class of unmatched uncertainties

Global and asymptotic stabilization of uncertain dynamical systems which violate the matching conditions is investigated. A class of unmatched uncertainties, called equivalently matched uncertainties, is defined. A robust controller is proposed to stabilize the uncertain system asymptotically in the large provided that the nominal system is uniformly asymptotically stable and that the unmatched uncertainties are equivalently matched. The required information about uncertain dynamics in the system is merely that the uncertainties are bounded in Euclidean norm by known functions of the system state.

Stabilization of uncertain dynamic systems including state delay

The authors consider a class of time-varying systems with time-varying state delay and uncertain input and parameters. They present sufficient conditions on the open loop to obtain an arbitrary-degree closed-loop stability. They construct a min-max controller from knowledge of the upper bound of the delay. >