Sufficient Conditions For Robust Stability Research Articles

Criteria for robust absolute stability of time-varying nonlinear continuous-time systems

The present paper establishes results for the robust absolute stability of a class of nonlinear continuous-time systems with time-varying matrix uncertainties of polyhedral type and multiple time-varying sector nonlinearities. By using the variational method and the Lyapunov Second Method, criteria for robust absolute stability are obtained in different forms for the given class of systems. Specifically, the parametric classes of Lyapunov functions are determined which define the necessary and sufficient conditions of robust absolute stability. The piecewise linear Lyapunov functions of the infinity vector norm type are applied to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations. Several simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for special cases. Two examples are presented as applications of the present results to a particular second-order system and to a specific class of systems with time-varying interval matrices in the linear part.

Robust stability of perturbed switching systems by matrix measure

The robust stability of switching systems is a very difficult problem because of the difficulty in separating the perturbations of the models and the reset maps. In this paper, we apply matrix measure to separate the perturbations of the models and the reset maps. Then the conditions for robust stability of perturbed switching systems with structured perturbations can be readily derived. Sufficient conditions for robust stability of switching systems are proposed, and bounds for structured perturbations based on nonnegative matrices that still ensure stability are derived.

Robust stability of singularly perturbed state feedback systems using unified approach

A method for decomposition of unified state-feedback singularly perturbed systems is presented. Based on the small-gain theorem, a sufficient condition for robust stability is derived such that the composite state feedback renders the closed-loop singularly perturbed system asymptotically stable. With this method there is no need to consider two different approaches for continuous-time and discrete-time domains. An example illustrates the methodology.

Design Method for Robust Supervisor Controller

We suggest a method to design supervisor which guarantees robust stability for linear time-invariant systems in the presence of the uncertainty via output feedback. The system is a composite of continuous-time uncertain plant and switched controller. The controller is defined by a collection of given controllers whose state is shared. Furthermore the proposed supervisor controller determines the mode of the controller to guarantee robust stability of the total closed-loop system. A sufficient condition of robust stability is given in terms of an algebraic Riccati equation and bilinear matrix inequalities(BMI).

Rank-One LMI Approach to Robust Stability of Polynomial Matrices

Necessary and sufficient conditions are formulated for checking robust stability of an uncertain polynomial matrix. Various stability regions and uncertainty models are handled in a unified way. The conditions, stemming from a general optimization methodology similar to the one used in μ-analysis, are expressed as a rank-one LMI, a non-convex problem frequently arising in robust control. Convex relaxations of the problem yield tractable sufficient LMI conditions for robust stability of uncertain polynomial matrices.

Finite-time control of linear systems subject to parametric uncertainties and disturbances

In this paper we consider finite-time control problems for linear systems subject to time-varying parametric uncertainties and to exogenous constant disturbances. The main result provided is a sufficient condition for robust finite-time stabilization via state feedback. It can be applied to problems with both non-zero initial conditions and unknown constant disturbances. This condition is then reduced to a feasibility problem involving linear matrix inequalities (LMIs). A detailed example is presented to illustrate the proposed methodology.

Robust reliable control for a class of uncertain nonlinear systems with time-varying multistate time delays

This paper deals with the problem of robust reliable controller design for a class of uncertain nonlinear systems. The system under consideration is subject to time-varying multistate time delays, parameter uncertainties, nonlinearities and even failures in prescribed subsets of auctuators. The nonlinearities are assumed to satisfy boundedness conditions. The parameter uncertainties and multistate time delays are time varying and structured. A systematic approach using linear matrix inequalities (LMIs) is proposed to solve the design problem of this controller. Sufficient conditions for robust stability in the presence of possible actuator failure are given in terms of the LMIs and the controller that stabilizes the system is computed using standard LMI techniques.

LMI-based robust H∞ control of uncertain linear jump systems with time-delays

This paper examines the robust mean square stabilization and robust H ∞ control problem for a class of linear systems. The systems under consideration involve time delays, parameter uncertainties, and jumping parameters which may be seen as, for example, failures that occur in the systems at certain probabilities. The uncertain parameters are assumed to be time-varying and satisfy boundness conditions. A systematic approach via linear matrix inequalities (LMI) is proposed to solve the problem of concern. Sufficient conditions for robust mean square stabilization and robust H ∞ control are given in the form of LMI and the control law is computed using standard LMI techniques.

Stability analysis for the systems with feedback norm constraint uncertainties

This paper considers the problem of Lp stability checking for a feedback uncertain system which consists of a linear plant in its forward loop and non-linear uncertain dynamics in its feedback loop. It is supposed that the non-linear part of the system is of dynamic uncertainties characterized by a certain set of constraints. The concepts used here are the well-posedness of the feedback system and the norm constraint for the feedback uncertainties, leading to a sufficient condition for robust stability. The result is illustrated to be efficient through an example.

Robust reliable control for a class of uncertain nonlinear systems with time-varying multistate time delays

This paper deals with the problem of robust reliable controller design for a class of uncertain nonlinear systems. The system under consideration is subject to time-varying multistate time delays, parameter uncertainties, nonlinearities and even failures in prescribed subsets of auctuators. The nonlinearities are assumed to satisfy boundedness conditions. The parameter uncertainties and multistate time delays are time varying and structured. A systematic approach using linear matrix inequalities (LMIs) is proposed to solve the design problem of this controller. Sufficient conditions for robust stability in the presence of possible actuator failure are given in terms of the LMIs and the controller that stabilizes the system is computed using standard LMI techniques.

Non-linear control of advanced direct drive robots: Theory and experiments

Direct drive robots are widely used to perform high precision positioning, and advanced control algorithms are applied to attain the desired performance. Small direct drive manipulators are widely used in aerospace, biotechnology, instrumentation, medicine, micro manufacturing, high-density integrated circuits and VLSI fabrication, etc. This paper studies non-linear modelling, analysis and control of direct drive robots to guarantee the desired level of performance. We are particularly interested in the physical laws to model mechanical-electromagnetic phenomena and effects, and the integration of electric motors is a key factor to design high performance direct drive robots. The non-linear mathematical model, found using the Lagrange equations of motion, is applied in non-linear analysis and design. The design method is described and validated. The design concept uses the Hamilton-Jacobi and Lyapunov theories to synthesize robust tracking controllers. It is illustrated that minimizing a non-quadratic performance functional, a bounded control law is analytically designed using necessary conditions for optimality. The sufficient conditions for robust stability are examined using the criteria imposed on positive-definite return functions. Results indicate that the reported method allows one to guarantee robust accurate tracking and disturbance attenuation. A two-degree-of-freedom prototype of direct drive manipulator is described, and the achieved performance is documented and discussed.

On the robustness of uncertain time-delay systems with structured uncertainties

This paper considers the robust stability and robust performance of uncertain delay systems subject to possibly structured uncertainties using structured singular values with phase information. Computationally efficient sufficient conditions for robust stability and performance problem are derived.

A framework for robustness analysis of constrained finite receding horizon control

A framework for robustness analysis of input-constrained finite receding horizon control is presented. Under the assumption of quadratic upper bounds on the finite horizon costs, we derive sufficient conditions for robust stability of the standard discrete-time linear-quadratic receding horizon control formulation. This is achieved by recasting conditions for nominal and robust stability as an implication between quadratic forms, lending itself to S-procedure tools which are used to convert robustness questions to tractable convex conditions. Robustness with respect to plant/model mismatch as well as for state measurement error is shown to reduce to the feasibility of linear matrix inequalities. Simple examples demonstrate the approach.

Robust stability and stabilization: From linear to nonlinear

Some ramifications of the Lyapunov-Krasovskii Theory are explored. First, for linear delay systems the resulting sufficient conditions for robust asymptotic stability are related to the Schur-Cohn and the Hurwitz stability of the system matrices. Second, ideas towards a constructive parameterization for linear time-invariant delay systems are presented. Third, new applications for pseudo-linear systems are explored. Finally, some recent advances in the stability and stabilization for nonlinear systems are surveyed, which may be exploited in the arena of delay systems.

On the Robust Stability of 2D Schur Polynomials

In this note, the problem of the robust stability for a two-dimensional (two-variable) Schur polynomial which is the characteristic polynomial of a discrete-time linear time-invariant system is investigated. A new approach based on the Rouche theorem is adopted. The extension to the robust stability for multidimensional (multivariable) polynomials is also provided. Interesting sufficient conditions for such robust stability are derived. A two-dimensional example is included to support the theoretical result.

Robustness of discrete periodically time-varying control under LTI unstructured perturbations

Investigates robustness of linear periodically time varying (LPTV) control of discrete linear time invariant (LTI) plants subject to LTI unstructured perturbations. The note first derives a necessary and sufficient condition for robust stability of an LPTV system subject to LTI perturbations, which is less conservative than the well known small gain condition. It then presents a quantitative analysis on the robustness of LPTV control under LTI unstructured perturbations in comparison with that of LTI control. It is shown that under the normal value of the controller period suggested in the previous literature, the stability margin is deteriorated by LPTV control if LTI unstructured perturbations are considered. Hence LTI control is superior to LPTV control in this respect.

Robust Vibration Control for SCARA Robots Using Adaptive Pole Placement.

SCARA robot motions generally consist of the following two motions : One is horizontally transportation of a load by actuating two revolute joints. The other is vertically transportation of a load by actuating a prismatic joint. This paper deals with an adaptive control for the former motion with vibration suppression of the prismatic joint. The controlled object is modeled as a doble-input quadruple-output system. Firstly, we show a non-linear control method that guarantees a load to be transported to the neighborhood of its desired position. The mass of the load is estimated by the recursive least squares algorithm during the transportation. Then, the dynamics of the robot is linearized at the neighborhood of the desired position of the load. By using the linearized dynamics and the estimated mass, a pole placement controller for the robot is designed. We derive a sufficient condition for robust stability with respect to the estimation error of the mass. Under the condition, a pole placement method for robust performance is also presented. Experimental results demonstrate the effectiveness of the proposed method.

Robust Stabilizing Solution of the Riccati Difference Equation

The control law for infinite horizon H 2 control oflinear discrete-systems with norm bounded time-varying uncertainties has been presented in the literature. In this paper, a sufficient condition for robust stability of this control is presented. Based upon the results obtained, a connection is made between the monotonic behavior of the Riccati difference equation (RDE) and the robust stability of “frozen” closed-loop systems. This connection then allows the derivation of a robust stability condition for “frozen” closed-loop systems, which depends only on the initial condition of the RDE. An example is included to illustrate the implementation of the proposed theory.

Robust stability for singularly perturbed systems with structured state space uncertainties

Robust Stabilization for Singularly Perturbed Systems with Structured State Space Uncertainties Hiroaki Mukaidani, Member (Hiroshima City University), Koichi Mizukami, Member (Hiroshima University) This paper considers the robust stability of singularly perturbed systems with structured state space uncertainties. By making use of Lyapunov stability criterion and combining with the Lyapunov equations, a new approach for deciding a robust stability for uncertain linear singularly perturbed systems is presented. Based on the assumption that the reduced nominal system is stable, we also derive some sufficient conditions for robust stability. Some analytical methods and the Bellman-Gronwall inequality are used to investigate such sufficient conditions. It is worth pointing out that in this paper, we do not need to investigate both the slow system and the fast system by using the singularly perturbation method because of the proposed method is very direct. Furthermore, we only assume that the uncertainties are a norm bounded. Therefore, the robust stability condition derived here is less conservative than those reported in the control literatures. An numerical example is given to demonstrate the validity of our new results.

Some new stability conditions for two-dimensional difference systems

This paper considers robust stability of two-dimensional (2D) difference systems in the Fornasini-Marchesini (F-M) state space setting. An approach based on the MacL aurine series expansion is presented. First, a new and less conservative sufficient condition for asymptotic stability of nominal 2D discrete systems is derived. Then sufficient conditions for robust stability of 2D discrete systems subjected to both structured and unstructured perturbations are given. Finally, new robust stability conditions for 2D discrete-delay systems are derived. Numerical examples are given to illustrate the results.