This paper deals with the problems of robust stability analysis and robust stabilization for uncertain nonlinear polynomial systems. The combination of a polynomial system stability criterion with an improved robustness measure of uncertain linear systems has allowed the formulation of a new criterion for robustness bound estimation of the studied uncertain polynomial systems. Indeed, the formulated approach is extended to involve the global stabilization of nonlinear polynomial systems with maximization of the stability robustness bound. The proposed method is helpful to improve the existing techniques used in the analysis and control for uncertain polynomial systems. Simulation examples illustrate the potentials of the proposed approach.
Being subject of considerable theoretical and practical significance, stability analysis and control of nonlinear dynamic systems have been attracting the interest of investigators for several decades [
In this paper, we are concerned with further developments of robust stability and feedback stabilization methods of a class of nonlinear polynomial systems with structured uncertainties. Our motivations for studying polynomial systems mainly comes from the fact that they provide a convenient unified framework for mathematical modelling of many physical processes and practical applications such as electrical machines and robot manipulators, and also the ability to approach any analytical nonlinear dynamical systems, since, any nonlinear system can be developed into a polynomial form by Taylor series expansions [
The purpose of this work is then to find a bound on the size of the parameters for which the perturbed model remains stable as well as design robustly state feedback controller for the above-mentioned systems. The presented research is built on the stability and stabilization conditions of nonlinear polynomial systems proposed in [
This article is organized as follows. Section
We use standard notations throughout this paper.
In this section, we propose some algebraic tools and definitions needed to demonstrate the robustness measure and the stabilizing control, of the studied systems, in the next sections.
The Kronecker power of order
The relation between the redundant and the nun-redundant Kronecker power of the vector
If
An important vector valued function of matrix denoted
For a polynomial vectorial function denoted
We define the for for
Consider the class of uncertain nonlinear polynomial systems with structured uncertainties described by
We note that the structure of the parametric uncertainties
According to (
The objective of this paper is to address the following problems: in the robust stability analysis, it is assumed that the nominally system described by
the robust control problem consist to design a nonlinear robust controller which ensures the global stability of the nominal nonlinear systems (
In the next sections, we will address the robustness analysis and synthesis problems for nonlinear polynomial systems with the above described admissible structured uncertainty set.
This section is devoted to the development of a robustness stability measure of the above class of nonlinear systems. Our main result in this work is based on the duality principle, presented in the sequel, between the computation of stability robustness bound approach of linear systems and a previous result about the stability analysis of certain polynomial systems. First, we beginning by presenting the following basic results.
The problem of robust stability of linear state-space systems, for both structured and unstructured parametric uncertainty involving state space models, has been an active area of research for quite some time for extensive discussions and references. Let the following linear dynamical system with linear structured perturbations:
As mentioned in the first section, different results have been obtained for determining the extent of uncertainty that the system can tolerate without becoming unstable. However, it remains challenging to develop methods for finding less conservative robustness bounds in the presence of structured perturbations. we present here two results which have been applied in our work.
Consider an uncertain linear system described by (
The uncertain linear system (
Next, we review an improved robustness algorithm proposed by Gardiner that gives a better bound than the methods obtained in [
The system (
Before applying these two results for general high-order nonlinear polynomial systems, we assess recent algebraic stability criterion for the polynomial systems in the certain case.
The robustness measure developed in this study is specially based on the pertinent results presented in the following theorem [
The nonlinear polynomial system defined by ( an arbitrary parameters
such that the
The proof elements and the parameters notation of Theorem
Refrying to the stability condition of certain polynomial system (
In the following we will apply some results on linear robustness measures determination for characterizing a robustness measure of nonlinear polynomial systems. In this way, two results will be presented. A previous result based on the Yedavalli linear robustness measure [
Let consider the uncertain nonlinear polynomial system (
The nonlinear uncertain polynomial system (
This theorem was proved in [
We present now the proposed robustness bound of uncertain polynomial systems based on the Gardiner's method.
The nonlinear uncertain polynomial system (
The stability of the polynomial system (
Let us remark that Theorem
Consider the following second-order autonomous uncertain polynomial system:
For testing the robustness bound gives above, we assume now that the model parameters are reversed compared to the nominal values, the simulation results, show that the perturbed system remains stable if the parameter uncertainties are not greater than the founded bound, which provide the validity of the presented approach. As compared to previously developed techniques like [
In this section, we are interested the problem of nonlinear robust control of polynomial systems. In order to stabilize the considered nonlinear systems, a state feedback control law is built, based in recent and pertinent results about the polynomial systems stabilization using the LMI optimization tools (Linear Matrix Inequalities approach) and the robustness stability measure ( the global stability of the nominal (without uncertainties) nonlinear polynomial system is ensured, and the stability of the controlled perturbed system is ensured for a maximum bound of uncertainties affecting the system parameters.
We consider now the uncertain polynomial system given by (
The controlled uncertain system is described by the state equation:
The robust stabilization technique presented in this work is specially related to a recent approach which deals with the global asymptotic stabilization of nonlinear polynomial systems within the framework of Linear Matrix Inequalities (LMIs) in the certain case, presented in [
Find gain matrices a real parameters
such that
Therefore, the proposed synthesis method combines the robust analysis result in the last section (Theorem
The controlled nonlinear uncertain polynomial system (
The above synthesis method can be seen as a nonlinear optimization procedure given by two step algorithm to found a global optimum of the problem (
One has the following steps: Initialize the algorithm by determining a feasible solution using LMI technique of the nominal polynomial system: Solving the feasibility problem to find a symmetric positive matrix Consider the matrices
As we indicated above, these two algorithm steps establish a nonlinear optimization problem under nonlinear constraints [
In this section, we present two simulation examples to demonstrate the validity of the proposed result such as numerical example and flexible manipulator with a single-link case study.
We consider the uncertain polynomial system described by the state equation (
The system controlled with the obtained polynomial control law was simulated for a perturbation of
Behavior of the state variables of both nominal and perturbed system.
Simulation curve of the proposed feedback controller.
To study the validity of the proposed approach in control of robot manipulator application, we consider a single-link manipulator with flexible joints and negligible damping represented by Figure
Model of single-link manipulator with flexible joint.
We model the dynamics of this system ignoring damping terms as
The considered system can be described by a fourth-order model of the form:
For simulation, the single-link manipulator parameters are considered in Table
Nominal values of the system parameters.
Link inertia, | 0.031 Kg.m2 |
Rotor inertia, | 0.004 Kg.m2 |
Rotor friction, | 0.007 N.m.sec/rad |
Nominal load, | 0.8 N.m |
Joint stiffness, | 31.0 N.m/rad |
Thus, according to the values of the considered system parameters indicated in Table
Evolution of state variables without controller for both real model (−) and polynomial model (
In order to investigate the performance of the proposed synthesis method, applying the optimization algorithm presented in the section above, we obtain the following optimal solution:
Closed-loop system response using the polynomial controller.
As it can be seen, the nonlinear state feedback controller (
Simulation of the stabilizing polynomial control (
In this paper, we have investigated the problems of robust stability analysis and robust stabilization for a class of uncertain nonlinear systems with structured uncertainties. New developments on the analysis and control of nonlinear polynomial systems have been presented. The combination of a polynomial system stability criterion with an improved robustness measure of uncertain linear systems has allowed the formulation of a new criterion for robustness bound estimation of the studied uncertain polynomial systems. When compared with the previous results, this developed criterion led to a wider robustness measure. It has been shown from the simulation results that the proposed robust control scheme is efficient and permits the rapid stabilization of the nominal system and the maximization of the uncertainties bound. A future work will be to establish the robust stability and stabilization of uncertain polynomial systems with both structured and unstructured uncertainties. Moreover, the problem of feedback
For the complete proof of Theorem
The matrix