Improved Results on Robust Stability Analysis and Stabilization for a Class of Uncertain Nonlinear Systems

Abstract

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.

1. Introduction

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 [1–3]. The essential aim of robust analysis and nonlinear robust control theory is to internally stabilize the nonlinear plant while maximizing the upper bound on the parametric perturbations, such that the perturbed nonlinear system remains stable, as described in [4–8]. However, each of the published approaches on this subject concerns particular classes of nonlinear uncertain systems and there is no standard method to investigate robust stability and stabilization of general high-order nonlinear systems [9–13]. Therefore, in deep contrast with linear analysis and control methods, which are flexible, efficient and allowing to solve a broad class of linear control problems, there are few practical methods in nonlinear control which can handle real engineering problems with similar comfort.

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 [14–17]. Moreover, the description of polynomial systems can be simplified using the Kronecker product and power of vectors and matrices [18]. The parameter uncertainties in the studied polynomial systems are considered as real structured perturbations which are assumed to depend linearly on a set of interval parameters.

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 [16, 19, 20] and the measure of stability robustness of linear systems [12], and makes use of a duality principle between the linear and nonlinear uncertain systems [21, 22]. The derived results are formulated in terms of nonlinear optimization problem which is easy to testify by using Matlab Tool-Box. Let's mention that previous works of the same research team [21, 22] had considered the robustness measure of polynomial systems using the Yedavalli robustness measure of linear uncertain systems [10]. In the present contribution we aim to improve these previous results by considering a less conservative robustness criterion and to apply the obtained bound formulation to the robust control synthesis.

This article is organized as follows. Section 2 contains the mathematical notations and preliminary material on algebraic properties of the Kronecker product and power of vectors and matrices. Section 3 formulates the problem and the description of the nonlinear studied systems. Sections 4 and 5 present our main results, which concern the development of the stability robustness measurement and the design of a robust polynomial state feedback controller with robustness maximization. Numerical examples illustrating the effectiveness of the proposed methods are provided. Finally, a conclusion ends the paper.

2. Mathematical Notations and Useful Tools

We use standard notations throughout this paper. ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, ℝ^𝑛×𝑚 is the set of all real 𝑚×𝑛 matrices, and ⊗ is the symbol of the Kronecker product. 0_𝑛×𝑚∶(𝑛×𝑚) zero matrix, 0: zero matrix of convenient dimension, 𝐴^𝑇 is the transpose of matrix 𝐴, 𝐴>0(𝐴≥0) is the symmetric positive definite (semidefinite) matrix. The 𝑘th row of a matrix such as 𝐴 is denoted 𝐴^𝑘 and the 𝑖𝑘 element of 𝐴 will be denoted 𝑎_𝑖𝑘. The identity matrix of order 𝑚 is denoted as 𝐼_𝑚 (or simply 𝐼 if no confusion arises). |⋅| stands for the matrix formed by taking the componentwise absolute value of the matrix. 𝜆(⋅) is the eigenvalue of the considered matrix. 𝜌(⋅)=|𝜆_max(⋅)| is the spectral radius of the indicated matrix.

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.

2.1. Kronecker’s Power of Vectors

The Kronecker power of order 𝑖, denoted 𝑋^[𝑖], of the vector 𝑋∈ℝ^𝑛 is defined as (see [18]):𝑋^[0]𝑋=1,_[𝑖]=𝑋^[𝑖−1]⊗𝑋=𝑋⊗𝑋^[𝑖−1]for𝑖≥1. (2.1) For example, considering the second order, the 𝑋^[2] vector is given by𝑋^[2]=𝑥²₁,𝑥₁𝑥₂,…,𝑥₁𝑥_𝑛,𝑥₂𝑥₁,…,𝑥₂𝑥_𝑛,…,𝑥_𝑛𝑥₁,…,𝑥²_𝑛^𝑇, (2.2)𝑋^[𝑖]_{𝑖=1,…,𝑟}∈ℝ^𝑛𝑖, 𝑛_𝑖=(_𝑖^{𝑛+𝑖−1}) is the nonredundant Kronecker power of the state vector 𝑋 defined as𝑋^[1]=𝑋^[1]𝑋=𝑋,∀𝑖≥2,^[𝑖]=𝑥^𝑖₁,𝑥₁^𝑖−1𝑥₂,…,𝑥₁^𝑖−1𝑥_𝑛,…,𝑥₁^𝑖−2𝑥²_𝑛,…,𝑥₁^𝑖−3𝑥³₂,…,𝑥^𝑖_𝑛^𝑇, (2.3) where the repeated components of the redundant 𝑖th-power 𝑋^[𝑖] are omitted.

The relation between the redundant and the nun-redundant Kronecker power of the vector 𝑋 can be stated as follows:∀𝑖∈ℕ∃!𝑇_𝑖∈ℝ^{𝑛𝑖×𝑛𝑖},𝑋^[𝑖]=𝑇_𝑖𝑋^[𝑖], (2.4) where 𝑇_𝑖 is called the transition matrix which a procedure of the determination is given in [23].

2.2. Mat-Function: 𝑚𝑎𝑡(⋅)

If 𝑉 is a vector of dimension 𝑝=𝑛⋅𝑚, then 𝑀=mat_(𝑛,𝑚)(𝑉) is the (𝑛×𝑚) matrix verifying 𝑉=vec(𝑀).

2.3. Vec-Function: vec(⋅)

An important vector valued function of matrix denoted vec(⋅) was defined in [18] as follows:𝒞=𝑐₁𝑐₂⋯𝑐_𝑞∈ℝ^𝑝×𝑞, (2.5) where for all 𝑖∈{1,…,𝑞}, 𝑐_𝑖∈ℝ^𝑝 are the columns of 𝒞vec(𝒞)=𝑐^𝑇₁𝑐^𝑇₂⋯𝑐^𝑇_𝑞^𝑇∈ℝ^𝑝𝑞. (2.6)

2.4. Polynomial Vectorial Function

For a polynomial vectorial function denoted 𝑎(𝑋) given as follows:𝑎(𝑋)=^𝑟_𝑖=1𝐴_𝑖𝑋^[𝑖], (2.7) where 𝑋∈ℝ^𝑛, 𝐴_𝑖 are (𝑛×𝑛^𝑖) constant matrices and 𝑟 is the polynomial order (considered odd) defined by 𝑟=2𝑠−1, with 𝑠∈ℕ^∗.

We define the (𝜐×𝜐) matrix ℳ(𝑎) as ℳ(𝑎)=𝑀₁₁𝐴₁𝑀₁₂𝐴₂0⋯00𝑀₂₂𝐴₃⋱⋱⋮⋮⋱⋱⋱0⋮⋱𝑀_{𝑠−1,𝑠−1}𝐴_2𝑠−3𝑀_{𝑠−1,𝑠}𝐴_2𝑠−20⋯⋯0𝑀_𝑠,𝑠𝐴_2𝑠−1, (2.8) where 𝜐=𝑛+𝑛²+⋯+𝑛^𝑠, and (i)for 𝑗=1,…,𝑠𝑀_𝑗,𝑗𝐴_2𝑗−1=mat_{(𝑛𝑗−1,𝑛𝑗)}𝐴^1𝑇_2𝑗−1mat_{(𝑛𝑗−1,𝑛𝑗)}𝐴^2𝑇_2𝑗−1⋮mat_{(𝑛𝑗−1,𝑛𝑗)}𝐴^𝑛𝑇_2𝑗−1, (2.9)(ii)for 𝑗=1,…,𝑠−1𝑀_𝑗,𝑗+1𝐴_2𝑗=mat_{(𝑛𝑗−1,𝑛𝑗)}𝐴^1𝑇_2𝑗mat_{(𝑛𝑗−1,𝑛𝑗)}𝐴^2𝑇_2𝑗⋮mat_{(𝑛𝑗−1,𝑛𝑗)}𝐴^𝑛𝑇_2𝑗, (2.10)𝐴_𝑘=𝐴_𝑘^1𝑇𝐴²_𝑘⋯𝐴_𝑘^𝑛𝑇. (2.11)

3. System Description and Problem Formulation

Consider the class of uncertain nonlinear polynomial systems with structured uncertainties described by 𝑋̇=𝑓(𝑋)+Δ𝑓(𝑋)+𝐵𝑈, (3.1) where 𝑓(𝑋)=^𝑟_𝑘=1𝐹_𝑘𝑋^{𝑘},Δ𝑓(𝑋)=^𝑟_𝑘=1Δ𝐹_𝑘𝑋^[𝑘], (3.2) with (i)𝑋=[𝑥₁,…,𝑥_𝑛]^𝑇∈ℝ^𝑛 is the state vector, (ii)𝑋^[𝑘]∈ℝ^𝑛𝑘 is the 𝑘th Kronecker power of the vector 𝑋, (iii)𝑈∈ℝ^𝑚 is the control input vector, (iv)𝐹_𝑘∈ℝ^{𝑛×𝑛𝑘} are constant matrices which describe the nominal system, (v)𝐵∈ℝ^𝑛×𝑚 is the constant control matrix, (vi)Δ𝐹_{𝑘,𝑘=1,…,𝑟}∈ℝ^{𝑛×𝑛𝑘} are the matrices of parametric uncertainties.

We note that the structure of the parametric uncertainties Δ𝐹_{𝑘,𝑘=1,…,𝑟} is multiple-parameter variations given by Δ𝐹_𝑘=^𝛾𝑘_𝑗=1𝜀_𝑘𝑗𝐸_𝑘𝑗, (3.3) where 𝐸_𝑘𝑗∈ℝ^{𝑛×𝑛𝑘} (for 𝑘=1,…,𝑟 and 𝑗=1,…,𝛾_𝑘) are numerically known matrices, 𝜀_𝑘𝑗 represent the uncertain parameters, and 𝛾_𝑘 represents the number of uncertain parameters of the matrix 𝐹_𝑘.

According to (3.1) and (3.3), let 𝐹_{𝑘,𝑘=1,…,𝑟} be some nominal matrices which present the parameters of the considered polynomial system. Define the perturbed terms which depend linearly on a number of parameters 𝜀_𝑘𝑗 which are unknown real scalars.

The objective of this paper is to address the following problems: (1)in the robust stability analysis, it is assumed that the nominally system described by 𝑋̇=𝑓(𝑋)=^𝑟_𝑘=1𝐹_𝑘𝑋^[𝑘] (3.4) is globally asymptotically stable, and we wish to find an upper bound 𝜂 of the parameters 𝜀_𝑘𝑗 (max(𝜀_𝑘𝑗)≤𝜂) such that the following nonlinear polynomial systems without controller (for 𝑈(𝑡)=0): 𝑋̇=𝑓(𝑋)+Δ𝑓(𝑋)=^𝑟_𝑘=1𝐹_𝑘+Δ𝐹_𝑘𝑋^[𝑘] (3.5) remains stable, and (2)the robust control problem consist to design a nonlinear robust controller which ensures the global stability of the nominal nonlinear systems (3.4) and maximizes the stability robustness of the controlled system. This means that the stability of the controlled system is ensured for a maximum value of the uncertainties affecting the system parameters.

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.

4. Robustness Measure

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.

4.1. Stability Robustness Bound for Linear System

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:𝑋̇=𝐴𝑋(𝑡)+^𝑝_𝑘=1𝜀_𝑘𝐸_𝑘𝑋(𝑡)=(𝐴+Δ𝐴)𝑋(𝑡), (4.1) where Δ𝐴=^𝑝_𝑘=1𝜀_𝑘𝐸_𝑘 (4.2) with 𝐴∈ℝ^𝑛×𝑛 is a nominal Hurwitz matrix with eigenvalues in the open left-half plane, Δ𝐴 presents the additive uncertainty with 𝐸_𝑘∈ℝ^𝑛×𝑛 are constant known matrices and 𝜀_{𝑘,𝑘=1,…,𝑝} denote the unknown real parameters. Note that the uncertain parameters enter the uncertainty matrix linearly.

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.

4.1.1. Robustness Measure of Yedavalli

Consider an uncertain linear system described by (4.1) with structured uncertainties given by (4.2), then it comes out the following Yedavalli result [10].

Theorem 4.1. The uncertain linear system (4.1)-(4.2) is asymptotically stable if: max𝜀_𝑘≤𝜌^𝑝_𝑘=1Γ_𝑘Φ⁻¹⁻¹, (4.3) with Γ_𝑘=𝐸_𝑘⊗𝐼_𝑛+𝐼_𝑛⊗𝐸_𝑘,Φ=𝐴⊗𝐼_𝑛+𝐼_𝑛⊗𝐴. (4.4)

4.1.2. Robustness Measure of Gardiner

Next, we review an improved robustness algorithm proposed by Gardiner that gives a better bound than the methods obtained in [9] and Yedavalli's works [10, 24, 25] presented above. Therefore, the following theorem provides am improved robustness bound of linear uncertain system [12].

Theorem 4.2. The system (4.1)-(4.2) is asymptotically stable if max𝜀_𝑘≤𝜌^𝑝_𝑘1^𝑘₌₁¹⁻¹_𝑘2=1Ξ_𝑘1𝐴₀⁻¹Ξ_𝑘2𝐴₀⁻¹+Ξ_𝑘2𝐴₀⁻¹Ξ_𝑘1𝐴₀⁻¹+^𝑝_𝑘=1Ξ_𝑘𝐴₀^−12−1/2, (4.5) with Ξ_𝑘=𝐸_𝑘⊗𝐼_𝑛+𝐼_𝑛⊗𝐸_𝑘,Ξ_𝑘𝑖=𝐸_𝑘𝑖⊗𝐼_𝑛+𝐼_𝑛⊗𝐸_𝑘𝑖,𝐴₀=𝐴⊗𝐼_𝑛+𝐼_𝑛⊗𝐴. (4.6)

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.

4.2. Stability Criterion of Nonlinear Polynomial Systems

The robustness measure developed in this study is specially based on the pertinent results presented in the following theorem [15, 16, 19], about the global stability condition of autonomous polynomial systems described by (3.4). This stability condition is based on the Lyapunov direct method with quadratic Lyapunov function 𝒱(𝑋)=𝑋^𝑇𝑃𝑋 (𝑃∈ℝ^𝑛×𝑛) [26].

Theorem 4.3. The nonlinear polynomial system defined by (3.4) is globally asymptotically stable if there exist: (i)an (𝑛×𝑛)-symmetric positive definite matrix P, and(ii)arbitrary parameters 𝜇_{𝑖,𝑖=1,…,𝛽}∈ℝ, such that the (𝜗×𝜗) symmetric matrix 𝒬 defined by 𝒬=𝜑𝑃,𝜇_{𝑖,𝑖=1,…,𝛽}=𝜏^𝑇𝒫ℳ𝑓+ℳ𝑓^𝑇𝒫𝜏+Π𝜇_{𝑖,𝑖=1,…,𝛽} (4.7) is negative definite.

The proof elements and the parameters notation of Theorem 4.3 are illustrated in appendix (see [15, 16, 19, 20] for more details).

4.3. Robustness Stability Measure of Uncertain Polynomial Systems

4.3.1. Duality Approach

Refrying to the stability condition of certain polynomial system (4.7), given by Theorem 4.1, the uncertain nonlinear polynomial system (3.5) is globally asymptotically stable, if there exist a positive definite matrix 𝑃 and real parameters 𝜇_{𝑖,𝑖=1,…,𝛽} such that:𝒬=𝒬+Δ𝒬=𝜑𝐹_𝑘+Δ𝐹_{𝑘,𝑘=1,…,𝑟)}<0, (4.8) where𝒬=𝜑𝐹_{𝑘,𝑘=1,…,𝑟},Δ𝒬=𝜑Δ𝐹_{𝑘,𝑘=1,…,𝑟}. (4.9) It can be verified that 𝜑Δ𝐹_{𝑘,𝑘=1,…,𝑟}=^𝑟𝛾_𝑘=1^𝑘_𝑗=1𝜀_𝑘𝑗Ξ_𝑘𝑗,Ξ_𝑘𝑗=𝜑Λ_{𝑖𝑗,𝑖=1,…,𝑟},Λ_𝑖𝑗=𝐸_𝑘𝑗if𝑖=𝑘,0if𝑖≠𝑘. (4.10) Indeed, the system (3.5) is globally asymptotically stable if the matrix 𝒬 is negative definite (i.e., when 𝜆_𝑖(𝒬)<0,𝑖=1,…,𝜗). Then, to ensure the negativity of 𝒬, it will be sufficient to guarantee the stability of the following linear uncertain system:𝒵̇=𝒬𝒵+^𝑟𝛾_𝑘=1^𝑘_𝑗=1𝜀_𝑘𝑗Ξ_𝑘𝑗𝒵 (4.11) said linear dual system of nonlinear system (3.5). So, the robustness measure of the nonlinear system (3.5) is equivalent to the robustness measure of the linear dual system (4.11).

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 [21] and a new improved result using the Gardiner linear robustness bound (4.5).

4.3.2. Yedavalli Based Robustness Measure of Polynomial Systems

Let consider the uncertain nonlinear polynomial system (3.5), We assume that the nominal system satisfies the stability criterion (4.7) of Theorem 4.3, then we obtain the following result formulated in Theorem 4.4 [21].

Theorem 4.4. The nonlinear uncertain polynomial system (3.5) is robustly stable, if 𝜂₁=max𝜀_𝑘𝑗≤𝜂^⋆₁ (4.12) with 𝜂^⋆₁=max_𝑝𝜌^𝑟𝛾_𝑘=1^𝑘_𝑗=1Ω_𝑘𝑗Θ⁻¹⁻¹, (4.13) where for 𝑘=1,…,𝑟, 𝑗=1,…,𝛾_𝑘Ω_𝑘𝑗=𝜀_𝑘𝑗⊗𝐼_𝜗+𝐼_𝜗⊗𝜀_𝑘𝑗,Θ=𝒬⊗𝐼_𝜗+𝐼_𝜗Ξ⊗𝒬,_𝑘𝑗=𝜑Λ_{𝑖𝑗,𝑖=1,…,𝑟};Λ_𝑖𝑗=𝐸_𝑘𝑗if𝑖=𝑘,0if𝑖≠𝑘,𝜗=^𝑠_𝑘=1𝑘.𝑛+𝑘−1 (4.14)

This theorem was proved in [21].

4.3.3. Gardiner Based Robustness Measure of Polynomial Systems

We present now the proposed robustness bound of uncertain polynomial systems based on the Gardiner's method.

Theorem 4.5. The nonlinear uncertain polynomial system (3.5) is robustly stable, if 𝜂₂=max𝜀_𝑘𝑗≤𝜂^⋆₂, (4.15) with 𝜂^⋆₂=max_{𝑃,𝜇𝑖,𝑖=1,…,𝛽}𝜌_{𝑘1,𝑗1𝑘2,𝑗2}Ω_𝑘1𝑗1Θ⁻¹Ω_𝑘2𝑗2Θ⁻¹+Ω_𝑘2𝑗2Θ⁻¹Ω_𝑘1𝑗1Θ⁻¹+_𝑘,𝑗Ω_𝑘𝑗Θ^−12−1/2, (4.16) where for 𝑘₁=1,…,𝑟, 𝑘₂=1,…,𝑘₁−1, 𝑗₁=1,…,𝛾_𝑘1 and 𝑗₂=1,…,𝛾_𝑘2Ω_𝑘𝑗=Ξ_𝑘𝑗⊗𝐼_𝜗+𝐼_𝜗⊗Ξ_𝑘𝑗,Ω_{𝑘𝑖𝑗𝑖}=Ξ_{𝑘𝑖𝑗𝑖}⊗𝐼_𝜗+𝐼_𝜗⊗Ξ_{𝑘𝑖𝑗𝑖},Θ=𝒬⊗𝐼_𝜗+𝐼_𝜗Ξ⊗𝒬,_𝑘𝑗=𝜑Λ_{𝑙𝑗,𝑙=1,…,𝑟};Λ_𝑙𝑗=𝐸_𝑘𝑗Ξif𝑙=𝑘,0if𝑙≠𝑘,_{𝑘𝑖𝑗𝑖}=𝜑Λ_{𝑙𝑗𝑖,𝑙=1,…,𝑟};Λ_𝑙𝑗𝑖=𝐸_{𝑘𝑖𝑗𝑖}if𝑙=𝑘_𝑖,0if𝑙≠𝑘_𝑖. (4.17)

Proof. The stability of the polynomial system (3.5) is equivalent to that of the dual linear system (4.11). Then, applying the above Gardiner criterion [12] leads to the result of Theorem 4.5, which completes the proof.

Let us remark that Theorem 4.5 leads an improved robustness bound, since it is based on the linear Gardiner measure which is proved larger than Yedavalli one. Moreover, the optimization of the new measure is made with regard to 𝑃 matrix and the 𝜇_𝑖-parameters while the previous measure given in Theorem 4.4 is optimized only with regard to the Lyapunov matrix 𝑃.

4.4. An Illustrative Numerical Example

Consider the following second-order autonomous uncertain polynomial system:𝑥̇₁=𝑎₁₁+Δ𝑎₁₁𝑥₁+𝑏₁₁+Δ𝑏₁₁𝑥²₁+𝑏₁₃+Δ𝑏₁₃𝑥²₂+𝑐₁₁+Δ𝑐₁₁𝑥³₁+𝑐₁₄+Δ𝑐₁₄𝑥³₂,𝑥̇₂=𝑎₂₂+Δ𝑎₂₂𝑥₂+𝑏₂₁+Δ𝑏₂₁𝑥²₁+𝑏₂₃+Δ𝑏₂₃𝑥²₂+𝑐₂₁+Δ𝑐₂₁𝑥³₁+𝑐₂₄+Δ𝑐₂₄𝑥³₂. (4.18) This system can be written in the following polynomial form:𝑋̇=𝐹₁+Δ𝐹₁𝑋+𝐹₂+Δ𝐹₂𝑋^[2]+𝐹₃+Δ𝐹₃𝑋^[3], (4.19) where𝐹₁=𝑎₁₁00𝑎₂₂,𝐹₂=𝑏₁₁00𝑏₁₃𝑏₂₁00𝑏₂₃,𝐹₃=𝑐₁₁000000𝑐₁₄c₂₁000000𝑐₂₄,Δ𝐹₁=Δ𝑎₁₁1000+Δ𝑎₂₂0001=Δ𝑎₁₁𝐸₁₁+Δ𝑎₂₂𝐸₁₂,Δ𝐹₂=Δ𝑏₁₁10000000+Δ𝑏₁₃00010000+Δ𝑏₂₁00001000+Δ𝑏₂₃00000001=Δ𝑏₁₁𝐸₂₁+Δ𝑏₁₃𝐸₂₂+Δ𝑏₂₁𝐸₂₃+Δ𝑏₂₃𝐸₂₄,Δ𝐹₃=Δ𝑐₁₁1000000000000000+Δ𝑐₁₄0000000100000000+Δ𝑐₂₁0000000010000000+Δ𝑐₂₄0000000000000001=Δ𝑐₁₁𝐸₃₁+Δ𝑐₁₄𝐸₃₂+Δ𝑐₂₁𝐸₃₃+Δ𝑐₂₄𝐸₃₄, (4.20) with: 𝑎₁₁=−0.839,𝑎₂₂=1.921,𝑏₁₁𝑏=−0.751,₁₃=−0.432,𝑏₂₁𝑏=−1.673,₂₃=−2.786,𝑐₁₁=−0.753,𝑐₁₄𝑐=−0.634,₂₁=−0.532,𝑐₂₄=−2.431. (4.21) Then, we have the following uncertain parameters:𝜀_𝑖𝑗=Δ𝑎_𝑖𝑗,𝑖=1,2;𝑗=1,2Δ𝑏_𝑖𝑗,𝑖=1,2;𝑗=1,3Δ𝑐_𝑖𝑗,𝑖=1,2;𝑗=1,4. (4.22) Using the optimization tools offered by the Matlab software as the “fmincon.m” [27] function which allows the resolution of the nonlinear optimization problem with inequalities constraints, given by the procedure of computational of the robustness measure, presented in Theorem 4.5, and after some manipulations concerning the initialization task of the optimization algorithm under Matlab, we obtain the following optimal solution: 𝑃=0.0124000.1001,𝜇₁=0.0167,𝜇₂=0.0314,𝜇₃=0.0100, (4.23) and the system (4.18) is robustly stable for all uncertainty parameters 𝜀_𝑘𝑗 such that |𝜀_𝑘𝑗|≤𝜂^⋆₂=1.71.

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 [21], the presented approach improve the robustness measure, which enlarges the uncertainties variation domain.

5. Stabilizing Control Synthesis with Robustness Maximization

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 (4.16), proposed by Theorem 4.5 in Section 4. An optimization algorithm allows to establish the following two conditions satisfied by the control law synthesis: (i)the global stability of the nominal (without uncertainties) nonlinear polynomial system is ensured, and(ii)the stability of the controlled perturbed system is ensured for a maximum bound of uncertainties affecting the system parameters.

5.1. Nonlinear State Feedback Controller

We consider now the uncertain polynomial system given by (3.1). Our purpose is to find a polynomial state feedback controller described by𝑈=𝑘(𝑋)=^𝑟_𝑘=1𝐾_𝑘𝑋^[𝑘], (5.1) with 𝐾_{𝑘,𝑘=1,…,𝑟} are constant gains matrices which stabilizes asymptotically and globally the equilibrium (𝑋=0) of the considered system (3.1) and to maximize the robustness measure.

The controlled uncertain system is described by the state equation:𝑋̇=^𝑟_𝑘=1𝐴_𝑘+Δ𝐴_𝑘𝑋^[𝑘], (5.2) where𝐴_𝑘=𝐹_𝑘+𝐵𝐾_𝑘,Δ𝐴_𝑘=Δ𝐹_𝑘=^𝛾𝑘_𝑗=1𝜀_𝑘𝑗𝐸_𝑘𝑗. (5.3)

5.2. Robust Synthesis for Uncertain Polynomial Systems

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 [20]. The stabilization of nominal polynomial systems can be formulated as the following matrix inequality problem:

Find (i)gain matrices 𝐾_{𝑖,𝑖=1,…,𝑟}, (ii)a (𝑛×𝑛) matrix 𝑃, and(iii)real parameters 𝜇_{𝑖,𝑖=1,…,𝛽},

such that𝒬𝑃>0,_𝑐=𝒬𝑃,𝜇_{𝑖,𝑖=1,…,𝛽},𝐾_{𝑘,𝑘=1,…,𝑟}=𝜏^𝑇𝒫(𝑃)ℳ𝑓+𝐵𝑘+ℳ𝑓+𝐵𝑘^𝑇𝒫(𝑃)𝜏+Π𝜇_{𝑖,𝑖=1,…,𝛽}<0. (5.4)(5.5) Let us note that this inequality (5.5) is a NLMI (Nonlinear Matrix Inequality) problem which can be turned into an LMI feasibility problem using specific transformations which are presented in [20].

Therefore, the proposed synthesis method combines the robust analysis result in the last section (Theorem 4.5) with the quadratic stabilization technique using the LMI approach in terms of a global optimization algorithm, presented above. Thus, our aim is to find a nonlinear state feedback controller (5.1) which stabilizes the closed loop system (5.2) and, at the same time, maximizes the bound on the perturbation parameters which the considered system can tolerate without going unstable. Then, our main result concerning the robust stabilization can be stated in the following theorem.

Theorem 5.1. The controlled nonlinear uncertain polynomial system (5.2) is robustly asymptotically stable, if 𝜂_𝑐=max𝜀_𝑘𝑗≤𝜂^⋆_𝑐, (5.6) with 𝜂^⋆_𝑐=max_{(𝑃,𝜇𝑖,𝐾𝑘)}𝜌_{𝑘1,𝑗1𝑘2,𝑗2}Ω_𝑘1𝑗1Ψ⁻¹Ω_𝑘2𝑗2Ψ⁻¹+Ω_𝑘2𝑗2Ψ⁻¹Ω_𝑘1𝑗1Ψ⁻¹+_𝑘,𝑗Ω_𝑘𝑗Ψ^−12−1/2, (5.7) where Ψ=𝒬_𝑐⊗𝐼_𝜗+𝐼_𝜗⊗𝒬_𝑐,𝒬_𝑐=𝒬𝑃,𝜇_{𝑖,𝑖=1,…,𝛽},𝐾_{𝑘,𝑘=1,…,𝑟}=𝜏^𝑇𝒫ℳ𝑓+𝐵𝑘+ℳ𝑓+B𝑘^𝑇𝒫𝜏+Π𝜇_{𝑖,𝑖=1,…,𝛽}. (5.8)

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 (5.7). It should be mentioned that the resolution of the nonlinear optimization problem, presented in Theorem 5.1, is very sensitive to the initialization of the optimization algorithm. This difficulty is overcame simply when using the systematic results given by the LMI stabilization approach of nominal controlled polynomial systems, as a starting point for the nonlinear optimization described by the following algorithm.

Algorithm 5.2. One has the following steps:Step 1 (Stabilization of the nominal system (Initialization)). 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 𝑃, arbitrary parameters 𝜇_{𝑖,𝑖=1,…,𝛽} and gain matrices 𝐾_{𝑘,𝑘=1,…,𝑟} which globally stabilize the closed-loop nominal system. The obtained solution constitues the initialization parameters 𝑃₀, 𝜇_𝑖0 and 𝐾_𝑘0 of the next step. The second step allows the adjustment of the control gains.Step 2 (Maximization of the robustness bound). Consider the matrices 𝐾_{𝑘,𝑘=1,…,𝑟}, 𝑃 and 𝜇_{𝑖,𝑖=1,…,𝛽} obtained in Step 1 as initial values (𝑃₀, 𝜇_𝑖0 and 𝐾_𝑘0), and determine the optimal solution ensuring the stability of the controlled system (5.2) such that the following conditions are verified: (i)𝒫=𝒫^𝑇 is positive definite, and(ii)𝒬_𝑐(𝐾_𝑘,𝑃,𝜇_𝑖)=𝜏^𝑇(𝒫ℳ(𝐴_{𝑘,𝑘=1,…,𝑟})+ℳ(𝐴_𝑘,𝑘=1,…,𝑟)^𝑇𝒫)𝜏+Π(𝜇_{𝑖,𝑖=1,…,𝛽}) is negative definite, and maximizing the robustness measure 𝜂^⋆_𝑐=max_{(𝑃,𝜇𝑖,𝐾𝑘)}𝜌_{𝑘1,𝑗1𝑘2,𝑗2}Ω_𝑘1𝑗1Ψ⁻¹Ω_𝑘2𝑗2Ψ⁻¹+Ω_𝑘2𝑗2Ψ⁻¹Ω_𝑘1𝑗1Ψ⁻¹+_𝑘,𝑗Ω_𝑘𝑗Ψ^−12−1/2, (5.9) with 𝒬𝑃>0,_𝑐=𝒬𝑃,𝐴_{𝑘,𝑘=1,…,𝑟},𝜇_{𝑖,𝑖=1,…,𝛽}=𝜏^𝑇𝒫ℳ𝐴_{𝑘,𝑘=1,…,𝑟}+ℳ𝐴_{𝑘,𝑘=1,…,𝑟}^𝑇𝒫𝜏+Π𝜇_{𝑖,𝑖=1,…,𝛽}<0. (5.10)As we indicated above, these two algorithm steps establish a nonlinear optimization problem under nonlinear constraints [27]. In the sequel, we give illustrative examples to show the effectiveness of the proposed algorithm.

5.3. Numerical Example and Single Link Manipulator Application

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.

5.3.1. Example  1: Numerical Example

We consider the uncertain polynomial system described by the state equation (5.11), which the global asymptotic stabilization of the nominal part is treated in [20]:𝑋̇=𝐹₁+Δ𝐹₁𝑋+𝐹₂+Δ𝐹₂𝑋^[2]+𝐹₃+Δ𝐹₃𝑋^[3]+𝐵𝑈, (5.11) where 𝐹₁=−11−11.5,𝐹₂=1100−1−0.500,𝐹₃=−110−10002−1−100.5000−2, (5.12) with 𝐵=(01)^𝑇, andΔ𝐹₁=Δ𝐹₁₁00Δ𝐹₁₄,Δ𝐹₂=Δ𝐹₂₁00Δ𝑏₂₄Δ𝐹₂₅00Δ𝑏₂₈,Δ𝐹₃=Δ𝐹₃₁000000Δ𝐹₃₈Δ𝐹₃₉000000Δ𝑐₄₆. (5.13) First, according to Step 1, we are interested with the stabilization of the nominal part (i.e., global asymptotic stabilization of the origin equilibrium (𝑋=0)) of the system (5.11) using an LMI stabilization method developed in [20]. The obtained solution as initial condition of the next step is 𝜇₁₀𝜇=−0.0319,₂₀𝜇=0.0246,₃₀𝑃=0.1928,⁰=0.02460.00630.00630.0071, (5.14) and the searched gain matrices which stabilizes the nominal closed-loop system: 𝐾₁₀=−9.8872−5.5518,𝐾₂₀=,𝐾2.10500.896600₃₀=−0.8541−4.2706−1.3702−8.80840.9371−1.82164.31521.3277. (5.15) In the second step, solving the optimization problem [27] formulated by Theorem 5.1, we obtain: 𝜇₁𝜇=0.005₂𝜇=0.005₃=0.0707𝑃=0.00540.0050.0050.0051. (5.16) The optimal control gain matrices are 𝐾₁=1.7811−2.5004,𝐾₂=,𝐾−1.21870.0500−0.78860.0227₃=−1.2343−6.9917−4.11032.476312.0761−4.58911.54240.3126. (5.17) The stability of the uncertain system is guaranteed for all uncertainty parameters 𝜀_𝑘𝑗 such that max(𝜀_𝑘𝑗)=𝜂_𝑐≤𝜂^⋆_𝑐=2.3542.

The system controlled with the obtained polynomial control law was simulated for a perturbation of 100% on the state variable 𝑥₁ and parametric perturbation corresponding to values of 𝜀_𝑘𝑗=2.3542. Figure 1 presents the state trajectories of the both nominal and perturbed polynomial system with the proposed nonlinear polynomial controllers. The simulation depicted in Figure 2 presents the evolution of the performed feedback stabilizing polynomial controller.

Figure 1 

Behavior of the state variables of both nominal and perturbed system.

Figure 2 

Simulation curve of the proposed feedback controller.

5.3.2. Example  2: Single-Link Manipulator Application

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 3 [28].

Figure 3 

Model of single-link manipulator with flexible joint.

We model the dynamics of this system ignoring damping terms as(𝐼+𝐽)𝑞̈+𝐵𝑞̇+𝑀𝑔𝑙sin𝑞=𝑢 (5.18) which can be expressed as𝐼𝑞̈₁+𝑀𝑔𝑙sin𝑞₁+𝑘𝑞₁−𝑞₂𝐽𝑞̈=0,₂−𝑘𝑞₁−𝑞₂−𝑢=0. (5.19) The parameters used in these equations are defined as follows: 𝐼 is the link inertia, 𝐽 represents the rotor inertia, 𝑞 is the vector of link positions, 𝑀𝑔𝑙 nominal load, 𝐵 rotor friction, 𝑘 denotes the joint stiffness and 𝑢 represents the input torque.

The considered system can be described by a fourth-order model of the form:𝑋̇=𝑓(𝑋)+𝐵𝑢, (5.20) where 𝑋 is the state vector defined by 𝑋=𝑞₁𝑞̇₁𝑞₂𝑞̇₂^𝑇=𝑥₁𝑥₂𝑥₃𝑥₄^𝑇,𝑓(𝑋)=𝑞̇₁−𝑎sin𝑞₁−𝑏𝑞₁−𝑞₂𝑞̇₂𝑐𝑞₁−𝑞₂, (5.21) and 𝐵=[000𝑑]^𝑇 with 𝑎=𝐼⁻¹𝑀𝑔𝑙,𝑏=𝐼⁻¹𝑘,𝑐=𝐽⁻¹𝑘,𝑑=𝐽⁻¹. (5.22) The nonlinear system (5.20) can be developed into polynomial form by a Taylor series expansions, then we have𝑋̇=𝐹₁+Δ𝐹₁𝑋+𝐹₂+Δ𝐹₂𝑋^[2]+𝐹₃+Δ𝐹₃𝑋^[3]+𝐵𝑈, (5.23) where 𝐹₁=−()0100𝑎+𝑏0𝑏00001𝑐0−𝑐0,𝐹₂=0_4×16, (5.24)𝐹₃(1,1)=𝑎/6; 𝐹₃(𝑖,𝑗)=0 for all 𝑖,𝑗(𝑖=1,…,4;𝑗=1,…,64), and Δ𝐹₁=0000Δ𝐹₁₁0Δ𝐹₁₂00000Δ𝐹₁₃0Δ𝐹₁₄0,Δ𝐹₂=0_4×16, (5.25)Δ𝐹₃(1,1)=Δ𝐹₃₁; Δ𝐹₃(𝑖,𝑗)=0 for all 𝑖,𝑗(𝑖=1,…,4;𝑗=1,…,64).

For simulation, the single-link manipulator parameters are considered in Table 1 [29].

Table 1 

Nominal values of the system parameters.

Thus, according to the values of the considered system parameters indicated in Table 1 and an initial condition is assumed to be 𝑞₁(0)𝑞̇₁(0)𝑞₂(0)𝑞̇₂(0)^𝑇=𝜋0030^𝑇. (5.26) The evolution of the state variables of the single-link manipulator without controller is shown in Figure 4. It appears in the simulations that the polynomial approach is a more objective of the real system, since, we can't distinguish the real system behavior and the polynomial one.

(a)

(b)

(c)

(d)

Figure 4 

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: 𝑃=0.00050.01720.00020.00070.01720.59580.00830.90660.00020.00830.00050.01890.00070.30660.01890.1582. (5.27) The obtained state feedback control law𝑈=𝐾₁𝑋+𝐾₂𝑋^[2]+𝐾₃𝑋^[3] (5.28) is builded with the following optimal gains matrices: 𝐾₁=,𝐾71.345−0.3219−69.4698−2.9487₂=[],𝐾−0.0033−0.0034−0.0678−0.00819−0.0123−0.4854−0.0086−0.0065−0.0345−0.0059−0.0295−0.0003−0.0045−0.0023−0.089−0.0196₃=[].73.1389−0.0039−0.0079−0.0057−0.9574−0.0049−0.0052−0.0084−68.1638−0.0052−0.0027−0.0026−2.3459−0.0023−0.0025−0.0035−0.00527871.4631−0.0023−0.0034−0.0010−0.1539−0.0015−0.0034−0.0026−69.9302−0.0012−0.0015−0.0014−2.2098−0.0034−0.0034−0.0024−0.004569.4388−0.0036−0.0035−0.0201−0.1123−0.0045−0.0088−0.0049−73.3322−0.0069−0.0062−0.0063−2.2398−0.0057−0.0057−0.0015−0.004565.5724−0.0035−0.0057−0.0047−0.5739−0.0046−0.0013−0.0013−64.5544−0.0024−0.0086−0.0024−2.3040 (5.29) The closed-loop system responses using the obtained polynomial controller are depicted in Figure 5.

(a)

(b)

(c)

(d)

Figure 5 

Closed-loop system response using the polynomial controller.

As it can be seen, the nonlinear state feedback controller (5.28) stabilizes the considered system. Moreover, the proposed robust controller, shown in Figure 6, can rapidly damp the oscillations of the manipulator and greatly enhance transient stability for all uncertainty parameters 𝜀_𝑘𝑗=Δ𝐹_𝑘𝑗≤1.0581.

Figure 6 

Simulation of the stabilizing polynomial control (5.28).

6. Conclusion

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 𝐻_∞ control for the studied class of nonlinear uncertain systems presents another interesting topic for future research.

Appendix

Proof Elements of Theorem 4.3 and Relative Notations
For the complete proof of Theorem 4.3, we can refer to [15, 16, 19]. We limit here to remind the fundamental principles, we consider the following quadratic lyapunov function 𝒱(𝑋)=𝑋^𝑇𝑃𝑋, then we can show that: 𝑑𝒱(𝑋)𝒳𝑑𝑡=−^T𝒬𝒳, (A.1) where 𝒳 is the 𝜗-vector defined by 𝒳=𝑋^𝑇𝑋^[2]𝑇⋯𝑋^{[𝑠]𝑇𝑇}, (A.2) and 𝒬 is the symmetric matrix defined in (4.7), where 𝒫=diag𝑃,𝑃⊗𝐼_𝑛,…,𝑃⊗𝐼_𝑛𝑠−1,𝒳=𝜏𝒳, (A.3) and 𝜏 is defined by: 𝜏=diag𝑇₁,𝑇₂,…,𝑇_𝑠, (A.4) with ℳ(𝑓) is the matrix of the system parameters, expressed by (2.8).
The matrix Π(𝜇_{𝑖,𝑖=1,…,𝛽}) is defined asΠ𝜇_{𝑖,𝑖=1,…,𝛽}=^𝛽_𝑖=1𝜇_𝑖mat_(𝜂,𝜂)𝐶_𝑖, (A.5) where the columns 𝐶_{𝑖,𝑖=1,…,𝛽} and the coefficients 𝜇_𝑖 and 𝛽 are detailed in [15].