Observer-based robust stabilization for a class of uncertain polynomial systems

Abstract

This paper focuses on the observer-based robust stabilization problem for a class of polynomial systems with norm-bounded time-varying uncertainties. The structural features of such systems guarantee the fulfilment of the separation principle between the reduced-order observer and the state feedback. The existence conditions of the observer-based robust stabilization controller are obtained by using Lyapunov stability theory. Furthermore, based on the polynomial sum of squares (SOS) theory, the above conditions are transformed into the corresponding SOS convex optimization constraints, for the avoidance of computing difficulties that exist widely in the control of non-linear systems. Finally, two numerical examples are given to illustrate the feasibility and effectiveness of the proposed approach.

Keywords

Non-linear robust stabilization observer polynomial systems sum of squares uncertainty

Introduction

Although the traditional design-via-linearization method is widely adopted in engineering, the controller works only in some neighbourhood of a single equilibrium. This limits the applications to some practical issues with the concern of large-scale mobility or manoeuvrability for a spacecraft or other vehicles. In order to realize increasingly complex engineering tasks and meet various performance requirements, the research on non-linear control theory is both fundamental and necessary. However, due to the inherent complexities of non-linear dynamic systems, no general methodologies and tools are available for their effective treatment, and therefore the computation problem is still one of the major difficulties in non-linear control design.

Fortunately, the recent exciting developments in the polynomial sum of squares (SOS), a convex relaxation technique to test non-negativity of polynomial functions, offer potentially effective ways to cope with non-linear control problems. Within the framework of SOS, many useful approaches can be generalized from linear systems to polynomial non-linear systems, and a lot of control problems with polynomial constraints can be converted into SOS convex programming problems, thereby overcoming, to a certain degree, the computational difficulties typically involved in the non-linear control design. Currently, three toolboxes can be used to solve the SOS programming problems: SOSTOOLS (Prajna et al., 2012), Yalmip (Lofberg, 2004) and Gloptipoly (Henrion et al., 2003).

So far, the remarkable progress of SOS technique has triggered a hot research on non-linear control. By introducing the density function and constructing Lyapunov functions dependent on partial states, Prajna et al. (2004a, 2004b) studied the non-linear state feedback synthesis problem. Based on an iterative SOS approach, Nguang et al. (2011) proposed a non-linear robust static output feedback control scheme. The simultaneous stabilization and robust performance control for a class of polynomial non-linear systems, and synthesis of non-linear discrete-time systems were addres-sed in Xu et al. (2007, 2009). For the discrete-time non-linear networked control systems, Chae et al. (2014) suggested a robust H_∞ fuzzy dynamic output feedback control method. In Narendra (2008), the switched control was investigated for the rigid satellite attitude manoeuvre. Aiming at a class of axisymmetric spacecrafts, Zheng and Wu (2009, 2011) presented a non-linear output feedback H_∞ control scheme. Apart from the applications mentioned above, SOS theory was also used to study other control problems, such as optimal control (Ichihara, 2009), mixed H₂/H_∞ control (Ma et al., 2012) and non-linear robust control of a hypersonic aircraft (Ataei and Wang, 2012). Generally, the output feedback approaches based on SOS technique are non-convex (Chae et al., 2014; Nguang et al., 2011; Zheng and Wu, 2009, 2011) and thus the computational difficulties are unavoidable. As a result, most of the existing results adopt the state feedback to design controllers.

In order to apply a state feedback control law, the system’s states should be known. Nevertheless, the states are not always measurable, or the technical cost of measurement is too high. One has to resort to an observer-based control scheme, which uses the system input and output signals to generate an estimate of the state. In general, the observer-based state feedback control does not satisfy the separation principle for a generic non-linear system, unless the controlled system has a special structure.

In practice, one often encounters some control systems (e.g. the attitude system of a flexible satellite, the Rossler chaotic systems) whose mathematical models have certain common characteristics: 1) the dynamics are governed by a linear-like differential equation; 2) the coefficient matrices are polynomial functions of partial states, which are measurable; and 3) the uncertainties enter the system from only the subsystem with measurable states. For these kinds of systems with the above specific structure, this paper focuses on the robust stabilization problem. With the aid of SOS, this paper presents a convex observer-based control approach and verifies the fulfilment of the separation principle. The main features of the paper are threefold. First, the observer-based non-linear control satisfies a separation principle, i.e. the reduced-order observer and the state feedback controller can be designed independently. Second, the robust stabilization problem is transformed into an SOS convex optimization problem, thus effectively avoiding the difficulties in constructing Lyapunov functions and performing the numerical simulations. Finally, the designed observer and controller are easy to implement in engineering because they are only the polynomial or rational functions of the measurable states as well as the estimated values of the immeasurable states.

The notations used are standard. $R^{n}$ is the set of n-dimensional real vectors; $R^{m \times n}$ the set of real matrices of dimension $m \times n$ ; $S^{n}$ ( $S_{+}^{n}$ ) the set of real symmetric (positive definite) matrices of dimension $n \times n$ ; I the identity matrix; * the symmetric block of a matrix; $He (A) : A + A^{T}$ for a square matrix A; $R {[x]}^{p \times q}$ the set of all $p \times q$ matrices with each entry being a real polynomial in $x \in R^{n}$ ; $S {[x]}^{p}$ the set of all $p \times p$ symmetric matrices with each entry being a real polynomial in $x \in R^{n}$ ; and $Φ_{sos}$ ( $Φ_{sos}^{+}$ ) the set of (positive) SOS polynomials. All matrices are assumed to have compatible dimensions.

Problem description and preliminaries

System description and problem statement

Consider the following uncertain polynomial system

\begin{matrix} \overset{\cdot}{x} = (A (x_{1}) + Δ A (x_{1})) x + (B (x_{1}) + Δ B (x_{1})) u \\ y = x_{1} \end{matrix}

(1)

x = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], A (x_{1}) = [\begin{matrix} A_{11} (x_{1}) & A_{12} (x_{1}) \\ A_{21} (x_{1}) & A_{22} (x_{1}) \end{matrix}], B (x_{1}) = [\begin{matrix} B_{1} (x_{1}) \\ B_{2} (x_{1}) \end{matrix}]

[\begin{matrix} Δ A (x_{1}) & Δ B (x_{1}) \end{matrix}] = [\begin{matrix} I \\ 0 \end{matrix}] [\begin{matrix} Δ A_{11} (x_{1}) & Δ A_{12} (x_{1}) & Δ B_{1} (x_{1}) \end{matrix}]

where $x \in R^{n}$ is the entire state with the measurable sub-state $x_{1} \in R^{p}$ and the immeasurable sub-state $x_{2} \in R^{q}$ , $q = n - p$ ; $u \in R^{m}$ is the control input; $y \in R^{p}$ is the measured output; $A (x_{1})$ and $B (x_{1})$ are known polynomial matrices in $x_{1}$ ; $Δ A (x_{1})$ and $Δ B (x_{1})$ represent the uncertainties satisfying the following assumption.

Assumption 1

\begin{array}{l} [\begin{matrix} Δ A_{11} (x_{1}) & Δ A_{12} (x_{1}) & Δ B_{1} (x_{1}) \end{matrix}] \\ = G_{1} (x_{1}) Δ (t) [\begin{matrix} H_{11} (x_{1}) & H_{12} (x_{1}) & H_{2} (x_{1}) \end{matrix}] \end{array}

where $G_{1} (x_{1})$ , $H_{11} (x_{1})$ , $H_{12} (x_{1})$ and $H_{2} (x_{1})$ are known polynomial matrices characterizing the structures of uncertainties, $Δ (t) \in R^{g \times h}$ is an unknown time-varying matrix satisfying $‖ Δ (t) ‖_{2} \leq 1$ .

Assumption 1 is a quite mild limitation imposed on the uncertainties, and thus is widely adopted in the robust control field (e.g. Asemani and Majd, 2013; Nguang, 1996; Nguang et al., 2011).

Remark 1

Many practical and theoretical research objects exhibit the similar structure as in system (1). Take the attitude system of a flexible satellite as an example. The entire state includes the sub-state of the rigid body (attitude angles and angular velocity) and sub-state of the flexible appendages (modal coordinates). The sub-state of the rigid body can be measured directly by some detection means, whereas the sub-state of the flexible body is immeasurable or difficult to measure. Suppose that the moment of inertia is uncertain and the other parameters are fixed, then the state-space description of the flexible satellite attitude model can be attributed to the form of system (1) (see Numerical examples for details).

For further use, let $x_{1} = {[\begin{matrix} x_{11} & x_{12} & \dots & x_{1 p} \end{matrix}]}^{T}$ ; $A_{11 j} (x_{1})$ , $Δ A_{11 j} (x_{1})$ and $G_{1 j} (x_{1})$ denote the jth row of $A_{11} (x_{1})$ , $Δ A_{11} (x_{1})$ and $G_{1} (x_{1})$ respectively; $J = {j_{1}, j_{2}, \dots, j_{k}}$ denotes the row indices of $[\begin{matrix} A_{12} (x_{1}) + Δ A_{12} (x_{1}) & B_{1} (x_{1}) + Δ B_{1} (x_{1}) \end{matrix}]$ whose corresponding row is equal to zero; $\tilde{J} = {{\tilde{j}}_{1}, {\tilde{j}}_{2}, \dots, {\tilde{j}}_{r}}$ denotes the row indices of $[\begin{matrix} Δ A_{11} (x_{1}) & A_{12} (x_{1}) + Δ A_{12} (x_{1}) B_{1} (x_{1}) + Δ B_{1} (x_{1}) \end{matrix}]$ whose corresponding row is equal to zero. Define $G_{J} (x_{1}) = [G_{1 j_{1}}^{T} (x_{1}) G_{1 j_{2}}^{T} (x_{1}) \dots G_{1 j_{k}}^{T} (x_{1})]^{T}, x_{J} = [x_{1 j_{1}} x_{1 j_{2}} \dots x_{1 j_{k}}]^{T}$ , ${\tilde{x}}_{J} = {[\begin{matrix} x_{1 {\tilde{j}}_{1}} x_{1 {\tilde{j}}_{2}} \dots x_{1 {\tilde{j}}_{r}} \end{matrix}]}^{T}$ .

Problem 1 (the observer-based robust stabilization problem)

For system (1), the control objective of this paper is to design an observer-based non-linear robust controller such that the zero equilibrium of the corresponding closed-loop system is asymptotically stable for all admissible uncertainties satisfying Assumption 1.

Preliminaries

Definition 1 (SOS; Prajna et al., 2012)

A polynomial $f (x)$ in $x \in R^{n}$ is a sum of squares, if there exist polynomials $f_{1} (x), f_{2} (x), \dots, f_{m} (x)$ such that

f (x) = \sum_{i = 1}^{m} f_{i}^{2} (x)

The above SOS condition is equivalent to the exis-tence of a positive semi-definite matrix Q, such that $f (x) = Z^{T} (x) QZ (x)$ , where $Z (x)$ is some properly chosen vector of monomials.

Obviously, $f (x) \in Φ_{sos}$ naturally implies $f (x) \geq 0$ for all $x \in R^{n}$ , whereas the converse is generally not true. Although, for a polynomial, the SOS condition is not necessary for its non-negativity, numerical experiments seem to indicate that the gap between them is not significant, for example, they are equivalent for the quadratic polynomials (Chesi et al., 2009; Prajna et al., 2012). Consequently, changing the non-negativity testing problem into the SOS convex optimization problem is now becoming a preferable way to handle the polynomial non-linear control problems.

To end this section, we list some important lemmas for further use in the subsequent section.

Lemma 1 (Prajna et al., 2004b)

Let $P (x) \in S {[x]}^{m}$ be non-singular for all $x \in R^{n}$ , then

\frac{\partial P (x)}{\partial x_{i}} = - P (x) \frac{\partial P^{- 1} (x)}{\partial x_{i}} P (x), i = 1, 2, \dots, n

Lemma 2 (S-procedure; Boyd et al., 1994)

For $σ_{1} (x) = x^{T} Q_{1} x \geq 0$ , if there is a point $\tilde{x} \in R^{n}$ satisfying $σ_{1} (\tilde{x}) > 0$ , then the following conditions are equivalent:

(i) For all non-zero $x \in R^{n}$ such that $σ_{1} (x) \geq 0$ , $x^{T} Q_{0} x > 0$ ;

(ii) There exists a constant $m \geq 0$ such that $Q_{0} - m Q_{1} > 0$ .

Lemma 3 (Zhou and Khargonekar, 1988)

For real matrices G, H with appropriate dimensions and a constant $ε > 0$ , the following inequality holds:

G^{T} H + H^{T} G \leq ε G^{T} G + ε^{- 1} H^{T} H

Lemma 4

Suppose $α \subset R^{n}$ is a bounded and closed subset, $δ > 0$ is a constant, $Ψ_{1} (x) \in S^{p}$ , $Ψ_{2} (x) \in S^{q}$ and $Ψ_{3} (x) \in R^{p \times q}$ are continuous function matrices in $x \in α$ , $Δ (t) = {(Δ_{ij} (t))}_{g \times h}$ is an unknown time-varying matrix satisfying $‖ Δ (t) ‖_{2} \leq δ$ , $Δ Ψ_{1} (x) \in S^{p}$ is a continuous function matrix in $x \in α$ and $Δ_{ij} (t)$ ( $i = 1, 2, \dots, g$ , $j = 1, 2, \dots, h$ ). The following statements are equivalent:

$Ψ_{1} (x) + Δ Ψ_{1} (x) < 0$ and $Ψ_{2} (x) < 0$ ;

For any $Ψ_{3} (x) \in R^{p \times q}$ , there exists a constant $μ > 0$ such that

[\begin{matrix} Ψ_{1} (x) + Δ Ψ_{1} (x) & * \\ Ψ_{3}^{T} (x) & μ Ψ_{2} (x) \end{matrix}] < 0

Proof. (ii)⇒(i) is trivial.

(i)⇒(ii). The value range of all $Δ_{ij} (t)$ belongs to a bounded and closed subset of $R^{g \times h}$ because $‖ Δ (t) ‖_{2} \leq δ$ .

Denote $Ψ_{4} = Ψ_{3}^{T} (x) {(Ψ_{1} (x) + Δ Ψ_{1} (x))}^{- 1} Ψ_{3} (x) \leq 0$ . Since the eigenvalues of $Ψ_{4}$ and $Ψ_{2} (x)$ are continuous functions in x or $Δ_{ij} (t)$ ( $i = 1, 2, \dots, g$ , $j = 1, 2, \dots, h$ ), when $x \in α$ and $‖ Δ (t) ‖_{2} \leq δ$ , the minimum eigenvalue of $Ψ_{4}$ denoted ${\underline{λ}}_{Ψ_{4}} \leq 0$ and the maximum eigenvalue of $Ψ_{2} (x)$ denoted ${\bar{λ}}_{Ψ_{2}} < 0$ exist and satisfy

{\underline{λ}}_{Ψ_{4}} I \leq Ψ_{4}, Ψ_{2} (x) \leq {\bar{λ}}_{Ψ_{2}} I

Let $μ_{0} = \frac{{\underline{λ}}_{Ψ_{4}}}{{\bar{λ}}_{Ψ_{2}}} \geq 0$ , for any $μ > μ_{0}$ , one has

μ Ψ_{2} (x) < μ_{0} Ψ_{2} (x) \leq {\underline{λ}}_{Ψ_{4}} I \leq Ψ_{3}^{T} (x) {(Ψ_{1} (x) + Δ Ψ_{1} (x))}^{- 1} Ψ_{3} (x)

By Schur complement, the above inequality implies that (ii) holds.□

Controller design based on observer

Before the controller design, we first construct a reduced-order state observer to estimate the immeasurable sub-state $x_{2}$ in system (1).

From system (1), we have

\begin{matrix} {\overset{\cdot}{x}}_{2} = A_{22} (x_{1}) x_{2} + A_{21} (x_{1}) y + B_{2} (x_{1}) u \\ \overset{\cdot}{y} - A_{11} (x_{1}) y - B_{1} (x_{1}) u = A_{12} (x_{1}) x_{2} \end{matrix}

(2)

Let $\bar{u} = A_{21} (x_{1}) y + B_{2} (x_{1}) u$ and $\bar{y} = \overset{\cdot}{y} - A_{11} (x_{1}) y - B_{1} (x_{1}) u$ , the system in (2) can be rewritten as

\begin{matrix} {\overset{\cdot}{x}}_{2} = A_{22} (x_{1}) x_{2} + \bar{u} \\ \bar{y} = A_{12} (x_{1}) x_{2} \end{matrix}

(3)

For the system in (3), we can construct the full-order state observer as follows, by regarding $\bar{u}$ and $\bar{y}$ as input and output, respectively

{\overset{\cdot}{\hat{x}}}_{2} = (A_{22} (x_{1}) - L ({\tilde{x}}_{J}) A_{12} (x_{1})) {\hat{x}}_{2} + L ({\tilde{x}}_{J}) \bar{y} + \bar{u}

(4)

where ${\hat{x}}_{2}$ is the estimate of $x_{2}$ , $L ({\tilde{x}}_{J})$ is the observer gain matrix to be designed to guarantee that the zero equilibrium of

\overset{\cdot}{e} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{\hat{x}}}_{2} = (A_{22} (x_{1}) - L ({\tilde{x}}_{J}) A_{12} (x_{1})) e

is asymptotically stable.

Substituting $\bar{u}$ and $\bar{y}$ into (4) yields

\begin{array}{l} {\dot{\hat{x}}}_{2} = (A_{22} (x_{1}) - L ({\tilde{x}}_{J}) A_{12} (x_{1})) {\hat{x}}_{2} + L ({\tilde{x}}_{J}) \\ (\dot{y} - A_{11} (x_{1}) y - B_{1} (x_{1}) u) + A_{21} (x_{1}) y + B_{2} (x_{1}) u \end{array}

(5)

In order to eliminate $\overset{\cdot}{y}$ in (5), we introduce the variable substitution

z = {\hat{x}}_{2} - L ({\tilde{x}}_{J}) y

(6)

then, from (5) and (6), the reduced-order observer of system (1) becomes

\begin{matrix} \overset{\cdot}{z} = (A_{21} (x_{1}) - L ({\tilde{x}}_{J}) A_{11} (x_{1}) - \sum_{j \in \tilde{J}} \frac{\partial L ({\tilde{x}}_{J})}{\partial x_{1 j}} (A_{11 j} (x_{1}) x_{1})) y + \\ (A_{22} (x_{1}) - L ({\tilde{x}}_{J}) A_{12} (x_{1})) {\hat{x}}_{2} + (B_{2} (x_{1}) - L ({\tilde{x}}_{J}) B_{1} (x_{1})) u \\ {\hat{x}}_{2} = z + L ({\tilde{x}}_{J}) y \end{matrix}

(7)

Remark 2

If $A (x_{1})$ , $B (x_{1})$ and $L ({\tilde{x}}_{J})$ reduce to constant mat-rices, the observer in (7) reduces to the Luenberger observer for the linear time-invariant continuous system (Zheng, 2002). Different from the usual Luenberger observer, the gain matrix $L ({\tilde{x}}_{J})$ is related to the partial state ${\tilde{x}}_{J}$ of system (1), thus cannot be obtained by assigning its eigenvalues. In the following, we will focus on the computation of the observer’s gain matrix and the observer-based controller by using SOS theory.

Based on the observer in (7), the following state feedback control law is adopted

u = K (x_{1}) \hat{x}

(8)

where $\hat{x} = [\begin{matrix} x_{1} \\ {\hat{x}}_{2} \end{matrix}] = x - Qe$ , $Q : [\begin{matrix} 0 \\ I \end{matrix}]$ and $K (x_{1})$ is the controller gain matrix to be designed.

Combining (1), (7) and (8), the closed-loop system can be expressed as

{\overset{\cdot}{x}}_{cl} = (A_{cl} (x_{1}) + G_{cl} (x_{1}) Δ (t) H_{cl} (x_{1})) x_{cl}

(9)

where

\begin{matrix} x_{cl} & = [\begin{matrix} x \\ e \end{matrix}], G_{cl} (x_{1}) = [\begin{matrix} G (x_{1}) \\ 0 \end{matrix}], G (x_{1}) : [\begin{matrix} G_{1} (x_{1}) \\ 0 \end{matrix}], \\ A_{cl} (x_{1}) & = [\begin{matrix} A (x_{1}) + B (x_{1}) K (x_{1}) & - B (x_{1}) K (x_{1}) Q \\ 0 & A_{22} (x_{1}) - L ({\tilde{x}}_{J}) A_{12} (x_{1}) \end{matrix}], \\ H_{cl} (x_{1}) & Ψ = [\begin{matrix} H_{1} (x_{1}) + H_{2} (x_{1}) K (x_{1}) & - H_{2} (x_{1}) K (x_{1}) Q \end{matrix}], \\ H_{1} (x_{1}) & : [\begin{matrix} H_{11} (x_{1}) & H_{12} (x_{1}) \end{matrix}] \end{matrix}

The observer-based robust stabilization problem is to design the reduce-order observer (7) and the state feedback controller (8) such that the zero equilibrium of closed-loop system (9) is asymptotically stable.

Using Lyapunov function method, a stability criterion of the above system is established as follows.

Lemma 5

For a given positive constant $ε$ , if there exists $P (x_{J}) \in S_{+}^{n + q}$ which is continuously differentiable with respect to $x_{J}$ , such that

[\begin{matrix} He (P (x_{J}) A_{cl} (x_{1})) + \\ \sum_{j \in J} \frac{\partial P (x_{J})}{\partial x_{1 j}} ((A_{11 j} (x_{1}) + Δ A_{11 j} (x_{1})) x_{1}) & * & * \\ G_{cl}^{T} (x_{1}) P (x_{J}) & - ε^{- 1} I & * \\ H_{cl} (x_{1}) & 0 & - ε I \end{matrix}] < 0

(10)

then the zero equilibrium of closed-loop system (9) is asymptotically stable.

Proof. Define the Lyapunov function as $V (x_{cl}) = x_{cl}^{T} P (x_{J}) x_{cl}$ . According to Lemma 3, it is obvious that the above conclusion holds.□

On the basis of Lemma 5, we first give the existence conditions of the reduced-order observer (7) and the state feedback controller (8).

Theorem 1

Suppose $x_{1}$ belongs to a bounded and closed subset of $R^{p}$ and $ε > 0$ is a given constant. For $P_{1} (x_{J}) \in S_{+}^{n}$ and $P_{2} ({\tilde{x}}_{J}) \in S_{+}^{q}$ , which are continuously differentiable with respect to $x_{J}$ , the following statements are equivalent:

There exist a constant $μ > 0$ , $P (x_{J}) = diag (P_{1}^{- 1} (x_{J}), μ P_{2} ({\tilde{x}}_{J}))$ , and continuous function matrices $K (x_{1})$ , $L ({\tilde{x}}_{J})$ such that (10) holds.

There exist continuous function matrices $Y (x_{1}) \in R^{m \times n}$ and $W ({\tilde{x}}_{J}) \in R^{q \times p}$ such that

Ξ_{1} (x_{1}) + Δ Ξ_{1} (x_{1}) < 0

(11)

Ξ_{2} (x_{1}) < 0

(12)

Ξ_{1} (x_{1}) : = [\begin{matrix} He (A (x_{1}) P_{1} (x_{J}) + B (x_{1}) Y (x_{1})) - \\ \sum_{j \in J} \frac{\partial P_{1} (x_{J})}{\partial x_{1 j}} (A_{11 j} (x_{1}) x_{1}) & * & * \\ G^{T} (x_{1}) & - ε^{- 1} I & * \\ H_{1} (x_{1}) P_{1} (x_{J}) + H_{2} (x_{1}) Y (x_{1}) & 0 & - ε I \end{matrix}]

\begin{array}{l} Ξ_{2} (x_{1}) : = He (P ({\tilde{x}}_{J}) A_{22} (x_{1}) - W ({\tilde{x}}_{J}) A_{12} (x_{1})) \\ + \sum_{j \in \tilde{J}} \frac{\partial P_{2} ({\tilde{x}}_{J})}{\partial x_{1 j}} (A_{11 j} (x_{1}) x_{1}) \end{array}

Δ Ξ_{1} (x_{1}) : = diag (- \sum_{j \in J} \frac{\partial P_{1} (x_{J})}{\partial x_{1 j}} (Δ A_{11 j} (x_{1}) x_{1}), 0, 0)

Furthermore, if (ii) holds, the zero equilibrium of the closed-loop system in (9) is asymptotically stable, and the controller and observer gain matrices are given by

K (x_{1}) = Y (x_{1}) P_{1}^{- 1} (x_{J})

(13)

L ({\tilde{x}}_{J}) = P_{2}^{- 1} ({\tilde{x}}_{J}) W ({\tilde{x}}_{J})

(14)

Proof. (i) ⇒ (ii). By (10), we have

[\begin{matrix} Ψ_{0} & * & * & * \\ {(- B (x_{1}) K (x_{1}) Q)}^{T} P_{1}^{- 1} (x_{J}) & μ Ψ_{2} (x_{1}) & * & * \\ G^{T} (x_{1}) P_{1}^{- 1} (x_{J}) & 0 & - ε^{- 1} I & * \\ H_{1} (x_{1}) + H_{2} (x_{1}) K (x_{1}) & - H_{2} (x_{1}) K (x_{1}) Q & 0 & - ε I \end{matrix}] < 0

(15)

where

\begin{matrix} Ψ_{0} = He (P_{1}^{- 1} (x_{J}) (A (x_{1}) + B (x_{1}) K (x_{1}))) \\ + \sum_{j \in J} \frac{\partial P_{1}^{- 1} (x_{J})}{\partial x_{1 j}} ((A_{11 j} (x_{1}) + Δ A_{11 j} (x_{1})) x_{1}), \\ Ψ_{2} (x_{1}) = He (P_{2} ({\tilde{x}}_{J}) (A_{22} (x_{1}) - L ({\tilde{x}}_{J}) A_{12} (x_{1}))) \\ + \sum_{j \in \tilde{J}} \frac{\partial P_{2} ({\tilde{x}}_{J})}{\partial x_{1 j}} (A_{11 j} (x_{1}) x_{1}) \end{matrix}

Denoting $W ({\tilde{x}}_{J}) = P_{2} ({\tilde{x}}_{J}) L ({\tilde{x}}_{J})$ , the above inequality im-plies that (12) holds by Schur complement.

Multiplying the matrix of (15) from the left-hand and right-hand sides, respectively, by $[\begin{matrix} P_{1} (x_{J}) & 0 & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & I \end{matrix}]$ and its transpose, and denoting $Y (x_{1}) = K (x_{1}) P_{1} (x_{J})$ , implies that (11) holds by Lemma 1.

(ii) ⇒ (i). Let $K (x_{1}) = Y (x_{1}) P_{1}^{- 1} (x_{J})$ and $L ({\tilde{x}}_{J}) = P_{2}^{- 1} ({\tilde{x}}_{J}) W ({\tilde{x}}_{J})$ , (11) and (12) can be respectively rewritten as

Ψ_{1} (x_{1}) + Δ Ψ_{1} (x_{1}) < 0, and Ψ_{2} (x_{1}) < 0

where

\begin{matrix} Ψ_{1} (x_{1}) = [\begin{matrix} He (P_{1}^{- 1} (x_{J}) (A (x_{1}) + B (x_{1}) K (x_{1}))) + \sum_{j \in J} \frac{\partial P_{1}^{- 1} (x_{J})}{\partial x_{1 j}} (A_{11 j} (x_{1}) x_{1}) & * & * \\ G^{T} (x_{1}) P_{1}^{- 1} (x_{J}) & - ε^{- 1} I & * \\ H_{1} (x_{1}) + H_{2} (x_{1}) K (x_{1}) & 0 & - ε I \end{matrix}], \\ Δ Ψ_{1} (x_{1}) = diag (\sum_{j \in J} \frac{\partial P_{1}^{- 1} (x_{J})}{\partial x_{1 j}} (Δ A_{11 j} (x_{1}) x_{1}), 0, 0) \end{matrix}

According to Lemma 4, there exists a constant $μ > 0$ satisfying

[\begin{matrix} Ψ_{1} (x_{1}) + Δ Ψ_{1} (x_{1}) & * \\ Ψ_{3}^{T} (x_{1}) & μ Ψ_{2} (x_{1}) \end{matrix}] < 0

(16)

where

Ψ_{3} (x_{1}) = {[\begin{matrix} - {(B (x_{1}) K (x_{1}) Q)}^{T} P_{1}^{- 1} (x_{J}) & 0 & - {(H_{2} (x_{1}) K (x_{1}) Q)}^{T} \end{matrix}]}^{T}

After exchanging the rows and columns, (16) yields exactly (10) with $P (x_{J}) = diag (P_{1}^{- 1} (x_{J}),$ $μ P_{2} ({\tilde{x}}_{J}))$ , and by Lemma 5, the zero equilibrium of closed-loop system (9) is asymptotically stable. This completes the proof.□

Remark 3

The deduction of Theorem 1 implies that the controller gain $K (x_{1})$ is only dependent on the inequality in (11), and the observer gain $L ({\tilde{x}}_{J})$ is only related to the inequality in (12). There is no cross variables in (11) and (12), therefore, the solvability of the two inequalities in (11) and (12) is independent with each other. This indicates that the observer-based robust stabilization problem satisfies a so-called separation principle, i.e. the reduced-order observer and the state feedback controller can be designed independently.

Although Theorem 1 provides a simplified condition to test the stability of the closed-loop system (9), the calculation of the gains of state feedback and observer is still difficult. In order to obtain the observer-based state feedback contro-ller, in the next theorem, we restrict $P_{1} (x_{J})$ , $P_{2} ({\tilde{x}}_{J})$ , $Y (x_{1})$ and $W ({\tilde{x}}_{J})$ to polynomial matrices, and then convert the conditions in (11) and (12) into the corresponding SOS feasibility conditions. Before introducing Theorem 2, we define a set as

α_{1} = {x_{1} | | x_{1 i} | \leq δ_{i}, i = 1, 2, \dots, p}

to confine the sub-state $x_{1}$ , where $δ_{i} > 0$ ( $i = 1, 2, \dots, p$ ) are given constants.

Theorem 2

Suppose $x_{1} \in α_{1}$ , for a constant $ε > 0$ and a given polynomial $ς (x_{1}) \in Φ_{sos}$ , if the following two conditions hold, then the observer-based robust stabilization problem is solvable and the state feedback and observer gain matrices can be constructed by (13) and (14), respectively.

There exist $P_{1} (x_{J}) \in S [x_{J}]^{n}$ , $Y (x_{1}) \in R [x_{1}]^{m \times n}, ε_{1} (x_{1})$ $\in Φ_{sos}^{+}$ , and constants $τ_{1} > 0$ , $p_{i} > 0$ ( $i = 1, 2, \dots, k$ ), $q_{i} > 0$ ( $i = 1, 2, \dots, p$ ) such that

v_{1}^{T} (P_{1} (x_{J}) - τ_{1} I) v_{1} - \sum_{i = 1}^{k} p_{i} (δ_{j_{i}}^{2} - x_{1 j_{i}}^{2}) ϑ_{i}^{2} \in Φ_{sos}

(17)

\begin{array}{l} - v_{2}^{T} (Ξ_{1} (x_{1}) + (ς (x_{1}) β (x_{1}) + ε_{1} (x_{1})) Q_{1}) v_{2} - \\ \sum_{i = 1}^{p} q_{i} (δ_{i}^{2} - x_{1 i}^{2}) φ_{i}^{2} \in Φ_{s o s} \end{array}

(18)

v_{3}^{T} [\begin{matrix} ς^{2} (x_{1}) {(v_{21}^{T} v_{21})}^{2} & * \\ G_{J}^{T} (x_{1}) Γ^{T} (x_{J}, v_{21}) & I \end{matrix}] v_{3} \in Φ_{sos}

(19)

where $v_{1} \in R^{n}$ , $v_{2} = {[\begin{matrix} v_{21}^{T} & v_{22}^{T} \end{matrix}]}^{T} \in R^{n + g + h}$ and $v_{21} \in R^{n}$ , $v_{3} \in R^{g + 1}$ , $ϑ_{i}$ ( $i = 1, 2, \dots, k$ ) are any k elements of $v_{1}$ , $φ_{i}$ ( $i = 1, 2, \dots, p$ ) are any p elements of $v_{2}$ , $Γ (x_{J}, v_{21}) = [\begin{matrix} v_{21}^{T} \frac{\partial P_{1} (x_{J})}{\partial x_{1 j_{1}}} v_{21} & v_{21}^{T} \frac{\partial P_{1} (x_{J})}{\partial x_{1 j_{2}}} v_{21} \dots v_{21}^{T} \frac{\partial P_{1} (x_{J})}{\partial x_{1 j_{k}}} v_{21} \end{matrix}],$ $β (x_{1}) = 1 + x_{1}^{T} H_{11}^{T} (x_{1}) H_{11} (x_{1}) x_{1}$ , $Q_{1} = diag (I, 0, 0)$ .

(ii) There exist $P_{2} ({\tilde{x}}_{J}) \in S [{\tilde{x}}_{J}]^{q}$ , $W ({\tilde{x}}_{J}) \in R [{\tilde{x}}_{J}]^{q \times p} ε_{2} (x_{1})$ , $\in Φ_{sos}^{+}$ , and constants $τ_{2} > 0$ , $g_{i} > 0$ ( $i = 1, 2, \dots, r$ ), $h_{i} > 0$ ( $i = 1, 2, \dots, p$ ) such that

v_{4}^{T} (P_{2} ({\tilde{x}}_{J}) - τ_{2} I) v_{4} - \sum_{i = 1}^{r} g_{i} (δ_{{\tilde{j}}_{i}}^{2} - x_{1 {\tilde{j}}_{i}}^{2}) θ_{i}^{2} \in Φ_{sos}

(20)

- v_{5}^{T} (Ξ_{2} (x_{1}) + ε_{2} (x_{1}) I) v_{5} - \sum_{i = 1}^{p} h_{i} (δ_{i}^{2} - x_{1 i}^{2}) ϕ_{i}^{2} \in Φ_{sos}

(21)

where $v_{4} \in R^{q}$ , $v_{5} \in R^{q}$ , $θ = {[\begin{matrix} θ_{1} & θ_{2} & \dots & θ_{r} \end{matrix}]}^{T}$ is a vector that consists of any r elements in $v_{4}$ , $ϕ = [\begin{matrix} ϕ_{1} & ϕ_{2} \end{matrix}$ ${\begin{matrix} \dots & ϕ_{p} \end{matrix}]}^{T}$ is a vector that consists of any p elements in $v_{5}$ .

Proof. Denote $ℓ (x_{1}, v_{2}) = - \sum_{i = 1}^{p} q_{i} (δ_{i}^{2} - x_{1 i}^{2}) φ_{i}^{2} - v_{2}^{T} (Ξ_{1} (x_{1}) + (ς (x_{1}) β (x_{1}) + ε_{1} (x_{1})) Q_{1}) v_{2}$ and it can be decomposed into

\begin{array}{l} ℓ (x_{1}, v_{2}) = H^{T} (x_{1}, v_{2}) G_{0} H (x_{1}, v_{2}) \\ - H^{T} (x_{1}, v_{2}) (\sum_{i = 1}^{p} q_{i} G_{i}) H (x_{1}, v_{2}) \end{array}

(22)

where $G_{i}$ ( $i = 0, 1, \dots, p$ ) are constant matrices and $H (x_{1}, v_{2})$ is a vector of monomials in $x_{1}$ and $v_{2}$ .

Obviously, from (18) and (22), we have

G_{0} - \sum_{i = 1}^{p} q_{i} G_{i} \geq 0

Therefore, according to Lemma 2, $- v_{2}^{T} (Ξ_{1} (x_{1}) + (ε_{1} (x_{1}) + ς (x_{1}) β (x_{1})) Q_{1}) v_{2} \geq 0$ when $x_{1} \in α_{1}$ , i.e.

v_{2}^{T} Ξ_{1} (x_{1}) v_{2} + v_{21}^{T} ς (x_{1}) β (x_{1}) v_{21} < 0

(23)

Note that (19) implies

Γ (x_{J}, v_{21}) G_{J} (x_{1}) G_{J}^{T} (x_{1}) Γ^{T} (x_{J}, v_{21}) \leq {(v_{21}^{T} ς (x_{1}) v_{21})}^{2}

thus

\begin{matrix} v_{2}^{T} Δ Ξ_{1} (x_{1}) v_{2} & = - Γ (x_{J}, v_{21}) G_{J} (x_{1}) Δ (t) H_{11} (x_{1}) x_{1} \\ \leq ‖ Γ (x_{J}, v_{21}) G_{J} (x_{1}) ‖_{2} \times ‖ Δ (t) ‖_{2} \\ \times ‖ H_{11} (x_{1}) x_{1} ‖_{2} \\ \leq v_{21}^{T} ς (x_{1}) v_{21} ‖ H_{11} (x_{1}) x_{1} ‖_{2} \\ \leq v_{21}^{T} ς (x_{1}) β (x_{1}) v_{21} \end{matrix}

(24)

Combining (23) and (24), implies that $Ξ_{1} (x_{1}) + Δ Ξ_{1} (x_{1}) < 0$ . Similarly, by (17), (20) and (21), one has $P_{1} (x_{J}) > 0$ , $P_{2} ({\tilde{x}}_{J}) > 0$ and $Ξ_{2} (x_{1}) < 0$ , respectively.

By Theorem 1 and Lemma 5, the observer-based robust stabilization problem is solvable and the state feedback and observer gain matrices can be given by (13) and (14). This completes the proof.□

Remark 4

If $A (x_{1})$ , $Δ A (x_{1})$ , $B (x_{1})$ , $Δ B (x_{1})$ , $L ({\tilde{x}}_{J})$ , $K (x_{1})$ and $P (x_{J})$ are constant matrices, the SOS constraints (17-21) will naturally reduce to the LMI conditions for the observer-based linear robust controller

u = K \hat{x}

(25)

which has been widely studied in the linear control field (e.g. Boyd et al., 1994; Lin et al., 2008). In other words, Theorem 2 provides an extension of the classical linear-robust-control theory to a class of uncertain polynomial systems.

Remark 5

In Theorem 2, both the elements of $P_{1} (x_{J})$ and $P_{2} ({\tilde{x}}_{J})$ should be polynomials whose highest degree is even ( $0, 2, 4 \dots$ ). This is the precondition to ensure that (17) and (20) hold.

Remark 6

If the condition (i) of above theorem holds with $ς (x_{1}) = 0$ , then $Γ (x_{J}, v_{21}) G_{J} (x_{1}) = 0$ . From (24), it means that there exists $P_{1} (x_{J}) \in S [x_{J}]^{n}$ such that $v_{2}^{T} Δ Ξ_{1} (x_{1}) v_{2} = 0$ , so the treatment of $v_{2}^{T} Δ Ξ_{1} (x_{1}) v_{2}$ does not have any conser-vativeness.

Remark 7

In the conditions (20) and (21), the relationships of $v_{4}$ and $θ$ , $v_{5}$ and $ϕ$ are given for only $q \geq p$ . When $r \leq q < p$ , $θ$ is still a vector that consists of any r elements in $v_{4}$ , and $v_{5}$ is a vector that consists of any q elements in $ϕ$ . When $q < r$ , $v_{4}$ and $v_{5}$ should be vectors that consist of any q elements in $θ$ and $ϕ$ , respectively.

Finally, from the proof of Theorem 2, it is easy to know that $ς (x_{1})$ plays an important role in the settlement of the uncertain term $Δ Ξ_{1} (x_{1})$ . Therefore, we present the following corollary to optimize $ς (x_{1})$ which can reduce the conservativeness of the above SOS conditions.

Corollary 1

Suppose $x_{1} \in α_{1}$ , for a constant $ε > 0$ and a given polynomial $ς (x_{1}) \in Φ_{sos}$ , if the following optimization conditions hold

$min ρ \geq 0$ , $s . t .$ (17), (20), (21), (26), (27)

- v_{2}^{T} (Ξ_{1} (x_{1}) + (ρ ς (x_{1}) β (x_{1}) + ε_{1} (x_{1})) Q_{1}) v_{2} - \sum_{i = 1}^{p} q_{i} (δ_{i}^{2} - x_{1 i}^{2}) φ_{i}^{2} \in Φ_{s o s}

(26)

v_{3}^{T} [\begin{matrix} ρ ς^{2} (x_{1}) {(v_{21}^{T} v_{21})}^{2} & * \\ G_{J}^{T} (x_{1}) Γ^{T} (x_{J}, v_{21}) & ρ I \end{matrix}] v_{3} \in Φ_{sos}

(27)

then the observer-based robust stabilization problem is solvable, and the state feedback and observer gain matrices are given by (13) and (14), respectively.

Proof. By (27) and Schur complement, one has

Γ (x_{J}, v_{21}) G_{J} (x_{1}) G_{J}^{T} (x_{1}) Γ^{T} (x_{J}, v_{21}) \leq {(v_{21}^{T} ρ ς (x_{1}) v_{21})}^{2}

The rest of the proof can be obtained by following the proof of Theorem 2.□

Remark 8

If the optimal value of $ρ$ is zero, we can obtain the same conclusion as in Remark 6.

Numerical examples

In this section, two examples are provided to illustrate the feasibility and effectiveness of the proposed observer-based robust stabilization approach. They are dealt with by Matlab toolbox SOSTOOLS.

Example 1: Rossler chaotic system

Consider the Rossler chaotic system (Lam, 2010) with the external disturbance w, whose dynamics can be described by

\begin{array}{l} [\begin{matrix} {\dot{x}}_{11} \\ {\dot{x}}_{12} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} 0 & - 1 & - 1 \\ 1 & a & 0 \\ b & 0 & x_{11} - c \end{matrix}] [\begin{matrix} x_{11} \\ x_{12} \\ x_{2} \end{matrix}] \\ + [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] u + [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] w, x_{1} = [\begin{matrix} x_{11} \\ x_{12} \end{matrix}] \end{array}

(28)

where $a = 0.34$ , $b = 0.4$ , $c = 4.5$ .

Assume that $y = x_{1}$ and a varies by $\pm 50 %$ of its nominal value ( $| Δ a | \leq 0.5 a$ ). Let $G_{1} (x_{1})$ $= {[\begin{matrix} 0 & a \end{matrix}]}^{T}$ , $Δ (t) = Δ a (0.5 a)^{- 1}$ , $H_{11} (x_{1}) = [\begin{matrix} 0 & 0.5 \end{matrix}]$ , $H_{12} (x_{1}) = H_{2} (x_{1}) = 0$ . In addition, for the above system, $J = {2}$ and $\tilde{J} = \emptyset$ , so $x_{J} = x_{12}$ , $P_{2} ({\tilde{x}}_{J})$ and $W ({\tilde{x}}_{J})$ are constant matrices.

In this example, a two-degree $P_{1} (x_{J})$ and a one-degree $Y (x_{1})$ are designed, and the related parameters are given by $ε = 10^{5}$ , $τ_{1} = 0.01$ , $τ_{2} = 1$ , $ε_{1} (x_{1}) = 10^{- 3}$ , $ε_{2} (x_{1}) = 0.1, δ_{1}$ = $δ_{2} = 5$ , $ς (x_{1}) = 10^{- 5}$ , $p_{2} = q_{1} = q_{2} = h_{1} = 1$ . According to Theorem 2, the observer-based robust stabilization problem of system (28) is solvable and the corresponding solutions are given in Appendix A.

Setting the initial condition at $x_{cl 0} = {[\begin{matrix} - 1 & 0.1 & 1 & 0 \end{matrix}]}^{T}$ . To show the robustness of the designed controller, the simulations are performed for the following three different situations (the resulting closed-loop response curves are in Figures 1 –4, and the curves of control input and external disturbance are in Figures 5 and 6 respectively):

Nominal system (NS), in which, $Δ = 0$ and $w = 0$ ;

Uncertain system (US), in which, $Δ = 1$ and $w = 0$ ;

Uncertain and disturbed system (UADS), in which, $Δ = 1$ and

w = {\begin{matrix} 1 + \sin (1.5 t) - 0.5 \cos t, & 0 < t \leq 10 \\ 0, & t > 10 \end{matrix}

The six figures suggest that the zero equilibrium of the closed-loop system is asymptotically stable for all three kinds of simulations, and the effects of uncertainty and disturbance or their mixture on the system gradually vanish as time increases. After 20s, the uncertain system with or without disturbance behaves like the nominal one. The simulations show that the non-linear stabilization controller proposed in this paper has good robustness against the parameter uncertainty as well as external disturbance.

Example 2: Flexible satellite attitude control

The flexible satellite is made up of a main rigid body and some flexible appendages, whose attitude is described by two sets of equations: the kinematic equations and the dynamic equations. The kinematic equations of the satellite determine the attitude of the rigid body and can be written in terms of the Rodrigues parameters (Shuster, 1993) as follows

\overset{\cdot}{σ} = ρ (σ) ω, ρ (σ) : \frac{1}{2} [\begin{matrix} 1 + σ_{1}^{2} & σ_{1} σ_{2} - σ_{3} & σ_{2} + σ_{1} σ_{3} \\ σ_{3} + σ_{1} σ_{2} & 1 + σ_{2}^{2} & σ_{2} σ_{3} - σ_{1} \\ σ_{1} σ_{3} - σ_{2} & σ_{2} σ_{3} + σ_{1} & 1 + σ_{3}^{2} \end{matrix}]

(29)

where $σ = {[\begin{matrix} σ_{1} & σ_{2} & σ_{3} \end{matrix}]}^{T}$ is the Rodrigues parameters and $ω = {[\begin{matrix} ω_{1} & ω_{2} & ω_{3} \end{matrix}]}^{T}$ is the angular velocity in the body fixed frame.

The dynamic equations of flexible satellite describe both the overall body motion and essential motions of flexible appendages, which can usually be modelled as (Guan et al., 2007; Sidi, 1997; Zhang, 1998)

\begin{matrix} I_{s} \overset{\cdot}{ω} + S (ω) I_{s} ω + F_{s} \overset{\cdot\cdot}{η} = T_{c} + T_{d} \\ \overset{\cdot\cdot}{η} + 2 ξ Ω \overset{\cdot}{η} + Ω^{2} η + F_{s}^{T} \overset{\cdot}{ω} = 0 \end{matrix}

(30)

where $S (ω) = [ω \times]$ is the skew symmetric matrix, $T_{d}$ is the external disturbance torque, $T_{c}$ is the control torque, $η = {[\begin{matrix} η_{1} & η_{2} & \dots & η_{N} \end{matrix}]}^{T}$ is the modal coordinates, N is the number of the flexible modes considered, and the meanings of the known system parameters are as follows: $I_{s}$ is the symmetric inertia matrix of the whole structure; ξ, $Ω$ the damping matrix and the natural frequency matrix; $F_{s}$ the coupling matrix between the flexible and rigid structures.

As only the attitude angles and angular velocity can be measured directly, combining (29) with (30), and denoting $R_{s} = (I_{s} - F_{s} F_{s}^{T})^{- 1}$ , $γ = \overset{\cdot}{η} + F_{s}^{T} ω$ , then the state-space des-cription of the flexible satellite attitude model can be formulated as

\begin{matrix} \overset{\cdot}{x} = A (x_{1}) x + Bu + Bw \\ y = x_{1} \end{matrix}

(31)

x = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], x_{1} : [\begin{matrix} σ \\ ω \end{matrix}], x_{2} : [\begin{matrix} η \\ γ \end{matrix}], u = T_{c}, w = T_{d}

\begin{array}{l} A (x_{1}) = [\begin{matrix} 0 & ρ (σ) & 0 & 0 \\ 0 & R_{s} (- S (ω) I_{s} - 2 F_{s} ξ Ω F_{s}^{T}) & R_{s} F_{s} Ω^{2} & 2 R_{s} F_{s} ξ Ω \\ 0 & - F_{s}^{T} & 0 & I \\ 0 & 2 ξ Ω F_{s}^{T} & - Ω^{2} & - 2 ξ Ω \end{matrix}] \\ B = {[\begin{matrix} 0 & R_{s}^{T} & 0 & 0 \end{matrix}]}^{T} \end{array}

Supposing there exists a perturbation in $I_{s}$ caused by temperature difference, fuel consumption, solar radiation pressure and so on, we have $x_{J} = {\tilde{x}}_{J} = σ$ , $G_{1} (x_{1}) = {[\begin{matrix} 0 & 0.5 R_{s}^{T} \end{matrix}]}^{T}$ , $H_{11} (x_{1}) =$ $[\begin{matrix} 0 & - S (ω) I_{s} - 2 F_{s} ξ Ω F_{s}^{T} \end{matrix}]$ , $H_{12} (x_{1}) = [F_{s} Ω^{2} 2 F_{s} ξ Ω]$ and $H_{2} (x_{1}) = I$ .

The proposed approach is applied to system (31) with specific data taken from Gennaro (2003). Considering only the first two modes of the flexible appendages, the related parameters are as follows

\begin{array}{l} I_{s} = [\begin{matrix} 350 & 3 & 4 \\ 3 & 280 & 10 \\ 4 & 10 & 190 \end{matrix}], F_{s} = [\begin{matrix} 6.45637 & - 1.25619 \\ 1.27814 & 0.91756 \\ 2.15629 & - 1.67264 \end{matrix}], \\ ξ = [\begin{matrix} 5.607 & 0 \\ 0 & 8.62 \end{matrix}] \times 10^{- 3}, Ω = [\begin{matrix} 0.7681 & 0 \\ 0 & 1.1038 \end{matrix}] \end{array}

In addition, the amplitude of the control torque is restricted to be no more than 20 Nm.

For the sake of simplicity in calculation, $P_{1} (x_{J})$ and $P_{2} ({\tilde{x}}_{J})$ are designed as constant matrices, $Y (x_{1})$ and $W ({\tilde{x}}_{J})$ are designed as two-degree polynomial matrices, and the other parameters are chosen as $ε = 5 \times 10^{6}$ , $τ_{1} = 10^{- 5}$ , $τ_{2} = 0.01$ , $ε_{1} (x_{1}) = ε_{2} (x_{1}) = 10^{- 3}$ , $q_{i} = 1$ ( $i = 1, 2, \dots, 6$ ), $δ_{i}$ $= 2$ ( $i = 1, 2, 3$ ), $δ_{3 + i} = h_{i} = 1$ ( $i = 1, 2, 3$ ). By Theorem 2, an observer-based non-linear robust controller is obtained. Due to space limitations, herein we only give the first row of $L ({\tilde{x}}_{J}) =$ ${(l_{ij} ({\tilde{x}}_{J}))}_{4 \times 6}$ and the first column of $K (x_{1}) = {(k_{ij} (x_{1}))}_{3 \times 10}$ (Appendix B). Besides, as a comparison of the proposed approach, an observer-based linear robust controller (25) is designed by approximately linearizing the flexible satellite attitude system (31) (Appendix C).

Suppose the flexible satellite is required to finish the large angle attitude stabilization with an principal angle of $160^{\circ}$ , i.e. the corresponding initial Rodrigues parameters is $σ (0) = [\begin{matrix} - 1.5157 & 4.5471 \end{matrix}$ ${- 3.0314]}^{T}$ . Moreover, we let $ω (0) = 0 rad / s$ , $η (0) = 0$ , $γ (0) = 0$ , $\hat{η} (0) = {[\begin{matrix} 1.5 & - 0.1 \end{matrix}]}^{T}$ and $\hat{γ} (0) = {[\begin{matrix} - 1.5 & 0.1 \end{matrix}]}^{T}$ . To show the advantages of the designed approach, we respectively impose the linear controller (25) and the non-linear controller (8) on the uncertain and disturbed system (31), in which, $Δ = I$ and

w = {\begin{matrix} {[\begin{matrix} 0.1 \cos (0.5 t) & 0.5 & - \sin (0.1 t) \end{matrix}]}^{T}, \begin{matrix} \begin{matrix} 0 < t \leq 200 \end{matrix} \end{matrix} \\ 0, \begin{matrix} t > 200 \end{matrix} \end{matrix}

The corresponding simulation results are presented in Figures 7 –14.

As seen from (ii) of Figures 7 –12, the two flexible modes and their first derivatives are observed with a satisfactory level, the non-linear controller is robust to the system parameter perturbations and suppresses the flexible vibration and external disturbance in the process of large angle attitude stabilization as well. From Figure 13, it can be found that both the linear control input and the non-linear control input meet the saturation constraint. However, the buffet of such linear control input is serious and thus it can hardly be realized in the engineering practice. By comparing (i) and (ii) in Figures 7 –12, we conclude that although the linear controller has better convergence property in the control of attitude angle and angular velocity, the non-linear controller can achieve better control effect with low overshoot, small vibration and stationary process. To sum up, the proposed observer-based non-linear robust control strategy is effective and has remarkable superiority over the classical linear robust control method.

Conclusions

This paper presents an observer-based robust stabilization approach for a class of uncertain polynomial systems. The major advantages of the method lie in that the design of the state feedback and the reduced-order observer satisfies a separation principle, and the robust stabilization problem is converted into the SOS convex optimization problem, thus effectively avoiding the computing difficulties that generally exist in the synthesis of non-linear control systems. Moreover, the designed observer-based non-linear robust controller is of easy implementation in engineering because it is only the polynomial or rational function of the system states. Simulation results show that the proposed approach not only guarantees the stability of the closed-loop system, but also possesses good robustness against the norm-bounded time-varying uncertainties as well as external disturbance.

Footnotes

Appendix

Conflict of interest statement

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to thank the Fundamental Research Funds for the Central Universities (No. 20720150177) and the National Natural Science Foundation of China (No. 61374037) for supporting this research.

References

Asemani

Majd

(2013) A robust H_∞ observer-based controller design for uncertain T–S fuzzy systems with unknown premise variables via LMI. Fuzzy Sets and Systems 212: 21–40.

Ataei

Wang

(2012) Non-linear control of an uncertain hypersonic aircraft model using robust sum-of-squares method. IET Control Theory and Applications 6(2): 203–215.

Boyd

Ghaoui

Feron

. (1994) Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: Society for In-dustrial and Applied Mathematics.

Chae

Nguang

(2014) SOS based robust H_∞ fuzzy dynamic output feedback control of nonlinear networked control systems. IEEE Transactions on Cybernetics 44(7): 1204–1213.

Chesi

Garulli

Tesi

. (2009) Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems. Berlin: Springer-Verlag.

Gennaro

(2003) Output stabilization of flexible spacecraft with active vibration suppression. IEEE Transactions on Aerospace and Electronic Systems 39(3): 747–759.

Guan

Liu

(2007) Variable structure attitude control of flexible satellite. Control Theory & Applications 24(3): 480–484.

Henrion

Lasserre

(2003) Gloptipoly: global optimization over polynomials with Matlab and SeDuMi. ACM Transactions on Mathematical Software 29(2): 165–194.

Ichihara

(2009) Optimal control for polynomial systems using matrix sum of squares relaxations. IEEE Transactions on Automatic Control 54(5): 1048–1053.

10.

Lam

(2010) Chaotic synchronisation using output/full state-feedback polynomial controller. IET Control Theory and Applications 4(11): 2285–2292.

11.

Lin

Wang

Lee

. (2008) Observer-based H_∞ fuzzy control design for T-S fuzzy systems with state delays. Automatica 44(3): 868–874.

12.

Lofberg

(2004) Yalmip: a toolbox for modeling and optimization in Matlab. In: The Computer Aided Control Systems Design Conference, Taipei, Taiwan, pp. 284–289.

13.

Zheng

Cai

. (2012). Mixed H₂/H_∞ control of spacecraft large angle attitude maneuvers. Journal of Harbin Institute of Technology 44(11): 105–111.

14.

Narendra

(2008) Switched control of satellites for global stabilization and local performance: a sum of squares approach. In: The 2008 American Control Conference, Washington, DC, pp. 2987–2992.

15.

Nguang

(1996) Robust non-linear H_∞ output feedback control. IEEE Transactions on Automatic Control 41(7): 1003–1007.

16.

Nguang

Saat

Krug

(2011) Static output feedback controller design for uncertain polynomial systems: an iterative sum of squares approach. IET Control Theory and Applications 5(9): 1079–1084.

17.

Prajna

Papachristodoulou

Seiler

. (2012) SOSTOOLS: sum of squares optimization toolbox for Matlab: user’s guide version 2.0. Available: http://www.cds.caltech.edu/sostool/.

18.

Prajna

Papachristodoulou

(2004b) Non-linear control synthesis by sum of squares optimization: a Lyapunov-based approach. In: The 5th Asian Control Conference, Melbourne, Australia, pp. 157–165.

19.

Prajna

Parrilo

Rantzer

(2004a) Non-linear control syn-thesis by convex optimization. IEEE Transactions on Automatic Control 49(2): 310–314.

20.

Shuster

(1993) A survey of attitude representations. Journal of the Astronautical Sciences 41(4): 439–517.

21.

Sidi

(1997) Spacecraft Dynamics and Control. Cambridge, UK: Cambridge University Press.

22.

Xie

Wang

(2007) Synthesis of discrete-time nonlinear systems: a SOS approach. In: The 2007 American Control Conference, New York, pp. 4829–4834.

23.

Xie

Wang

(2009) Simultaneous stabilization and robust control of polynomial non-linear systems using SOS techniques. IEEE Transactions on Automatic Control 54(8): 1892–1897.

24.

Zhang

(1998) Satellite Orbital Dynamics and Attitude Control. Beijing: Beijing University of Aeronautics and Astronautics Press.

25.

Zheng

(2002) Linear System Theory. Beijing: Tsinghua University Press.

26.

Zheng

(2009) Non-linear H_∞ control design with axisymmetric spacecraft control. Journal of Guidance, Control, and Dynamics 32(3): 850–859.

27.

Zheng

(2011) Generalized non-linear H_∞ synthesis condition with its numerically efficient solution. International Journal of Robust and Non-linear Control 21(18): 2079–2100.

28.

Zhou

Khargonekar

(1988) Robust stabilization of linear systems with norm-bounded time-varying uncertainty. Systems & Control Letters 10(1): 17–20.