Adaptive finite-time control for non-linear delay systems via non-fragile state feedback

Abstract

The problem of adaptive finite-time control is addressed in this paper for a class of non-linear delay systems. First, the concepts of adaptive finite-time stability and adaptive finite-time boundedness are defined, respectively. Then, by resorting to the Lyapunov–Krasovskii functional technique, some new delay-dependent criteria guaranteeing adaptive finite-time boundedness and adaptive finite-time stability are developed, respectively. An explicit expression for the desired non-fragile state feedback controller is also presented. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed results.

Keywords

Adaptive finite-time stability adaptive finite-time boundedness non-fragile state feedback controller time delay

Introduction

The concept of finite-time stability, which was first introduced to the control field in the 1960s (Dorato,1961), is that, given a bound on the initial condition, the system’s state does not exceed a certain threshold during a specified time interval. Unlike Lyapunov asymptotical stability defined on an infinite time interval, finite-time stability shows the behaviour of the system over a fixed finite-time interval. In fact, in many practical applications, such as chemical processes, manufacturing systems, and power systems, the main concern is the transient response of system states over a finite-time horizon. In Amato et al. (1999), the authors further extended the definition of finite-time stability to the concept of finite-time boundedness. Since then, the problems of finite-time stability and finite-time boundedness have been extensively studied for various systems (Amato et al., 2001, 2006; Du et al., 2009; Feng et al., 2005). Note that many meaningful results on finite-time stability and finite-time boundedness have been developed in the past decade. However, it should be pointed out that, to date, the systems considered in the previous references do not refer to unknown parameters. When the systems contain unknown parameters, the above-mentioned methods may not guarantee system finite-time stability and finite-time boundedness. Hence, it is necessary to develop some new conditions to solve the problem of finite-time stability and finite-time boundedness for systems with unknown parameters.

In addition, time delays often present in many practical systems such as chemical processes and networked control systems as well as long transmission lines in pneumatic systems. The existence of time delays may degrade the system performance, cause oscillations, and lead to instability. In the field of engineering, in the past several decades, considerable attention has been paid to the study of the stability and finite-time stability of time-delay systems. Some useful results have been reported in the literature (Hou et al., 2012a, 2012b; Wang et al., 2009a, 2009b, 2009c; Wu et al., 2009; Zhang and Yu, 2009; Zong et al., 2008, 2011).

In the above references, there is an assumption that the controller will be implemented exactly. However, in practice, the controller exhibits a certain degree of fragility due to inaccuracies or uncertainties in the implementation of a controller design. Hence, some researchers have developed non-fragile controller design algorithms in common linear or non-linear uncertain systems (Hu et al., 2013; Liu et al., 2014). To date, to the best of the authors’ knowledge, the problems of adaptive finite-time control for non-linear delay systems have not been fully investigated, which motivated us to carry out the present study.

In this paper, attention is focused on solving the adaptive finite-time control problem for a class of non-linear delay systems. Because of the existence of time delays and unknown parameters, this problem is apparently hard to tackle. The main contributions in this paper can be illustrated from the following three aspects. First, the definitions of adaptive finite-time boundedness and adaptive finite-time stability are expanded for the considered systems, respectively. Second, a new non-fragile adaptive finite-time controller is proposed for the controlled plant with uncertain parameters and external disturbances. Third, on the basis of this non-fragile adaptive controller, some new delay-dependent criteria guaranteeing adaptive finite-time boundedness and adaptive finite-time stability are developed for the non-linear delay systems. By virtue of the cone complementary linearization (CCL) technique and the linear matrix inequality approach, the gain of the desired non-fragile adaptive controller can be obtained. Finally, a numerical example is proposed to demonstrate the effectiveness of the obtained results.

Notation. The notation of this paper is standard. For symmetric matrices P, the notation $P > 0$ (respectively, $P \geq 0$ ) means that the matrix is positive-definite (respectively, semi-definite); T denotes the transpose; $R^{n}$ represents the n-dimensional Euclidean space; I is the identity matrix with appropriate dimension; ∥·∥ denotes the Euclidean norm of a vector and its induced norm of a matrix; * stands for the symmetric part in a block symmetric matrix; matrices, if not explicitly stated, are assumed to have compatible dimensions.

Problem formulation and preliminaries

Consider the following non-linear delay system

\begin{matrix} \overset{\cdot}{x} (t) = & Ax (t) + A_{d} x (t - τ) + B (u (t) + φ^{T} (x) θ) \\ + Ff (t, x (t)) + E ω (t), \end{matrix}

(1a)

\overset{\cdot}{ω} (t) = G ω (t),

(1b)

x (s) = ϕ (s), s \in [- τ, 0]

where $x (t) \in R^{n}$ is the state, $u (t) \in R^{m}$ is the controlled input, $ω (t) \in R^{p}$ is the disturbance generated by the exogenous system, equation (1b), $τ$ is the time delay, and $ϕ (s)$ is the initial value function. $φ (x) \in R^{1 \times m}$ is a known function. $θ$ is an unknown constant. $A \in R^{n \times n}$ , $A_{d} \in R^{n \times n}$ , $B \in R^{n \times m}$ , $E \in R^{n \times p}$ , $F \in R^{n \times q}$ , $G \in R^{p \times p}$ are constant matrices of appropriate dimensions. $f (t, x (t)) \in R^{q}$ is a non-linear function, which satisfies the following assumption.

Assumption 1. $(i) f (t, 0) = 0,$

(ii) | | f (t, x_{1} (t)) - f (t, x_{2} (t)) | | \leq | | U (x_{1} (t) - x_{2} (t)) | |

(2)

where U is a given constant weighting matrix.

For system (1a) with $ω (t) = 0$ , we first present the following definition.

Definition 1. System (1a) with $ω (t) = 0$ is said to be adaptive finite-time stable (AFTS) with respect to $(c_{1}$ , $c_{2}$ , $c_{3}$ , T, $R_{1}$ , $R_{3})$ if

\begin{matrix} \max_{- τ \leq s \leq 0} ϕ^{T} (s) R_{1} ϕ (s) \leq c_{1}; \\ R_{3} θ^{2} \leq c_{2} \end{matrix}} \Rightarrow x^{T} (t) R_{1} x (t) < c_{3}, \forall t \in [0, T]

where $c_{3} > c_{1} > 0$ , $T > 0$ is a given time-constant, $c_{2} > 0$ and $R_{3} > 0$ are constants, and $R_{1} > 0$ is a positive-definite matrix.

The general idea of finite-time boundedness presents the boundedness of the state of systems over a finite-time interval given both some initial conditions and an external disturbance working on the systems (Amato and Ariola 2005; Amato et al., 1999, 2001, 2006). With the aid of these definitions, in the sequel, we present the definition of adaptive finite-time boundedness for the case of non-linear delay systems.

Definition 2. System (1) is said to be adaptive finite-time bounded (AFTB) with respect to $(c_{1}$ , $c_{0}$ , $c_{2}$ , $c_{3}$ , T, $R_{1}$ , $R_{2}$ , $R_{3})$ if

\begin{matrix} \max_{- τ \leq s \leq 0} ϕ^{T} (s) R_{1} ϕ (s) \leq c_{1}; \\ R_{3} θ^{2} \leq c_{2}; \\ ω^{T} (0) R_{2} ω (0) \leq c_{0} \end{matrix}} \Rightarrow x^{T} (t) R_{1} x (t) < c_{3}, \forall t \in [0, T]

where $c_{3} > c_{1} > 0$ , $T > 0$ is a given time-constant, $c_{0} > 0$ , $c_{2} > 0$ , and $R_{3} > 0$ are constants, and $R_{1} > 0$ and $R_{2} > 0$ are positive-definite matrices.

Remark 1. When systems do not have unknown parameters, the definitions of adaptive finite-time boundedness and adaptive finite-time stability are reduced to those in Amato et al. (2001).

Lemma 1. (Petersen and Hollot, 1986) Given matrices $Q,$ D, E and F of appropriate dimensions withQ symmetrical, then

Q + DFE + E^{T} F^{T} D^{T} \leq 0

for all F satisfying $F^{T} F \leq I,$ if and only if there exists $ε > 0$ such that

Q + ε D D^{T} + ε^{- 1} E^{T} E \leq 0

The non-fragile adaptive controller is designed as

u = (K + Δ K) x - φ^{T} (x) \hat{θ},

(3a)

\overset{\cdot}{\hat{θ}} = ρ φ (x) B^{T} P_{1} x + \frac{β}{4} \hat{θ}, \hat{θ} (0) = 0

(3b)

where $\hat{θ}$ is the estimation of $θ$ , $Δ K = M Θ (t) N$ with M and N being known constant matrices with appropriate dimensions, and $Θ^{T} (t) Θ (t) \leq I$ . $P_{1} > 0$ , $ρ > 0$ , and $β > 0$ . The feedback connection between system (1) and controller (3) leads to the following closed-loop system

\overset{\cdot}{x} (t) = \bar{A} x + A_{d} x (t - τ) + Ff (t, x (t)) + E ω (t) + B φ^{T} (x) \tilde{θ},

(4a)

\overset{\cdot}{ω} (t) = G ω (t)

(4b)

where

\bar{A} = A + B (K + Δ K), \tilde{θ} = θ - \hat{θ}

Given six positive scalars $c_{1}$ , $c_{0}$ , $c_{2}$ , $c_{3}$ , T, and $R_{3}$ , and two positive-definite symmetric matrices $R_{1},$ $R_{2}$ , the purpose of this paper is to find a non-fragile adaptive state feedback controller (3) such that system (4) is AFTB with respect to $(c_{1}$ , $c_{0}$ , $c_{2}$ , $c_{3}$ T, $R_{1}$ , $R_{2}$ , $R_{3}$ ), and system (4a) is AFTS with respect to $(c_{1}$ , $c_{2}$ , $c_{3}$ , T, $R_{1}$ , $R_{3})$ when $ω (t) = 0$ .

Main results

Adaptive finite-time boundedness and stability

In the sequel, the adaptive finite-time boundedness and finite-time stabilization ( $ω (t) = 0$ ) will be presented for system (1), respectively.

Theorem 1. System (4) is AFTB with respect to $(c_{1}$ , $c_{0}$ , $c_{2}$ , $c_{3}$ T, $R_{1}$ , $R_{2}$ , $R_{3})$ if there exist positive scalars $α$ , $ρ$ , $λ$ and positive-definite matrices $P_{1} \in R^{n \times n}$ , $P_{2} \in R^{n \times n}$ , $P_{3} \in R^{p \times p}$ , such that the following inequalities hold

Ω = [\begin{matrix} Ω_{1} & P_{1} A_{d} & P_{1} F & P_{1} E \\ * & - e^{α τ} P_{2} & 0 & 0 \\ * & * & - \frac{1}{λ^{2}} I & 0 \\ * & * & * & Ω_{2} \end{matrix}] < 0,

(5a)

λ_{1} < \frac{c_{3} λ_{\min} ({\tilde{P}}_{1}) - \frac{ρ^{- 1}}{8 α} c_{2} β (e^{α T} - 1)}{c_{1} + c_{0} + c_{2} + \frac{c_{1}}{α} (e^{α τ} - 1) e^{α T}}

(5b)

where

\begin{matrix} Ω_{1} = P_{1} \bar{A} + {\bar{A}}^{T} P_{1} + P_{2} + \frac{1}{λ^{2}} U^{T} U - α P_{1}, \\ Ω_{2} = P_{3} G + G^{T} P_{3} - α P_{3}, \\ λ_{1} = \max {λ_{\max} ({\tilde{P}}_{1}), ρ^{- 1}, λ_{\max} ({\tilde{P}}_{3}), λ_{\max} ({\tilde{P}}_{2})} \end{matrix}

(6)

and ${\tilde{P}}_{1} = R_{1}^{- 1 / 2} P_{1} R_{1}^{- 1 / 2},$ ${\tilde{P}}_{2} = R_{1}^{- 1 / 2} P_{2} R_{1}^{- 1 / 2},$ ${\tilde{P}}_{3} = R_{2}^{- 1 / 2} P_{3} R_{2}^{- 1 / 2},$ and $\tilde{ρ} = R_{3} ρ .$

Proof. Choose a Lyapunov–Krasovskii functional candidate as

V (t) = V_{1} (t) + V_{2} (t) + V_{3} (t) + V_{4} (t) + V_{5} (t)

(7)

where

V_{1} (t) = x^{T} (t) P_{1} x (t),

V_{2} (t) = \int_{t - τ}^{t} e^{α (t - s)} x^{T} (s) P_{2} x (s) d s, α \geq β > 0,

V_{3} (t) = ρ^{- 1} \tilde{θ} {(t)}^{2},

V_{4} (t) = \frac{1}{λ^{2}} \int_{0}^{t} e^{α (t - s)} (| | Ux (s) | |^{2} - | | f (s, x (s)) | |^{2}) d s,

V_{5} (t) = ω^{T} (t) P_{3} ω (t)

Calculating the derivative of $V (t)$ along the trajectory of systems (1a) and (1b), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) = & {\overset{\cdot}{x}}^{T} (t) P_{1} \overset{\cdot}{x} (t) + x^{T} (t) P_{1} \overset{\cdot}{x} (t) \\ = & {(\bar{A} x (t) + A_{d} x (t - τ) + Ff (t, x (t)) + E ω (t) + B φ^{T} (x) \tilde{θ} (t))}^{T} P_{1} x (t) \\ + x^{T} (t) P_{1} (\bar{A} x (t) + A_{d} x (t - τ) + Ff (t, x (t)) + E ω (t) + B φ^{T} (x) \tilde{θ} (t)) \\ - α V_{1} (t) + α V_{1} (t) \\ = & x^{T} (t) (P_{1} \bar{A} + {\bar{A}}^{T} P_{1} - α P_{1}) x (t) \\ + x^{T} (t) P_{1} A_{d} x (t - τ) + x^{T} (t) P_{1} E ω (t) + x^{T} (t) P_{1} B φ^{T} (x) \tilde{θ} (t) \\ + x^{T} (t - τ) A_{d}^{T} P_{1} x (t) + ω^{T} (t) E^{T} P_{1} x (t) + {\tilde{θ}}^{T} (t) φ (x) B^{T} P_{1} x (t) \\ + f^{T} (t, x (t)) F^{T} P_{1} x (t) + x^{T} (t) P_{1} Ff (t, x (t)) + α V_{1} (t), \end{matrix}

(8a)

{\overset{\cdot}{V}}_{2} (t) = x^{T} (t) P_{2} x (t) - e^{α τ} x^{T} (t - τ) P_{2} x (t - τ) + α V_{2} (t),

(8b)

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) & = - 2 ρ^{- 1} \tilde{θ} (t) \overset{\cdot}{\hat{θ}} (t) \\ = - 2 ρ^{- 1} \tilde{θ} (t) (ρ φ (x) B^{T} P_{1} x (t) + \frac{β}{4} \hat{θ} (t)) \\ = - 2 \tilde{θ} (t) φ (x) B^{T} P_{1} x (t) - \frac{β}{2} ρ^{- 1} \tilde{θ} (t) \hat{θ} (t) \\ = - 2 \tilde{θ} (t) φ (x) B^{T} P_{1} x (t) + \frac{β}{2} ρ^{- 1} \tilde{θ} {(t)}^{2} - \frac{β}{2} ρ^{- 1} \tilde{θ} θ (t) \end{matrix}

(8c)

{\overset{\cdot}{V}}_{4} (t) = \frac{1}{λ^{2}} (| | Ux (t) | |^{2} - | | f (t, x (t)) | |^{2}) + α V_{4} (t),

(8d)

\begin{matrix} {\overset{\cdot}{V}}_{5} (t) & = ω^{T} (t) (G^{T} P_{3} + P_{3} G) ω (t) - α V_{5} (t) + α V_{5} (t) \\ = ω^{T} (t) (G^{T} P_{3} + P_{3} G - α P_{3}) ω (t) + α V_{5} (t) \end{matrix}

(8e)

Note that

- \frac{β}{2} ρ^{- 1} \tilde{θ} (t) θ (t) \leq \frac{β}{2} ρ^{- 1} \tilde{θ} {(t)}^{2} + \frac{β}{8} ρ^{- 1} θ^{2}

(9)

Combining equations (8c) and (9), yields

{\overset{\cdot}{V}}_{3} (t) \leq - 2 {\tilde{θ}}^{T} (t) φ (x) B^{T} P_{1} x (t) + β V_{3} (t) + \frac{β}{8} θ^{T} (t) ρ^{- 1} θ (t)

(10)

In view of equations (5a), (8), and (10), there holds

\overset{\cdot}{V} (t) \leq η^{T} (t) Ω η (t) + α V (t) + \bar{θ} \leq α V (t) + \bar{θ}

(11)

where

η (t) = {[\begin{matrix} x^{T} (t) & x^{T} (t - τ) & f^{T} (t, x (t)) & ω^{T} (t) \end{matrix}]}^{T},

\bar{θ} = \frac{β}{8} ρ^{- 1} θ^{2}

Solving the inequality (11), we obtain

V (t) \leq (\bar{θ} \int_{0}^{t} e^{- α s} d s + V (0)) e^{α t}

(12)

For any $t \in [0, T]$ , we have

V (t) \leq \frac{\bar{θ}}{α} (e^{α T} - 1) + V (0) e^{α T}

(13)

By equation (7), we derive

\begin{matrix} V (t) & \geq x^{T} (t) P_{1} x (t) = x^{T} (t) R_{1}^{\frac{1}{2}} R_{1}^{- \frac{1}{2}} P_{1} R_{1}^{- \frac{1}{2}} R_{1}^{\frac{1}{2}} x (t) \\ \geq λ_{\min} (R_{1}^{- \frac{1}{2}} P_{1} R_{1}^{- \frac{1}{2}}) x^{T} (t) R_{1} x (t) \\ = λ_{\min} ({\tilde{P}}_{1}) x^{T} (t) R_{1} x (t) \end{matrix}

(14)

and

\begin{matrix} V (0) & = x^{T} (0) P_{1} x (0) + \int_{- τ}^{0} e^{- α s} x^{T} (s) P_{2} x (s) d s + ω^{T} (0) P_{3} ω (0) + ρ^{- 1} {\tilde{θ}}^{2} (0) \\ \leq λ_{\max} ({\tilde{P}}_{1}) x^{T} (0) R_{1} x (0) + ρ^{- 1} R_{3}^{- 1} R_{3} θ^{2} + λ_{\max} ({\tilde{P}}_{3}) ω^{T} (0) R_{2} ω (0) \\ + λ_{\max} ({\tilde{P}}_{2}) \max_{- τ \leq s \leq 0} ϕ (s) R_{1} ϕ (s) \int_{- τ}^{0} e^{- α s} d s \\ = λ_{\max} ({\tilde{P}}_{1}) x^{T} (0) R_{1} x (0) + ρ^{- 1} R_{3}^{- 1} R_{3} θ^{2} + λ_{\max} ({\tilde{P}}_{3}) ω^{T} (0) R_{2} ω (0) \\ + λ_{\max} ({\tilde{P}}_{2}) \max_{- τ \leq s \leq 0} ϕ (s) R_{1} ϕ (s) \frac{1}{α} (e^{α τ} - 1) \\ \leq λ_{\max} ({\tilde{P}}_{1}) c_{1} + ρ^{- 1} R_{3}^{- 1} c_{2} + λ_{\max} ({\tilde{P}}_{3}) c_{0} + λ_{\max} ({\tilde{P}}_{2}) c_{1} \frac{1}{α} (e^{α τ} - 1) \\ \leq \max {λ_{\max} ({\tilde{P}}_{1}), {\tilde{ρ}}^{- 1}, λ_{\max} ({\tilde{P}}_{3}), λ_{\max} ({\tilde{P}}_{2})} (c_{1} + c_{2} + c_{0} + \frac{c_{1}}{α} (e^{α τ} - 1)) \end{matrix}

(15)

Then, from equations (13)–(15), the following inequality holds

\begin{matrix} x^{T} (t) R_{1} x (t) \leq \frac{\frac{\bar{θ}}{α} (e^{α T} - 1) + V (0) e^{α T}}{λ_{\min} ({\tilde{P}}_{1})} \\ \leq \frac{\max {λ_{\max} ({\tilde{P}}_{1}), {\tilde{ρ}}^{- 1}, λ_{\max} ({\tilde{P}}_{3}), λ_{\max} ({\tilde{P}}_{2})} (c_{1} + c_{2} + c_{0} + \frac{c_{1}}{α} (e^{α τ} - 1)) e^{α T}}{λ_{\min} ({\tilde{P}}_{1})} \\ + \frac{ρ^{- 1} c_{2} β (e^{α T} - 1)}{8 α λ_{\min} ({\tilde{P}}_{1})} \\ = \frac{λ_{1} (c_{1} + c_{2} + c_{0} + \frac{c_{1}}{α} (e^{α τ} - 1)) e^{α T}}{λ_{\min} ({\tilde{P}}_{1})} + \frac{ρ^{- 1} c_{2} β (e^{α T} - 1)}{8 α λ_{\min} ({\tilde{P}}_{1})} \end{matrix}

which, combining with equation (5b), further implies that

x^{T} (t) R_{1} x (t) < c_{3}

(16)

This completes the proof of Theorem 1.□

Remark 2. If the time delay in system (1) is the time-varying delay $τ (t)$ and it satisfies $0 \leq \underline{τ} \leq τ (t) \leq \bar{τ}$ , then in the proof of Theorem 1, the Lyapunov function will be related not only to the time-varying delay $τ (t)$ , but also to $\underline{τ}$ and $\bar{τ}$ , and the corresponding results contain $\underline{τ}$ and $\bar{τ}$ , which may have less conservativeness. In the future study, we will focus upon the problem of adaptive finite-time stability and adaptive finite-time boundedness for non-linear switched systems with time-varying delay and mode-dependent time-varying delays step-by-step.

When $ω (t) = 0,$ choose a Lyapunov–Krasovskii functional candidate as

V (t) = V_{1} (t) + V_{2} (t) + V_{3} (t) + V_{4} (t)

(17)

For system (1a) with $ω (t) = 0$ , we derive the following result.

Theorem 2. System (1) ( $ω (t) = 0$ ) is AFTS with respect to $(c_{1}$ , $c_{2}$ , $c_{3}$ , T, $R_{1}$ , $R_{3})$ if there exist positive scalars $α$ , $λ$ , $ρ$ , positive-definite matrices $P_{1} \in R^{n \times n}$ and $P_{2} \in R^{n \times n}$ , such that the following inequalities hold

[\begin{matrix} Ω_{1} & P_{1} A_{d} & P_{1} F \\ * & - e^{α τ} P_{2} & 0 \\ * & * & - \frac{1}{λ^{2}} I \end{matrix}] < 0,

(18a)

λ_{2} < \frac{c_{3} λ_{\min} ({\tilde{P}}_{1}) - \frac{ρ^{- 1}}{8 α} c_{2} β (e^{α T} - 1)}{c_{1} + c_{2} + \frac{c_{1}}{α} (e^{α τ} - 1) e^{α T}}

(18b)

where $Ω_{1}$ and $λ_{2}$ are defined as equation (6).

Non-fragile adaptive finite-time controller design

Now, we are in a position to present a solution to the problem of non-fragile adaptive finite-time control for system (4). By Theorem 1, we can derive the following result.

Theorem 3. System (4) is AFTB with respect to $(c_{1}$ , $c_{0}$ , $c_{2}$ , $c_{3}$ T, $R_{1}$ , $R_{2}$ , $R_{3})$ if there exist positive scalars $α$ , $τ$ , $ρ$ , $β$ , positive-definite matrices $Q_{1}$ , $P_{2}$ , $P_{3},$ and a matrix $\bar{K}$ such that the following inequalities hold

[\begin{matrix} \tilde{A} & A_{d} Q_{1} & F Q_{1} & E Q_{1} & U Q_{1} & Q_{1} & Q_{1}^{T} N^{T} \\ * & - e^{α τ} P_{2} & 0 & 0 & 0 & 0 & 0 \\ * & * & - \frac{1}{λ^{2}} I & 0 & 0 & 0 & 0 \\ * & * & * & P_{3} G + G^{T} P_{3} - α P_{3} & 0 & 0 & 0 \\ * & * & * & * & - λ^{2} I & 0 & 0 \\ * & * & * & * & * & - P_{2}^{- 1} & 0 \\ * & * & * & * & * & * & - ε I \end{matrix}] < 0

(19)

λ_{2} < \frac{c_{3} λ_{\max} ({\tilde{Q}}_{1}) - \frac{ρ^{- 1}}{8 α} c_{2} β (e^{α T} - 1)}{c_{1} + c_{0} + c_{2} + \frac{c_{1}}{α} (e^{α τ} - 1) e^{α T}}

(20)

where

\tilde{A} = A Q_{1} + Q_{1} A^{T} + B \bar{K} + {\bar{K}}^{T} B^{T} - α Q_{1} + ε M M^{T},

λ_{2} = \max {λ_{\min} ({\tilde{Q}}_{1}), ρ^{- 1}, λ_{\max} ({\tilde{P}}_{3}), λ_{\max} ({\tilde{P}}_{2})}

and ${\tilde{Q}}_{1} = {\tilde{P}}_{1}^{- 1} .$ Moreover, the gain of the non-fragile adaptive controller is given by $K = \bar{K} Q_{1}^{- 1} .$

Proof. Multiplying equation (5a) by $diag {Q_{1}, I, I, I}$ and $diag {Q_{1}, I, I, I}$ on the left and on the right, respectively, leads to

[\begin{matrix} Ω_{3} & A_{d} Q_{1} & F Q_{1} & E Q_{1} \\ * & - e^{α τ} P & 0 & 0 \\ * & * & - \frac{1}{λ^{2}} I & 0 \\ * & * & * & Ω_{2} \end{matrix}] < 0

(21)

where

\begin{matrix} Ω_{3} = & A Q_{1} + Q_{1} A^{T} + B \bar{K} + {\bar{K}}^{T} B^{T} - α Q_{1} \\ + Q_{1} P_{2} Q_{1} + \frac{1}{λ^{- 2}} Q_{1} U^{T} U Q_{1} \\ + M Θ (t) N Q_{1} + Q_{1} N^{T} Θ^{T} (t) M^{T}, \bar{K} = K Q_{1}, Q_{1} = P_{1}^{- 1} \end{matrix}

Based on the Schur complement formula, the following inequality holds

\begin{matrix} [\begin{matrix} Ω_{4} & A_{d} Q_{1} & F Q_{1} & E Q_{1} & U Q_{1} & Q_{1} \\ * & - e^{α τ} & 0 & 0 & 0 & 0 \\ * & * & - \frac{1}{λ^{2}} I & 0 & 0 & 0 \\ * & * & * & P_{3} G + G^{T} P_{3} - α P_{3} & 0 & 0 \\ * & * & * & * & - λ^{2} I & 0 \\ * & * & * & * & * & - P_{2}^{- 1} \end{matrix}] \\ + [\begin{matrix} M \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] Θ (t) [\begin{matrix} N Q_{1} & 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ + [\begin{matrix} Q_{1} N^{T} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] Θ^{T} (t) [\begin{matrix} M^{T} & 0 & 0 & 0 & 0 & 0 \end{matrix}] < 0 \end{matrix}

(22)

where

Ω_{4} = A Q_{1} + Q_{1} A^{T} + B \bar{K} + {\bar{K}}^{T} B^{T} - α Q_{1}

Using Lemma 1 and the Schur complement formula, equation (19) can be obtained. Replacing $λ_{\max} ({\tilde{P}}_{1})$ , $λ_{\min} ({\tilde{P}}_{1})$ with $λ_{\min} ({\tilde{Q}}_{1})$ , $λ_{\max} ({\tilde{Q}}_{1})$ , respectively, yields equation (20). This completes the proof of Theorem 3.□

It is noted that equation (19) is not a linear matrix inequality due to the existence of the terms $P_{2}$ and $P_{2}^{- 1}$ . In order to obtain the desired adaptive state feedback controller (3), in the sequel, we resort to the CCL method (Ghaoui et al., 1997).

We introduce the new variable $S_{1},$ such that $P_{2}^{- 1} \geq S_{1} .$ Then we have the following result.

Theorem 4. System (4) is AFTB with respect to $(c_{1}$ , $c_{0}$ , $c_{2}$ , $c_{3}$ T, $R_{1}$ , $R_{2}$ , $R_{3})$ if there exist positive scalars $α$ , $τ$ , $ρ$ , $β$ , positive-definite matrices $Q_{1}$ , $P_{2}$ , $P_{3},$ and a matrix $\bar{K}$ such that equation (20) and the following inequalities hold

[\begin{matrix} \tilde{A} & A_{d} Q_{1} & F Q_{1} & E Q_{1} & U Q_{1} & Q_{1} & Q_{1}^{T} N^{T} \\ * & - e^{α τ} P_{2} & 0 & 0 & 0 & 0 & 0 \\ * & * & - \frac{1}{λ^{2}} I & 0 & 0 & 0 & 0 \\ * & * & * & P_{3} G + G^{T} P_{3} - α P_{3} & 0 & 0 & 0 \\ * & * & * & * & - λ^{2} I & 0 & 0 \\ * & * & * & * & * & - S_{1} & 0 \\ * & * & * & * & * & * & - ε I \end{matrix}] < 0,

(23)

P_{2}^{- 1} \geq S_{1}

(24)

Moreover, the non-fragile adaptive controller gain is given by equation (3) with

K = \bar{K} Q_{1}^{- 1}

(25)

The inequality (24) shows that the conditions in Theorem 4 are not strict linear matrix inequalities. With the help of the CCL method, the non-convex feasibility problem formulated by equations (20), (23), and (24) can be transformed into the following non-linear minimization problem

Minimize $Tr (S_{1} S_{2} + P_{2} Q_{2})$

subject to equations (20) and (23) and

[\begin{matrix} S_{2} & I \\ I & Q_{2} \end{matrix}] \geq 0, [\begin{matrix} S_{1} & I \\ I & S_{2} \end{matrix}] \geq 0, [\begin{matrix} P_{2} & I \\ I & Q_{2} \end{matrix}] \geq 0

(26)

If the solution of the above minimization problem is $2 n$ , then system (4) is AFTB with respect to $(c_{1}$ , $c_{0}$ , $c_{2}$ , $c_{3}$ T, $R_{1}$ , $R_{2}$ , $R_{3})$ via the controller (3) with $K = \bar{K} Q_{1}^{- 1}$ .

Based on Theorems 2 and 4, we have the following result.

Theorem 5. System (1a) ( $ω (t) = 0$ ) is AFTS with respect to $(c_{1}$ , $c_{2}$ , $c_{3}$ , T, $R_{1}$ , $R_{3})$ if there exist positive scalars $α$ , $λ$ , $ρ$ , and positive-definite matrices $P_{1} \in R^{n \times n}$ and $P_{2} \in R^{n \times n}$ , such that equation (26) and the following inequalities hold

[\begin{matrix} \tilde{A} & A_{d} Q_{1} & F Q_{1} & U Q_{1} & Q_{1} & Q_{1}^{T} N^{T} \\ * & - e^{α τ} P_{2} & 0 & 0 & 0 & 0 \\ * & * & - \frac{1}{λ^{2}} I & 0 & 0 & 0 \\ * & * & * & - λ^{2} I & 0 & 0 \\ * & * & * & * & - S_{1} & 0 \\ * & * & * & * & * & - ε I \end{matrix}] < 0,

(27a)

λ_{2} < \frac{c_{3} λ_{\max} ({\tilde{Q}}_{1}) - \frac{ρ^{- 1}}{8 α} c_{2} β (e^{α T} - 1)}{c_{1} + c_{2} + \frac{c_{1}}{α} (e^{α τ} - 1) e^{α T}}

(27b)

Moreover, the non-fragile adaptive controller gain is given by equation (3) with equation (25).

Remark 3. In the process of solving the controller, the terms $P_{2}$ and $P_{2}^{- 1}$ often occur simultaneously, which leads to the non-linear conditions. In order to get the controller gain, the non-linear matrix inequality must be transformed to a linear matrix inequality. In fact, the term $P_{2}^{- 1}$ can be replaced by $- 2 I + P_{2}$ based on $(P_{2}^{- 1} - I) P_{2} (P_{2}^{- 1} - I) \geq 0$ . However, from Zong et al. (2009), we know that using the CCL method produces less conservative results than the above inequality. Hence, in this paper, we employ the CCL method to transform the non-convex feasibility problem to an optimization problem, though it is more difficult and complex than other inequality techniques.

A numerical example

In this section, an example is presented to show the effectiveness of the main results in this paper.

Consider system (1) with the following parameters

A = [\begin{matrix} - 0.02 & - 0.01 \\ 0 & 0.07 \end{matrix}], A_{d} = [\begin{matrix} 0.1 & - 0.1 \\ 0.3 & 0.1 \end{matrix}], B = [\begin{matrix} - 1 \\ 0.8 \end{matrix}],

E = [\begin{matrix} 0 & - 1.2 \\ 1.2 & 0 \end{matrix}], F = [\begin{matrix} 0.1 & - 0.5 \\ 0.2 & 0.1 \end{matrix}], N = [\begin{matrix} - 0.01 & 0.05 \\ - 0.02 & 0.01 \end{matrix}],

U = [\begin{matrix} - 0.1 & 0.5 \\ - 0.2 & 0.1 \end{matrix}], G = [\begin{matrix} - 0.1 & 0.5 \\ - 0.2 & - 0.1 \end{matrix}], M = [\begin{matrix} - 0.5 & 0.2 \end{matrix}]

Let $c_{0} = 0.1$ , $c_{1} = 0.1$ , $c_{2} = 10,$ $c_{3} = 20$ , $R_{1} = I$ , $R_{2} = I$ , $R_{3} = 0.1$ , $β = 0.01$ , $T = 3$ , $ρ = 1$ , $λ = 0.1$ , $φ (x) = \sin (x_{1} + x_{2})$ .

Then, applying Theorem 4 and solving the corresponding matrix inequalities, we can obtain the parameters of the non-fragile adaptive controller (3) as follows

K = [\begin{matrix} 2.9431 & 0.6640 \end{matrix}]

We choose the initial functions $ϕ (s) = {[\begin{matrix} 1 & 0.2 \end{matrix}]}^{T}$ and $ω (0) = {[\begin{matrix} 0.2 & 0.2 \end{matrix}]}^{T}$ . Figure 1 presents the state $x (t)$ , and Figure 2 shows the state trajectory of system (4) over 0–3 s. From Figure 3 we can see that system (4) is AFTB with respect to (0.1, 0.1, 10, 20, 3, I, I, 0.1).

Conclusions

In this paper, the problem of adaptive finite-time control via non-fragile state feedback has been studied for a class of delay systems. The concepts of adaptive finite-time stability and adaptive finite-time boundedness have been defined, respectively. Some sufficient criteria have been developed to solve the problem of adaptive finite-time boundedness, adaptive finite-time stability and adaptive finite-time non-fragile state feedback control. A feasible non-fragile state feedback controller has also been given. Finally, a numerical example has been provided to show the effectiveness of the results.

Footnotes

Funding

This work was supported in part by the National Natural Science Foundation of China (grant numbers 61304059, 61304153, and 61403227), the Natural Science Foundation of Shandong Province (grant number ZR2013FQ016), the Specialized Research Fund for the Doctoral Program of Higher Education (grant number 20133705120004), and the Dr startup funds of Qufu Normal University (grant number 2012024), the Youth Foundation of Humanities and Social Sciences from the Ministry of Education of China (No. (15YJCZH204)) and the Tianjin City High School Science and Technology Fund Planning Project (No. (20141001)).

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