Finite-time sliding mode control synthesis under explicit output constraint

doi:10.1016/j.automatica.2015.11.037

Automatica

Volume 65, March 2016, Pages 111-114

https://doi.org/10.1016/j.automatica.2015.11.037 Get rights and content

Abstract

This work deals with the input–output finite-time stabilization problem for a class of nonlinear systems by employing sliding mode control (SMC) approach. A suitable SMC law is designed to ensure that the state trajectories can be driven onto the specified sliding surface during the assigned finite-time interval. Moreover, some parameters-dependent sufficient conditions are derived such that the input–output finite-time stability (IO-FTS) during both reaching phase and sliding motion phase are attained. Simulation results are provided to illustrate the proposed approach.

Introduction

Input–output finite-time stability (IO-FTS) of a system concerns the quantitative behavior of the output variables over an assigned finite-time interval. That is, a system is said to be input–output finite-time stable if its output (weighted) norm does not exceed an assigned threshold $β$ (i.e., an explicit output constraint condition) during the specified time interval $[0, T]$ (Amato et al., 2010, Amato et al., 2012, Amato et al., 2014, Song et al., 2015).

Sliding mode control (SMC), as an effective robust control strategy for systems subjected to parameter uncertainties and external disturbances, has attracted considerable attention, see Basin and Rodríguez-Ramírez (2011), Chen, Niu, and Zou (2013), Wu, Su, and Shi (2012), Xia, Lu, and Zhu (2013) and the references therein. However, it should be pointed out that, in almost all aforementioned works on SMC, the behavior of sliding mode dynamics was considered within a sufficiently long (in principle infinite) time interval and there was no any constraint on transient dynamics. Apparently, this case is not always true in some practical applications, such as when considering IO-FTS, in which the specified finite-time interval $T$ and the explicit output constraint scalar $β$ should be taken into account simultaneously. Specifically, the following two questions should be answered for the input–output finite-time stabilization via SMC:

(1)
For any specified finite (possibly short) interval $T$ , is it possible to design a SMC law such that the state trajectories can be driven onto the sliding surface in a finite-time $T^{*}$ with $T^{*} \leq T$ ; if so, how to design?
(2)
For an assigned threshold $β$ , how to guarantee that the output (weighted) norm does not exceed $β$ during both reaching phase and sliding motion phase?

In this technical communique, we focus on addressing the input–output finite-time stabilization problem for a class of uncertain nonlinear systems by using SMC approach. The above two questions will be answered in the following design (see Theorem 1, Theorem 2, respectively).

Notation

$λ_{\max} (\cdot)$ denotes the maximum eigenvalue of the corresponding matrix. $‖ \cdot ‖$ and $| \cdot |$ denote, respectively, the Euclidean norm and 1-norm of a vector (sum of absolute values) or its induced matrix norm. For a real matrix, $A^{T}$ represents the transpose of $A$ , and we denote $He {A} = A + A^{T}$ . The shorthand “ $diag {\cdot}$ ” denotes a block diagonal matrix. In symmetric block matrices, the symbol “ $⋆$ ” is used as an ellipsis for terms induced for symmetry.

Section snippets

System descriptions and definition

Consider the nonlinear system described by $Σ : {\begin{matrix} \dot{x} (t) = f (x (t), w (t)) + B u (t), \\ y (t) = C x (t), \end{matrix}$ where $x (t) \in R^{n}$ is the state; $u (t) \in R^{m}$ is the input; $w (t) \in R^{r}$ is the disturbance input. The unknown function $f (x (t), w (t))$ satisfies the conic sector constraint (ElBsat & Yaz, 2013): $‖ f (x (t), w (t)) - [\bar{A} (t) x (t) + F w (t)] ‖ \leq ‖ {\bar{A}}_{r} (t) x (t) + F_{r} w (t) ‖,$ where $\bar{A} (t) = A + A_{Δ} (t)$ and ${\bar{A}}_{r} (t) = A_{r} + A_{r Δ} (t)$ . Matrices $A$ , $B$ , $C$ , $F$ , $A_{r}$ , and $F_{r}$ are assumed to be known. $A_{Δ} (t)$ and $A_{r Δ} (t)$ are parameter uncertainties satisfying the following norm-bounded

Sliding surface design

In this work, our aim is to cope with the IO-FTS problem of SMC for nonlinear system $\bar{Σ}$ , wherein the sliding function $s (t)$ is chosen as $s (t) = L x (t) - \int_{0}^{t} L (A + B K) x (τ) d τ,$ where the matrix $K$ will be designed later (in Theorem 2), and the matrix $L$ is chosen so that $L B$ is nonsingular, which can be attained by choosing $L = B^{T} X$ with $X > 0$ , since $B$ is assumed to be of full column rank.

Reachability with $T^{*} \leq T$

In this section, a suitable sliding mode controller is designed to drive the state trajectories onto the specified sliding surface $s (t) = 0$ in a finite time $T^{*}$ and then are maintained there over the rest time interval $[T^{*}, T]$ .

Theorem 1

Consider system ( $Σ$ ) in (1), (2), (3). The reachability of the specified sliding surface (8) can be ensured in a finite time $T^{*}$ with $T^{*} \leq T$ by the SMC law: $u (t) = K x (t) - η (t) sgn (s (t)),$ where the robust term $η (t)$ is given by $η (t) = ς + v ‖ w (t) ‖ + d ‖ x (t) ‖,$ in which $v = ‖ {(B^{T} X B)}^{- 1} B^{T} X ‖ (‖ F ‖ + ‖ F_{r} ‖$

IO-FTS over reaching phase $[0, T^{*}]$

In the sequel, it will be shown that the closed-loop system is IO-FTS during reaching phase, i.e., in the time interval $[0, T^{*}]$ . By substituting SMC law (9) into (4), the closed-loop system over $[0, T^{*}]$ is obtained by: ${\tilde{Σ}}_{[0, T^{*}]} : {\begin{matrix} \dot{x} = \hat{A} x + F w + g (x, w) - B η_{s}, \\ y = C x, \end{matrix}$ where $\hat{A} = \bar{A} + B K$ and $η_{s} ≜ η \cdot sgn (s)$ .

Lemma 1

Given a feasible scalar $α > 0$ . The resulting closed-loop system ${\tilde{Σ}}_{[0, T^{*}]}$ in (15) is IO-FTS with respect to $(β, [0, T^{*}], R, W)$ , if there exist matrices $K \in R^{m \times n}$ , $W > 0$ and $P > 0$ , and scalars $ϵ > 0$ and $γ > 0$ , satisfying the

Illustrative example

Consider a closed-loop Chua’s circuit in form of (4) with $A = [\begin{matrix} - α_{c} (1 + b) & α_{c} & 0 \\ 1 & - 1 & 1 \\ 0 & - β_{c} & - μ \end{matrix}], B = [\begin{matrix} 2 \\ 5 \\ 2 \end{matrix}], F = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}],$ $C = {[\begin{matrix} 0.41 \\ 0.41 \\ 0.83 \end{matrix}]}^{T}, M = M_{r} = [\begin{matrix} 0.03 \\ 0.02 \\ 0.04 \end{matrix}], N = {[\begin{matrix} 0.05 \\ 0.02 \\ 0.02 \end{matrix}]}^{T},$ $g (x, w) = {[\begin{matrix} 0.5 α_{c} (a - b) (| x_{1} + 1 | - | x_{1} - 1 |) & 0 & 0 \end{matrix}]}^{T} .$ We could rewrite $g (x, w)$ in form of (5) with $A_{r} = diag {α_{c} (a - b), 0, 0}$ and $F_{r} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T}$ . The parameters, i.e., $α_{c} = 9.1$ , $β_{c} = 16.5811$ , $μ = 0.138083$ , $a = - 1.3659$ , and $b = - 0.7408$ , are taken from the example in ElBsat and Yaz (2013).

Let $β = 11$ , $T = 2$ , $R = 0.5 I$ , and $δ = 0.5$ . We choose $L = B^{T} X$ with $X = I$ . With $x (0) = {[\begin{matrix} - 1.2 & 2.1 & 4.7 \end{matrix}]}^{T}$ , the

Conclusions

This work has investigated the input–output finite-time stabilization for a class of nonlinear system via SMC approach. The IO-FTS during both reaching stage and sliding motion stage have been analyzed, which is just the key feature of IO-FTS via SMC.

References (9)

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Input–output finite time stabilization of linear systems
Automatica
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Input–output finite-time stability of linear systems: necessary and sufficient conditions
IEEE Transactions on Automatic Control
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F. Amato et al.
Input–output finite-time stabilisation of linear systems with input constraints
IET Control Theory & Applications
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M.V. Basin et al.
Sliding-mode filtering design for linear systems with unmeasured states
IEEE Transactions on Industrial Electronics
(2011)

There are more references available in the full text version of this article.

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^☆: This work was supported in part by NNSF of China under grants 61273073, 61374107, and 61203051. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Michael V. Basin under the direction of Editor André L. Tits.

¹: Corresponding author. Tel.: +86 2164253794; fax: +86 2164253794.

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Technical communiqueFinite-time sliding mode control synthesis under explicit output constraint☆

Abstract

Introduction

Section snippets

System descriptions and definition

Sliding surface design

Reachability with T∗≤T

IO-FTS over reaching phase [0,T∗]

Illustrative example

Conclusions

Input–output finite time stabilization of linear systems

Automatica

Input–output finite-time stability of linear systems: necessary and sufficient conditions

IEEE Transactions on Automatic Control

Input–output finite-time stabilisation of linear systems with input constraints

IET Control Theory & Applications

Sliding-mode filtering design for linear systems with unmeasured states

IEEE Transactions on Industrial Electronics

Technical communique
Finite-time sliding mode control synthesis under explicit output constraint☆

Reachability with $T^{*} \leq T$

IO-FTS over reaching phase $[0, T^{*}]$