Sampled-data static output-feedback control for nonlinear systems in T–S form via descriptor redundancy

doi:10.1016/j.neucom.2018.06.048

Neurocomputing

Volume 318, 27 November 2018, Pages 1-6

https://doi.org/10.1016/j.neucom.2018.06.048 Get rights and content

Abstract

This paper deals with a static output-feedback stabilization problem within a sampled-data framework for nonlinear systems in Takagi–Sugeno form. We investigate this problem on the basis of the descriptor-redundancy scheme. The resulting features are that: (i) the sampled-data synthesis does not involve (even approximate) discrete-time models of nonlinear systems; (ii) the sampled-data stability is analyzed in the continuous-time Lyapunov sense; and (iii) the design algorithm consists of a single-stage linear matrix inequality problem without linear matrix equality constraints.

Introduction

From the viewpoint of feedback control, a typical foible of Takagi–Sugeno (T–S) fuzzy methodologies is that time-continuous measurements of a state are sometimes onerous in practical applications [1]. An immediate substitute can be to take measurements in a sampling manner, which yields time-hybrid closed-loop systems. The model is converted into a time-homogeneous one to design a controller in the discrete-time or continuous-time domain. For the former, the direct discrete-time design—to determine a discrete-time controller for the (approximate) discrete-time model of a continuous-time plant, and then to emulate it—can be utilized [2], [3]. Although this technique is promising, since the discrete-time modeling of nonlinear systems is necessarily only approximate, further consideration is required to guarantee the sampled-data closed-loop stability [4], [5], [6]. For the latter, the sampled-data dynamics is transformed into a continuous-time model with input delay, by which a sampled-data fuzzy tracking controller was derived [7]. It was extended to the robustification against time delays and uncertainties [8], the fuzzy sampled-data filtering [9], and the robust finite-time non-fragile stabilization against stochastic actuator faults [10].

Another vulnerability in feedback control is that it is not that common for all state variables to be measurable. The static output feedback (SOF) as an alternative of state feedback is known to be a fascinating yet challenging strategy in T–S fuzzy model-based design. The fundamental burden on the fuzzy SOF problem is caused by the fact that the Lyapunov inequality formulated in terms of linear matrix inequalities (LMI) is non-convex. Solutions to this difficulty mainly fall into two categories: the multiple-stage LMI-based method [11] and the single-stage method entailing additional constraints [12], [13], [14], [15], [16].

We note that extensive efforts have been researched for the single-stage method. In [12], the Lyapunov inequality is convexified by similarity-transforming the output matrices, as long as they are common and have no uncertainties. In [13], LMI conditions are proposed for the SOF problem with fuzzy measured outputs without uncertainties, based on the “P-problem” (or “W-problem”) of [17]. In [14], it is addressed that the work in [13] implicitly requires common output matrices and no uncertainties therein. It is worthwhile to mention that [15] reveals that the P-problem-based approach may admit distinct output matrices. The uncertain fuzzy output case is analyzed in [16]. Recently, a novel SOF synthesis was proposed in [18], where the design is accomplished in a single LMI step excluding the linear matrix equality (LME) constraint.

This paper continues efforts to investigate the SOF fuzzy controller design problem, especially in the sampled-data framework that few SOF studies have focused on. We pay attention to the descriptor redundancy as a new tool for the concerned problem in that it can simultaneously represent a combined set of differential (i.e., plant) and algebraic (i.e., SOF and sampled-data) equations into a single model [19], thereby reducing the excessive design involved in existing methods [20]. The contributions of this paper are that: (i) differing from the previous works, this sampled-data design does not require approximate [4] nor exact [6] discrete-time models of T–S fuzzy systems; (ii) the sampled-data stability is analyzed in the continuous-time Lyapunov sense, while the input delay does not appear; and (iii) the design conditions are derived in the format of a single-stage LMI problem without accompanying LME constraints.

Notation: x and x_kT stand for x(t) and x(kT). He{X} and * are the shorthands for $X + X^{T}$ and the transposed element in the symmetric positions.

Section snippets

Preliminaries

A class of physical nonlinear systems can be modeled in the following T–S form [1], [21]: $\begin{matrix} {\begin{matrix} \dot{x} = \sum_{i = 1}^{r} θ_{i} (z) (A_{i} x + B_{i} u) \\ y_{k T} = \sum_{i = 1}^{r} θ_{i} (z_{k T}) C_{i} x_{k T} \end{matrix} \end{matrix}$ where $x \in R^{n}$ is the state; $u \in R^{m}$ is the control input; $y \in R^{p}$ is the output; $z = (z_{1}, \dots, z_{q}) \in R^{q}$ is the vector containing the premise variables in the fuzzy IF–THEN rules, which is injectively mapped from y; and θ_i is the firing strength satisfying the following properties: $\begin{matrix} θ_{i} (z) \in R_{[0, 1]}, \sum_{i = 1}^{r} θ_{i} (z) = 1 . \end{matrix}$

Assumption 1

Only y_kT is available for feedback.

By taking into account Assumption 1,

Main results

Our main contribution is summarized as follows:

Theorem 1

For given compatible matrices X₁ and X₂, the T–S fuzzy system (1) is asymptotically stabilizable via the sampled-data SOF fuzzy controller (2), if there exist $\begin{matrix} P : = [\begin{matrix} P_{11} & 0 & 0 \\ P_{21} & P_{22} & P_{23} \\ P_{31} & P_{32} & P_{33} \end{matrix}] ∋ P_{11} = P_{11}^{T} ≻ 0 \end{matrix}$ and $R = R^{T} ≻ 0$ , and Z_ijh, $L_{ij} = L_{ji}^{T}$ , such that $\begin{matrix} M_{ijj} + {TZ}_{ijj} - L_{ii} ≺ 0, (i, j) \in I_{R} \times I_{R} \end{matrix}$ $\begin{matrix} M_{ijh} + M_{ihj} + {TZ}_{ijh} + {TZ}_{ihj} - He {L_{jh}} ≺ 0 \end{matrix}$ $\begin{matrix} [\begin{matrix} R & * \\ P^{T} {\tilde{C}}_{h} & Z_{i j h} \end{matrix}] ≽ 0, (i, j, h) \in I_{R} \times I_{J} \times I_{R} \end{matrix}$ $\begin{matrix} {[\begin{matrix} L_{ij} \end{matrix}]}_{r \times r} ≺ 0 \end{matrix}$ where $\begin{matrix} M_{i j h} : = [\begin{matrix} (\begin{matrix} He {A_{i}^{T} P_{11} \\ + C_{h}^{T} P_{21}} \\ + T A_{i}^{T} R A_{i} \end{matrix}) & * & * \\ (\begin{matrix} P_{22}^{T} C_{h} \\ - P_{21} \\ + F_{j}^{T} X_{1} \end{matrix}) & (\begin{matrix} He {F_{j}^{T} X_{2} \\ - P_{22}} \end{matrix}) & * \\ (\begin{matrix} B_{i}^{T} P_{11} \\ + P_{23}^{T} C_{h} \\ - P_{33} X_{1} \\ + T B_{i}^{T} R A_{i} \end{matrix}) & (\begin{matrix} F_{j} \\ - P_{} \end{matrix} \end{matrix} \end{matrix}$

Robustification

This section robustifies the development in the earlier section for the following uncertain T–S fuzzy system: $\begin{matrix} {\begin{matrix} \dot{x} = \sum_{i = 1}^{r} θ_{i} (z) (A_{i} x + B_{i} u + B_{w_{i}} w) \\ y_{k T} = \sum_{i = 1}^{r} θ_{i} (z_{k T}) (C_{i} + Δ C_{i}) x_{k T} \end{matrix} \end{matrix}$ where ΔC_i is a function matrix representing parametric uncertainties and $w \in R^{s}$ is the external disturbance belonging to $L_{2}$ . The closed-loop system of (13) with (2) is $\begin{matrix} E \dot{ξ} & = \sum_{i, j, h = 1}^{r} θ_{i} (z) θ_{j} (z_{k T}) θ_{h} (z_{k T}) \\ \times ({\hat{A}}_{i j h} ξ - {\hat{C}}_{h} \int_{k T}^{t} \dot{x} (τ) d τ + {\hat{B}}_{w_{i}} w) \end{matrix}$ where $\begin{matrix} {\hat{A}}_{i j h} = [\begin{matrix} A_{i} & 0 & B_{i} \\ C_{h} + Δ C_{h} & - I & 0 \\ 0 & K_{j} & - I \end{matrix}], {\hat{C}}_{h} = [\begin{matrix} 0 \\ C_{h} + Δ C_{h} \\ 0 \end{matrix}] \\ {\hat{B}}_{w_{i}} = [\begin{matrix} B_{w_{i}} \\ 0 \\ 0 \end{matrix}] . \end{matrix}$

Assumption 2

[12], [13], [14] There exist known compatible

Example

Consider the following smooth-air-gap permanent magnet synchronous motor (PMSM) [24] with external disturbance w but without the load torque: $\begin{matrix} {\begin{matrix} {\dot{i}}_{d} = - \frac{R}{L} i_{d} + n_{p} i_{q} ω + v_{d} \\ {\dot{i}}_{q} = - \frac{R}{L} i_{q} - n_{p} i_{d} ω - \frac{ψ_{r}}{L} ω + v_{q} \\ \dot{ω} = \frac{ψ_{r}}{L} i_{q} - \frac{β}{J} ω + 2 w \end{matrix} \end{matrix}$ where i_d and i_q are the d- and the q-axis current, respectively; ω is the motor angular velocity; v_d and v_q are the d- and the q-axis voltage, respectively; $R = 0.9 Ω$ is the stator winding resistance; $L = 0.01425 H$ is the inductance for both d- and the q-axis stators; $J = 4.7 \times 10^{- 5} {Kg m}^{2}$ is the polar moment of

Conclusions

In this paper, we developed a sampled-data SOF fuzzy controller design technique for nonlinear systems in T–S form in the continuous-time domain by using the descriptor redundancy method. The advantages herein are the non-use of an (approximate) discrete-time model of a T–S fuzzy system and the single-stage LMI problems without additional LME constraints. A numerical PMSM example verified the theoretical discussions. Future research efforts will be devoted to extending the synthesis to

Acknowledgements

This work was supported by INHA UNIVERSITY Research Grant (2018).

Hyoung Bin Kang received his B.S. and M.S. degrees from the Department of Electronic Engineering, Inha University, Incheon, Korea, in 2016 and 2018, respectively. His research interests include sampled-data control of fuzzy systems and singularly perturbed systems, and autonomous underwater vehicles.

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Ho Jae Lee received his B.S., M.S., and Ph. D. degrees from the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 1998, 2000, and 2004, respectively. In 2005, he was a Visiting Assistant Professor with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX. Since 2006, he has been with the Department of Electronic Engineering, Inha University, Incheon, Korea, where he is currently a Professor. His research interests include fuzzy control systems, sampled-data systems, hybrid dynamical systems, large-scale systems, digital redesign, and underwater gliders.

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Sampled-data static output-feedback control for nonlinear systems in T–S form via descriptor redundancy

Abstract

Introduction

Section snippets

Preliminaries

Main results

Robustification

Example

Conclusions

Acknowledgements

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Automatica

Fuzzy Sets Syst.

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Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

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Fuzzy Sets Syst.

Fuzzy model-based output-tracking control for 2 degree-of-freedom helicopter

J. Electr. Tech.

Direct discrete-time design approach to robust H∞ sampled-data observer-based output-feedback fuzzy control

Int. J. Syst. Sci.

Direct discrete-time design approach to robust $H_{\infty}$ sampled-data observer-based output-feedback fuzzy control