H∞ observer design for uncertain one-sided Lipschitz nonlinear systems with time-varying delay

doi:10.1016/j.amc.2020.125066

Applied Mathematics and Computation

Volume 375, 15 June 2020, 125066

https://doi.org/10.1016/j.amc.2020.125066 Get rights and content

Highlights

•
An H_∞ observer design for uncertain one-sided Lipschitz nonlinear systems with time-varying delay is proposed.
•
Both quadratic inner-bounded requirement and one-sided Lipschitz condition are considered.
•
A class of relaxed integral inequalities are adopted to reduce computation burden and achieve better results.
•
Illustrative examples are given to show the validity of the present results.

Abstract

The H_∞ observer design for uncertain one-sided Lipschitz nonlinear systems with time-varying delay is investigated in this paper. By considering the quadratic inner-bounded requirement and the one-sided Lipschitz condition concurrently and employing a class of relaxed integral inequalities, the H_∞ observer for nominal Lipschitz systems is designed firstly. Then, the results are extended to the one-sided Lipschitz systems with norm-bounded uncertainties. Finally, the proposed results are demonstrated via two simulation examples.

Introduction

In the field of control, many researches have been carried out around nonlinear systems [1], [2], [3], [4], [5], [6], [7], [8], [9]. The design of state observer has been an important topic for nonlinear systems. The observer can help to complete the state reconstruction and replace the real state with the reconstruct state, and achieve the required state feedback finally. Lipschitz system is a class of nonlinear systems with great importance. With respect to the observer design of Lipschitz system, many achievements have been made in recent years [10], [11], [12], [13], [14], [15], [16]. Some basic ideas on the observer design problem are proposed for the Lipschitz systems and sufficient and necessary conditions are obtained to ensure the observer stable asymptotically in [17]. It is proved in [18] that the existence condition of the full order observer can guarantee that the reduced order observer exists. A new H_∞ observer design is introduced in [19] which popularizes the Lipschitz observer used in the past and gives generalized sufficient condition to guarantee the state estimation to be convergent asymptotically.

In contrast to traditional lipschitz systems, the one-sided Lipschitz nonlinear systems is more general. Recently, the limitation of Lipschitz condition is relaxed by introducing one-sided Lipschitz functions in [20]. In allusion to nonlinear systems, sufficient conditions of designing observer are given separately in [21], [22], [23]. We have designed the H_∞ observer for one-sided Lipschitz discrete-time singular systems in [24] under the condition of the time-varying delays and disturbance inputs. We also use an extended reciprocal convexity inequality to obtain an effective reduced-order observer design by combining the quadratically inner-bounded requirement and free-weighting matrix technique in [25]. As far as the one-sided Lipschitz system is concerned, an observer of the system with disturbances and constant time-delay is designed in [26]. We extend the results in [26] to the time-varying systems in [27].

Physical systems usually involve uncertainties, caused by system modelling errors, unknown system parameters, measurement error and external interference etc. So far, a lot of work have been done on observer design for uncertain systems [14], [28], [29], [30], [31]. However, H_∞ observer design for uncertain one-sided Lipschitz nonlinear systems with time-varying delay it has not been fully studied.

Motivated by the above observation, we investigate the H_∞ observer design for the delayed one-sided Lipschitz systems with uncertainties. The nonlinear function is assumed to satisfy both quadratic inner-boundedness criterion and one-sided Lipschitz condition. The time-varying uncertainties are supposed to be norm-bounded. The relaxed inequality can help to reduce computation and achieve better results. And we aim to design the H_∞ observer for uncertain one-sided Lipschitz nonlinear systems with time-varying delay. In the process of our research work, the difficulty is that we need to consider multiple factors simultaneously, such as perturbations, uncertainties, and nonlinear functions that satisfy the quadratic inner-bounded requirement and the one-sided Lipschitz condition concurrently. We are trying to find a way to get better results while reducing the amount of computation. We design the H_∞ observer for the nominal system firstly, and then extend the results to the uncertain one-sided Lipschitz nonlinear systems. We can save a lot of computation in this way and obtain better results which is reflected in larger upper bound with time delay. Finally, two simulation examples demonstrating the effectiveness of the proposed observer design are presented.

This paper is organized as follows. Section 2 is devoted to the formulation and preliminaries of main problem. A design of the H_∞ observer for the nominal one-sided Lipschitz systems is presented in Section 3 firstly, and then the results are generalized to the uncertain one-sided Lipschitz systems. The validity of the proposed observers is verified by introducing two numerical examples in Section 4.

Notation: ‘ $A^{- 1}$ ’ and ‘A^T’ represent the inverse and the transpose of matrix A. The symbol * denotes a block matrix inferred by symmetry. The inner product is shown by $〈 a, b 〉 = a^{T} b,$ for vectors a, b. $R^{n}$ and $R^{n \times m}$ represent the real n-dimensional Euclidean vector space and the space of real matrices with dimension n × m respectively. $s y m {X} = X + X^{T}$ . ‖·‖ means the Euclidean norm, and the L₂ norm of vector x is denoted by ${∥ x ∥}_{L_{2}} = {(\int_{0}^{\infty} ∥ x^{2} ∥ d t)}^{\frac{1}{2}},$ where $L_{2} = {x : {∥ x ∥}_{L_{2}} < + \infty}$ . P > 0 and P < 0 denote that P is positive and negative definite matrix respectively. If the dimension of matrices in algebraic operations are not specified, we assume they have compatible dimensions.

Section snippets

Preliminaries

The uncertain one-sided Lipschitz nonlinear system studied in our paper is described by $\begin{matrix} \dot{x} (t) & = & (A + Δ A) x (t) + (A_{h} + Δ A_{h}) x (t - h (t)) + B_{1} f_{1} (F_{1} x (t, u (t))) + B_{2} f_{2} (F_{2} x (t - h (t)), u (t - h (t))) \\ + B_{u} u (t) + G_{1} w (t), \\ y (t) & = & C x (t) + C_{h} x (t - h (t)) + G_{2} w (t), \\ x (t) & = & ϕ (t), t \in [- h_{2}, 0] \end{matrix}$ where $x (t) \in R^{n}$ is the state, $u (t) \in R^{s}$ is the input, $w (t) \in R^{p}$ is the disturbance input, $y (t) \in R^{q}$ is the output. Moreover, $A \in R^{n \times n},$ $A_{d} \in R^{n \times n},$ $B_{1} \in R^{n \times m_{1}},$ $B_{2} \in R^{n \times m_{2}},$ $B_{u} \in R^{n \times s},$ $G_{1} \in R^{n \times p},$ $G_{2} \in R^{q \times p},$ $F_{1} \in R^{m_{1} \times n},$ $F_{2} \in R^{m_{2} \times n},$ $C \in R^{q \times n},$ $C_{h} \in R^{q \times n}$ are known constant matrices, $f_{1} (F_{1} x (t, u (t))) \in R^{m_{1}},$ $f_{2}$

Main results

To simplify the matrix representation, we define the following notations. $\begin{matrix} e_{j} & = & [\begin{matrix} 0_{n \times (n - 1)} & I_{n} & 0_{n \times (8 - i) n} & 0_{n \times (m_{1} + m_{2} + q)} \end{matrix}] \in R^{n \times (8 n + m_{1} + m_{2} + q)}, j = 1, . . ., 8, \\ e_{9} & = & [\begin{matrix} 0_{m_{1} \times 8 n} & I_{m_{1}} & 0_{m_{1} \times (n + q)} \end{matrix}] \in R^{m_{1} \times (8 n + m_{1} + m_{2} + q)}, \\ e_{10} & = & [\begin{matrix} 0_{m_{2} \times 8 n} & 0_{m_{2} \times m_{1}} & I_{m_{2}} & 0_{m_{2} \times q} \end{matrix}] \in R^{m_{2} \times (8 n + m_{1} + m_{2} + q)}, \\ e_{11} & = & [\begin{matrix} 0_{q \times (8 n + m_{1} + m_{2})} & I_{q} \end{matrix}] \in R^{q \times (8 n + m_{1} + m_{2} + q)} . \end{matrix}$

We now present the main results of this paper.

Theorem 1

Under Assumptions 2 and 3, for ρ > 0 and given scalar $υ > 0,$ if there exist 3n × 3n matrix P > 0, n × n matrices W₁ > 0, W₂ > 0, Q₁ > 0, Q₂ > 0, Q₃ > 0, Q₄ > 0, 2n × 2n matrices $M_{i} (i = 1, 2),$ n × q

Numerical examples

In this section, we provide two simulation examples to show the validity of superior results obtained in our paper.

Example 1

We consider the nominal system (6), in which the parameters come from the well-known single-link flexible joint robot system in the literature [35] given as $\begin{matrix} A = [\begin{matrix} 0 & 1 & 0 & 0 \\ - 48.6 & - 1.25 & 48.6 & 0 \\ 0 & 0 & 0 & 1 \\ 19.5 & 0 & - 19.5 & 0 \end{matrix}], A_{h} = [\begin{matrix} 0 & 0.03 & 0 & 0 \\ - 0.2 & 0.01 & - 0.2 & 0 \\ 0 & 0 & 0 & 0.01 \\ 0.1 & 0 & - 0.1 & 0 \end{matrix}], \\ B_{1} = [\begin{matrix} 0 \\ 0 \\ 0 \\ - 1 \end{matrix}], B_{2} = [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}], B_{u} = [\begin{matrix} 0 \\ 21.6 \\ 0 \\ 0 \end{matrix}], G_{1} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}], \\ G_{1} = [\begin{matrix} 0 \\ 1 \end{matrix}], C = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}], C_{h} = [\begin{matrix} 0.01 & 0 & 0.2 & 0 \\ 0 & - 0.31 & 0 & 0 \end{matrix}], \\ F_{1} = [\begin{matrix} 0 & 0 & 1 & 0 \end{matrix}], F_{2} = [\begin{matrix} 0 & 0 & 0 & 1 \end{matrix}], \\ f_{1} (F_{1} x (t)) = 3.33 \sin (x_{3} (t)), f_{2} (F_{2} x (t - h (t)) \end{matrix}$

Conclusion

The H_∞ observer design for the uncertain one-sided Lipschitz systems is studied in this paper. We consider multiple factors simultaneously, such as perturbations, uncertainties, and nonlinear functions satisfying the quadratic inner-bounded requirement and the one-sided Lipschitz condition concurrently. Some useful tools such as Wirtinger-based inequality and relaxed integral inequalities are applied to obtain a new design method which has merits over some existing designs. We design the H_∞

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The topic of nonfragile observer design for a class of one-sided Lipschitz nonlinear systems with quantized inputs is investigated in this study. The one-sided Lipschitz and quadratically inner-bounded conditions were used to obtain less conservative synthesis requirements for the observer design. Sufficient conditions, stated in terms of linear matrix inequalities, are supplied using auxiliary variables and a decoupling method to ensure the resilience of the quantized closed-loop. The controller and observer gains may then be found in the same process. An example generated from a nonlinear DC motor model is provided to demonstrate the efficiency of the suggested technique.
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^☆: The work is supported in part by the National Natural Science Foundation of China (61673227, 61873137) and the Taishan Scholar Project and the Natural Science Foundation of Shandong Province (ZR2016FM06).

View full text

H∞ observer design for uncertain one-sided Lipschitz nonlinear systems with time-varying delay☆

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Main results

Numerical examples

Conclusion

Appl. Math. Comput.

J. Franklin Instit.

Syst. Control Lett.

Automatica

Syst. Control Lett.

Syst. Control Lett.

Neurocomputing

Appl. Math. Comput.

Comput. Math. Appl.

Syst. Control Lett.

Syst. Control Lett.

Automatica

J. Franklin Instit.

Finite-time adaptive fuzzy control for nonlinear systems with full state constraints

IEEE Trans. Syst. Man Cybern.

Command filter-based adaptive fuzzy control for nonlinear systems with unknown control directions

IEEE Trans. Syst. Man Cybern.

Generalised dissipative asynchronous output feedback control for markov jump repeated scalar non-linear systems with time-varying delay

IET Control Theory Appl.

Observer-based dissipative control for networked control systems: a switched system approach

Complexity

Non-fragile observer-based control for discrete–time systems using passivity theory

Circuits Syst. Signal Process.

H_∞ observer design for uncertain one-sided Lipschitz nonlinear systems with time-varying delay☆