Parametric control to a type of descriptor quasi-linear high-order systems via output feedback

doi:10.1016/j.ejcon.2020.09.002

European Journal of Control

Volume 58, March 2021, Pages 223-231

https://doi.org/10.1016/j.ejcon.2020.09.002 Get rights and content

Abstract

This study presents a parametric approach to design output feedback for descriptor quasi-linear high-order systems. Utilizing the solution to the high-order generalized Sylvester equations (HGSEs), the more unified parametric form of output feedback is developed. With the proposed approach, the closed-loop is transformed into a linear constant system with an expected eigenstructure. Meanwhile, the regularity of the closed-loop system is maintained easily and effectively. Finally, a synchronization problem of Genesio-Tesi and Coullet systems is proposed to illustrate the effectiveness of the parametric approach.

Introduction

In the real world, many dynamical models of practical applications are quasi-linear, including spacecraft rendezvous [5], attitude control of combined spacecraft [18], robotic systems [28], and so on. Simultaneously, quasi-linear systems maintain strong coupling and highly nonlinear characteristics but in a linear format, which can be regarded as the link and bridge between linear systems and general nonlinear systems. In the past, the quasi-linear system has been intensively considered in the literature (see [2], [14], [15], [20], [22], [25], [26], [32] and the references therein).

Note that quasi-linear high-order systems can be commonly used for the modeling of synchronization of chaotic systems [12], jerk system [21], wind-induced vibrating comfort [27], etc., which has great value in engineering applications, however, descriptor quasi-linear high-order system, which can represent static and dynamic processes of practical systems, has been not investigated in the recent researches.

Gu and his team present an effective approach to design the static output feedback and the dynamic compensator for linear time-varying systems [9], [13] and quasi-linear systems [5], [6], [7], [8], [10], [11], [12], which provides the fundamental to deal with the parametric design of output feedback for descriptor quasi-linear high-order systems.

In this study, a parametric approach, based on the solution of HGSEs, is proposed to design output feedback for descriptor quasi-linear high-order systems such that the completely parameterized solution of output feedback is established. With the proposed parametric approach, the closed-loop system yields a linear constant form with an expected eigenstructure. Simultaneously, it is more simple and easy to maintain the regularity of closed-loop systems. Additionally, the parametric approach can offer free parameters to provide flexibility and degrees of design freedom in the design process. Compared with the related result in [31], this paper focuses on descriptor quasi-linear high-order systems and obtains the more explicit and unified expressions of output feedback gain matrices, it also effectively maintains the regularity of the closed-loop system.

The remaining part of this paper is respectively organized as follows. Section 2 presents the problem formulation of descriptor quasi-linear high-order systems with output feedback, and provides some preliminary results. In Section 3, parametric approach is proposed to establish the complete parameterization form of output feedback for two cases. A synchronization problem of Genesio–Tesi and Coullet systems is given to prove the effectiveness of parametric approach in Section 4. Section 5 draws the conclusions.

Section snippets

System description

In this paper, a type of descriptor quasi-linear high-order systems is considered as follows ${\begin{matrix} \sum_{i = 0}^{m} A_{i} (θ, y) q^{(i)} = B (θ, y) u, \\ y_{i} = C_{i} (θ, y) q^{(i)}, i = 0, 1, \dots, m - 1, \end{matrix}$ where $q \in R^{n},$ $u \in R^{r},$ $y_{i} \in R^{p_{i}},$ $i = 0, 1, \dots, m - 1$ are the state vector, the control vector and the measured output, and $y = {[\begin{matrix} y_{0}^{T} & y_{1}^{T} & \dots & y_{m - 1}^{T} \end{matrix}]}^{T},$ the matrices A_i(θ, y)  ∈  $R^{n \times n},$ $i = 0, 1, \dots, m,$ B(θ, y)  ∈  $R^{n \times r}$ and C_i(θ, y)  ∈  $R^{p_{i} \times n},$ $i = 0, 1, \dots, m - 1,$ $\sum_{i = 0}^{m - 1} p_{i} = p$ are the coefficient matrices of system (1), and are also piecewise continuous functions of y and θ. θ is a

F is an arbitrary matrix

With the above discussion, we present the following Theorem 1 regarding to Problem 1.

Theorem 1

Let N(θ, y, s) and D(θ, y, s) be a pair of polynomial matrices satisfying RCF (16).

1. Problem 1 has a solution if and only if there exists an arbitrary parameter matrix $Z \in C^{r \times n_{c}}$ satisfying $\det (V_{c} (θ, y)) \neq 0,$ where $V (θ, y) = \sum_{i = 0}^{σ} N_{i} (θ, y) Z F^{i},$ and $\begin{matrix} V_{R \infty} (θ, y) = {[\begin{matrix} 0_{(n - n_{0}) \times (m - 1) n}^{T} & V_{\infty}^{T} {(θ, y)}_{(n - n_{0}) \times n} \end{matrix}]}^{T}, \\ A_{m} (θ, y) V_{\infty} (θ, y) = 0 . \end{matrix}$

2. When the above condition (18) holds, the coefficient matrices of static output feedback controller (2) can be

System description

The Genesio–Tesi system is as follows $\overset{⃛}{α} + 0.45 \ddot{α} + 1.1 \dot{α} + α - α^{2} = 0,$ and the Coullet system is as follows $\overset{⃛}{β} + 0.45 \ddot{β} + 1.1 \dot{β} - 0.8 β + β^{3} = 0$ based on the existing results in [7], [10], the model of the synchronization problem of Genesio–Tesi and Coullet systems can be rewritten in the form of the descriptor quasi-linear high-order system as $A_{3} (θ, y) \overset{⃛}{q} + A_{2} (θ, y) \ddot{q} + A_{1} (θ, y) \dot{q} + A_{0} (θ, y) q = u,$ where $q = {[\begin{matrix} β & e \end{matrix}]}^{T},$ $e = α - β$ is the tracking error, and $\begin{matrix} A_{3} (θ, y) & = & [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], A_{2} (θ, y) = [\begin{matrix} 0 & 0.45 \\ 0 & 0 \end{matrix}], \\ A_{1} (θ, y) & = & [\begin{matrix} 0 & 1.1 \\ 0 & 0 \end{matrix}], A_{0} (θ, y) = [\begin{matrix} θ_{1} - β & 1 - θ_{2} \\ 0 & 0 \end{matrix}], \end{matrix}$ $θ_{1} = 1.8 - β^{2},$ $θ_{2} = 2 β$

Conclusions

In this research, a parametric approach has been investigated to design the output feedback controller for descriptor quasi-linear high-order systems. The presented parametric approach has provided the completely parametric solution of the output feedback controller such that the closed-loop system has the expected eigenstructure. Note that the arbitrary parameter Z can provide the flexibility and degrees of design freedom to reduce the computation load and realize systems optimization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (32)

Y.F. Cai et al.
Robust partial pole assignment problem for high order control systems
Automatica
(2012)
F. Castillo et al.
Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control
Automatica
(2013)
J.F. Dos Santos et al.
Robust pole placement under structural constraints
Syst. Control Lett.
(2018)
D.K. Gu et al.
Parametric control to second-order linear time-varying systems based on dynamic compensator and multi-objective optimization
Appl. Math. Comput.
(2020)
D.K. Gu et al.
Parametric control to a type of quasi-linear high-order systems via output feedback
Eur. J. Control
(2019)
C.K. Han et al.
Quasi-linear systems of PDE of first order with Cauchy data of higher codimensions
J. Math. Anal. Appl.
(2015)
L. Jadachowski et al.
Backstepping observers for periodic quasi-linear parabolic PDEs
IFAC Proc. Vol.
(2014)
U. Konigorski
Pole placement by parametric output feedback
Syst. Control Lett.
(2012)
D. Rotondo et al.
Model reference switching quasi-LPV control of a four wheeled omnidirectional robot
IFAC Proc. Vol.
(2014)
V.L. Rozenberg
Dynamical input reconstruction problem for a quasi-linear stochastic system
IFAC-PapersOnLine
(2018)

X.T. Wang et al.

Partial eigenvalue assignment with time delay in high order system using the receptance

Linear Algebra Appl.

(2017)

G.R. Duan

Generalized Sylvester Equations—Unified Parametric Solutions

(2014)

D.K. Gu et al.

Parametric control to a type of quasi-linear second-order systems via output feedback

Int. J. Control

(2019)

D.K. Gu et al.

Parametric control to a type of descriptor quasi-linear systems based on dynamic compensator and multi-objective optimisation

IET Control Theory Appl.

(2020)

D.K. Gu et al.

Parametric control to a type of quasi-linear descriptor systems via proportional plus derivative feedback

Circt. Syst. Signal Process.

(2020)

D.K. Gu et al.

Parametric control to quasi-linear systems based on dynamic compensator and multi-objective optimization

Kybernetika

(2020)

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^☆: This work was supported in part by the National Natural Science Foundation of China [grant numbers 61690212, 61690210].

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Parametric control to a type of descriptor quasi-linear high-order systems via output feedback☆

Abstract

Introduction

Section snippets

System description

F is an arbitrary matrix

System description

Conclusions

Declaration of Competing Interest

Automatica

Automatica

Syst. Control Lett.

Appl. Math. Comput.

Eur. J. Control

J. Math. Anal. Appl.

IFAC Proc. Vol.

Syst. Control Lett.

IFAC Proc. Vol.

IFAC-PapersOnLine

Linear Algebra Appl.

Generalized Sylvester Equations—Unified Parametric Solutions

Parametric control to a type of quasi-linear second-order systems via output feedback

Int. J. Control

Parametric control to a type of descriptor quasi-linear systems based on dynamic compensator and multi-objective optimisation

IET Control Theory Appl.

Parametric control to a type of quasi-linear descriptor systems via proportional plus derivative feedback

Circt. Syst. Signal Process.

Parametric control to quasi-linear systems based on dynamic compensator and multi-objective optimization

Kybernetika