Output feedback based simultaneous stabilization of two Port-controlled Hamiltonian systems with disturbances

doi:10.1016/j.jfranklin.2019.02.039

Journal of the Franklin Institute

Volume 356, Issue 15, October 2019, Pages 8154-8166

https://doi.org/10.1016/j.jfranklin.2019.02.039 Get rights and content

Abstract

In motor system control design, a single controller is usually employed to simultaneously control two or more motors for saving costs, which also achieves the computational simplification of control. In practical Hamiltonian systems control, more systems also need to be stabilized by a single controller under some working conditions. Thus, this paper studies simultaneous stabilization problem of two nonlinear Port-controlled Hamiltonian (PCH) systems with disturbances by a composite controller. Based on the Hamiltonian structure properties, two PCH systems are combined together to generate an augmented PCH system by utilizing output feedbacks firstly. Then, to estimate disturbances effectively, it is essential to design a nonlinear disturbance observer (NDOB) and the estimate is employed to feedforward compensate the effects of disturbances. Next, combining the output feedback part and the disturbance compensation part together, a simultaneous stabilization controller is developed. Subsequently, it is proved that the closed-loop system under the proposed controller is asymptotically stable. Finally, an example with simulations reveals that the proposed method is effective.

Introduction

Since the Port-controlled Hamiltonian (PCH) systems were proposed by Maschke and Van Der Schaft [16], [25], they have been well studied [3], [24], [29]. As the sum of kinetic energy and potential energy in engineering systems, the Hamilton function usually is used as a Lyapunov function candidate for many engineering systems [17], [19], [21]. Due to the special PCH system structure with clear physical meaning, the PCH system has attracted considerable attention [1], [6], [10], [23]. In fact, in classical mechanics, biological engineering and aerospace science, the PCH system widely exists. The PCH system control theory has become a very important research topic in today’s nonlinear control theory, and some energy-based control methods have been developed for power and electronic systems [5], [22], [27] up to now.

In control designs of engineering systems, the problem of simultaneous stabilization is often considered [13], [18], [26]. It means that a set of systems can be stabilized simultaneously under a single controller designed. For example, in motor system control, a single controller is usually employed to simultaneously control two or more motors for saving costs, which also achieves the computational simplification of control. As a significant research project in the field of robust control, the simultaneous stabilization problem has attracted a great deal of attention and there are some significant results for the problem up to now [14], [29], [34]. The simultaneous H_∞ stabilization problem for a family of linear norm bounded uncertain systems is studied, and a sufficient and necessary condition is given for the existence of the desired controller in [14]. Simultaneous stabilization problem of a class of nonlinear PCH systems with disturbances is investigated in [29], and an L₂ disturbance attenuation control law is designed. Paper [34] investigates a simultaneous H₂/H_∞ stabilization problem for the chemical reaction systems. These control methods can be divided into state feedback control and output feedback control [35], [36]. In fact, the output feedback is easier to implement than the state feedback in real systems. Meanwhile, the above methods belong to the robust control method, which can make systems have good robustness against disturbances. However, the robust control is a worst case based design and the nominal control performance of closed-loop systems is generally sacrificed for robustness.

In order to overcome the shortcoming of the above control method and enhance the control systems’ disturbance rejection performance, papers [2], [7], [8], [9], [12] develops the advanced control methods based on the disturbance observer. The disturbance observer technique is an efficient disturbance estimation technique and the estimates are employed to compensate the effects of disturbances [15], [20], [30], [31], [32], [33]. Via the disturbance observer based control (DOBC), a class of composite control framework has been given, which contains feedforward compensation and baseline feedback control part. With the help of disturbance feedforward compensation, the composite control deals with disturbances more promptly and directly compared with pure feedback control. Hence, the DOBC method is effective for the control problems of disturbed systems. In particular, for the simultaneous stabilization of nonlinear Hamiltonian disturbed systems, there are relatively fewer results.

In this paper, the simultaneous stabilization problem of two nonlinear PCH disturbed systems is studied. Firstly, based on the Hamiltonian structure properties and using output feedbacks, an augmented PCH disturbed system is generated by combining two PCH disturbed systems. Then, to estimate disturbances effectively, a NDOB is designed and the estimate is employed to compensate the effects of disturbances. Finally, a composite simultaneous stabilization controller is designed by integrating the output feedback and the disturbance compensation part together. By means of the Lyapunov stability theorems, it is proved that the closed-loop PCH disturbed system under the proposed controller is asymptotically stable. An example with simulations shows that the proposed method is effective. The main merits can be divided into two aspects. (1) The proposed control method can make systems have good robustness against disturbances and better control performance, such as smaller overshoots and shorter settling time. (2) In the absence of disturbances, the proposed control method can still work well for PCH systems, i.e., the proposed method does not sacrifice its nominal control performance.

The remainder of this paper contains four sections. Section 2 gives some preliminaries and problem formulation. Section 3 is the main result, which consists of the output feedback controller design, the NDOB design and the asymptotic stability analysis of the closed-loop PCH disturbed system. Section 4 presents an illustrative example, which is followed by the conclusions in Section 5.

Notations: $R$ is the set of real numbers. $R^{n}$ denotes the set of n-dimensional vectors. $R^{n \times l}$ is the set of n × l real matrices. The notation R ≥ 0 (R > 0) means that the matrix R is positive semi-definite (positive definite). λ_min(R) stands for the minimum eigenvalue of matrix R.

Section snippets

Preliminaries and problem statement

Consider the following two nonlinear PCH disturbed systems $\begin{matrix} {\begin{matrix} \dot{x} = (J_{1} (x) - R_{1} (x)) \nabla H_{1} (x) + G_{1} (x) U + {\tilde{G}}_{1} (x) d (t), \\ y = G_{1}^{T} (x) \nabla H_{1} (x), \end{matrix} \end{matrix}$ $\begin{matrix} {\begin{matrix} \dot{η} = (J_{2} (η) - R_{2} (η)) \nabla H_{2} (η) + G_{2} (η) U + {\tilde{G}}_{2} (η) d (t), \\ ξ = G_{2}^{T} (η) \nabla H_{2} (η), \end{matrix} \end{matrix}$ where $x, η \in R^{n}$ are the states, $y, ξ \in R$ are the outputs, $U \in R^{m}$ is the control input, $J_{j}^{T} (x) = - J_{j} (x) \in R^{n \times n},$ $0 \leq R_{j} (x) \in R^{n \times n},$ $H_{j} (x) \in R$ is the Hamilton function, $\nabla H_{j} (x) \in R^{n},$ denoting $\frac{\partial H_{j} (x)}{\partial x},$ is the gradient of the Hamilton function H_j(x), $G_{j} (x) \in R^{n \times m}$ is a full column rank matrix, (m ≤ n), $(R_{j}, G_{j}) \in R^{n \times (n + m)}$ is a full row rank matrix, ${\tilde{G}}_{j} (x) \in R^{n}$

Main results

The section includes three parts. A simultaneous feedback controller u is first designed. Then the feedforward compensation control v is developed based on NDOB. Finally a composite simultaneous controller is proposed, and the asymptotic stability analysis of the closed-loop augmented Hamiltonian system is given.

Simulations

An example with simulations is provided to illustrate that the proposed control method is effective.

Example: Consider two PCH systems subject to disturbances $\begin{matrix} {\begin{matrix} \dot{x} = (J_{1} (x) - R_{1} (x)) \nabla H_{1} (x) + G_{1} U + {\tilde{G}}_{1} d (t), \\ y = G_{1}^{T} \nabla H_{1} (x), \end{matrix} \end{matrix}$ $\begin{matrix} {\begin{matrix} \dot{η} = (J_{2} (η) - R_{2} (η)) \nabla H_{2} (η) + G_{2} U + {\tilde{G}}_{2} d (t), \\ ξ = G_{2}^{T} \nabla H_{2} (η), \end{matrix} \end{matrix}$ where $x = {(x_{1}, x_{2})}^{T} \in R^{2},$ $η = {(η_{1}, η_{2})}^{T} \in R^{2},$ $d \in R$ is the disturbance, $U \in R$ is the control input, $H_{1} (x) = x_{1}^{4} + x_{2}^{2},$ $H_{2} (η) = η_{1}^{2} + η_{2}^{2},$ $\begin{matrix} J_{1} (x) = (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}), J_{2} (η) = (\begin{matrix} 0 & 2 \\ - 2 & 0 \end{matrix}), \end{matrix}$ $\begin{matrix} R_{1} (x) = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}), R_{2} (η) = (\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}), \end{matrix}$ $\begin{matrix} G_{1} = {\tilde{G}}_{1} = (\begin{matrix} 0 \\ 2 \end{matrix}), a n d G_{2} = {\tilde{G}}_{2} = (\begin{matrix} 1 \\ 0 \end{matrix}) . \end{matrix}$ Assumption 1 is satisfied with $ϕ = - 1$ . According to

Conclusions

In this paper, the simultaneous stabilization problem of two nonlinear PCH systems subject to disturbances has been investigated by a composite control strategy. Based on the Hamiltonian structure properties and the nonlinear disturbance observer technique, a composite simultaneous stabilization controller has been developed by combining the output feedback part and the disturbance compensation part. The study on an example has been given to demonstrate that the composite control method is

Acknowledgments

This work was supported by the National Nature Science Foundation of China under Grants 61473080, 61633003, 61627810 and 61873060.

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Output feedback based simultaneous stabilization of two Port-controlled Hamiltonian systems with disturbances

Abstract

Introduction

Section snippets

Preliminaries and problem statement

Main results

Simulations

Conclusions

Acknowledgments

Automatica

Automatica

ISA Trans.

Automatica

J. Frankl. Inst. Eng. Appl. Math.

Appl. Math. Comput.

Automatica

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IEEE Trans. Ind. Electron.

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IEEE Trans. Autom. Control

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