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Augmented error system approach to control design for a class of neutral systems
- Yujie Xu^{1},
- Jing Zhang^{1}Email author and
- Fucheng Liao^{2}
https://doi.org/10.1186/s13662-015-0596-2
© Xu et al. 2015
Received: 10 February 2015
Accepted: 6 August 2015
Published: 3 September 2015
Abstract
In this paper, control design based on target tracking for a class of neutral systems is investigated by augmented error system approach. The dynamic equation of tracking error is derived by use of the desired output which is assumed to be known. Based on the main thought of preview control, the augmented error system is constructed. A criterion to guarantee the asymptotic stability of the corresponding nominal system is established. Through control design of the augmented error system, a delay-dependent control and a delay-independent control are respectively presented for the original neutral system. Control design in this paper contains integrators, which can effectively reduce static errors. Examples are given to illustrate the efficiency of the proposed method.
Keywords
- neutral system
- augmented error system
- asymptotic stability
- preview control
- target tracking
1 Introduction
Time-delay is often encountered in many practical systems such as process control systems, neural network systems, and nuclear reactor systems (see [1, 2]). It is well known that time-delay is a major source of instability and oscillation. Therefore, stability analysis and control design of time-delay systems are important in theory and practice (see [3–6]).
As a special class of time-delay systems, neutral systems contain time-delay both in the states and in the derivatives of states, which can reflect the systems’ operation more comprehensively and accurately. It has been shown that a number of practical systems can be modeled by neutral systems such as heat exchangers, population ecology, and partial element equivalent circuit (PEEC) (see [7, 8]). Additionally, there exist some time-delay systems which can be transformed into neutral systems by model transformation, including lossless transmission model, standard delay systems, and standard distributed delay systems (see [9]). Due to the particularity of time-delay, stability issues of neutral systems are proved to be more complex. They have attracted much attention and hence a lot of research has been done over recent years (see [10, 11]).
To the best of the authors’ knowledge, though some frequency domain methods, such as the spectral decomposition theory and semi-group theory, are applied to investigate the stability of neutral systems, they are not as popular as Lyapunov-Krasovskii functional (LKF). Through LKF approach, many stability criteria have been proposed (see [11–13]). Recently, LMIs and free weighting matrices have been widely used in LKF for the stability of neutral systems (see [14–16]). For example, by constructing a new LKF with free weighting matrices, stability criteria for uncertain neutral systems were investigated in [11]. Moreover, free weighting matrices and model transformations were both employed in [14] for cross terms. Then a less conservative stability criterion based on LMIs was derived for stochastic neutral systems. However, control design based on these stability criteria is usually the state feedback, which is delay-independent or delay-dependent.
Actually, it is required that the output should track the target in some control systems such as robot routine control systems. \(H_{\infty}\) output tracking control, where the output of a control system tracks the output of a given reference model well in the sense, is often adopted as the main way of tracking control for neutral systems (see [6, 17, 18]). Augmented error system approach is usually used in tracking control by combining the error signal and the state. Sometimes, LKF uses augmented vector to simplify the results (see [1, 19]).
This paper utilizes the known desired output, but not the reference model, to construct an augmented error system which contains the state vector, the error signal, the desired output, and their derivatives. Thus, control design in the paper contains not only state feedback, but also an integrator, which can effectively reduce static errors and is not similar to output tracking control (see [20, 21]). The main thought comes from the theory of preview control, which belongs to tracking control. In this paper, we keep time-delay in the augmented error system, which is different from the ordinary method of preview control (see [20–22]).
Notations
Throughout this paper, the following notations will be used. \(\mathbf{R}^{n} \) denotes an n-dimensional Euclidean space. \(\mathbf{R}^{n \times m} \) is the set of all \(n \times m\) real matrices. ∗ refers to the symmetric part of a matrix. \(P > 0\) represents that P is a symmetric positive definite matrix. I denotes an identity matrix with appropriate dimensions.
2 Problem statements
The basic problem considered in this paper is to design control for system (1) such that the output \(y(t)\) tracks the desired output \(r(t)\). For system (1), we assume the following.
Assumption 1
\(\vert {\lambda_{i} (G)} \vert < 1\) (\(i = 1,2, \ldots,n\)), where \(\lambda_{i} (G)\) is the ith characteristic value of G.
Assumption 2
The desired output \(r(t) \in\mathbf{R}^{p} \) is piecewise differentiable with finite discontinuity points. Except its discontinuity points, \(r(t)\) is differentiable up to the sth order, i.e., \(r'(t), r''(t), \ldots, r^{(s - 1)}(t)\) are all continuous and \(r^{(s)} (t) = r^{(s+1)} (t)=\cdots= 0\).
Remark 1
The stability of \({\mathcal{D}}x(t)\) can be replaced by the Schur-Cohn stability of G. It means that if all characteristic values of G are in unite circle, the operator \({\mathcal{D}}x(t)\) is stable (see [9]). So Assumption 1 insures that \({\mathcal{D}}x(t)\) is stable.
Remark 2
System (1) is described by differential equations. So it is necessary that the derivatives of desired output are taken to construct an augmented error system. Assumption 2 can make sure the method is suitable for more extensive desired output such as staircase signal which is mostly common in signal processing (see [23]). Additionally, at discontinuity points of the desired output, we can take left or right derivatives as their derivatives for values of limited points do not affect the system’s performance much.
Remark 3
If \(r(t)\) is derivable for any order, its first to sth derivatives are selected within error range, which is similar to the truncation method of signal procession (see [23]). Some desired output may be expanded into power series with limited terms if necessary.
The following well-known lemma will be used for providing the main results in the sequel.
Lemma 1
(Schur complement in [24])
3 Main results
In this section, main results on both delay-dependent and delay-independent control design will be presented for system (1). Firstly, an augmented error system is constructed. And then, the stability criterion of its corresponding nominal system is proposed.
3.1 Construction of an augmented error system
In order to make better use of the desired output \(r(t)\), we consider combining the tracking error \(e(t)\) with system (1).
Therefore, the problem tackled in this paper is to design control for system (6).
3.2 Asymptotic stability analysis of the nominal system
Theorem 1
Proof
Based on Assumption 1 and \(0<\alpha<1 \), it is easy to get \(\vert {\lambda_{i} (\bar{G})} \vert < 1\) (\(i = 1,2, \ldots,2n + p\)). Therefore, we conclude that \(\vert {\lambda_{j} (\tilde{G})} \vert < 1\) (\(j = 1,2, \ldots,2n + p + sp\)), which ensures \({\mathcal{D}}X(t) = X(t) - \tilde{G}X(t - h)\) is stable (see [9]).
As a foundation, Theorem 1 provides a sufficient condition for the asymptotic stability of system (7). In the following sections, the proposed delay-independent and delay-dependent controllers are calculated not only to stabilize the closed-loop neutral system, but also to fully utilize the desired output \(r(t)\).
3.3 Delay-independent control for system (1)
Theorem 2
Theorem 3
Remark 4
From (15), we can find that control \(u(t)\), which is calculated from \(\dot{u}(t)\), not only contains state feedback and the integral of state vector \(x(t)\), but also relates to the initial condition \(\phi(t)\). Furthermore, both the integrator on tracking error signal \(e(t)\) and the integral of desired output \(r(t)\) are contained in control \(u(t)\). It is well known that introduction of integrators to control design is helpful to eliminate static errors (see [21]).
3.4 Delay-dependent control for system (1)
From Theorem 1, replacing Ã and \(\tilde{A}_{1} \) of system (7) respectively with \(\tilde{A} + \tilde{B}L\) and \(\tilde{A}_{1} + \tilde{B}W\) presents the following theorem to guarantee the asymptotic stability of system (16).
Theorem 4
Theorem 5
4 Examples
In this section, two examples are given to show the effectiveness of the proposed control design method.
Example 1
From Figures 1-4, we find that when the control is based on target tracking, the state curve and the output response are much smoother, and the curve amplitude is much smaller.
Comparing these figures, we find that for this system, the amplitude of the state curve and output response is much smaller when the control \(u(t)\) is relevant to time delay.
Example 2
5 Conclusions
In this paper, control design based on target tracking is investigated. The properties of the desired output are taken to construct an augmented error system. Based on the asymptotic stability theory of the nominal system, we present the feedback control for the augmented error system. The corresponding control design, which contains integrator on the tracking error signal, is deduced for the original system in Theorems 3 and 5. Two examples dealing with different kinds of desired output have been given to show the effectiveness of the proposed method. Control design in the paper contains not only the state feedback, but also the integrator on the tracking error signal, the integrals of the state and the desired output. We hope that the proposed method in the paper can provide an approach to control design for a class of neutral systems described as system (1).
Declarations
Acknowledgements
The work was supported by ‘New Start’ Academic Research Projects of Beijing Union University (Grant No. Zk10201503), the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (Grant No. IDHT201304089). We are indebted to Professor Wansheng Tang of Tianjin University for many useful comments. We are grateful to the anonymous reviewers for helpful advices that improved the presentation of the article.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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