Stabilization of neutral-type indirect control systems to absolute stability state
- Andriy Shatyrko^{1},
- Ronald RP van Nooijen^{2}Email author,
- Alla Kolechkina^{2} and
- Denys Khusainov^{1}
https://doi.org/10.1186/s13662-015-0405-y
© Shatyrko et al.; licensee Springer. 2015
Received: 16 September 2014
Accepted: 5 February 2015
Published: 26 February 2015
Abstract
This paper provides sufficient conditions for absolute stability of an indirect control Lur’e problem of neutral type. The conditions are derived using a Lyapunov-Krasovskii functional and are given in terms of a system of matrix algebraic inequalities. From these matrix inequalities a sufficient condition for linear state feedback stabilizability follows.
Keywords
MSC
1 Introduction
The problem of absolute stability is often encountered in engineering practice. One specific form of this problem is the indirect control Lur’e problem, where the system to be controlled is linear, but the control action is the output of a nonlinear scalar system that itself receives output feedback. The special case where the output of the controller is a nonlinear function of one variable whose graph lies between two lines in the first and third quadrants of the coordinate plane is usually studied. Initially only systems of ordinary differential equations were considered; see for example [1–6]. A historical overview of the absolute stability problem can be found in [7] or in the introduction of [8].
In practical control processes time delays are common and they often cause instabilities, as a result, the absolute stability problem of nonlinear control systems with delay has attracted a lot of interest [5, 6, 9–11]. Nonlinear systems of neutral type with indirect control are considered in [10–15]. Sufficient conditions for absolute stability for such systems are derived in [13, 14] by the direct Lyapunov method using Lyapunov-Krasovskii functionals, these conditions are given in Theorem 1 of this paper. The functionals are constructed by taking the sum of a quadratic form of the current coordinates, integrals over the delay of quadratic forms of the state and its derivative, and the integral of the nonlinear components of the considered system [5, 6, 9–11]. All results from [11, 13, 14] can be put into a unified form in terms of matrix algebraic inequalities. A very different approach is given in [16, 17] or [18, 19], where integral operators are used.
In this paper we also consider what to do if absolute stability of the system under investigation cannot be established using the result given in Theorem 1. There are two obvious options: either change the method of investigation or change the Lyapunov function or functional. But there is a third option: we can try to add a linear state feedback to stabilize the closed loop system for the previously chosen Lyapunov function or functional. There are some interesting papers devoted to the investigation of stability and stabilization tasks [20–22].
The present article is a direct extension of [22]. The remainder of this paper is organized as follows. In Section 2 the absolute stability problem of neutral type indirect nonlinear control system is formulated, some notation is defined, and a result from [13, 14] is stated. Section 3 introduces the concepts of stability and stabilization with respect to a given functional for the case of a linear control system with delay. In Section 4 the scalar case of a neutral system with nonlinear indirect control is treated. The indirect control system of neutral type in the general matrix form is considered in Section 5. Finally, some conclusion are drawn in Section 6.
2 Problem formulation and preliminaries
Definition 1
A pair \((x,\sigma )\in C ( [t_{0}-\tau,\infty ),\mathbb{R}^{n} )\times C ( [t_{0},\infty ),\mathbb{R} )\) is a solution of (1), (2), (3) on \([t_{0},\infty )\) if x satisfies ( 3 ) and the pair satisfies the system (1) and (2).
Evidently, as discussed in [23], p.169, there are obviously two families of metrics or measures for stability in this case, one based on x alone and another based on x and its derivative. A general theory of stability in two metrics or measures was first given by [24] and extended by [25]; see also [26, 27]. We use the definition of measure given in [26].
Definition 2
Note the large difference in meaning conveyed by the subtle difference in terminology between a ‘measure in X’ and a ‘measure on X’.
Definition 3
Definition 4
Definition 5
Definition 6
Definition 7
We call the zero solution \(x:t\mapsto0_{n\times1}\), \(\sigma:t\mapsto0\) of (1), (2) stable if it is \((h_{0},h )\) stable for \(h_{0} (t,\phi )=\Vert \phi \Vert _{\infty}\) and \(h (t, \langle x_{t},\sigma_{t} \rangle )=\sqrt{\vert x_{t} (0 )\vert ^{2}+\vert \sigma _{t} (0 )\vert ^{2}}\).
Definition 8
We call the zero solution \(x:t\mapsto0_{n\times1}\), \(\sigma:t\mapsto0\) of (1), (2) asymptotically stable if it is \((h_{0},h )\) asymptotically stable for \(h_{0} (t,\phi )=\Vert \phi \Vert _{\infty}\) and \(h (t, \langle x_{t},\sigma_{t} \rangle )=\sqrt{\vert x_{t} (0 )\vert ^{2}+\vert \sigma _{t} (0 )\vert ^{2}}\).
Definition 9
We call the zero solution \(x:t\mapsto0_{n\times1}\), \(\sigma:t\mapsto0\) of (1), (2) globally asymptotically stable if it is \((h_{0},h )\) globally asymptotically stable for \(h_{0} (t,\phi )=\Vert \phi \Vert _{\infty}\) and \(h (t, \langle x_{t},\sigma_{t} \rangle )=\sqrt{\vert x_{t} (0 )\vert ^{2}+\vert \sigma _{t} (0 )\vert ^{2}}\).
Definition 10
Definition 11
The system (1), (2) is called absolutely stable if the zero solution of the system (1), (2) is globally asymptotically stable for an arbitrary function \(f (\sigma )\) that satisfies (4).
Theorem 1
Let \(\vert D\vert <1\), \(\rho,\tau>0\) and suppose that there exist positive definite matrices \(G_{1}\), \(G_{2}\), H, and constants \(\zeta>0\), \(\beta>0\) such that the matrix \(S [A_{1},A_{2},b,c,\rho,\tau,H,G_{1},G_{2},\beta,\zeta ]\) is positive definite. In that case the system (1), (2) is absolutely stable in metric with respect to the metric defined earlier for \(\mathcal{C}^{1}\).
Corollary 1
Let \(\vert D\vert <1\), \(\rho,\tau>0\) and suppose that there exist positive definite matrices \(G_{1}\), \(G_{2}\), H, and constants \(0<\lambda<1\), \(\beta>0\) such that the matrix \(\tilde{S} [A_{1},A_{2},b,c,\rho,\tau ,H,G_{1},G_{2},\beta,\lambda ]\) given by \(S_{ij}\) for \((i,j)\notin\{ (2,2), (3,3) \}\) and \(\tilde{S}_{22} = \lambda G_{1}-A_{2}^{T}G_{2}A_{2}\), \(\tilde{S}_{33} = \lambda G_{2}-D^{T}G_{2}D\) is positive definite. In that case the system (1), (2) is absolutely stable in metric in metric \(\mathcal{C}^{1}\) for all finite delays τ.
Proof
For each τ this follows from Theorem 1 by taking \(\zeta= \tau^{-1} \log\lambda\). □
Note 1
In this corollary there are no conditions on the delay other than \(\tau>0\).
Definition 12
A system is stable with respect to the functional V with exponent \(\gamma>0\) if inequality (10) holds for the total derivative of the functional \(V [x,t ]\) along any solution of \(x:t\mapsto x (t )\) of the system.
For some systems it can be profitable to examine the possibility of stabilizing the system by allowing a specific type of linear state feedback.
Definition 13
A system is stabilizable with respect to functional V and state feedback of a given type if the adding state feedback of that type results in a system that is stable with respect to the functional V with exponent \(\gamma>0\).
To illustrate the use of these definitions in the next two sections we will apply these definitions first in the case of a linear system with delay and then in the case of a scalar nonlinear neutral system with indirect control.
3 A Lyapunov-Krasovkii functional approach to a linear problem with delay
Theorem 2
Proof
Example 1
Corollary 2
Proof
This follows immediately from Theorem 2. □
Corollary 3
Proof
If we apply the change of basis \(y (t )=R^{-1}x (t )\), then this corollary follows immediately from the previous corollary. □
4 A scalar Lur’e system of neutral type with indirect control
Theorem 3
If there exist constants \(h>0\), \(g_{1}>0\), \(g_{2}>0\), \(\beta>0\), \(\zeta>0\) such that the matrix \(S [a_{1},a_{2},b,c,\rho,h,g_{1},g_{2},\beta,\zeta ]\) is positive definite, then the system (18), (19) is absolutely stable.
Proof
The proof of this theorem follows directly from Theorem 1. □
Example 2
Lemma 1
Proof
See [30], Theorem 1.12. □
Now we can use this to formulate another set of stability conditions.
Theorem 4
Proof
According to Lemma 1, S is positive definite if and only if \(W_{11}\) and \(W_{22}-W_{12}^{T}W_{11}^{-1}W_{12}\) are positive definite. This completes the proof. □
5 Stabilization
Theorem 5
The sufficient conditions of absolute stability of neutral-type indirect control system (1), (2) are the existence of the positive definite matrices \(W_{11}\) and \(W_{22}- (W_{12} )^{T} (W_{11} )^{-1}W_{12}\).
Proof
According to Lemma 1 the condition imposed on the matrices \(W_{11}\) and \(W_{22}- (W_{12} )^{T} (W_{11} )^{-1}W_{12}\) implies that \(S_{3}\) is positive definite. Theorem 1 now implies that the system is stable. □
Therefore, the absolute stability investigation problem is reduced to the task of checking of positive definiteness for two matrices, one of which is 2n-dimensional and the other is \(n+1\)-dimensional. Note that we can use Lemma 1 to reduce the proof of positive definiteness of the 2n-dimensional case to positive definiteness of two n-dimensional matrices.
Example 3
Theorem 6
Suppose that there are matrices \(C_{1}\), \(C_{2}\), and \(C_{3}\), such that the matrix \(S_{4}\) is positive definite. In that case the system (1), (2) is stabilizable with respect to the state feedback shown in (43), (44), and the functional (7).
Proof
The proof follows immediately from Theorem 1. □
Using the results of Lemma 1, we can replace verification of positive definiteness of matrix \(S_{4}\) by verification of positive definiteness of two matrices of lower dimensionality.
Theorem 7
6 Conclusions
We discussed the stabilization problem for an indirect control Lur’e system of neutral type. Based on the direct Lyapunov method (Lyapunov-Krasovskii approach) several stabilization criteria were given in terms of a set of matrix algebraic inequalities. A sufficient condition for absolutely stability of the closed loop system was presented.
Declarations
Acknowledgements
The authors would like to thank for their support: the first and the fourth authors were supported by the budget program 2201250 of the Ministry of Education and Science of Ukraine ‘Study and training of students, scientist and teachers on abroad’ of 2012.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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