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Stabilization of Lur’etype nonlinear control systems by LyapunovKrasovskii functionals
Advances in Difference Equations volume 2012, Article number: 229 (2012)
Abstract
The paper deals with the stabilization problem of Lur’etype nonlinear indirect control systems with timedelay argument. The sufficient conditions for absolute stability of the control system are established in the form of matrix algebraic inequalities and are obtained by the direct Lyapunov method.
MSC:34H15, 34K20, 93C10, 93D05.
1 Introduction
One of the problems of stability motion is the problem of absolute stability. The problems of absolute stability of nonlinear control systems arise in solving practical tasks. In technical control systems, the control function is the function of one variable located between two lines in the first and third quarters of the coordinate plane. The stability of the control system with a control function located in this sector is referred to, for example, in [1–6]. Originally, the control systems of ordinary differential equations were considered. The systems with aftereffect, that better describe the real processes, become an object of study later, e.g., in [3, 4, 7, 8]. Some nonlinear systems with indirect regulation and delay argument are considered in [9, 10]. The sufficient conditions of absolute interval stability are derived in the papers [3, 6] by LyapunovKrasovskii functionals in the form of the sum of the quadratic form and the integral of nonlinear components of the considered system, and by the socalled Sprogram the coefficients of the exponential decay of solutions are calculated. But, in the case when the conditions of the theorems quoted there are not met, the linear feedback method is used to stabilize the system.
The main goal of the paper is to solve the problem of stabilization of an indirect control system. The sufficient conditions for absolute stability of the control system are obtained using LyapunovKrasovskii functionals which contain an exponential multiplier.
Throughout the paper we will use the following notation. Let $\mathcal{S}$ be a real symmetric square matrix. Then the symbol ${\lambda}_{min}(\mathcal{S})$ (${\lambda}_{max}(\mathcal{S})$) will denote the minimal (maximal) eigenvalue of $\mathcal{S}$. We will also use the following vector norms:
where $x={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{T}$ and ξ is a real parameter.
The paper is organized as follows. Since for the onedimensional process it is possible to get simple explicit criteria, Section 2 deals with the stabilization of onedimensional processes described by two scalar equations with delay. Then the indirect control system in the general matrix form is considered in Section 3.
2 Stabilization of onedimensional processes
Let us consider an indirect control system described by a system of two scalar equations with delay argument in the form
where $t\ge {t}_{0}\ge 0$, x is the state function, σ is the control defined on $[{t}_{0},\mathrm{\infty})$, ${a}_{1}$, ${a}_{2}$, b, c, $\tau >0$, $\rho >0$ are constants, $f(\sigma )$ is a continuous nonlinear function on ℝ satisfying the socalled sector condition. It means there exist constants ${k}_{1}$, ${k}_{2}$, ${k}_{2}>{k}_{1}>0$ such that inequalities
are satisfied.
Definition 1 The continuous vector function $(x,\sigma ):[{t}_{0}\tau ,\mathrm{\infty})\to {\mathbb{R}}^{2}$ is said to be a solution of (1), (2) on $[{t}_{0},\mathrm{\infty})$ if $(x,\sigma )$ is continuously differentiable on $[{t}_{0},\mathrm{\infty})$ and satisfies the system (1), (2) on $[{t}_{0},\mathrm{\infty})$.
Definition 2 The system (1), (2) is called absolutely stable if the trivial solution $(x,\sigma )=(0,0)$ of the system (1), (2) is globally asymptotically stable for an arbitrary function $f(\sigma )$ satisfying (3).
In the investigation of absolute stability of the control systems with delay, we will use LyapunovKrasovskii functionals which contain, in addition to the quadratic form and the integral of the nonlinear component of the considered system, an exponential multiplier, i.e.,
where h, g, β, ξ are positive constants, $(x,\sigma )$ is a solution of (1), (2), and $t\ge {t}_{0}$. It is easy to see that the last term in (4) is always nonnegative due to the lefthand part of ‘sector condition’ (3). Define, using the coefficients of the functional (4), auxiliary numbers
and a matrix
Our first result is the theorem on absolute stability for the considered system (1), (2).
Theorem 1 Suppose that there exist constants $g>0$, $h>0$, $\beta >0$, and $\xi >0$ such that the matrix ${S}_{1}(g,h,\beta ,\xi )$ is positive definite. Then the system (1), (2) is absolutely stable.
Proof Compute the full derivative of the functional $V[x(t),\sigma (t)]$ defined by (4) along trajectories of the system (1), (2). Then
Using (3) we get
From this inequality and the estimates
where the last term can be derived using the righthand part of ‘sector condition’ (3), we deduce the absolute stability of the system (1), (2) (we also refer to a theorem by Krasovskii in [[11], Theorem 2, p.145]). □
The crucial assumption in Theorem 1 is the assumption of positive definiteness of the matrix ${S}_{1}(g,h,\beta ,\xi )$. If we cannot find suitable constants g, h, β, and ξ to ensure positive definiteness, or such constants do not exist, Theorem 1 is not applicable. In such a case, we can modify the control function in (1) by adding a linear combination of the values of the state function at the moments t and $t\tau $ and we will consider a modified system
where
${c}_{1}$ and ${c}_{2}$ are suitable constants, and $t\ge {t}_{0}\ge 0$.
Then we can apply the following result.
Theorem 2 Let $g>0$, $h>0$, $\beta >0$, and $\xi >0$ be fixed. Then the system (5), (6) is absolutely stable if the constants ${c}_{1}$, ${c}_{2}$ in the control function (7) fulfill the inequality
Proof We employ the same functional (4) and the scheme of the proof of Theorem 1. Tracing the proof of Theorem 1, we get that for the absolute stability of the system (5), (6), it is sufficient that the matrix
is positive definite. Applying the known positivity criterion (Sylvester criterion) [[12], p.260], [13] to the matrix ${S}_{2}$, we require the positivity of its main diagonal minors, i.e.,
The inequality (9) can be rewritten as
By a simple modification of inequality (10), we have
Hence, taking into account that ${s}_{22}^{1}={e}^{\xi \tau}g>0$, we get a more suitable relationship,
The inequality (11) can be modified into the form
Finally, with regard to the assumptions
the inequality (14) can be written in the form (8). If this inequality holds, then obviously (12) and (13) hold as well. Moreover, it is easy to see that it is possible to find parameters ${c}_{1}$ and ${c}_{2}$ such that the inequality (8) is fulfilled. □
3 Stabilization of the indirect control systems with matrix coefficients
Our goal in this section is to extend the considerations developed in Section 2 to the study of stabilization of the indirect control systems whose coefficients are expressed in a matrix form. It means we will consider an ndimensional process x described by the system of $(n+1)$ equations,
where $t\ge {t}_{0}\ge 0$, $x={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{T}$ is the ndimensional column vector function of the state, σ is the scalar function of the control defined on $[{t}_{0},\mathrm{\infty})$, A and B are $n\times n$ constant matrices, $b={({b}_{1},{b}_{2},\dots ,{b}_{n})}^{T}$ is an ndimensional constant column vector, $c=({c}_{1},{c}_{2},\dots ,{c}_{n})$ is an ndimensional constant row vector, $\tau >0$ and $\rho >0$ are constants, and $f(\sigma )$ is a continuous nonlinear function on ℝ satisfying sector condition (3).
To investigate the system (15), (16) we use a LyapunovKrasovskii functional, generalizing the functional (4), in the form
where H and G are $n\times n$ constant positive definite symmetric matrices, and ξ and β are positive constants.
We give a generalization of Theorem 1 to the case of the control system (15), (16). For it, we define the matrices
and
where $\theta ={(\theta ,\theta ,\dots ,\theta )}^{T}$ is an ndimensional zero column vector.
Theorem 3 Suppose that there exist positive definite symmetric matrices H, G and constants $\beta >0$, $\xi >0$ such that the matrix ${S}_{3}(G,H,\beta ,\xi )$ is positive definite. Then the system (15), (16) is absolutely stable.
Proof The scheme of the proof repeats the proof of Theorem 1. Compute the full derivative of the functional $V[x(t),\sigma (t)]$ defined by (17) along trajectories of the system (15), (16). Then
Using (3) we get
From this inequality and the estimates
where the last term can be derived using the righthand part of sector condition (3), we deduce the absolute stability of the system (15), (16) (we also refer to a theorem by Krasovskii in [[11], Theorem 2, p.145]). □
It may happen that it is not easy to find suitable positive definite symmetric matrices H, G and constants $\beta >0$, $\xi >0$ such that the matrix ${S}_{3}(G,H,\beta ,\xi )$ will be positive definite, or such matrices and constants do not exist. In such a case, we can modify the control function in the system (15), (16) by adding a linear combination of the values of the state function at the moments t and $t\tau $. Therefore, instead of the system (15), (16), we will consider a modified system
where
${C}_{1}$ and ${C}_{2}$ are $n\times n$ constant matrices (the socalled control matrices), and $t\ge {t}_{0}\ge 0$. Our task is to find conditions on the matrices ${C}_{1}$, ${C}_{2}$ such that the system (18), (19) will be absolutely stable.
We will need some auxiliary results from the theory of matrices.
Lemma 1 [13]
Let A be a regular $n\times n$ matrix, B be an $n\times q$ matrix, and C be a $q\times q$ regular matrix. Let a Hermitian matrix S be represented as
Then the matrix S is positive definite if and only if the matrices A and $C{B}^{\ast}{A}^{1}B$ are positive definite.
Lemma 2 [[12], Frobenius formula]
Let A be a regular $n\times n$ matrix, D be a $q\times q$ matrix, B be an $n\times q$ matrix, and C be a $q\times n$ matrix, and the matrix
be regular. Then the matrix $R=DC{A}^{1}B$ is regular and
Theorem 4 Suppose that there exist positive definite symmetric matrices H and G, control matrices ${C}_{1}$ and ${C}_{2}$, and constants $\beta >0$ and $\xi >0$ such that

(1)
The matrices
$${\mathrm{\Delta}}_{1}^{4}:={S}_{11}^{3}{C}_{1}^{T}HH{C}_{1},$$(21)
are positive definite.

(2)
The number
$${\mathrm{\Delta}}_{3}^{4}:=\beta \rho {\left({S}_{13}^{3}\right)}^{T}[{\left({S}_{11}^{4}\right)}^{1}+{\left({S}_{11}^{4}\right)}^{1}{S}_{12}^{4}{\mathcal{R}}^{1}{\left({S}_{12}^{4}\right)}^{T}{\left({S}_{11}^{4}\right)}^{1}]{S}_{13}^{3},$$(23)
where
is positive.
Then the system (18), (19) is absolutely stable.
Proof The philosophy of the proof is the same as in the proof of Theorem 2, only the calculations will be more complicated, because now we work with the matrix case. In accordance with Theorem 3, the system (15), (16) is absolutely stable if the matrix ${S}_{3}(G,H,\beta ,\xi )$ is positive definite. Define the auxiliary matrix
The matrix ${S}_{4}$ plays the same role for the system (18), (19) as the matrix ${S}_{3}(G,H,\beta ,\xi )$ for the system (15), (16). Therefore, the system (18), (19) is absolutely stable if the matrix ${S}_{4}(G,H,{C}_{1},{C}_{2},\beta ,\xi )$ is positive definite. It follows from Lemma 1 that the matrix ${S}_{4}(G,H,{C}_{1},{C}_{2},\beta ,\xi )$ is positive definite if and only if the matrix
is positive definite and the inequality
holds.
The matrix ${M}_{4}$ is positive definite (we use Lemma 1 again) if and only if the matrices
are positive definite. The matrix ${S}_{11}^{4}$ is positive definite due to (21). The matrix
is positive definite due to (22).
We compute the inverse matrix to the matrix ${M}_{4}$ using Lemma 2. We get
where
Therefore, the inequality (25) can be rewritten as
and is valid due to (23).
Consequently, the system with control of the form (18), (19) is absolutely stable if there exist matrices ${C}_{1}$, ${C}_{2}$ in (20) such that conditions (21)(23) are valid. □
Remark 1 Let us recall the wellknown facts that for the validity of Theorem 1, it is necessary that ${a}_{1}<0$ and for the validity of Theorem 3, it is necessary that all characteristic values of the matrix A have negative real parts.
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Acknowledgements
The authors would like to thank the following for their support: The first and the third authors were supported by the National Scholarship Program of the Slovak Republic (SAIA), the second author was supported by Grant P201/11/0768 of the Czech Grant Agency (Prague), the fourth author was supported by the Grant Agency of the Slovak republic (VEGA 1/0090/09). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.
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Shatyrko, A., Diblík, J., Khusainov, D. et al. Stabilization of Lur’etype nonlinear control systems by LyapunovKrasovskii functionals. Adv Differ Equ 2012, 229 (2012). https://doi.org/10.1186/168718472012229
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Keywords
 Lyapunov functional
 absolute stability
 timedelay argument
 stabilization