Exponentially practical stability of discrete time singular system with delay and disturbance
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Abstract
In this paper, we consider exponentially practical stability of a discrete time singular system with disturbance. By using Lyapunov–Krasovskii stability theory, some criteria for exponentially practical stability of such a system are derived. Moreover, by using a Razumikhin-type technique, the criteria for exponentially practical stability of a discrete time singular system with delay and disturbance are also obtained. Some numerical examples are given to show the success of our theoretical results.
Keywords
Discrete time system Singular system Delay Exponentially practical stability Disturbance1 Introduction
Singular systems, which are also called descriptor systems, implicit systems, or generalized systems, have been investigated extensively in many areas [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Generally, the systems can be described using algebraic and differential equations. Such systems are natural presentations of several dynamic systems which are better than regular systems, such as economical systems, chemical systems, robotic systems, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 20, 21, 22]. Moreover, singular systems are very complicated because we have to consider the stability of the systems as well as the regularity and also impulse free (in case of continuous singular systems) or causality (in case of discrete singular systems) [2, 11, 18]. In addition, a discrete time system is often represented in the real world systems such as population models and switched systems. There are several studies on the stability of a discrete time system [2, 3, 4, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 22, 23, 24].
In real world systems, the variation of systems’ current status often depends not only on the current state but also on the past state of the systems; such systems are called time delay systems. Examples of time delay systems are population dynamic models, mechanical transmissions, and digital control systems [4, 7, 8, 9, 10, 12, 13, 14]. It is well known that time delay may cause instability, oscillation, and poor performance of systems. For the above-mentioned reasons, time delay systems have been extensively discussed in many literature works [2, 4, 6, 7, 8, 9, 10, 13, 14, 15, 16, 18, 19, 22, 23]. As is known, most common approaches to studying stability analysis of a time delay system are Lyapunov–Krasovskii functional approach and Razumikhin-type technique. In the case of Lyapunov–Krasovskii functional method, it requires that a candidate Lyapunov–Krasovskii functional is decreasing on the whole state space. Meanwhile, the Razumikhin-type technique has an advantage that the Lyapunov–Krasovskii functional is not required to be decreasing on the whole state space. In general, disturbance inputs often occur in modeling of phenomena and engineering systems which may be due to data transformation, unknown disturbances, or measurement errors [4, 10, 24, 25]. Therefore, it is important to study the stability of a discrete time singular system with delay and disturbance.
Considering asymptotic stability, it is more desirable to consider exponential stability criterion for dynamical systems [1, 6, 7, 8, 9, 10, 18, 19, 20, 21, 24, 25]. For exponential stability, it is required that all solutions starting near an equilibrium point not only stay nearby, but tend to the equilibrium point very fast with exponential decay rate. In practice, we may only need to stabilize a system into the region of a phase space where the system may oscillate near the state in which the implementation is still acceptable. This concept is called practical stability [18, 22, 26, 27, 28, 29] which is very useful for studying the asymptotic behavior of the system in which the origin is not necessarily an equilibrium point. In this case, practical stability is an important concept to analyze the asymptotic behavior of solutions with respect to a small neighborhood of the origin. Recently, there have been several studies on practical stability of continuous time systems with delay, see [26, 27, 28, 29]. However, there have been few studies on practical stability of discrete time systems with delay [22, 23]. In [3], the authors studied discrete time singular systems with disturbance and obtained some stability criteria by using Lyapunov stability theory. In [23], the authors used the Razumikhin-type technique to derive the exponentially practical stability condition for impulsive discrete time systems with delay. Motivated by the above discussions, we propose to study exponentially practical stability of a discrete time singular system with delay and disturbance. We shall derive a new criterion for exponentially practical stability of the system, namely the solutions tend to the origin state with exponential decay rate in the early stage (but eventually oscillate in a neighborhood of the origin), in which the performance is still acceptable.
This work is organized as follows. In Sect. 2, some notations and definitions are introduced. In Sect. 3, we present some criteria for exponentially practical stability of a discrete time system with disturbance, exponentially practical stability of a discrete time singular system with disturbance, exponentially practical stability of a discrete time system with delay and disturbance, and exponentially practical stability of a discrete time singular system with delay and disturbance; definitions and assumption will be used in the proof our result. Some numerical examples are given to show the effectiveness of our theoretical result in Sect. 4. The last section concludes the work.
2 Preliminaries
3 Main results
In this section, we consider exponentially practical stability problems for the following four cases: a discrete time system with disturbance, a discrete time singular system with disturbance, a discrete time system with delay and disturbance, and a discrete time singular system with delay and disturbance.
Definition 3.1
Theorem 3.1
- (i)
\(c_{1} \Vert x(k) \Vert ^{p}\leq V(k,x(k))\leq c_{2} \Vert x(k) \Vert ^{p}+a\),
- (ii)
\(\Delta V(k,x(k))=V(k+1,x(k+1))-V(k,x(k))\leq-c_{3} \Vert x(k) \Vert ^{p}+ \rho( \Vert w(k) \Vert )\).
Then system (3.1) is exponentially practically stable in thepth-moment with\(\eta= \frac{c_{2}}{c_{1}}, \lambda=1-\frac{c_{3}}{c_{2}}\), and\(r=\frac{a}{c_{1}}+\frac{a_{1}}{c_{1} ( 1-\sigma ) }\), where\(a_{1}=\frac{c_{3}a}{c_{2}}+\rho_{1}\)and\(\rho_{1}=\sup_{w(k)\in\Omega }\lbrace\rho( \Vert w(k) \Vert ) \rbrace\).
Proof
Definition 3.2
Theorem 3.2
- (i)
\(c_{1} \Vert Ex(k) \Vert ^{p}\leq V(k,x(k))\leq c_{2} \Vert Ex(k) \Vert ^{p}+a\),
- (ii)
\(\Delta V(k,x(k))=V(k+1,x(k+1))-V(k,x(k))\leq-c_{3} \Vert x(k) \Vert ^{p}+ \rho( \Vert w(k) \Vert )\)hold for some positive constants\(a, c_{1}, c_{2}, c_{3}, p; c_{3}< c_{2}\)and aK-functionρ.
Proof
- (i)
\(\overline{c}_{1} \Vert \overline{y_{1}}(k) \Vert ^{p}\leq V(k,\overline {y_{1}}(k))\leq\overline{c}_{2} \Vert \overline{y_{1}}(k) \Vert ^{p}+a\),
- (ii)
\(\Delta V(k,\overline{y_{1}}(k))=V(k+1,\overline {y_{1}}(k+1))-V(k,\overline{y_{1}}(k))\leq-c_{3} \Vert \overline{y_{1}}(k) \Vert ^{p}+ \rho( \Vert w(k) \Vert )\).
From \({ME}x_{0}=MEN y_{0}= \bigl({\scriptsize\begin{matrix}{}I_{r}& 0\cr 0&0 \end{matrix}}\bigr) y_{0}\) and \(y_{0}= \bigl({\scriptsize\begin{matrix}{}y_{10}\cr 0 \end{matrix}}\bigr) \), it follows that \(\Vert y_{10} \Vert \leq \Vert M \Vert \Vert E \Vert \Vert x_{0} \Vert \).
Therefore, the singular system (3.2) is exponentially practically stable with respect to \(w(k)\) with \(\eta=\eta_{6}, \lambda =\lambda_{2}\), and \(r=r_{7}\). □
Remark 3.1
Definition 3.3
Theorem 3.3
- (i)
\(c_{1} \Vert x(k) \Vert ^{p}\leq V(k,x(k))\leq c_{2} \Vert x(k) \Vert ^{p}+a\),
- (ii)If\(V(k+s,x(k+s))\leq qV(k+1,x(k+1))\)with\(s \in\mathbb {N}_{k_{0}-\tau}\), thenThen system (3.13) is exponentially practically stable in thepth-moment with\(\eta=\frac{c_{2}}{c_{1}}, q^{-\frac{1}{\tau+1}}\leq \lambda<1\), and\(r=\frac{a}{c_{1}}\).$$ \Delta V\bigl(k,x(k)\bigr)=V\bigl(k+1,x(k+1)\bigr)-V\bigl(k,x(k)\bigr)\leq-\beta V\bigl(k,x(k)\bigr)+ \rho \bigl( \bigl\Vert w(k ) \bigr\Vert \bigr). $$
Proof
Finally, we consider exponentially practical stability for system (2.1) with delay and disturbance.
Definition 3.4
In order to proceed with the main result on exponentially practical stability in the pth-moment of a singular system with delay and disturbance (2.1), we make the following assumption; an explanation for this assumption is given in Remark 3.2.
Assumption 3.1
There exist nonsingular matrices \(M,N\) with appropriate dimensions such that \(MEN= \bigl({\scriptsize\begin{matrix}{}I_{r}& 0\cr 0&0_{n-r} \end{matrix}}\bigr) \), \({MAN}= \bigl({\scriptsize\begin{matrix}{} A_{1}& 0\cr0&I_{n-r} \end{matrix}}\bigr) \), \({MBN}= \bigl({\scriptsize\begin{matrix}{}B_{1}& 0\cr B_{3}&B_{4} \end{matrix}}\bigr) \), and \({MG}= \bigl({\scriptsize\begin{matrix}{} G_{1}\cr G_{2} \end{matrix}}\bigr) \), where \(\Vert B_{4} \Vert <1\).
Remark 3.2
For a physical meaning of Assumption 3.1, it implies that there is a plant (\(y_{1}\)) in this situation which does not dynamically depend on the other plant (\(y_{2}\)). For future investigation, we propose to study a more general case in which \({MBN}= \bigl({\scriptsize\begin{matrix}{} B_{1}& B_{2}\cr B_{3}&B_{4} \end{matrix}}\bigr) \), where \(B_{2}\) may not be zero; namely, all plants dynamically interact, and that \(\Vert B_{4} \Vert \) may not be less than 1.
Theorem 3.4
- (i)
\(c_{1} \Vert Ex(k) \Vert ^{p}\leq V(k,x(k))\leq c_{2} \Vert Ex(k) \Vert ^{p}+a\),
- (ii)If\(V(k+s,x(k+s))\leq qV(k+1,x(k+1))\)with\(s \in\mathbb {N}_{k_{0}-\tau}\), thenhold for some positive constants\(a, c_{1}, c_{2}, p,q, \beta\), where\(q>1, 0<\beta<1\), and someK-functionρ.$$ \Delta V\bigl(k,x(k)\bigr)=V\bigl(k+1,x(k+1)\bigr)-V\bigl(k,x(k)\bigr)\leq-\beta V\bigl(k,x(k)\bigr)+ \rho \bigl( \bigl\Vert w(k ) \bigr\Vert \bigr) $$
Proof
- (i)
\(\bar{c_{1}} \Vert \overline{y_{1}}(k) \Vert ^{p}\leq V(k,\overline {y_{1}}(k))\leq\bar{c_{2}} \Vert \overline{y_{1}}(k) \Vert ^{p}+a\),
- (ii)If \(V(k+s,\overline{y_{1}}(k+s))\leq qV(k+1,\overline{y_{1}}(k+1))\) with \(s \in\mathbb{N}_{k_{0}-\tau}\), then$$ \Delta V\bigl(k,\overline{y_{1}}(k)\bigr)=V\bigl(k+1, \overline{y_{1}}(k+1)\bigr)-V\bigl(k,\overline {y_{1}}(k) \bigr)\leq-\beta V\bigl(k,\overline{y_{1}}(k)\bigr)+ \rho\bigl( \bigl\Vert w(k ) \bigr\Vert \bigr). $$
Therefore, the discrete time singular system (2.1) is exponentially practically stable with respect to \(w(k)\) with \(\eta=\eta_{8}, \lambda =\lambda_{2}\), and \(r=r_{8}\). □
Remark 3.3
From the method of proof of Theorem 3.4, it is clear that this method can be applied for a discrete time singular system with disturbance and time varying delay \(\tau(k)\) with \(0\leq\tau(k) \leq\tau, \tau>0\).
Remark 3.4
As is known, Razumikhin techniques only require less restrictive assumptions, namely they employ a type of Lyapunov–Krasovskii functional which is required to decrease only if a certain condition on the past state trajectory and the current state is satisfied. However, such Razumikhin-type techniques usually lead to a delay-independent criterion which is less conservative than a delay-dependent result, especially for constant delay systems. To deal with this conservativeness, several mathematical approaches have been considered in recent works, e.g., the LMI approach and the time-dependent Lyapunov functional method. Recently, in [31, 32], the Razumikhin technique was expressed by utilizing the LMI approach and the time-invariant Lyapunov functional method which avoid the conservativeness of the Razumikhin-type techniques. It is our future investigation to apply the above mentioned approaches to obtain less conservative criteria for exponentially practical stability of discrete time singular systems with delay and disturbance.
Remark 3.5
Obviously, exponential stability implies exponentially practical stability but not conversely. However, in several practical applications, it only needs to stabilize a system into the region of a phase space, namely the system may oscillate near the equilibrium point, in which the performance is still acceptable. To the best of our knowledge, the present work is the first result on exponentially practical stability of a discrete time singular system with delay and disturbance. Moreover, compared to [22] which proposed asymptotically practical stability criteria for a discrete time system with delay, we derive an exponentially practical stability condition which is more desirable.
4 Numerical examples
Remark 4.1
- 1.
First, we choose an appropriate Lyapunov functional or Lyapunov–Krasovskii functional candidate according to the assumptions of Theorem 3.2 or Theorem 3.4, respectively. Then, we estimate the values of \(c_{1},c_{2}\), and a which satisfy condition (i) of the corresponding theorems.
- 2.
From the estimations of \(c_{1}, c_{2}\), and a obtained in 1, we choose appropriate \(q, \beta\), \(c_{3}\), and ρ which satisfy condition (ii) of the corresponding theorems.
Example 4.1
- (i)
\(\Vert Ex(k) \Vert ^{2} \leq V(k,x(k)) \leq \Vert Ex(k) \Vert ^{2}+a\),
- (ii)$$\begin{aligned} &\Delta V\bigl(k,x(k)\bigr)\\ &\quad=V\bigl(k+1,x(k+1)\bigr)-V\bigl(k,x(k)\bigr) \\ &\quad= x^{T}(k+1)E^{T}Ex(k+1)+a-x^{T}(k)E^{T}Ex(k)-a \\ &\quad= x^{T}(k)A^{T}Ax(k)+2x^{T}(k)A^{T}Gw(k)+w^{T}(k)G^{T}Gw(k)-x^{T}(k)E^{T}Ex(k) \\ &\quad= \begin{bmatrix}x_{1}(k)\\x_{2}(k) \end{bmatrix} ^{T} \begin{bmatrix}0.5& 0\\0.5&1 \end{bmatrix} \begin{bmatrix}0.5& 0.5\\0&1 \end{bmatrix} \begin{bmatrix}x_{1}(k)\\x_{2}(k) \end{bmatrix} +2 \begin{bmatrix}x_{1}(k)\\x_{2}(k) \end{bmatrix} ^{T} \begin{bmatrix}0.5& 0\\0.5&1 \end{bmatrix} \begin{bmatrix}1\\1 \end{bmatrix} w(k) \\ &\qquad{}+w^{T}(k) \begin{bmatrix}1&1 \end{bmatrix} \begin{bmatrix}1\\1 \end{bmatrix} w(k)- \begin{bmatrix}x_{1}(k)\\x_{2}(k) \end{bmatrix} ^{T} \begin{bmatrix}1& 0\\0&0 \end{bmatrix} \begin{bmatrix}1& 0\\0&0 \end{bmatrix} \begin{bmatrix}x_{1}(k)\\x_{2}(k) \end{bmatrix} \\ &\quad= 0.25x_{1}^{2}(k)+0.5x_{1}(k)x_{2}(k)+1.25x_{2}^{2}(k)+x_{1}(k)w(k)+3x_{2}(k)w(k) \\ &\qquad{}+2w(k)^{2}-x_{1}(k)^{2} \\ &\quad= -0.75x_{1}^{2}(k)+1.25w^{2}(k)+0.5x_{1}(k)w(k)-w(k)^{2} \\ &\quad\leq-0.75x_{1}^{2}(k)+0.25x_{1}^{2}(k)+0.25w^{2}(k)+0.25w(k)^{2} \\ &\quad=-0.75x_{1}^{2}(k)+0.25x_{1}^{2}(k)+0.25x_{2}^{2}(k)+0.25w(k)^{2} \\ &\quad = -0.5x_{1}^{2}(k)+0.25x_{2}^{2}(k)+0.25w(k)^{2} \\ &\quad= -0.5x_{1}^{2}(k)-0.5x_{2}^{2}(k)+0.75x_{2}^{2}(k)+0.25w(k)^{2} \\ &\quad= -0.5 \bigl( x_{1}^{2}(k)+x_{2}^{2}(k) \bigr) +0.75w_{2}^{2}(k)+0.25w(k)^{2} \\ &\quad = -0.5 \bigl\Vert x(k) \bigr\Vert ^{2}+w^{2}(k). \end{aligned}$$
The trajectory of solution of system (3.2)
The trajectory of solution of system (3.2) without disturbance
Example 4.2
- (i)
\(\Vert Ex(k) \Vert \leq V(k,x(k))= \vert x_{1}(k) \vert +a \leq \Vert Ex(k) \Vert +a\),
- (ii)If \(V(k+s,x(k+s))\leq qV(k+1,x(k+1))\) with \(s\in\mathbb{N}_{-\tau }\), then we haveThus,$$\begin{aligned} &x_{1}(k+1)=-0.05x_{1}(k)+0.4x_{1}(k- \tau)+0.5w(k), \\ &\bigl\vert x_{1}(k+1) \bigr\vert \leq0.05 \bigl\vert x_{1}(k) \bigr\vert +0.4 \bigl\vert x_{1}(k-\tau ) \bigr\vert + 0.5 \bigl\vert w(k) \bigr\vert , \\ &\bigl\vert x_{1}(k+1) \bigr\vert +a-a \leq0.05 \bigl\vert x_{1}(k) \bigr\vert +0.05 a-0.05 a+ 0.4 \bigl\vert x_{1}(k-\tau) \bigr\vert + 0.4 a- 0.4 a \\ &\phantom{\bigl\vert x_{1}(k+1) \bigr\vert +a-a \leq}{}+ 0.5 \bigl\vert w(k) \bigr\vert , \\ &\bigl\vert x_{1}(k+1) \bigr\vert +a \leq0.05 \bigl( \bigl\vert x_{1}(k) \bigr\vert +a \bigr)+ 0.4 \bigl( \bigl\vert x_{1}(k-\tau) \bigr\vert +a \bigr)+0.55a + 0.5 \bigl\vert w(k) \bigr\vert , \\ &V\bigl(k+1,x(k+1)\bigr)\leq0.05V\bigl(k,x(k)\bigr)+0.4V\bigl(k-\tau,x(k-\tau) \bigr)+0.55a + 0.5 \bigl\Vert w(k) \bigr\Vert \\ &\phantom{V\bigl(k+1,x(k+1)\bigr)}\leq0.05V\bigl(k,x(k)\bigr)+0.4qV\bigl(k+1,x(k+1)\bigr)+0.55a + 0.5 \bigl\Vert w(k) \bigr\Vert \\ &\phantom{V\bigl(k+1,x(k+1)\bigr)}\leq\frac{0.05}{1-0.4q}V\bigl(k,x(k)\bigr)+\frac{0.55a + 0.5 \Vert w(k) \Vert }{1-0.4q}. \end{aligned}$$where \(\beta\leq1-\frac{0.05}{1-0.4q}\) and \(\rho( \Vert w(k) \Vert )=\frac{0.55a + 0.5 \Vert w(k) \Vert }{1-0.4q}\).$$\begin{aligned} \Delta V\bigl(k,x(k)\bigr)&=V\bigl(k+1,x(k+1)\bigr)-V\bigl(k,x(k)\bigr) \\ &\leq\frac{0.05}{1-0.4q}V\bigl(k,x(k)\bigr)+\frac{0.55a + 0.5 \Vert w(k) \Vert }{1-0.4q}-V\bigl(k,x(k)\bigr)\\ &=- \biggl( 1-\frac{0.05}{1-0.4q} \biggr)V\bigl(k,x(k)\bigr)+\frac{0.55a + 0.5 \Vert w(k) \Vert }{1-0.4q} \\ &=-\beta V\bigl(k,x(k)\bigr)+\rho\bigl( \bigl\Vert w(k) \bigr\Vert \bigr), \end{aligned}$$
The trajectory of solution of system (2.1)
The trajectory of solution of system (2.1) without disturbance
5 Conclusion
In this paper, exponentially practical stability of a discrete time singular system with delay and disturbance has been investigated. For systems with disturbance but without delay, by using Lyapunov stability theory, we obtained a criterion for exponentially practical stability of a general discrete time system and a linear discrete time singular system, respectively. For systems with delay and disturbances, by using the Razumikhin-type technique, we derived exponentially practical stability criteria for a general discrete time system and a linear singular system, respectively. Numerical examples were given to show effectiveness of our theoretical results.
Notes
Acknowledgements
The first author is supported by student scholarship from the Human Resources Development in Science Project (Science Achievement Scholarship of Thailand SAST). The second author is supported by Chiang Mai University. This research is also (partially) supported by the Center of Excellence in Mathematics, The Commission on Higher Education, Thailand.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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