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Stability analysis of discretetime systems with variable delays via some new summation inequalities
Advances in Difference Equations volume 2016, Article number: 95 (2016)
Abstract
This paper proposes an improved stability condition of discretetime systems with variable delays. Based on some mathematical techniques, a series of new summation inequalities are obtained. These new inequalities are less conservative than the Jensen inequality. Based on these new summation inequalities and the reciprocally convex combination inequality, a novel sufficient criterion on asymptotical stability of discretetime systems with variable delays is obtained by constructing a new LyapunovKrasovskii functional. The advantage of the proposed inequality in this paper is demonstrated by a classical example from the literature.
Introduction
Time delay is usually encountered in many practical situations such as signal processing, image processing etc. There has been an increasing research activity on timedelay systems during the past years [1–16]. The problem of the delaydependent stability analysis of timedelay systems has become a hot research topic in the control community [17, 18] due to the fact that stability criteria can provide a maximum admissible upper bound of time delay. The maximum admissible upper bound can be regarded as an important index for the conservatism of stability criteria [19–23]. To our knowledge, Jensen’s inequality has been mostly used as a powerful mathematical tool in the stability analysis of timedelay systems. However, Jensen’s inequality neglected some terms, which unavoidably introduced some conservatism. In order to investigate the stability of a linear discrete systems with constant delay, Zhang and Han [24] established the following Abel lemmabased finitesum inequality, which improved the Jensen inequality to some extent.
Theorem A
[24]
For a constant matrix \(R\in R^{n\times n}\) with \(R=R^{T}>0\), and two integers \(r_{1}\) and \(r_{2}\) with \(r_{2}r_{1}>1\), the following inequality holds:
where \(\eta(j)=x(j+1)x(j)\), \(\nu_{1}=x(r_{2})x(r_{1})\), \(\nu_{2}=x(r_{2})+x(r_{1})\frac{2}{r_{2}r_{1}1}\sum_{j=r_{1}+1}^{r_{2}1}x(j)\), \(\rho_{1}=r_{2}r_{1}\), \(\rho_{2}=r_{2}r_{1}1\), \(\rho_{3}=r_{2}r_{1}+1\).
Seuret et al. [25] also obtained a new stability criterion for the discretetime systems with timevarying delay via the following novel summation inequality.
Theorem B
[25]
For a given symmetric positive definite matrix \(R \in R^{n\times n}\) and any sequence of discretetime variables z in \([h, 0]\cap Z\rightarrow R^{n}\), where \(h\geq1\), the following inequality holds:
where \(y(i)=z(i)z(i1)\), \(\Theta_{0}=z(0)z(h)\), \(\Theta_{1}=z(0)+z(h)\frac{2}{h+1}\sum_{i=h}^{0}z(i)\).
In fact, Theorem A is equivalent to Theorem B. These two summation inequalities encompass the Jensen inequality. It is worth mentioning that Theorem A and Theorem B can be regarded as a discrete time version of the Wirtingerbased integral inequality, which was proved in [26].
Recently, Park et al. [27] developed a novel class of integral inequalities for quadratic functions via some intermediate terms called auxiliary functions which improved the Wirtingerbased integral inequality. Based on the novel inequalities, some new stability criteria are presented for systems with timevarying delays by constructing some appropriate LyapunovKrasovskii functionals in [27].
The LyapunovKrasovskii functional method is the most commonly used method in the investigation of the stability of delayed systems. The conservativeness of this approach is mainly from the construction of the LyapunovKrasovskii functional and the estimation of its time derivative. In order to get less conservative results, Jensen’s integral inequality, Wirtinger’s integral inequality, and a freematrixbased integral inequality are proposed to obtain a tighter upper bound of the integrals occurring in the time derivative of the LyapunovKrasovskii functional. Many papers have focused on integral inequalities and their applications in stability analysis of continuoustimedelayed systems. However, only a few papers have studied the summation inequalities and their application in stability analysis of discretetime systems with variable delays. The summation inequalities in Theorem A and Theorem B are used to obtain a bound for \(\sum_{j=r_{1}}^{r_{2}1}\eta^{T}(j)R\eta(j)\) or \(\sum_{i=h+1}^{0}y^{T}(i)Ry(i)\).
Motivated by the above works, in order to provide a tighter bound for \(\sum_{j=r_{1}}^{r_{2}1}\eta^{T}(j)R\eta(j)\) or \(\sum_{i=h+1}^{0}y^{T}(i)Ry(i)\), this paper is aimed at establishing some novel summation inequalities as the discretetime versions of the integral inequalities obtained in [27]. In this paper, we will extend the two summation inequalities given in [24, 25]. Some new summation inequalities are proposed to provide a sharper bound than the summation inequalities in [24, 25]. The inequalities in Theorem A and Theorem B are a special case of Corollary 6 in our paper. Moreover, a novel estimation to the double summation as \(\sum_{i=h+1}^{0}\sum_{k=i}^{0}\Delta x(k)^{T}R \Delta x(k)\) is also given in this paper. Based on these new summation inequalities, the reciprocally convex combination inequality, and a new LyapunovKrasovskii functional, a less conservative sufficient criterion on asymptotical stability of discretetime systems with variable delays is obtained.
Notations
Throughout this paper, \(R^{n}\) and \(R^{n\times m}\) denote, respectively, the ndimensional Euclidean space and the set of all \(n\times m\) real matrices. For real symmetric matrices X and Y, the notation \(X\geq Y\) (or \(X>Y\)) means that the matrix \(XY\) is a positive semidefinite (or positive definite). The symbol ∗ within a matrix represents the symmetric term of the matrix.
Novel summation inequalities
Theorem 1
For a positive definite matrix \(R>0\), any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), and any sequence of discretetime variables \(p:[h, 0]\cap Z\rightarrow R\) satisfying \(\sum_{k=h+1}^{0}p(k)=0\), the following inequality holds:
where \(\Theta_{0}= \sum_{k=h+1}^{0}y(k)\).
Proof
Let \(z(i)=y(i)\frac{1}{h}\Theta_{0}p(i)v \), where \(v\in R^{n}\) is to be defined later. Then find a vector v̂ to minimize the following energy function \(J(v)\):
Obviously,
If \(\sum_{k=h+1}^{0}p^{2}(k)>0\), solving the equation \(J'(v)=0\) gives
Substituting v̂ for v in \(J(v)\), we get
By the nonnegative characteristic of the energy function \(J(v)\), we have
If \(\sum_{k=h+1}^{0}p^{2}(k)=0\), obviously, inequality (3) holds.
This completes the proof of Theorem 1. □
By choosing an appropriate sequence \(p(k)\), we get the following corollaries.
Corollary 1
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), the following inequality holds:
where \(\Theta_{0}=\sum_{k=h+1}^{0}y(k)\), \(\Omega_{1}=\sum_{s=h+1}^{0}y(s) \frac{2}{h+1}\sum_{k=h+1}^{0}\sum_{s=h+1}^{k}y(s)\).
Proof
Let \(p(k)=h1+2k\), then \(\sum_{k=h+1}^{0}p(k)=0\) and \(\sum_{i=h+1}^{0}p^{2}(i)=\frac{(h1)h(h+1)}{3}\),
By using Theorem 1, inequality (9) holds. □
Let \(\Omega_{1}^{*}=\sum_{s=h+1}^{0}y(s) \frac{2}{h+1}\sum_{k=h+1}^{0}\sum_{s=k}^{0}y(s)\). Due to
we have
Hence, Corollary 1 is equivalent to Corollary 2.
Corollary 2
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), the following inequality holds:
where \(\Theta_{0}= \sum_{k=h+1}^{0}y(k)\), \(\Omega_{1}^{*}=\sum_{s=h+1}^{0}y(s) \frac{2}{h+1}\sum_{k=h+1}^{0}\sum_{s=k}^{0}y(s)\).
Corollary 3
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(x:[h, 0]\cap Z\rightarrow R^{n}\), the following inequality holds:
where \(\Delta x(i)=x(i)x(i1)\), \(\Theta_{0}=x(0)x(h)\), \(\Theta_{1}=x(0)+x(h)\frac{2}{h+1}\sum_{k=h}^{0}x(k)\).
Proof
Let \(y(i)=\Delta x(i)=x(i)x(i1)\) in Corollary 1. Then we have
and
Using Corollary 1, we have completed the proof of Corollary 3. □
Remark 1
Corollary 1 or Corollary 2 in this paper can be regard as a discrete version of Wirtingerbased integral inequality proved in [26]. Corollary 3 is a special case of Corollary 1 or Corollary 2. In fact, Corollary 3 is equivalent to Theorem A and Theorem B. So Corollary 1 or Corollary 2 in this paper implies Theorem A and Theorem B.
Generally, we have the following result which includes Corollary 3 as a special case.
Corollary 4
For a positive definite matrix \(R>0\), any sequence of discretetime variables \(x:[h, 0]\cap Z\rightarrow R^{n}\), and any sequence of discretetime variables \(p:[h, 0]\cap Z\rightarrow R\) satisfying \(\sum_{k=h+1}^{0}p(k)=0\), the following inequality holds:
where \(\Theta_{0}=x(0)x(h)\), \(\Delta x(i)=x(i)x(i1)\).
To go a step further, suppose that \(p_{1}(i)=h1+2i\), \(p_{2}(i)=i^{2}+(h1)i+\frac{(h1)(h2)}{6}\), then

(1)
\(\sum_{i=h+1}^{0}p_{m}(i)=0\), \(m=1, 2\),

(2)
\(\sum_{i=h+1}^{0}p_{1}(i)p_{2}(i)=0\),

(3)
\(\sum_{i=h+1}^{0}p_{2}^{2}(i)=\frac {(h2)(h1)h(h+1)(h+2)}{180}\).
Noting that
and
Then we get
Let \(p(k)=p_{2}(k)\) in Theorem 1, we have the following theorem.
Theorem 2
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\) the following inequality holds:
where \(\Omega_{2}=\sum_{i=h+1}^{0}y(i)\frac{6}{h+1}\sum_{i=h+1}^{0} \sum_{k=i}^{0}y(k) +\frac{12}{(h+1)(h+2)}\sum_{i=h+1}^{0}\sum_{k=i}^{0} \sum_{m=k}^{0}y(m)\), \(\Theta_{0}=\sum_{k=h+1}^{0}y(k)\).
Remark 2
Theorem 2 gives a new form of summation inequality and the idea which stimulates our interests in establishing a novel combinational summation inequality underlying quadrature rules. Based on Theorem 1 and Theorem 2, an improved summation inequality can be obtained as follows.
Theorem 3
For a positive definite matrix \(R>0\), any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), and any two sequences of discretetime variables \(p_{i}:[h, 0]\cap Z\rightarrow R\) satisfying \(\sum_{k=h+1}^{0}p_{i}(k)=0\), \(i=1, 2\), \(\sum_{k=h+1}^{0}p_{1}(k)p_{2}(k)=0\), then the following inequality holds:
where \(\Theta_{0}= \sum_{k=h+1}^{0}y(k)\).
Proof
Let \(z(i)=y(i)\frac{1}{h}\Theta_{0} \frac{p_{1}(i)}{\sum_{i=h+1}^{0}p_{1}(i)^{2}}\sum_{i=h+1}^{0}y(i)p_{1}(i)\). Based on the proof of Theorem 1, we have
Let \(x(i)=z(i)\frac{p_{2}(i)}{\sum_{i=h+1}^{0}p_{2}(i)^{2}}\sum_{i=h+1}^{0}z(i)p_{2}(i)\). Similarly, we have
So
Since \(\sum_{k=h+1}^{0}p_{i}(k)=0\) (\(i=1, 2\)) and \(\sum_{k=h+1}^{0}p_{1}(k)p_{2}(k)=0\), we obtain
This completes the proof of Theorem 3. □
Noting that \(\sum_{i=h+1}^{0}p_{m}(i)=0\) (\(m=1, 2\)) and \(\sum_{i=h+1}^{0}p_{1}(i)p_{2}(i)=0\), combining Theorem 3 with Corollary 2 and Theorem 2 gives the following result.
Corollary 5
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), the following inequality holds:
where \(\Theta_{0}= \sum_{k=h+1}^{0}y(k)\), \(\Omega_{1}^{*}=\sum_{s=h+1}^{0}y(s) \frac{2}{h+1}\sum_{k=h+1}^{0}\sum_{s=k}^{0}y(s)\), \(\Omega_{2}=\sum_{i=h+1}^{0}y(i) \frac{6}{h+1}\sum_{i=h+1}^{0}\sum_{k=i}^{0}y(k) +\frac{12}{(h+1)(h+2)}\sum_{i=h+1}^{0}\sum_{k=i}^{0}\sum_{m=k}^{0}y(m)\).
Corollary 6
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), the following inequality holds:
where \(\Theta_{0}= x(0)x(h)\), \(\Omega_{1}=x(0)+x(h)\frac{2}{h+1}\sum_{k=h}^{0}x(k)\), \(\Omega_{2}=x(0)x(h)+\frac{6h}{(h+1)(h+2)} \sum_{i=h}^{0}x(i) \frac{12}{(h+1)(h+2)}\sum_{i=h+1}^{0}\sum_{k=i}^{0}x(k)\).
Proof
Let \(y(i)=\Delta x(i)=x(i)x(i1)\), \(\Omega_{1}=\Omega_{1}^{*}\) in Corollary 5. Then \(\Theta_{0}=\sum_{k=h+1}^{0}y(k)=x(0)x(h)\). Simple computation leads to
and
An identical transformation leads to
This completes the proof of Corollary 6. □
Remark 3
The righthand side of summation inequality in Corollary 5 (or Corollary 6) contains a term \(\frac{5(h+1)(h+2)}{(h2)(h1)h}\Omega _{2}^{T}R\Omega_{2}\). However, the summation inequality in Theorem A or Theorem B neglects this term. If \(h>2\) and \(\Omega_{2}\neq0\), then \(\frac {5(h+1)(h+2)}{(h2)(h1)h}\Omega_{2}^{T}R\Omega_{2}>0\). Since a positive quantity is added in the righthand side of the inequality, the summation inequality in Corollary 5 (or Corollary 6) can provide a sharper bound for \(\sum_{i=h+1}^{0}y^{T}(i)Ry(i)\) than the summation inequalities in [24, 25].
As we have mentioned before, Jensen’s inequality has mostly been used as a powerful mathematical tool in dealing with the difference of LyapunovKrasovskii functionals, single or double. In the case of a double, just like \(\sum_{i=h+1}^{0}\sum_{k=i}^{0} y(k)^{T}Ry(k)\), Jensen’s inequality may neglect some terms, which unavoidably introduces conservatism. Then we will give some improved double summation inequalities.
Theorem 4
For a positive definite matrix \(R>0\), any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), and any nonzero sequence of discretetime variables \(p:[h, 0]\cap Z\rightarrow R\) satisfying \(\sum_{i=h+1}^{0}\sum_{k=i}^{0}p(k)=0\), the following inequality holds:
where \(E_{0}=\sum_{i=h+1}^{0}\sum_{k=i}^{0}y(k)\), \(E_{1}=\sum_{i=h+1}^{0}\sum_{k=i}^{0}y(k)p(k)\).
Proof
Define the energy function as \(J(v)=\sum_{i=h+1}^{0}\sum_{k=i}^{0}z(k)^{T}Rz(k)\) and \(z(i)=y(i)\frac{2}{h(h+1)}E_{0}p(i)v \). Similar to the proof of Theorem 1, we are now proceeding to find a vector v̂ to minimize the energy function \(J(v)\).
If \(\sum_{i=h+1}^{0}\sum_{k=i}^{0}p(k)^{2}>0\) and \(\sum_{i=h+1}^{0}\sum_{k=i}^{0}p(k)=0\), then
The solution v̂ of \(J'(v )=0\) can be found as
In this case, we have
This completes the proof of Theorem 4. □
Specially, the choice of \(p(k)\) in Theorem 4 as \(p_{3}(k)=3k+h1\) satisfying
yields
and
Let \(\Omega_{3}=\sum_{i=h+1}^{0}\sum_{k=i}^{0}y(k) \frac{3}{h+2}\sum_{i=h+1}^{0}\sum_{k=i}^{0}\sum_{m=k}^{0}y(m)\). Then the following inequality based on Theorem 4 holds:
Furthermore, we have the following corollary.
Corollary 7
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), the following inequality holds:
where \(\Omega_{3}=\sum_{i=h+1}^{0}\sum_{k=i}^{0}y(k)\frac {3}{h+2}\sum_{i=h+1}^{0}\sum_{k=i}^{0}\sum_{m=k}^{0}y(m)\).
Corollary 8
For a positive definite matrix \(R>0\) and any sequence of discretetime variables \(y:[h, 0]\cap Z\rightarrow R^{n}\), the following inequality holds:
where \(\Omega_{4}=[x(0)+\frac{2}{(h+1)}\sum_{i=h}^{0}x(i)\frac {6}{(h+1)(h+2)}\sum_{i=h}^{0}\sum_{k=i}^{0}x(k)]\).
Proof
Let \(y(i)=\Delta x(i)=x(i)x(i1)\) in Corollary 7, we have
and
So
Replacing \(y(i)\) by \(\Delta x(i)\) in Corollary 7 leads to
where \(\Omega_{4}=[x(0)+\frac{2}{(h+1)}\sum_{i=h}^{0}x(i) \frac{6}{(h+1)(h+2)}\sum_{i=h}^{0}\sum_{k=i}^{0}x(k)]\).
This completes the proof of Corollary 8. □
Remark 4
The double Jensen inequality is often used to estimate a upper bound of \(\sum_{i=h+1}^{0}\sum_{k=i}^{0} y(k)^{T}Ry(k)\) in the difference of LyapunovKrasovskii functionals. In this paper, we have extended the double Jensen inequality. Some improved double summation inequalities are presented in Corollary 7 (or Corollary 8). Since these improved double summation inequalities contain \(\frac {16(h+2)}{(h1)h(h+1)}\Omega_{3}^{T}R\Omega_{3}\), they can provide a tighter bound for \(\sum_{i=h+1}^{0}\sum_{k=i}^{0} y(k)^{T}Ry(k)\). Therefore, these improved double summation inequalities can be used to establish less conservative stability conditions for the discretetime systems with variable delays.
Application in stability analysis
In this section, we will consider the following linear discrete system with timevarying delay:
where \(x(k)\in R^{n}\) is the state vector, φ is the initial value, A and B are \(n\times n\) constant matrices. The delay \(h(k)\) is assumed to be a positive integervalued function, for some integers \(h_{2}\geq h_{1}>1\), \(h(k)\in[h_{1},h_{2}]\), \(\forall k\geq0\).
Based on the above summation inequalities, we will establish a new criterion on asymptotical stability for system (45).
First, the following notations are needed:
Theorem 5
For given integers \(h_{1}\), \(h_{2}\) satisfying \(1< h_{1}\leq h_{2}\), system (45) is asymptotically stable for \(h_{1}\leq h(k)\leq h_{2}\), if there are positive define matrices \(P\in R^{4n\times4n}\), \(Z_{1}\in R^{n\times n}\), \(Z_{2}\in R^{n\times n}\), \(Z_{3}\in R^{n\times n}\), \(Q_{1}\in R^{n\times n}\), \(Q_{2}\in R^{n\times n}\), and any matrix \(X\in R^{2n\times2n}\) such that the following LMIs are satisfied:
Proof
Choose a Lyapunov functional candidate as follows:
where
Next, we calculate the difference of \(V(k)\). For \(V_{1}(k)\) and \(V_{2}(k)\), we have
and
Calculating \(\Delta V_{3}(k)\) gives
By Corollary 6, we get
Under the condition of \(Z_{20}>0\), by Corollary 6 and the lower bounded lemma, we get
Then we have
Calculating \(\Delta V_{4}(k)\) gives
By Corollary 8, we have
Then we have
Hence
If \(\Xi<0\), then \(\Delta V(k)<0\).
This completes the proof of Theorem 5. □
Remark 5
Theorem 5 gives a sufficient condition for asymptotical stability criterion for discretetime system (45) with variable delay. The freeweighting matrix method was developed and was applied to the stability analysis of systems with timevarying delays [18]. However, the computational burden will increase because of the introduction of freeweighting matrices. Different from the freeweighting matrix method, some new sharper summation inequalities are developed via auxiliary functions. By employing these improved inequalities and the reciprocally convex combination inequality method, a less conservative result is derived. The conditions in Theorem 5 are described in terms of two matrix inequalities, which can be realized by using the linear matrix inequality algorithm proposed in [28].
Numerical example
In this section, to demonstrate the effectiveness of our proposed method, we consider the following example, which is widely used in the delaydependent stability analysis of discretetime systems with time delay.
Example 1
Consider the discretetime system
Since the system addressed in [24] is a discretetime system with constant delay, the stability criterion obtained cannot be applied to this system. For different \(h_{1}\), the maximum allowable upper bounds of \(h(k)\) guaranteeing this system to be asymptotically stable are given in Table 1 [18–23, 25]. From Table 1, Theorem 5 in our paper can provide larger feasible region than those of [18–21]. For the same \(h_{1}\), the maximum allowable upper bound of \(h(k)\) obtained in this paper is the same as that in [25]. Although more decision variables are needed in our stability criterion, the new summation inequality in Corollary 6 is sharper than that in [25].
Conclusions
In this paper, by the construction of an appropriate auxiliary function, some new summation inequalities are established. As an application of the summation inequality, an asymptotic stability analysis of discrete linear systems with time delay is carried out. Finally, a numerical example is provided to illustrate the usefulness of the theoretical results.
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Acknowledgements
This work is partly supported by NSFC under grant nos. 61271355 and 61375063 and the ZNDXYJSJGXM under grant no. 2015JGB21.
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Keywords
 Jensen’s inequality
 summation inequality
 stability
 discretetime system