 Research
 Open Access
 Published:
Robust finitetime control of descriptor Markovian jump systems with impulsive
Advances in Difference Equations volume 2019, Article number: 196 (2019)
Abstract
In this paper, we obtain sufficient conditions ensuring stability of the robust finitetime for descriptor Markovian jump systems with impulsive effects and timevarying normbounded disturbance, especially, when the system is in actuator saturation. Using the theory of Lyapunov functions and the concept of convex hullbased representation of saturation function. We design the state feedback controller and obtain estimation of domain of attraction, extending the results to convex optimization problems; the solvability condition of the controller can be equivalent to a feasibility problem of coupled linear matrix inequalities (LMIs). Finally, we present some numerical examples showing the effectiveness of the obtained theoretical results.
Introduction
Descriptor systems, also called singular systems, generalized statespace systems, or differentialalgebraic systems, have been widely used in many scientific areas because they can better describe the actual system. Descriptor system theory has become an important field in the study of modern control theory. When the structural parameters of a system are randomly mutated, it is naturally modeled as a Markovian jump system or a semiMarkov jump system. Markovian jump systems can be regarded as an extension of singlemode systems to multimodal systems with essentially more complex structure than singlemode systems. During the last decades, Markovian jump systems have attracted great attention in the field of control because they are more suitable for dynamic systems with random changes in the structure of model than singlemode systems. They are widely used in some practical systems, such as manufacturing systems, power systems, economic systems, spare systems, and many other systems [1, 2]. Hence a great number of fundamental notions and substantive results are also emerging. The authors in [3] investigated the stochastic admissibility problems for descriptor Markovian jump systems with partially unknown transition rates, descriptor Markovian jump systems with timevarying delay, and nonlinear descriptor Markovian jump systems with time delay. The problems of the robust exponential stability of uncertain singular Markovian jump timedelay systems were studied in [4]. Shen, Su, and Park [5, 6] extended passive and nonfragile fault detection filtering problem, which is investigated for a class of discretetime singular Markov jump systems (SMJs) with timevarying delays. Their attention is focused on the design of a general filter that contains the modeindependent and modedependent parts to address the filtering issue and on the design of a modedependent nonfragile fault detection filter to guarantee the fault detection system to be stochastically admissible with an \({H_{\infty }}\) performance index for all admissible uncertainties. In [7] a reliable filtering is designed so that the considered filtering error system in the presence of a timevarying delay and sensor failures is meansquare exponentially admissible with a specified decay rate and simultaneously satisfies an \({H_{\infty }}\) performance.
In addition, finitetime stability means that the system is stable in finite or short time, which is first mentioned in [8]. Based on the Lyapunov theory, the transient performance of the finitetime interval internal system is dealt with. Some researchers [9,10,11,12] give a definition of finitetime stabilization and finitetime boundedness. The timedomain stability is a special form of timedomain boundedness, and timedomain boundedness is an extended concept of timedomain stability. They are interrelated and different from each other. During the last decades, people paid more attention to the bounded problem of the system state within a limited time. With the advance of time and linear matrix inequality (LMI) technology, scholars had a deeper understanding of the stability of the time domain of a dynamic system and obtained some meaningful results about the stability of the time domain. In particular, [13] focuses on the problem of robust finitetime stabilization for one family of uncertain singular Markovian jump systems. Sufficient conditions for singular stochastic finitetime boundedness are obtained for a class of singular stochastic systems with parametric uncertainties and timevarying normbounded disturbance.
Impulsive systems are a kind of discontinuous systems. The impulsive phenomenon exists in different fields of nature and evolutionary processes, which states sudden changes at some points. It is a transient change of state at a certain time in the actual system. The impulsive effect can better describe the evolution process of the system state. From the control point of view, its influence on the stability of the pulse can be divided into two categories, namely, suppression of the stability of an unstable pulse and improvement of the stability of a stable pulse. It is worth mentioning that there have been some important results in timedomain stability for Markov jump systems with impulses. In [14] a new concept of stochastic finitetime stability for a class of nonlinear Markovian switching systems with impulsive effects is introduced. In [15] a stochastic finitetime stability (SFTS) and control synthesis for a class of nonlinear Markovian jump stochastic systems with impulsive effects is proposed. The impulsive of a system can be better described by introducing a timevarying stochastic Lyapunov function with discontinuities at impulse times.
Actuator saturation [16,17,18,19,20] means that if the input of the system actuator reaches a certain limit, then it enters the saturation state. Because further increasing the input cannot affect the output of the actuator, the saturation of the actuator reduces the dynamic performance of the system and even leads to instability of the closedloop system. Therefore, it is necessary to study the saturation problem. In [21] the robust stochastic problem for discretetime uncertain singular Markov jump systems with actuator saturation is considered. In [22] the problem of robust exponential stabilization for uncertain impulsive bilinear timedelay systems with saturating actuators is investigated. In [23] the problems of robust linear feedback stabilization and estimation of domain of attraction for a class of uncertain impulsive systems with saturating actuator are investigated.
It can be seen from analysis of the previous literature that, in spite of many studies about the finitetime stability of Markovian jump systems, there are no papers on finitetime stability of systems with actuator saturation and impulse and disturbance effects, which is important and significant in engineering applications. Motivated by these, in this paper, we consider the finitetime stability of systems with simultaneous impulse and saturation effects. Sufficient conditions for the timedomain stability of the system are given by Lyapunov function theory, free weight matrix, LMI, and Sprocedure. Based on the previous conditions, a state feedback controller is designed so that the resultant closedloop system is finitetime stable. Finally, an example is given to solve the problem by MATLAB.
Notations
Throughout the paper, for real symmetric matrices X and Y, the notation \(X > Y\) means that the matrix \(X  Y\) is positive definite; I is the identity matrix of appropriate dimension; the superscript T represents the transpose; \(\operatorname{diag} \{ \cdots \}\) denotes a blockdiagonal matrix. For a symmetric block matrix, we use ∗ as an ellipsis for the terms that are introduced by symmetry; \(\mathrm{E} \{ \cdot \}\) denotes the expectation operator with respect to given probability measure P.
Modeling
Given a complete probability space \(( {\varOmega ,F,P} )\), the continuoustime descriptor Markovian jump impulsive system is described by
where \(x ( t ) \in {R^{n}}\) is the state vector, \(y(t) = {R^{m}}\) is the control output, \(u ( t ) \in {R^{m}}\) is the control input, \(E \in {R^{n \times n}}\) is a descriptor matrix with \(\operatorname{rank}(E) = r \le n\), \(A(r(t))\), \(B(r(t))\), \(C(r(t))\), \(D(r(t))\), \(M(r(t))\), and \(G(r(t))\) are known matrices of appropriate dimensions depending on \(r(t)\), where \(\{ {r ( t ),t \ge 0} \}\) is a continuoustime Markovian stochastic process defined on a probability space and taking values in a finite space; its transition probabilities from mode i at time t to mode j at time \(t + 1\) are described as
where \(\delta > 0\), \(\lim_{t \to 0} ({o ( \delta )}/ \delta ) = 0\), \({\gamma _{ij}} \ge 0\) (\(i,j \in S\), \(j \ne i \)) is the transition rate from i to j, and \({\gamma _{ii}} =  \sum_{j \in S,j \ne i} {{\gamma _{ij}}} \). The saturating function \(\operatorname{sat}:{R^{p}}  {R^{p}}\) is defined as
In the case of unity saturation level, that is, \(\operatorname{sat} ( {{u_{i}}(t)} ) \le 1\), \(i = 1,2,\ldots, p \); \(\Delta A ( {r(t)} )\) and \(\Delta B ( {r(t)} )\) are matrix functions with timevarying uncertainties. Further,
where \({H_{e}}(i)\), \({F_{a}}(i)\), \({F_{b}}(i)\) are the known real constant matrices of appropriate dimensions, and \(\Delta (t,i)\) is an unknown analytic function matrix with Lebesguemeasurable elements satisfying
If (2) and (3) are established, then \(\Delta A ( {r(t)} )\) is called the structural robust uncertainty, and \(\Delta B ( {r(t)} )\) is said to be permissible.
Moreover, the disturbance \(\omega (t) \) satisfies
For a matrix, we denote the jth row of \(F(i)\) as \({f_{ij}}\) and define \(L ( {F(i)} )\) as
Let \(P \in {R^{n \times n}} \) be a symmetric matrix such that \({E^{T}}PE \ge 0 \) and define the set
Let D be the set of \(p \times p\) diagonal matrices whose diagonal elements are either 1 or 0. Suppose that each element of D is labeled as \({D_{l}}\), \(l = 1,2,\ldots,{2^{p}}\) and denote \(D_{l}^{} = I  {D_{l}} \), Clearly, if \({D_{l}} \in D\), then \(D_{l}^{} \in D \).
Definition 1

1.
The continuoustime system (1) is said to be uniformly regular if there is a constant s such that the characteristic polynomial \(\det ( {sE  A(r(t))} )\) is not identically 0 for any \(t \in [ {0,T} ]\).

2.
The continuoustime system (1) is said to be impulse free in the time interval \([ {0,T} ]\) if \(\deg ( {\det (sE  A(r(t)))} ) = \operatorname{rank}(E)\) for all \(t \in [ {0,T} ]\).
Definition 2
Given three positive scalars \({c_{1}}\), \({c_{2}}\), T with \({c_{1}} < {c_{2}}\), positive definite matrices \({R_{i}}\), \(i \in S\), and positive definite matrixvalued functions \({\varGamma _{i}}\), a descriptor Markovian jump impulsive system is finitetime stable with respect to \(( {{c_{1}},{c_{2}},T,{R_{i}},{\varGamma _{i}}} )\) if
for all admissible uncertainties satisfying (2).
Definition 3
([13])
Let \(V ( {x(t),r(t),t} )\) be a stochastic Lyapunov function of a closedloop SMJS. We define the operator J by
Lemma 1
([24])
Let \(F(i),H(i) \in {R^{p \times n}}\). Then for any \(x ( t ) \in L(H(i))\),
or, equivalently,
where co stands for the convex hull, \({a_{l}}\), \(l = 1,2,\ldots, {2^{p}}\), are some scalars satisfying \(0 \le {a_{l}} \le 1\) and \(\sum_{l = 1}^{{2^{p}}} {{a_{l}} = 1} \).
Lemma 2
([25])
Given a set of suited dimension real matrices \({T_{1}}\), \({T_{2}}\), and \(F(t)\) is a timevarying matrix with \(F{(t)^{T}}F(t) \le I\), Then, we have the following:

(1)
For any scalar \(\varepsilon > 0\),
$$\begin{aligned} {T_{1}}F(t){T_{2}} + {T_{2}}^{T}F{(t)^{T}} {T_{1}}^{T} \le \varepsilon {T_{1}} {T_{1}}^{T} + {\varepsilon ^{  1}} {T_{2}}^{T}{T_{2}}. \end{aligned}$$ 
(2)
For any positive definite matrix G,
$$\begin{aligned} {T_{1}} {T_{2}} + {T_{2}}^{T}{T_{1}}^{T} \le {T_{1}}G{T_{1}}^{T} + {T_{2}}^{T}{G^{  1}} {T_{2}}. \end{aligned}$$
Lemma 3
([22])
Let \(v(t)\) be a nonnegative function such that
for some constants \(a,b \ge 0\). Then we have the following inequality:
In this paper, we consider the state feedback controller
such that the closedloop system is defined by
For convenience, denote the matrix \(A(r(t))\) as \({A_{i}}\) and
Main results
Robust finitetime stabilization
Theorem 1
Consider the closedloop system (5) for \(t \in [ {0,T} ]\). Let \({c_{1}}\), \({c_{2}}\), T be three positive scalars with \({c_{1}} < {c_{2}}\), let \({R_{i}}\), \(i \in S\) be positive definite matrices, and let \({\varGamma _{i}}\) be positive definite matrixvalued functions, Suppose that there exist a scalar \(\alpha \ge 0\), a set of nonsingular matrices \({P_{i}} \in {R^{n \times n}}\), two sets of symmetric positive definite matrices \({Q_{2}}(i) \in {R^{d \times d}}\), \(i \in S\), and \({Q_{1}}(i) \in {R^{n \times n}}\), \(i \in S\), such that the following hold:
where
for all \(i,j = 1,2,\ldots,s\), \(l = 1,2,\ldots,{2^{n}}\), and \(\varOmega ( {{E^{T}}{X_{i}}E} ) \subset L ( {{H_{i}}} )\). Then the closedloop system (5) with respect \(( {{c_{1}},{c_{2}},T,R,d,{\varGamma _{i}}} )\) is robust finitetime stable within \(\bigcap_{i = 1}^{N} {\varOmega ( {{E^{T}} {X_{i}}E} )} \).
Proof
Firstly, we prove that the closedloop systems (5) is regular and impulsefree in the time interval \([0,T]\). By the Schur complement and condition (7) we obtain
Choose nonsingular matrices M and N such that
Then, according to (6) and (12), it is not difficult to prove that \({P_{12}}(i) = 0\) and \(\det ( {{P_{22}}(i)} ) \ne 0\). Pre and postmultiplying (11) by \({N^{T}}\) and N, we get
We can easily obtain that \(A_{22}^{T}{P_{22}}(i) + P_{22}^{T}{A_{22}}(i) < 0\) and \({A_{22}}(i)\) is nonsingular, which implies that the closedloop SMJS is regular and impulsefree in the time interval [0,T].
Construct the following Lyapunov function: \(V ( {x(t),i} ) = {x^{T}}(t){E^{T}}{P_{i}}x(t)\).
When \(t \ne {t_{k}}\), using Definition 3, we obtain that
where $z(t)=\left[\begin{array}{c}x(t)\\ \omega (t)\end{array}\right]$, so that
By Lemma 1, Lemma 2, and (2), this formula is equivalent to
According the last formula and (7), we can obtain that \(\ell V(x(t),r(t) = i,t) < 0\) and
When the system depends on the state to jump, applying (8), we have
So we derive that \(V(t,x)\) is strictly decreasing on T.
Integrating (13) from 0 to t and using Lemma 3, we have
where \({Q_{1}}(i) = {E^{  T}}R_{i}^{  \frac{1}{2}}{E^{T}}{P_{2}}R _{i}^{\frac{1}{2}}{E^{  1}} \), and \({\lambda _{\max }} ( {{Q_{1}}(i)} )\) and \({\lambda _{\min }} ( {{Q_{1}}(i)} )\) are the maximum and minimum eigenvalues of \({Q_{1}}(i)\). Thus
On the other hand,
Therefore
This proves that system (5) is robust finitetime stable. □
Theorem 2
Let (5) be a closedloop system with \(\omega (t) = 0\) for \(t \in [ {0,T} ]\), let \({c_{1}}\), \({c_{2}}\), T be three positive scalars with \({c_{1}} < {c_{2}}\), let \({R_{i}}\), \(i \in S\), be positive definite matrices, and let \({\varGamma _{i}}\) be positive definite matrixvalued functions. Suppose that there exist a scalar \(\alpha \ge 0\), a set of nonsingular matrices \({P_{i}} \in {R^{n \times n}}\), and two sets of symmetric positive definite matrices \({Q_{2}}(i) \in {R^{d \times d}}\), \(i \in S\), and \({Q_{1}}(i) \in {R^{n \times n}}\), \(i \in S\), such that the following hold:
for all \(i,j = 1,2,\ldots,s\), \(l = 1,2,\ldots,{2^{n}}\), and \(\varOmega ( {{E^{T}}{X_{i}}E} ) \subset L ( {{H_{i}}} )\). Then the closedloop system (5) with respect \(( {{c_{1}},{c_{2}},T,R,d,{\varGamma _{i}}} )\) is robust finitetime stable within \(\bigcap_{i = 1}^{N} {\varOmega ( {{E^{T}} {X_{i}}E} )} \).
This result can be proved in much the same way as Theorem 1.
Robust state feedback controller
Theorem 1 gives a set of conditions to judge if some initial state is in the domain of attraction in mean square sense. To further facilitate the synthesis procedure, we will state these conditions in terms of LMIs.
Theorem 3
Suppose that, for each mode \(i \in S\) and given scalars \({\varepsilon _{i}} > 0\), there exist a positive definite symmetric matrix \({X_{i}} > 0\), matrices \({Y_{i}}\) and \({H_{i}}\) for \(i = 1,2,\ldots, {2^{n}}\), and \(\varOmega ( {{E^{T}}{P_{i}}E} ) \subset L ( {{H_{i}}} )\). Then for all uncertainties satisfying (2) and (3), the closedloop system is robust finitetime stable within \(\bigcap_{i = 1}^{N} {\varOmega ( {{E^{T}}{X_{i}}E} )} \), and the state feedback controller gain matrix is given by \({K_{i}} = {L_{i}}X_{i}^{  1}\), \(i = 1,2,\ldots, N\).
Let \({X_{R}} \subset {R^{n}}\) be a prescribed bounded convex set containing origin that can be represented as the polyhedron \({X_{R}} = \operatorname{co} \{ {x_{0}^{1},x_{0}^{2},\ldots,x _{0}^{q}} \}\), where \({x_{0}^{1},x_{0}^{2},\ldots,x_{0}^{q}}\) are a priori given initial states in \({R^{n}}\). To see if the initial states \({x_{0}} \subset {R^{n}}\) are in the domain of attraction in the mean square sense, we can formulate the following optimization problem:
where \({h_{iq}}\) denotes the qth row of \({H_{i}}\). If \(\max \vert \alpha \vert > 1 \), then \({x_{0}} \in \varOmega ( {{E^{T}}{P_{i}}E} )\). Noticing that the optimization problem is nonconvex, we need to formulate this problem into a convex optimization problem.
Let \({P_{i}} = X_{i}^{  1}\), \({L_{i}} = {K_{i}}{X_{i}}\), \({E^{T}} {P_{i}} = {P_{i}}^{T}E = P_{i}^{  1}H(i)P_{i}^{  T}\). Condition (i) is equivalent to
By the Schur complement it can be converted to
Letting \(\beta = {\alpha ^{  2}}\), (15) can be rewritten as
Inequality (7) can be transformed into
where
Pre and postmultiplying (17) by te diagonal matrix \(\operatorname{diag} \{ {{X_{i}},I,I} \}\), we obtain
This matrix can be transformed into
where
Let \({Q_{i}} = Q_{1}^{  1}(i) = R_{i}^{  1/2}{X_{i}}R_{i}^{  1/2}\) and
Thus (10) can be rewritten as
or, equivalently,
Since \(\varOmega ({E^{T}}{P_{i}}E) \subset L ( {H(i)} )\), we have
which is equivalent to
Using the Schur complements, we have
and the optimization problem can be transformed into the following linear matrix inequality problem:
where \({\varepsilon _{i}} > 0\) is a given scalar. If \(\min \beta < 1\), then \({x_{0}} \in \varOmega ( {{E^{T}}{P_{i}}E} )\). The state feedback controller gain \({K_{i}} = {L_{i}}X_{i}^{  1}\) can be obtained by solving the linear matrix inequality problem directly.
Simulation example
Let us consider the robust finitetime stability for system (1) with the following coefficient matrices:
mode 1:
mode 2:
Let \({c_{1}} = 1\), \({c_{2}} = 11\), \(T = 3\), \(\alpha = 0.1\),
We use the LMI toolbox of MATLAB software to solve the optimization problem \(\beta = 0.218 < 1\). The gain of the state feedback controller is:
Then the MATLAB simulation is shown in the following image.
One possible realization of Markovian jumping mode is given in Fig. 1. The simulation results under this modedependent controller are shown in Fig. 2. We see that the state responses are satisfactory when the saturation appears. The corresponding state trajectory is shown in Fig. 3.
Conclusion
The robust finitetime control for descriptor Markov jump systems with impulsive effects, actuator saturation, and timevarying normbounded disturbance have been investigated. A sufficient condition for the finitetime stability of systems is given according to Lyapunov function theory and LIMs. Based on the conditions above, a state feedback controller is designed such that the resultant closedloop system is finitetime stable. The simulation results are given by MATLAB. The results in this paper can be applied in communication engineering and other fields. It has important theoretical significance for further study of some problems of descriptor Markovian jump systems (also, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]).
References
 1.
Feng, J.E., Lam, J., Shu, Z.: Stabilization of Markovian systems via probability rate synthesis and output feedback. IEEE Trans. Autom. Control 55, 773–777 (2010)
 2.
Wang, Z., Liu, Y., Liu, X.: Exponential stabilization of a class of stochastic system with Markovian jump parameters and modedependent mixed timedelays. IEEE Trans. Autom. Control 55, 1656–1662 (2010)
 3.
Zhang, Q., Liu, C., Zhang, X.: Bifurcations and control in singular biological economic model with stage structure. In: Complexity, Analysis and Control of Singular Biological Systems. Lecture Notes in Control and Information Sciences, vol. 421, pp. 43–66. Springer, London (2012)
 4.
ZhengGuang, W.U., HongYe, S.U., Jian, C.H.U.: Robust exponential stability of uncertain singular Markovian jump timedelay systems. Acta Autom. Sin. 36, 558–563 (2010)
 5.
Shen, H., Su, L., Park, J.H.: Extended passive filtering for discretetime singular Markov jump systems with timevarying delays. Signal Process. 128, 68–77 (2016)
 6.
Shen, H., Su, L., Park, J.H.: Robust nonfragile \(H_{\infty}\) fault detection filter design for delayed singular Markovian jump systems with linear fractional parametric uncertainties. Hybrid Syst. 32, 65–78 (2019)
 7.
Liu, G.B., Xu, S.Y., Park, J.H., Zhang, G.M.: Reliable exponential filtering for singular Markovian jump systems with time varying delays and sensor failures. Int. J. Robust Nonlinear Control 28, 4230–4245 (2018)
 8.
Kamenkov, G.V.: On stability of motion over a finite interval of time. Akad. Nauk SSSR Prikl. Mat. Meh. 17, 529–540 (1953)
 9.
Dorato, P.: Short time stability in linear timevarying systems. In: Proceedings of the IRE International Convention Record, pp. 83–87 (1961)
 10.
Kablar, N.A., Debelikovic, D.L.: Finitetime stability of timevarying linear singular systems. Am. Control Conf. IEEE 4, 3831–3836 (1994)
 11.
Garcia, G., Tarbouriech, S., Bernussou, J.: Finitetime stabilization of linear timevarying continuous systems. IEEE Trans. Autom. Control 54, 364–369 (2009)
 12.
Amato, F., Ariola, M., Cosentino, C.: Finitetime stability of linear timevarying systems: analysis and controller design. IEEE Trans. Autom. Control 55, 1003–1008 (2010)
 13.
Zhang, Y., Liu, C., Mu, X.: Robust finitetime stabilization of uncertain singular Markovian jump systems. Appl. Math. Model. 36, 5019–5121 (2012)
 14.
Jia, X., Sun, J., Dong, Y.: Stochastic finitetime stability of nonlinear Markovian switching systems with impulsive effects. J. Dyn. Syst. Meas. Control 134, 011 (2012)
 15.
Chen, W.H., Wei, C., Lu, X.: Stochastic finitetime stabilization for a class of nonlinear Markovian jump stochastic systems with impulsive effects. J. Dyn. Syst. Meas. Control 137, 044 (2015)
 16.
Fridman, E., Pila, A., Shaked, U.: Regional stabilization and H1 control of timedelay systems with saturating actuators. Int. J. Robust Nonlinear Control 13, 885–907 (2003)
 17.
Zhang, L.X., Boukas, E.K., Haidar, A.: Delayrangedependent control synthesis for timedelay systems with actuator saturation. Automatica 44, 2691–2695 (2008)
 18.
Gomes Da Silva, J.M., Tarbouriech, S.: Local stabilization of discretetime linear systems with saturating controls: an LMIbased approach. IEEE Trans. Autom. Control 46, 119–125 (2001)
 19.
Cao, Y.Y., Lin, Z.: Stability analysis of discretetime systems with actuator saturation by a saturationdependent Lyapunov function. Automatica 39, 1235–1241 (2003)
 20.
Alamo, T., Cepeda, A., Limon, D.: Estimation of the domain of attraction for saturated discretetime systems. J. Astronaut. 69, 274–280 (1998)
 21.
Ma, S., Zhang, C., Zhu, S.: Robust stability for discretetime uncertain singular Markov jump systems with actuator saturation. IET Control Theory Appl. 5, 255–262 (2011)
 22.
Yang, S.J., Shi, B., Zhang, Q., et al.: Robust exponential stabilization of uncertain impulsive bilinear timedelay systems with saturating actuators, 12, 261–265 (2010)
 23.
Li, Y., Chen, W.H.: Robust stabilization of uncertain impulsive systems with saturating actuator. J. Northwest Normal Univ. Nat. Sci. 6, 10–15 (2013)
 24.
Hu, T., Lin, Z., Chen, B.M.: Analysis and design for discretetime linear systems subject to actuator saturation. In: Decision and Control, 2001. Proceedings of the, IEEE Conference on, pp. 97–112. IEEE (2002)
 25.
Last, E.: Linear matrix inequalities in system and control theory, SAM. In: Proceedings of the IEEE, vol. 86, pp. 2473–2474 (1994)
 26.
Yan, Z., Zhang, G., Wang, J.: Finitetime stability and stabilization of linear stochastic systems. In: China Control Conference, pp. 1115–1120 (2010)
Acknowledgements
We would like to thank the anonymous referees very much for their valuable comments and suggestions.
Funding
This work was supported by National Natural Science Foundation of China under Grant No. 61074005 and the Talent Project of the High Education of Liaoning province (LR2012005).
Author information
Affiliations
Contributions
The authors have achieved equal contributions. Both authors read and approved the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Su, X., Zhao, X. Robust finitetime control of descriptor Markovian jump systems with impulsive. Adv Differ Equ 2019, 196 (2019). https://doi.org/10.1186/s1366201920521
Received:
Accepted:
Published:
Keywords
 Descriptor Markovian jump systems
 Actuator saturation
 Finitetime control
 Impulsive systems