Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

Li, Zhao-Yan; Zhou, Bin; Lam, James; Wang, Yong

doi:10.1016/j.amc.2011.01.031

Cited by 28 publications

(36 citation statements)

References 36 publications

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“…Then by Lemma 1 in [12], the inequality in (49) has a solution Q ∈ S n×n + if and only if ρ (L ) < 1. However, similar to (10)- (11) in [11], we can show that…”

Section: Theoremmentioning

confidence: 99%

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Stability Analysis of Integral Delay Systems With Multiple Delays

Zhou

2016

IEEE Trans. Automat. Contr.

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This note is concerned with stability analysis of integral delay systems with multiple delays. To study this problem, the well-known Jensen inequality is generalized to the case of multiple terms by introducing an individual slack weighting matrix for each term, which can be optimized to reduce the conservatism. With the help of the multiple Jensen inequalities and by developing a novel linearizing technique, two novel Lyapunov functional based approaches are established to obtain sufficient stability conditions expressed by linear matrix inequalities (LMIs). It is shown that these new conditions are always less conservative than the existing ones. Moreover, by the positive operator theory, a single LMI based condition and a spectral radius based condition are obtained based on an existing sufficient stability condition expressed by coupled LMIs. A numerical example illustrates the effectiveness of the proposed approaches.

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“…Then by Lemma 1 in [12], the inequality in (49) has a solution Q ∈ S n×n + if and only if ρ (L ) < 1. However, similar to (10)- (11) in [11], we can show that…”

Section: Theoremmentioning

confidence: 99%

“…The multiple Jensen inequalities (11) and (12) are clearly less conservative than the inequalities in (10) and (9) since the later ones can be obtained immediately by setting…”

Section: Now Every Integrationmentioning

confidence: 99%

Stability Analysis of Integral Delay Systems With Multiple Delays

Zhou

2016

IEEE Trans. Automat. Contr.

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“…Various kinds of matrix equations have received much attention in the literature (see, for example, [4], [7], [8], [11], [12], [13], [15], [14], [16], [19], [27], [28], [29], [34], [38], [39], [40], and the references therein). Especially, the problem of finding fixed points of the nonlinear matrix equation X + A * X −1 A = Q where A and Q > 0 are given and X is unknown, has been extensively studied in the last two decades.…”

Section: Introductionmentioning

confidence: 99%

Positive definite solutions of the nonlinear matrix equation X+AHX¯-1
Zhou
¹
,
Cai
²
,
Lam
³

2013
Applied Mathematics and Computation
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32
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19
0
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This paper is concerned with the positive definite solutions to the matrix equation X + A H X −1 A = I where X is the unknown and A is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation W + B T W −1 B = I which has been extensively studied in the literature, where B is a real matrix and is uniquely determined by A. It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix A. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed.

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“…The parallel algorithm for discretetime coupled Lyapunov equations has been investigated in [13]. And, two iterative approaches based on positive operator have been given in [14] for to solve the continuous-time and discrete-time stochastic coupled algebraic Lyapunov equations associated with the stability for stochastic systems with multiplicative noise and Markovian jumps. By using a convex optimization approach, the authors in [15] have studied the H 2 /H ∞ control problem for systems with multiplicative noise and Markovian jumps.…”

mentioning

confidence: 99%

“…Inspired by the implicit iterative algorithm in [14], an off-line iterative algorithm is first proposed for solving the stochastic CARE. Furthermore, by invoking the ST technique and integral reinforcement learning (IRL) approach [37], a data-driven policy iteration algorithm is developed to converge the solutions of the stochastic CARE based on sampling the states of the N decomposed linear subsystems.…”

mentioning

confidence: 99%

Data‐driven policy iteration algorithm for optimal control of continuous‐time Itô stochastic systems with Markovian jumps

Song

Liu

et al. 2016

IET Control Theory & Applications

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This paper studies the infinite horizon optimal control problem for a class of continuous-time systems subjected to multiplicative noises and Markovian jumps by using a data-driven policy iteration algorithm.The optimal control problem is equivalent to solve a stochastic coupled algebraic Riccatic equation (CARE).An off-line iteration algorithm is first established to converge the solutions of the stochastic CARE, which is generalized from a implicit iterative algorithm. By applying subsystems transforation (ST) technique, the off-line iterative algorithm is decoupled into N parallel Kleinman's iterative equations. To learn the solution of the stochastic CARE from N decomposed linear subsystems data, a ST-based data-driven policy iteration algorithm is proposed and the convergence is proved. Finally, a numerical example is given to illustrate the effectiveness and applicability of the proposed two iterative algorithms.

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Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

Cited by 28 publications

References 36 publications

Stability Analysis of Integral Delay Systems With Multiple Delays

Stability Analysis of Integral Delay Systems With Multiple Delays

Positive definite solutions of the nonlinear matrix equation X+AHX¯-1
Zhou
¹
,
Cai
²
,
Lam
³

2013
Applied Mathematics and Computation
Self Cite
32
0
19
0
View full text Add to dashboard Cite

Data‐driven policy iteration algorithm for optimal control of continuous‐time Itô stochastic systems with Markovian jumps

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Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

Cited by 28 publications

References 36 publications

Stability Analysis of Integral Delay Systems With Multiple Delays

Stability Analysis of Integral Delay Systems With Multiple Delays

Positive definite solutions of the nonlinear matrix equation X+AHX¯-1Zhou1, Cai2, Lam3 2013Applied Mathematics and ComputationSelf Cite320190View full textAdd to dashboardCite

Data‐driven policy iteration algorithm for optimal control of continuous‐time Itô stochastic systems with Markovian jumps

Contact Info

Product

Resources

About

Positive definite solutions of the nonlinear matrix equation X+AHX¯-1
Zhou
¹
,
Cai
²
,
Lam
³

2013
Applied Mathematics and Computation
Self Cite
32
0
19
0
View full text Add to dashboard Cite