Computation of the stabilizing solution of game theoretic Riccati equation arising in stochastic H  ∞  control problems

Abstract

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H ∞ control problem for a class of stochastic systems affected by state dependent and control dependent white noise. The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.

The Iterative Solution to Discrete-Time H∞ Control Problems for Periodic Systems

“…The performed numerical experiments are supportive to the fact that the new procedure is faster than the others, furthermore, it uses less RAM memory. We should note that our study is based on ideas developed in [9][10][11], where new iterative procedures for the numerical computation of a stabilizing solution in the time-invariant case has been proposed.…”

On computing the stabilizing solution of a class of discrete‐time periodic Riccati equations

“…In this paper, we propose an iterative procedure for the numerical computation of the stabilizing solution of . This method extends to the discrete‐time time‐varying case, the method developed in for the deterministic continuous‐time time‐invariant case, and the method developed in for the stochastic continuous‐time case.…”

“…Note that similar algorithms for the solution of H ∞ ‐type algebraic Riccati equations have been developed by for the time‐invariant case. Our main result can be viewed as an extension to the discrete‐time time‐varying case of the results in for the deterministic continuous‐time time‐invariant case and in for the stochastic continuous‐time time‐invariant case. We will particularly show that our algorithm shares similar desirable characteristics with its time‐invariant counterparts, namely simple initialization, global convergence, and local quadratic rate of convergence.…”

“…Hence, it is clear that the main advantages of the proposed algorithm are not related to the computational load saving. This issue has been discussed in detail in for the time‐invariant case, where the authors pointed out that the main advantage of their proposed algorithms relies on their high numerical reliability when facing ill‐conditioned problems. We believe that such a highly desirable property is still valid in the time‐varying case for our algorithm.…”

<i>H</i><sub>&#8734</sub> Optimal Control Problems for Jump

“…The paper of Lanzon et al [1] is the first where is investigated an algebraic Riccati equation with an indefinite quadratic part in the deterministic case. Further on, the Lanzon's approach has been extended and applied to the algebraic Riccati equations of different types [2]- [5] and for the stochastic case [6]. Many situations in management, economics and finance [7]- [9] are characterized by multiple decision makers/players who can enforce the decisions that have enduring consequences.…”