An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term

Abstract

In this paper, an iterative algorithm to solve Algebraic Riccati Equations (ARE) arising from, for example, a standard H ∞ control problem is proposed. By constructing two sequences of positive semidefinite matrices, we reduce an ARE with an indefinite quadratic term to a series of AREs with a negative semidefinite quadratic term which can be solved more easily by existing iterative methods (e.g. Kleinman algorithm in [2]). We prove that the proposed algorithm is globally convergent and has local quadratic rate of convergence. Numerical examples are provided to show that our algorithm has better numerical reliability when compared with some traditional algorithms (e.g. Schur method in [5]). Some proofs are omitted for brevity and will be published elsewhere.

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

“…Moreover, we will show that these sequences are convergent and we have Let us remark that in the special case of (1) with A l = 0, B l = 0, 1 ≤ l ≤ r, the algorithm described by (10) and (11) recovers the iterative procedure proposed in [21,22] to solve the game theoretic Riccati equations in the deterministic framework.…”

“…Let W ∈ A(γ ) and letX W be the stabilizing solution of the algebraic Riccati equation (16) with F replaced by W. IfĽ,L are linear operators associated to the triple (X, Z , W) via (26) and (27), then the following assertions hold: Proof First, we remark that if X, Z verified (28) then (22) reduces to:…”

“…The proposed procedure extends to this general framework the one from [21,22] in the deterministic case and [13] in the stochastic case. It is mainly based on the defect correction method proposed in [23].…”

On an iterative algorithm to compute the positive stabilizing solution of generalized algebraic Riccati equations

“…Compared with traditional H ∞ algebraic Riccati equations (ARE) in deterministic systems, equation (1) includes an additional disturbance termà T Πà which comes from state-dependent noises in the underlying systems. Equation (1) arises in characterizing the solution of a differential game problem for the following system Σ…”

“…Our interest is in providing a new type of solution algorithm to solve GARE (1), built on recent developments for solving algebraic Riccati equations (AREs) with an indefinite quadratic term (see [1,2]). Such problems arise in some applications such as launch vehicle ascent (see [15]), diesel engine control (see [16]).…”