A Game Theoretic Algorithm to Solve Riccati and Hamilton—Jacobi—Bellman—Isaacs (HJBI) Equations in H ∞ Control

Anderson, Brian D. O.; Feng, Yantao; Chen, Weitian

doi:10.1007/978-0-387-89496-6_15

Cited by 2 publications

(3 citation statements)

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“…where x ∈ E n , y ∈ E m ; x 0 and y 0 are given in (1)- (2). Consider the set U of all functions u = u(z, t) : E n+m × [0, +∞) → E m , which are measurable w.r.t.…”

Section: Main Definitions Let Us Introduce the Block Vectorsmentioning

confidence: 99%

“…Due to Theorem 4.4 and Lemma 3.4, in order to construct the saddlepoint equilibrium sequence of the SDG and obtain the value of this game, one has to solve the lower dimension regular RDG, and calculate two gain matrices P * 20 and P * 30 using the equations (34) and (30). Various methods and algorithms, applicable for computing the stabilizing solution P * 10 to the Riccati matrix algebraic equation (31), can be found in [2] and references therein. The Riccati matrix equation (31) becomes the scalar one 7P 2 10 + 6P 10 − 9 = 0.…”

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confidence: 99%

“…Various methods and algorithms, applicable for computing the stabilizing solution P * 10 to the Riccati matrix algebraic equation (31), can be found in [2] and references therein. The Riccati matrix equation (31) becomes the scalar one 7P 2 10 + 6P 10 − 9 = 0. This equation has two solutions, positive and negative: P 10,1 = (6 √ 2 − 3)/7, P 10,2 = −(6 √ 2 + 3)/7.…”

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confidence: 99%

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Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence

Glizer¹,

Kelis²

2017

Numerical Algebra, Control &Amp; Optimization

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We consider an infinite horizon zero-sum linear-quadratic differential game in the case where the cost functional does not contain a control cost of the minimizing player (the minimizer). This feature means that the game under consideration is singular. For this game, novel definitions of the saddle-point equilibrium and game value are proposed. To obtain these saddlepoint equilibrium and game value, we associate the singular game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. Using the solvability conditions, the solution of the cheap control game is reduced to solution of a Riccati matrix algebraic equation with an indefinite quadratic term. This equation is perturbed by a small parameter. Subject to a proper assumption, an asymptotic expansion of a stabilizing solution to this equation is constructed and justified. Using this asymptotic expansion, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.

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“…where x ∈ E n , y ∈ E m ; x 0 and y 0 are given in (1)- (2). Consider the set U of all functions u = u(z, t) : E n+m × [0, +∞) → E m , which are measurable w.r.t.…”

Section: Main Definitions Let Us Introduce the Block Vectorsmentioning

confidence: 99%

mentioning

confidence: 99%