Adaptive control : stability, convergence, and robustness by Shankar Sastry

By Shankar Sastry

This quantity surveys the main effects and strategies of study within the box of adaptive keep watch over. concentrating on linear, non-stop time, single-input, single-output platforms, the authors provide a transparent, conceptual presentation of adaptive equipment, permitting a serious evaluate of those recommendations and suggesting avenues of extra improvement. 1989 variation

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Example text

Discretization yields non-linear difference equations. Most research in reinforcement learning is conducted for systems that operate in discrete time. Therefore, we cover discrete-time dynamical systems here. The application of reinforcement learning techniques to continuous-time systems is significantly more involved and is the topic of the remainder of the book. RL policy iteration and value iteration methods have been used for many years to provide methods for solving the optimal control problem for discrete-time (DT) dynamical systems.

G, where each entry is a function mk (x) : X ! U; k ¼ 0,1, . .. Stationary deterministic policies are independent of time so that p ¼ fm, m, . g. Select a fixed stationary policy p(x, u) ¼ Prfujxg. Then the ‘closed-loop’ MDP reduces to a Markov chain with state space X. That is, the transition probabilities between states are fixed with no further freedom of choice of actions. The transition probabilities of this Markov chain are given by p Px;x0  Px;x 0 ¼ X u Prfx0 jx; ugPrfujxg ¼ X u pðx; uÞPx;x 0 ð2:2Þ u where the Chapman–Kolmogorov identity is used.

That is, reinforcement learning allows the solution of the algebraic Riccati equation online without knowing the system A matrix. d. 38) becomes P jþ1 ¼ ðA À BKÞT P j ðA À BKÞ þ Q þ K T RK ð2:50Þ This recursion converges to the solution to the Lyapunov equation P ¼ (A À BK)T P(A À BK) þ Q þ K T RK if (A – BK) is stable, for any choice of initial & value P0 . 16) of using the current policy pj (x, u). 38) until convergence. 39). 38) in their value update step. Usually, policy iteration converges to the optimal value in fewer steps j since it does more work in solving equations at each step.

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