This paper studies the stabilization for a kind of linear and impulse control systems in finite-dimensional spaces, where impulse instants appear periodically. We present several characterizations on the stabilization; show how to design feedback laws; and provide locations for impulse instants to ensure the stabilization. In the proofs of these results, we set up a discrete LQ problem; derived a discrete dynamic programming principle, built up a variant of Riccati's equation; applied repeatedly the Kalman controllability decomposition; and used a controllability result built up in [17].
About general theory for impulse systems, we refer readers to @cite_6 @cite_22 @cite_23 and the references therein.
@cite_23 @cite_22 @cite_6
2045019399 2076467525 2953326374 2061079066
For a linear impulsive system, the set of states that are reachable from the origin when the initial time, impulse times, and final time are fixed is contained in an invariant subspace determined by the system data. It is known that reversibility of the system is sufficient to yield, for a specified initial time, the existence of some impulse time set and final time for which the reachable set equals the invariant subspace. In this paper, we relax the reversibility requirement and present a condition that is necessary as well as sufficient under which this property holds. This new condition involves the property of achieving reversibility via feedback and admits an explicit geometric characterization. Moreover, this feedback-reversibility property only needs to hold for the subsystem defined as the full system restricted to the invariant subspace. We further show that feedback-reversibility of the restricted system ensures that the reachable set equals the invariant subspace for almost any impulse time set and final time for which the number of impulse times contained in the underlying time interval exceeds a lower bound. This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials. As a step towards developing zero-shot task generalization capabilities in reinforcement learning (RL), we introduce a new RL problem where the agent should learn to execute sequences of instructions after learning useful skills that solve subtasks. In this problem, we consider two types of generalizations: to previously unseen instructions and to longer sequences of instructions. For generalization over unseen instructions, we propose a new objective which encourages learning correspondences between similar subtasks by making analogies. For generalization over sequential instructions, we present a hierarchical architecture where a meta controller learns to use the acquired skills for executing the instructions. To deal with delayed reward, we propose a new neural architecture in the meta controller that learns when to update the subtask, which makes learning more efficient. Experimental results on a stochastic 3D domain show that the proposed ideas are crucial for generalization to longer instructions as well as unseen instructions. There has been a great deal of theoretical and experimental work in computer science on inductive inference systems, that is, systems that try to infer general rules from examples. However, a complete and applicable theory of such systems is still a distant goal. This survey highlights and explains the main ideas that have been developed in the study of inductive inference, with special emphasis on the relations between the general theory and the specific algorithms and implementations. 154 references.