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Register a new userOn optimal multiplexing of an ensemble of discrete-time constrained control systems on matrix Lie groups
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Added by
Chinmay Maheshwari
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We study a constrained optimal control problem for an ensemble of control systems. Each sub-system (or plant) evolves on a matrix Lie group, and must satisfy given state and control action constraints pointwise in time. In addition, certain multiplexing requirement is imposed: the controller must be shared between the plants in the sense that at any time instant the control signal may be sent to only one plant. We provide first-order necessary conditions for optimality in the form of suitable Pontryagin maximum principle in this problem. Detailed numerical experiments are presented for a system of two satellites performing energy optimal maneuvers under the preceding family of constraints.
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In contrast to the Euler-Poincar{e} reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is a velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. Consider now a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group, and assume that the mass geometry of the system may change under the action of internal control forces. Such system can also be reduced to the Lie group. With no controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and thus its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions, under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We show that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold.
Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ..., S(A_N)$ in the sense of finding the Euclidean least-squares distance $$min {|X_1+ ... + X_N - A_0|: X_j in S(A_j), j = 1, >..., N}.$$ Connections of the results to different pure and applied areas are discussed.
In this paper we study multi-agent discrete-event systems where the agents can be divided into several groups, and within each group the agents have similar or identical state transition structures. We employ a relabeling map to generate a template structure for each group, and synthesize a scalable supervisor whose state size and computational process are independent of the number of agents. This scalability allows the supervisor to remain invariant (no recomputation or reconfiguration needed) if and when there are agents removed due to failure or added for increasing productivity. The constant computational effort for synthesizing the scalable supervisor also makes our method promising for handling large-scale multi-agent systems. Moreover, based on the scalable supervisor we design scalable local controllers, one for each component agent, to establish a purely distributed control architecture. Three examples are provided to illustrate our proposed scalable supervisory synthesis and the resulting scalable supervisors as well as local controllers.
The problem of time-constrained multi-agent task scheduling and control synthesis is addressed. We assume the existence of a high level plan which consists of a sequence of cooperative tasks, each of which is associated with a deadline and several Quality-of-Service levels. By taking into account the reward and cost of satisfying each task, a novel scheduling problem is formulated and a path synthesis algorithm is proposed. Based on the obtained plan, a distributed hybrid control law is further designed for each agent. Under the condition that only a subset of the agents are aware of the high level plan, it is shown that the proposed controller guarantees the satisfaction of time constraints for each task. A simulation example is given to verify the theoretical results.
The purpose of this paper is to describe explicitly the solution for linear control systems on Lie groups. In case of linear control systems with inner derivations, the solution is given basically by the product of the exponential of the associated invariant system and the exponential of the associated invariant drift field. We present the solutions in low dimensional cases and apply the results to obtain some controllability results.
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