Deepak U. Patil

Computation of Time Optimal Feedback Control Using Groebner Basis

The synthesis of time optimal feedback control of a single input continuous time linear time invariant (LTI) system is considered. If the control input u(t) is constrained to obey |u(t)| ≤ 1, then it is known that the optimal input switches between the extreme values ±1, according to some “switching surfaces” in the state space. It is shown that for systems with non-zero, distinct and rational eigenvalues, these switching surfaces are semi-algebraic sets and a method to compute them using Groebner basis, is proposed. In the process the null-controllable region for such systems is characterized and computed. Numerical simulations illustrate the proposed computational methods.

Computation of feedback control for time optimal state transfer using Groebner basis

Abstract Computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not limited to the origin of state-space, but is allowed to be in the set of constrained controllable states. Switching surfaces are formulated as semi-algebraic sets using Groebner basis based elimination theory. Using these semi-algebraic sets, a nested switching logic is synthesized to generate the time optimal feedback control. However, the optimal control law enforces an unavoidable limit cycle in the time-optimal trajectory for most non-origin target points. The time-period of this limit-cycle is dependent on the target position. This dependence is algebraically characterized and a method to compute the time-period of the limit-cycle is provided. As a natural extension, the set of constrained controllable states is also computed.

Switching surfaces and null-controllable region of a class of LTI systems using Gröbner basis

The problem of time optimal feedback control of a single input, continuous time, linear time invariant (LTI) system is considered. The control input is constrained to obey |u(t)| ≤ 1. It is known that the solution to this problem is bang-bang with the input switching between the extreme values ±1 according to some “switching surfaces” in the state space. It is shown that for a class of LTI systems, these switching surfaces can be represented as semi-algebraic sets and a method to construct these switching surfaces using Gröbner basis, is proposed. Numerical simulations demonstrate the effectiveness of this feedback control strategy.

Distributed computation of minimum time consensus for multi-agent systems

The problem of computing the minimum time to consensus of multiple identical double-integrator agents is considered. A distributed algorithm for computing the final consensus target state is proposed. Local feedback time optimal control laws are synthesized to drive each agent to the computed final consensus target state. Each agent is assumed to know the states of every other agent. As a part of the algorithm, every possible triplet of agents compute their mutual minimum time to consensus. The maximum value among these triplet minimum times is shown to be the required minimum time to consensus for the entire group. Since the required computation can be performed for each triple separately, it can be distributed evenly among the agents.

Gröbner basis computation of feedback control for time optimal state transfer

The synthesis of time-optimal feedback control of a single input, continuous time, linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not necessarily the origin of the state space. Semi-algebraic representations of switching surfaces corresponding to the bang-bang time-optimal control are computed using a Gröbner basis based elimination algorithm. These surfaces are then used to synthesize a nested switching logic for time-optimal feedback control. Non-origin target points introduce unavoidable limit cycles in time optimal trajectories, whose period depends on the target position. This dependence is algebraically characterized and a method to compute the periods of the limit cycles is proposed. As a natural extension, we also provide a semi-algebraic characterization of the set of all points reachable with constrained inputs.

Stochastic discrete higher order sliding mode control

This paper defines the stochastic discrete higherorder sliding mode. A control input is designed for an uncertain stochastic system with partial state information such that stochastic discrete higher-order sliding mode takes place. The proposed definition and control scheme is validated through the simulation of a rectilinear plant.

Switch observability for a class of inhomogeneous switched DAEs

Necessary and sufficient conditions for switching time and switch observability of a class of inhomogeneous switched differential algebraic equations (DAEs) are obtained. A characterization of initial states and inputs for which switched DAEs are switch unobservable is also provided by using the zeros of an augmented system obtained by combining the output of two modes suitably.

Computation of the target state and feedback controls for time optimal consensus in multi-agent systems

ABSTRACT N identical agents with bounded inputs aim to reach a common target state (consensus) in the minimum possible time. Algorithms for computing this time-optimal consensus point, the control law to be used by each agent and the time taken for the consensus to occur, are proposed. Two types of multi-agent systems are considered, namely (1) coupled single-integrator agents on a plane and, (2) double-integrator agents on a line. At the initial time instant, each agent is assumed to have access to the state information of all the other agents. An algorithm, using convexity of attainable sets and Hellys theorem, is proposed, to compute the final consensus target state and the minimum time to achieve this consensus. Further, parts of the computation are parallelised amongst the agents such that each agent has to perform computations of O(N2) run time complexity. Finally, local feedback time-optimal control laws are synthesised to drive each agent to the target point in minimum time. During this part of the operation, the controller for each agent uses measurements of only its own states and does not need to communicate with any neighbouring agents.

An O(N 2 ) algorithm for computation of the minimum time consensus

The problem of achieving minimum time consensus for an N-agent system, with double integrator agents having bounded inputs, is considered. At the initial time instant, each agent has access to the state information about all the other agents. An algorithm, of O(N2) complexity, is proposed to compute the final consensus target state and minimum time to achieve this consensus. Further, local control laws are synthesized to drive each agent to the target point in the computed minimum time to consensus.