Takuya Ikeda

Value function in maximum hands-off control for linear systems

In this brief paper, we study the value function in maximum hands-off control. Maximum hands-off control, also known as sparse control, is the L 0 -optimal control among the feasible controls. Although the L 0 measure is discontinuous and non-convex, we prove that the value function, or the minimum L 0 norm of the control, is a continuous and strictly convex function of the initial state in the reachable set, under an assumption on the controlled plant model. We then extend the finite-horizon maximum hands-off control to model predictive control (MPC), and prove the recursive feasibility and the stability by using the continuity and convexity properties of the value function.

Continuity of the value function in sparse optimal control

The purpose of this article is to show the continuity of the value function of the sparse optimal (or L0-optimal) control problem. The sparse optimal control is a control whose support is minimum among all admissible controls. Under the normality assumption, it is known that a sparse optimal control is given by L1 optimal control. Furthermore, the value function of the sparse optimal control problem is identical with that of the L1-optimal control problem. From these properties, we prove the continuity of the value function of the sparse optimal control problem by verifying that of the L1-optimal control problem.

Discrete-Valued Control of Linear Time-Invariant Systems by Sum-of-Absolute-Values Optimization

In this paper, we propose a new design method of discrete-valued control for continuous-time linear time-invariant systems based on sum-of-absolute-values (SOAV) optimization. We first formulate the discrete-valued control design as a finite-horizon SOAV optimal control, which is an extended version of

Maximum hands-off control without normality assumption

{\mathbb{L}^{1}}

Computation of maximum hands-off control

optimal control. We then give simple conditions that guarantee the existence, discreteness, and uniqueness of the SOAV optimal control. Also, we show that the value function is continuous, by which we prove the stability of infinite-horizon model predictive SOAV control systems. We provide a fast algorithm for the SOAV optimization based on the alternating direction method of multipliers (ADMM), which has an important advantage in real-time control computation. A simulation result shows the effectiveness of the proposed method.

Consensus by maximum hands-off distributed control with sampled-data state observation

Maximum hands-off control is a control that has the minimum L0 norm among all feasible controls. It is known that the maximum hands-off (or L0-optimal) control problem is equivalent to the L1-optimal control under the assumption of normality. In this article, we analyze the maximum hands-off control for linear time-invariant systems without the normality assumption. For this purpose, we introduce the Lp-optimal control with 0 <; p <; 1, which is a natural relaxation of the L0 problem. By using this, we investigate the existence and the bang-off-bang property (i.e. the control takes values of ±1 and 0) of the maximum hands-off control. We then describe a general relation between the maximum hands-off control and the L1-optimal control. We also prove the continuity and convexity property of the value function, which plays an important role to prove the stability when the (finite-horizon) control is extended to model predictive control.

A frameless, armless navigational system for computer-assisted neurosurgery. Technical note

Maximum hands-off control is a control that has the minimum L0 norm among all feasible controls. So far, we have proved that the maximum hands-off control is equivalent to the L1-optimal control under the normality assumption and is in general equivalent to the Lp-optimal control with 0 <; p <; 1. In this paper, by utilizing these results we give a numerical optimization method for the maximum hands-off control. We adopt a time discretization approach. As the complexity of the approximated problem then grows exponentially, we instead solve the equivalent L1 or Lp-optimization. Under the normality assumption we apply the alternating direction method of multipliers (ADMM) for the maximum hands-off control, and otherwise we apply the successive linearization algorithm (SLA).

Discrete-Valued Control by Sum-of-Absolute-Values Optimization.

In this paper, we propose a distributed control algorithm for consensus of dynamical multi-agent systems based on maximum hands-off control with sampled-data state observation. Maximum hands-off control is a control that maximizes the time duration on which the control is exactly zero among the feasible controls, which can reduce fuel or electricity consumption while the control signals take the value of zero. We give theorems for feasibility, characterization and stability for the proposed control that reaches consensus. A simulation result is shown to illustrate the effectiveness of the proposed control.