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Stabilizing a linear system with Saturation Through optimal control

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1 Author(s)

We construct a continuous feedback for a saturated system x(t)=Ax(t)+Bσ(u(t)). The feedback renders the system asymptotically stable on the whole set of states that can be driven to 0 with an open-loop control. The trajectories of the resulting closed-loop system are optimal for an auxiliary optimal control problem with a convex cost and linear dynamics. The value function for the auxiliary problem, which we show to be differentiable, serves as a Lyapunov function for the saturated system. Relating the saturated system, which is nonlinear, to an optimal control problem with linear dynamics is possible thanks to the monotone structure of saturation.

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IEEE Transactions on Automatic Control  (Volume:50 ,  Issue: 5 )