For linear systems with time-varying input delay and additive disturbances we show that the basic predictor feedback control law is inverse optimal, with respect to a meaningful differential game problem, and establish its robustness to constant multiplicative perturbations appearing at the system input. Both of these properties of the basic predictor feedback controller have not been established so far, even for the constant-delay case. We then show that the basic predictor feedback controller, when applied through a low-pass filter, is again inverse optimal and study its input-to-state stability as well as its robustness, to the low-pass filter time constant properties. All of the stability and inverse optimality proofs are based on the infinite-dimensional backstepping transformation, which allows us to construct appropriate Lyapunov functionals. A numerical example is also provided.

This article deals with event-triggered filter design of positive systems subject to state saturation. An event-triggering condition in the form of 1-norm is established based on the error and the measurable output. Under the presented event-triggering condition, the filter systems are transformed into interval uncertain systems. The positivity of the filter and error dynamics is achieved by considering the lower bound of the interval systems. Using a linear copositive Lyapunov function, the stability is guaranteed by considering the upper bound of the interval systems. By virtue of linear programming, an $\ell _1$ -gain filter is designed for the systems subject to exogenous disturbance. Then, multiplicative and additive gain fluctuations are introduced. Two classes of nonfragile filters are proposed, and a cone is constructed to guarantee that the states of the systems with state saturation can keep inside it. Finally, two examples are given to illustrate the effectiveness of the proposed approach.