A. Steinberg

Feedback control of uncertain systems: robustness with respect to neglected actuator and sensor dynamics†

Feedback control of a class of imperfectly known dynamical systems is considered. On the basis of known functional properties and bounds relating to the uncertain elements in the generic system, and initially neglecting control actuator and state sensor dynamics, a feedback structure is determined (using established techniques) which guarantees uniform ultimate boundedness of all motions of the system. The effects on performance of the introduction of (i) uncertain actuator dynamics, and (ii) uncertain sensor dynamics, with respective parameters μs≥ 0 and μs≥ 0 which (inversely) reflect the ‘fastness’ of these dynamics, are subsequently investigated, with main emphasis on case (i). Under appropriate assumptions, in case (i) the existence of a calculable threshold value μs∗ > 0 is established such that the property of global uniform ultimate boundedness is retained by the system for all parameter values μa e (0, μs∗). Finally, additional conditions are imposed on the sensor structure which enable the resul...

Stabilization of nonlinear systems with a dither control

Abstract In this paper a new approach is taken to analyze stabilization of a general nonlinear system with a dither input. Given the original system with a control, an autonomous relaxed system is constructed. It is shown that if the relaxed system is stable, then the original system with dither control would be stable in the finite time. An algorithm is given for constructing the dither control. The technique used here is general and does not have the limitations of the Dual Input Describing Function technique. Furthermore, in many cases it is possible to guarantee global contractive stability as well. Two examples are solved in detail using computer simulations for demonstration of the technique.

Output stabilizing robust control for discrete uncertain systems

Abstract A comprehensive solution to the problem of the robust output min–max control for discrete uncertain MIMO systems with matched uncertainties is presented in this paper. In the literature, state-dependent Lyapunov and Riccati min–max controllers were treated independently and the existence problem of the output min–max controllers has not been fully resolved. In this paper, both Lyapunov- and Riccati-based min–max output control laws are treated, and a simple criterion for the existence of a min–max output controller is provided. It is shown that when the criterion is satisfied, both Riccati- and Lyapunov-based output min–max controllers exist. The first controller is applied directly to the given system, while the latter controller is applied to the given system closed with a suitable feedback. Moreover, it is shown that both overall controllers are identical. In addition, an alternative criterion for the existence of a Lyapunov-based min–max output control, in terms of discrete positive real properties of the system, is derived.

On existence of optimal controls

AbstractAn optimal control is shown to exist for a system

Application of relaxed solutions to minimum sensitivity optimal control

\dot x = f(x,t,u)

The equivalent nonlinearity for a nonlinear system with dither

when the Hamiltonian is a strictly convex function of the control. It is proven that a system satisfying this condition must have state equations that are linear in the control and a cost functional whose integrand is strictly convex in the control.

On relaxed control and singular solutions

Stability of singularly perturbed systems with a periodic input is analysed by deriving the reduced system and using the relaxed method for analysing stability of the reduced system.

Numerical solution of a relaxed control problem by the epsilon method

Relaxed variational techniques are applied to a minimum sensitivity control problem. Sensitivity of a trajectory is minimized to perturbations in initial conditions. Rather than using the optimal control that does indeed exist and that satisfies the final conditions exactly, a suboptimal control is used that transfers the system from the given initial state to an arbitrary small neighborhood of the given final state, and that results in a considerably better performance than the optimal solution. The suboptimal control is constructed using the optimal controls of the relaxed problem.