Discrete feedback stabilization of semilinear control systems

Abstract

Abstract. F or continuous time semilinear control systems with constrained control values stabilizing discrete feedback c o n trols are discussed. It is shown that under an accessibility condition exponential discrete feedback stabilizability is equivalent to open loop exponential asymptotic null controllability. A n umerical algorithm for the computation of discrete feedback c o n trols is presented and a numerical example is discussed.

Input‐to‐state stability, numerical dynamics and sampled‐data control

“…In fact, this robust stability property is at the heart of many stability proofs for sampled-data systems in the literature, even if rarely explicitly mentioned. The respective ISS property for the inflated system is often ensured via Lyapunov functions, as in [9], [25] or [23], or via optimal value functions as in [4]. Our result shows that these conditions are not only sufficient for practical asymptotic stability but in fact necessary and sufficient for the exact sampled-data system (7.3) being practically ISS w.r.t.…”

“…4 We illustrate the theorem with a slight variation of a classical example from [3]. Consider the two-dimensional ordinary differential equation given bẏ It is easily seen that for the original system each disc…”

“…When proceeding this way we face the problem that the discrete time system Φ u h induced by (7.3) is hardly ever known exactly, because it is the solution of a nonlinear ordinary differential equation. Therefore, typically one has to design the feedback controller for a family of approximations Φ u k , k ∈ N, to Φ u h , see, e.g., [4,5,9,10,11,12,18,21,25,27,26] for examples of such design methods. Since a feedback law constructed on basis of an approximation Φ u k will depend on k, we need to consider sequences of feedback laws u k in the following definition of the approximating systems:…”

Homogeneous State Feedback Stabilization of Homogenous Systems

“…They generalize homogeneous bilinear and semilinear systems (see e.g. 5,6,9]). One interpretation of this structure is that the control a ects parameters in the system rather that representing some force acting on the system, cf.…”

Optimization-Based Stabilization of Sampled-Data Nonlinear Systems via Their Approximate Discrete-Time Models

“…An important set of stabilization methods is based on optimization techniques, such as receding horizon control (RHC) or model predictive control (MPC) (see [14,7] and references defined therein). In optimization based stabilization methods one can either compute control signals on-line, like in MPC algorithms, or off-line, like in [8,9,13]. In either case, it is common to implement the controller using a computer with A/D and D/A converters (sampler and zero-order hold) which leads to investigation of sampled-data nonlinear systems.…”