Issues on Nonlinear Digital Control

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

“…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:…”

“…Alternatively, in the literature often Fliess series approximations to Φ u k h are used as approximate models, e.g. in [11,21,25]. Convergence of these schemes follows if the systems Φ k (x, u) satisfy the Lipschitz condition (2.8) for a Lipschitz constant L > 0 independent of k, however, if the feedback laws u k are discontinuous, then (2.8) cannot hold and convergence may fail.…”

“…Of course, the regularity of the maps u k strongly depends on the sampled data feedback design method employed. While for series based (re-)design methods as presented, e.g., in [11,21,25] or for the Euler backstepping method [27,18] the Lipschitz property can be expected or even rigorously proved, optimal control based design methods as in [5,9,10,12,26] typically yield discontinuous feedback laws u k , for which convergence (7.6) cannot be expected in general.…”

Global stability results for systems under sampled‐data control

“…* considering the closed-loop system as a discrete-time system, i.e. ignoring inter-sample behavior (see [1,28]). As mentioned earlier, this point of view generally requires the knowledge of the solution map, which is rarely available for the nonlinear case.…”

Model Predictive Control for Nonlinear Sampled-Data Systems

“…Therefore, at the moment it is up to simulations to find a good set of parameters and a good initial guess of v [0,M −1] . A promising alternative approach and topic of future research is whether some of the methods developed in [9,12] can be used in order to construct the initial guess, an approach that would lead to a predictor-corrector type algorithm in which the MPC strategy plays the role of the corrector.…”