On Stabilization of Nonlinear Time–Delay Systems via Quantized Sampled–Data Dynamic Output Feedback Controllers

On the design and the digital implementation of observer‐based controllers for tracking of nonlinear time‐delay systems

“…The following proof is based on the results recently provided in the work by Di Ferdinando et al 24 where the stabilization in the sample-and-hold sense theory [20][21][22][23][24] is used as a tool in order to show that: there exists a suitably small sampling and an accurate quantization of the input/output channels such that the digital implementation of DOSDFs (continuous or not) guarantees the semi-global practical stability property of the related quantized sampled-data closed-loop system, with arbitrarily small final target ball of the origin (see Theorem 1 in the work by Di Ferdinando et al 24 ). It is here highlighted that, the proof of Theorem 1 provided in the work by Di Ferdinando et al 24 cannot be directly applied here due to new considerations regarding: (i) the case of observer-based tracking controllers; (ii) the problems related to the possible non-availability in the buffer of suitable past values of the output signal required for the correct implementation of a proposed delay-dependent observer-based tracking controller. Then, a new devoted proof is required to cope with tracking control problems and the use of spline methodologies for the approximation of the infinite dimensional output signal y z,t which, in the work by Di Ferdinando et al, 24 are not considered.…”

“…In particular, under suitable conditions, it is proved that there exist a suitably fast sampling and an accurate quantization of the input/output channels such that the digital implementation of the continuous-time observer-based tracking controller (41) (see also (48)) ensures the semi-global practical stability of the related quantized sampled-data closed-loop tracking error system, with arbitrarily small final target ball of the origin. The notion of DOSDF 22,24 and the stabilization in the sample-and-hold sense theory will be used as tools to provide the sufficient conditions for the digital implementation of the continuous-time observer-based tracking controller (41) (see also (48)). First, we notice that system (37) is in the following form…”

“…We denote here with  the set of Lyapunov-Krasovskii functionals V ∶  2n → R + with the following properties 22,24 :…”

“…In the following, the stabilization in the sample-and-hold theory [20][21][22][23][24] and the notion of DOSDF 22,24 are used as tools in order to prove the results. First, for the reader's convenience, the notion of DOSDF is recalled.…”

“…As a second contribution of this article, we provide sufficient Lyapunov-Krasovskii like conditions for the existence of a suitably fast sampling and of an accurate quantization of the input/output channels such that the digital implementation of the proposed continuous-time observer-based tracking controller ensures the semi-global practical stability property of the related quantized sampled-data closed-loop tracking error system with arbitrarily small final target ball of the origin. The stabilization in the sample-and-hold sense theory [20][21][22][23] and the notion of dynamic output steepest descent feedbacks (DOSDFs) 22,24 are used as a tool to prove the results. Time-varying sampling periods as well as the nonuniform quantization of the input/output channels are taken into account.…”