Fractional interval observers and initialization of fractional systems

Romig

2021

Sensors

Linear matrix inequalities (LMIs) have gained much importance in recent years for the design of robust controllers for linear dynamic systems, for the design of state observers, as well as for the optimization of both. Typical performance criteria that are considered in these cases are either H2 or H∞ measures. In addition to bounded parameter uncertainty, included in the LMI-based design by means of polytopic uncertainty representations, the recent work of the authors showed that state observers can be optimized with the help of LMIs so that their error dynamics become insensitive against stochastic noise. However, the joint optimization of the parameters of the output feedback controllers of a proportional-differentiating type with a simultaneous optimization of linear output filters for smoothening measurements and for their numeric differentiation has not yet been considered. This is challenging due to the fact that the joint consideration of both types of uncertainties, as well as the combined control and filter optimization lead to a problem that is constrained by nonlinear matrix inequalities. In the current paper, a novel iterative LMI-based procedure is presented for the solution of this optimization task. Finally, an illustrating example is presented to compare the new parameterization scheme for the output feedback controller—which was jointly optimized with a linear derivative estimator—with a heuristically tuned D-type control law of previous work that was implemented with the help of an optimized full-order state observer.

Section: Linear Output Filteringmentioning

confidence: 99%

Section: Linear Output Filteringmentioning

confidence: 99%

See 1 more Smart Citation

Linear Matrix Inequalities for an Iterative Solution of Robust Output Feedback Control of Systems with Bounded and Stochastic Uncertainty

Romig

2021

Sensors

“…Previous work for an interval-based state estimation of such systems has accounted for a cooperativity preserving or cooperativity enforcing design of observers [3,11]. These interval observers exploit specific monotonicity properties of positive dynamic systems and provide lower and upper bounding trajectories for all pseudo state variables 1 as soon as suitable initialization functions for the fractional dynamic system model are specified.…”

Section: Introductionmentioning

confidence: 99%

Quantification of Time-Domain Truncation Errors for the Reinitialization of Fractional Integrators

Malti

2022

Acta Cybern

Self Cite

In recent years, fractional differential equations have received a significant increase in their use for modeling a wide range of engineering applications. In such cases, they are mostly employed to represent non-standard dynamics that involve long-term memory effects or to represent the dynamics of system models that are identified from measured frequency response data in which magnitude and phase variations are observed that could be captured either by low-order fractional models or high-order rational ones. Fractional models arise also when synthesizing CRONE (Commande Robuste d'Ordre Non Entier) and/or fractional PID controllers for rational or fractional systems. In all these applications, it is frequently required to transform the frequency domain representation into time domain. When doing so, it is necessary to carefully address the issue of the initialization of the pseudo state variables of the time domain system model. This issue is discussed in this article for the reinitialization of fractional integrators which arises among others when solving state estimation tasks for continuous-time systems with discrete-time measurements. To quantify the arising time-domain truncation errors due to integrator resets, a novel interval observer-based approach is presented and, finally, visualized for a simplified battery model.

“…If these latter properties are guaranteed to be satisfied, it becomes possible to evaluate lower and upper bounding trajectories independently during the numerical simulation. Such properties are often exploited during the design of interval observers which can analogously be derived for both, integer-order and fractional-order system representations [3,8,20,31].…”

Section: Introductionmentioning

confidence: 99%

Toward the Development of Iteration Procedures for the Interval-Based Simulation of Fractional-Order Systems

Kersten

2020

Acta Cybern

In many fields of engineering as well as computational physics, it is necessary to describe dynamic phenomena which are characterized by an infinitely long horizon of past state values. This infinite horizon of past data then influences the evolution of future state trajectories. Such phenomena can be characterized effectively by means of fractional-order differential equations. In contrast to classical <em>linear</em> ordinary differential equations, <em>linear</em> fractional-order models have frequency domain characteristics with amplitude responses that deviate from the classical integer multiples of ±20 dB per frequency decade and, respectively, deviate from integer multiples of ±π/2 in the limit values of their corresponding phase response. Although numerous simulation approaches have been developed in recent years for the numerical evaluation of fractional-order models with point-valued initial conditions and parameters, the robustness analysis of such system representations is still a widely open area of research. This statement is especially true if interval uncertainty is considered with respect to initial states and parameters. Therefore, this paper summarizes the current state-of-the-art concerning the simulation-based analysis of fractional-order dynamics with a restriction to those approaches that can be extended to set-valued (interval) evaluations for models with bounded uncertainty. Especially, it is shown how verified simulation techniques for integer-order models with uncertain parameters can be extended toward fractional counterparts. Selected linear as well as nonlinear illustrating examples conclude this paper to visualize algorithmic properties of the suggested interval-based simulation methodology and point out directions of ongoing research.