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

Abstract

FDEs are powerful modeling tools in many engineering applications in which non-standard dynamics, characterized by infinite horizon states, can be observed. An example for such applications is modeling the charging and discharging dynamics of batteries. Previous work for an interval-based state estimation of such systems has accounted for a cooperativity preserving or cooperativity enforcing design of observers. These interval observers exploit specific monotonicity properties of positive dynamic systems and provide lower and upper bounding trajectories for all pseudo-state variables as soon as suitable initialization functions are specified. Moreover, interval-valued iteration procedures have been developed for a verified simulation of such systems. The latter, based on Mittag-Leffer function parameterizations of the pseudo-state enclosures, are not a priori restricted to cooperative models but are applicable also to nonlinear systems with interval parameters.

Full text

Quantification of Time-Domain Truncation Errors for the
Reinitialization of Fractional Integrators
Andreas Rauh1and Rachid Malti2
1ENSTA Bretagne, Lab-STICC, 29806 Brest, France
2IMS Laboratory, University of Bordeaux, 33405 Talence, France
Andreas.Rauh@interval-methods.de, rachid.malti@ims-bordeaux.fr
Keywords: Fractional differential equations (FDEs), Observer design, Uncertain cooperative
dynamics, Temporal truncation errors, State estimation
FDEs are powerful modeling tools in many engineering applications in which non-standard
dynamics, characterized by infinite horizon states, can be observed. An example for such ap-
plications is modeling the charging and discharging dynamics of batteries [1]. Previous work
for an interval-based state estimation of such systems has accounted for a cooperativity pre-
serving or cooperativity enforcing design of observers [1, 3]. These interval observers exploit
specific monotonicity properties of positive dynamic systems and provide lower and upper
bounding trajectories for all pseudo-state variables as soon as suitable initialization functions
are specified. Moreover, interval-valued iteration procedures have been developed [5] for a
verified simulation of such systems. The latter, based on Mittag-Leffer function parameteriza-
tions of the pseudo-state enclosures, are not a priori restricted to cooperative models but are
applicable also to nonlinear systems with interval parameters.
However, the evaluation of observer-based pseudo-state estimation procedures for continuous-
time fractional models supposes that measurements are also available in a continuous-time
form or at least at each sampling period [3]. For many practical applications, this is not the
case, so that continuous-time pseudo-state predictions need to be performed between the dis-
crete time instants at which measurements are available. Then, the measured pseudo-state
information (described by intervals to represent bounded measurement errors) can be inter-
sected with the predicted state information to enhance the knowledge of the actual system
dynamics. However, this intersection demands reinitializing the integration of the fractional
model. Similar requirements are discussed in [5], where temporal sub-slices were considered
to reduce the overestimation of interval-based simulation approaches.
Due to the infinite horizon memory property of fractional systems, the reinitialization of
time-domain simulations requires a rigorous consideration of the arising truncation errors.
Guaranteed outer bounds for these errors were derived in [4]. These bounds are the basis for
a novel error refinement strategy between discrete reinitialization points in an observer-based
setting. In this contribution, we discuss the following aspects:
1. Expressing non-constant pseudo-state initializations from a bounded past time window
in terms of uncertain initial conditions at a single point with a conservative interval-
valued correction of the FDE model;
2. Implementation of an observer-based quantification of truncation errors for simulations
with periodic reinitialization (e.g. based on the floating point MATLAB routines from [2]);
3. Performing an interval contractor-based state estimation of a continuous-time battery
model [1] with discrete-time measurements;
4. Describing possible interfaces with the verified simulation routines
from [5] as an outlook for future work.
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References
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