Verification Techniques for Sensitivity Analysis and Design of Controllers for Nonlinear Dynamic Systems with Uncertainties

Abstract

Control strategies for nonlinear dynamical systems often make use of special system properties, which are, for example, differential flatness or exact input-output as well as input-to-state linearizability. However, approaches using these properties are unavoidably limited to specific classes of mathematical models. To generalize design procedures and to account for parameter uncertainties as well as modeling errors, an interval arithmetic approach for verified simulation of continuoustime dynamical system models is extended. These extensions are the synthesis, sensitivity analysis, and optimization of open-loop and closed-loop controllers. In addition to the calculation of guaranteed enclosures of the sets of all reachable states, interval arithmetic routines have been developed which verify the controllability and observability of the states of uncertain dynamic systems. Furthermore, they assure asymptotic stability of controlled systems for all possible operating conditions. Based on these results, techniques for trajectory planning can be developed which determine reference signals for linear and nonlinear controllers. For that purpose, limitations of the control variables are taken into account as further constraints. Due to the use of interval techniques, issues of the functionality, robustness, and safety of dynamic systems can be treated in a unified design approach. The presented algorithms are demonstrated for a nonlinear uncertain model of biological wastewater treatment plants.

Parameter identification and observer-based control for distributed heating systems– the basis for temperature control of solid oxide fuel cell stacks

“…The rank of Q(x) yields sufficient information about the dimension of the observable manifold of the dynamical system [33,35,36].…”

“…First, uncertainties and robustness requirements can be expressed directly in the constraints (18). In addition, the verified approach can also handle differentially non-flat systems if stability of the internal dynamics can be guaranteed [32,33]. For most of these non-flat systems, the output y(t) does not coincide exactly with y d (t).…”

Nonlinear state observers and extended Kalman filters for battery systems

“…Using techniques for symbolic formula manipulation, it can be shown that the observability mapping (Rauh et al, 2009;Isidori, 1995) contains additional information that is not present in the cases in which the system model is linearized first. Moreover, the Jacobian of this observability mapping is a matrix of full rank, corresponding to a sufficient criterion for local observability.…”

Sensitivity Analysis of Uncertain Dynamic Systems Using Set-Valued Integration

“…Despite their widespread use in such areas as uncertainty quantification, experimental design, parameter estimation, optimization, and control, very little work has been devoted to set-valued integration for sensitivity analysis so far. In [39], the parametric ODEs and their first-order sensitivities are bounded as one combined set of ODEs. But more efficient staggered or simultaneous corrector methods have not been investigated in this context to date.…”

“…The nonlinear term N ϕ k (resp., N η k ) is so chosen that (37) and (40) (resp., (39) and (41) Moreover, we shall assume for now that nonlinearity bounders Ω ϕ k (t), Ω η0 , Ω η k ∈ IR ny can be constructed such that…”