Guaranteed State Estimation Using a Bundle of Interval Observers with Adaptive Gains Applied to the Induction Machine

Abstract

The scope of this paper is the design of an interval observer bundle for the guaranteed state estimation of an uncertain induction machine with linear, time-varying dynamics. These guarantees are of particular interest in the case of safety-critical systems. In many cases, interval observers provide large intervals for which the usability becomes impractical. Hence, based on a reduced-order hybrid interval observer structure, the guaranteed enclosure within intervals of the magnetizing current’s estimates is improved using a bundle of interval observers. One advantage of such an interval observer bundle is the possibility to reinitialize the interval observers at specified timesteps during runtime with smaller initial intervals, based on previously observed system states, resulting in decreasing interval widths. Thus, unstable observer dynamics are considered so as to take advantage of their transient behavior, whereby the overall stability of the interval estimation is maintained. An algorithm is presented to determine the parametrization of reduced-order interval observers. To this, an adaptive observer gain is introduced with which the system states are observed optimally by considering a minimal interval width at variable operating points. Furthermore, real-time capability and validation of the proposed methods are shown. The results are discussed with simulations as well as experimental data obtained with a test bench.

Interval observer design for nonlinear systems using simplified contraction theory

“…. This allows us to conclude that Equations ( 9) and ( 10) is a framer for Equation (8). Usually, it is not very difficult to achieve the framer property, which is the notion of providing intervals in which state variables stay, if one does not care the length of estimated intervals.…”

“…We introduce the following theorem as guidelines for selecting K in Lemma 2 to make framer Equations ( 9) and (10) become interval observer (i.e. framer satisfies a convergence property) for Equation (8).…”

“…Consider the system ( 8), ( 9), (10) with d (t ) = 0, d and d = 0. Let us assume that the origin is the equilibrium point of system (8). Suppose there exists the control u(y, z ), u(0, 0) = 0, u as a general smooth nonlinear function, such that the derivative of vector valued distance along virtual dynamics of the system (8) follows…”

“…In fact, they have a major advantage over classical observers in the sense that they can easily cope up with large uncertainties and have been successfully applied in biological fields [4] where large uncertainties are present. Moreover, in the recent years, interval observers have also been applied for several industrial and engineering applications [5][6][7][8][9]. The structure of interval observer comprises of two dynamical equations that estimate upper and lower bounds (interval) of the state vector at all times when the initial conditions are unknown and there are uncertainties present in the system, however, these initial conditions and uncertainties are bounded.…”

A Combined Model and Data-Driven Approach for the Determination of Rotor Temperature in an Induction Machine

“…The time constant of a first-order thermal model is computed based on a polynomial function of the speed and stator temperature of a PMSM [ 20 ]. A design procedure for guaranteed state estimation of unmeasurable states with practical application to induction machines is discussed [ 21 ]. Related to the approach involving particle filter we mention a few remarks.…”