State estimation of system with bounded uncertain parameters: Interval multimodel approach

Abstract

The objective of this study is the analysis of dynamic systems represented by a multimodel expression with variable parameters. Changes in these parameters are unknown but bounded. Since it is not possible to estimate these parameters over time, the simulation of such systems requires the consideration of all possible values taken by these parameters. More precisely, the goal is to determine, at any moment, the smallest set containing all the possible values of the state vector simultaneously compatible with the state equations and with a priori known bounds of the uncertain parameters. This set will be characterized by two trajectories corresponding to the lower and upper limits of the state at every moment.This characterization can be realized by a direct simulation of the system, given the bounds of its parameters. It can also be implemented with a Luenberger-type observer, fed with the system measurements. KEYWORDSbounded uncertainties, interval observers, multimodel technique, uncertain systems INTRODUCTIONAs widely known, one of the main difficulties in system simulation, control, or estimation is dealing with uncertainties. These uncertainties may affect the input or output signals of the system (eg, unknown input, disturbance, measurement noises, etc) as well as the system model itself (eg, nonmodeled dynamics, unknown parameter, etc). The uncertainties may also be of different natures: total lack of information (unknown value of a parameter) or partial knowledge (upper and lower bounds, statistical properties, etc). This paper deals with systems with uncertain parameters.Even a low magnitude change in some parameters may have a significant impact on the system behavior and, namely, on the system state trajectory. Uncertain parameters can be considered from two main points of view: the stochastic and the deterministic ones. In the first approach, the uncertain parameters are assumed to be the results of random process realizations. It then needs to choose the probability density functions and their parameters describing the system's uncertain parameters. In the deterministic approach, no statistical models of the parameters are assumed to be available, and only the upper and lower bounds of the parameter values are known. This approach, also known as the interval approach, is adopted in this paper.The search for trajectories, which are solutions of differential inclusions, 1 were the starting point of the large amount of works on stability, stabilization, and state estimation. In the latter topic, our communication follows the pioneering work of Gouzé et al, 2 which proposed the synthesis of the interval-type observer to reconstruct the system state 480 ;32:480-493. ICHALAL ET AL. 481from measurements of inputs and outputs. Since then, many results were published on designing observers adapted to nonlinear systems with uncertain but bounded parameters. [3][4][5][6] The model-based state estimation techniques in the context of bounded uncertainties can be classified into two categories. The first one is ba...

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

“…If simplified sparse solutions for H according to (11) shall be optimized by a minimization of µ ∞ ≥ 0 according to the constraint (21) with the help of the H ∞ design described above, the LMIs M (Σ) ≺ 0 in (19) with (20)…”

“…O2 Determine optimal gains κ i > 0 in (11) as a solution of the LMIs (23) with (24) which minimize the H ∞ performance criterion (21) for the output (15). In this case, the inequality constraint (22) is already satisfied due to the predefined structure of H. Due to multiplicative couplings of Q O and K in (24), the solution is again determined iteratively.…”

“…However, a reduction of the number of boxes is inevitable to obtain system models that can be evaluated in real time by the bank of interval observers. Moreover, it has to be noted that a naive replacement of all resulting boxes (either after 10 · 10 4 or 15 · 10 4 subdivisions) is not feasible due to the fact that the upper bounding matrix turns into an unstable model due to the mutual couplings between both vectors p and p. Such a coarse enclosure is also not further investigated because it can be shown that the resulting error dynamics cannot be stabilized with the help of the structurally predefined sparse observer gain matrix according to (11). For the optimization of the observer bank with L = 362 parameter boxes, the outcomes for the options O1, O2, and O3 are not only compared for the raw measured data but also with measurements artificially corrupted by measurement noise according to the bounded tolerance of ±0.75 K. The comparison shows the excellent attenuation of noise and moreover highlights the fact that for the application at hand all three options provide quite similar estimation results.…”

“…At least for the last two decades, interval observers have been designed for several different types of dynamic system models. Such system models can be characterized into continuous-and discrete-time state-space representations of systems with finite-dimensional dynamics as well as into special types of partial differential equations (PDEs) [11,14,18,19,34,35,43]. Especially for the case of finite-dimensional systems, linear time-invariant, linear parameter-varying, linear time-varying and (special types) of nonlinear dynamics have been accounted for [6,24,40,41].…”

Adaptive control of feedforward nonlinear systems with uncertain time‐varying parameters and an unknown time‐varying delay in the input

“…Meanwhile, uncertain parameters are often encountered in many real systems 12‐16 and they can affect the stability/performance of the controlled systems. In this regard, various control methods for nonlinear systems with uncertain constant parameters have been studied 3,12,14 . For example, in Reference 12, they consider nonlinear systems with uncertain parameters where they assume that upper bounds of the uncertain parameters are known.…”