Interval observers for continuous-time LPV systems with L1/L2 performance

Abstract

An approach to interval observer design for Linear Parameter-Varying (LPV) systems is proposed. It is assumed that the vector of scheduling parameters in LPV models is not available for measurement. Two different interval observers are constructed for nonnegative systems and for a generic case. Stability conditions are expressed in terms of matrix inequalities, which can be solved with respect to the observer gains using standard numerical solvers. Applying L1/L2 framework the robustness and estimation accuracy with respect to model uncertainty are analyzed. The efficiency of interval estimation for LPV models is demonstrated through numerical experiments for a microfluidic system and an academic example. I. INTRODUCTIONState estimation is an important issue in many engineering fields [1], [2], [3]. Estimated states may be required, for example, for control design or fault detection. This problem has been widely studied in the literature and many solutions already exist for linear systems and a number of nonlinear structures. For the latter situation, the observer design problem is solvable if the system model can be transformed into a canonical form, which may be a hard assumption to satisfy in many applications. To solve the problem, an appealing approach is based on the LPV transformation of the nonlinear system [4], [5], [6], [7].Note that Takagi-Sugeno decomposition can be another alternative solution to deal with nonlinear systems and to obtain the equivalent representation by a compact set of linear state space models with nonlinear weighting functions satisfying the convex sum property [8], [9]. In the presence of uncertainty (unknown parameters or/and external disturbances) the design of a conventional estimator, converging to the ideal value of the state, cannot be realized. However, an interval estimation may still remain feasible: an observer can be constructed that, using input-output information, evaluates the set of admissible values (interval) for the state at each instant of time. The interval length has to be minimized and it is proportional to the size of the model uncertainty. Despite such a formulation looks like a simplification of the state estimation problem, in fact it is an improvement since the interval mean can be used as the state pointwise estimate, while the interval limits give the admissible deviations from that value (thus, an interval estimator provides a simultaneous accuracy evaluation for bounded uncertainty, which may not have a known statistics). There are several approaches to design interval/set-membership estimators [10], [11], [12], [13]. This paper continues the trend of interval observer design based on the monotone systems theory [12], [13], [14], [15], [16]. In such a way the main restriction for the interval observer design consists in providing cooperativity of the interval estimation error dynamics by a proper design. Such a complexity has been recently overcame in [17], [15], [18] for LTI systems. In those studies, it has beenshown that under some mild condition...

Interval observers design for singularly perturbed systems

“…The proof is similar to the one of Theorem 1 in [24]. Consider the dynamics of interval estimation errors e = x s − x s and e = x s − x s , one can write: (14), (15) and (16), e(t) and e(t) are nonnegative, let's show that the variables x s (t) and x s (t) stay bounded for all t ≥ 0 in (17).…”

Robust state estimation for singularly perturbed systems

“…Then x s (t) and x s (t) stay bounded for all t ≥ t 0 , and the transfer δ → Zξ has an L ∞ gain less than γ [5].…”

Interval Prediction for Continuous-Time Systems with Parametric Uncertainties

“…In such a way the main difficulty for synthesis consists in ensuring cooperativity of the interval error dynamics by a proper design of the algorithm. As it has been shown in [15], [13], [16], such a complexity of the design can be handled by applying Edouard an additional transformation of coordinates to map a stable system into a stable and monotone one, see also [17], [7].…”