H_∞‐optimal observer design for systems with multiple delays in states, inputs and outputs: A PIE approach

Abstract

This article investigates the H ∞ -optimal estimation problem of a class of linear system with delays in states, disturbance input, and outputs. The estimator uses an extended Luenberger estimator format which estimates both the present and history states. The estimator is designed using an equivalent Partial Integral Equation (PIE) representation of the coupled nominal system. The advantage of the resulting PIE representation is compact and delay free-obviating the need for commonly used bounding technique such as integral inequalities which typically introduces conservatism into the resulting optimization problem. The H ∞ -optimal estimator synthesis problem is then reformulated as a Linear Partial Inequality (LPI)-a form of convex optimization using operator variables and inequlities. Such LPI-based optimization problems can be solved using semidefinite programming via the PIETOOLS toolbox in Matlab. Compared with previous work, the proposed method simplifies the analysis and computation process and resulting in observers which are non-conservtism to 4 decimal places when compared with Pad é-based ODE observer design methodologies. Numerical examples and simulation results are given to illustrate the effectiveness and scalability of the proposed approach.

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“…In view of ( 20)-( 22), (97) and the fact that E i 0, Y i 0, the DD matrices in (18) can be denoted as…”

“…with H i (τ ), E i (τ ) in ( 23) and κ i , µ i in Proposition 1. Applying ( 28)-( 30) to (18) in light of Lemma 3 produces…”

“…The construction of Krasovskiȋ functionals (KF) in timedomain [16] has increasingly become an effective strategy for the stability analysis and state estimation of LTDSs [17], [18], where the stability (synthesis) conditions are denoted as semidefinite programming (SDPs) problems [19]. The conservatism of this approach is mainly affected by the complexity of the underlying delay systems, the generality of the predetermined form of KFs [16], and the integral inequalities employed to construct them.…”

“…To the best of our knowledge, we may state that no effective solutions exist in the literature to the state and output estimation problem of LTDSs when both pointwise and general DDs 0 −r F (τ )x(t+τ )dτ are considered with dissipative constraints [22], [26]. Almost all existing methods [9], [18], [27]- [30] on this topic only consider pointwise delays, require delays to be commensurate [31], or restrict the number of delays to one [27], [28], [30]. As matrix integral kernel F (•) may contain general functions and 0 −r F (τ )e τ s dτ may not yield a closedform expression in s ∈ C, an effective solution to the mentioned problem needs to handle the mathematical complexity associated with all delays and dissipativity simultaneously.…”

“…As matrix integral kernel F (•) may contain general functions and 0 −r F (τ )e τ s dτ may not yield a closedform expression in s ∈ C, an effective solution to the mentioned problem needs to handle the mathematical complexity associated with all delays and dissipativity simultaneously. A noteworthy approach is proposed in [18] for the H ∞ state estimation of an LTDS with an unlimited number of pointwise delays, using the sums-of-squares approach [32] together with a coupled PDE-ODE estimator. However, when the PDE component is implemented using various discretization [33], the robustness of the closed-loop system (CLS) with the PDE-ODE estimator may not be guaranteed, which can lead to potential instability issues.…”