A New Algorithm to Design Minimal Multi‐Functional Observers for Linear Systems

Abstract

Designing minimum possible order (minimal) observers for multi-input multi-output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi-functional observers for linear time invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.

Minimum-order filter design for partial state estimation of linear systems with multiple time-varying state delays and unknown input delays

“…After substituting T from into and , and considering the relation , it can be obtained for i ={1,2,3} and j ={2,3} that …”

“…The main advantages that a functional observer brings, compared with a typical full‐order or reduced‐order observer, are acquiring observers of lower orders and relaxing the observability/detectability requirements of the system to less conservative conditions . For example, in the static output‐feedback controller design problem, u ( t ) = K x ( t ) should be calculated instead of the whole set of the states of the system.…”

“…However, if the system is not detectable, and for control aims the designer needs more information on the states of the system, rather than only the output measurements, then functional observers can be utilized as an effective solution (see, e.g., ). Mohajerpoor et al , propose a recursive algorithm to incrementally increase the order of the observer in the minimum required number to attain an asymptotic FO for ordinary LTI systems. This algorithm is extended to the unknown‐input FO design problem in .…”

Design Procedure for Linear Unknown Input Functional Observers

“…Namely, these methods consist in looking for a R(ρ×n) matrix with ν = l+ρ and P = I ν such that (3) is a Darouach observer of v(t) = R L . Nevertheless, as it is underlined in [11], to find R is an "intriguing and challenging problem" and some attempts have been proposed in [19], [20], [22], [33], [34], [35]. As it will be explained in the next section, our design principle differs a lot from this standpoint.…”

Minimal unknown-input functional observers for multi-input multi-output LTI systems

“…Functional observers are a generalized version of ordinary fullorder or reduced-order Luenberger observers that are aimed at reconstructing a single or multiple functions of the states of the system [12][13][14][15]. This class of estimators has superiority in giving observers with lower order than the ordinary reduced-order observers.…”