A New Approach to Design Interval Observers for Linear Systems

Abstract

Interval observers are dynamic systems that provide upper and lower bounds of the true state trajectories of systems. In this work we introduce a technique to design interval observers for linear systems affected by state and measurement disturbances, based on the Internal Positive Representations (IPRs) of systems, that exploits the order preserving property of positive systems. The method can be applied to both continuous and discrete time systems.Index Terms-Linear system observers, positive systems, uncertain systems. I. INTRODUCTIONThe properties of positive linear systems have been a subject of study in system theory for long time [11], [14]. Some of these properties, for example that the trajectories of positive systems are ordered with respect to the initial conditions and the forcing input, can be exploited in the problem of state estimation.This technical note proposes a framework for the state observation problem of linear systems in presence of bounded input/output uncertainties which is based on the use of positive systems to design interval observers, that is, a system of observers that provides a confident region that contains the trajectory of the observed system [13], [16]-[18]. A key tool to apply the positive properties to general (i.e., non necessarily positive) systems is the internally positive realization (IPR) [4]-[6], [12]. Some recent works on interval observers based on positive systems is reported in [2], [3], [20], where positive observed systems are considered. The case of stable linear systems is considered in [16], where the authors propose an approach based on a time-varying change of coordinates that transforms an autonomous system into a positive one. This approach is used to design a Luenberger observer with a positive error dynamics, on which an interval observer can be built. This idea has subsequently been extended to complex intervals in [7], and, using a time invariant change of coordinates, to a class of nonlinear systems in [18], to linear time-invariant and time-varying discrete-time systems in [8], [9], and to continuous-time time-varying systems in [10].Essentially, the idea behind these approaches is to design a stable observer by means of an appropriate gain and then to find a coordinate change such that the resulting error dynamics is positive. In this technical note we propose to reverse the approach: a positive observer of the system is initially built, and the observer gain is subsequently chosen so as to have a stable error dynamics. We show that this is always possible for observable systems. The design technique is very Manuscript

Stability andL1× ℓ1-to-L1× ℓ1performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

“…1 T S c = µ c ) and S d := diag(µ d ). Then, multiply from the right the conditions (7) and (8) by col(x(t k + τ ), w c,∆ (t k + τ ), w c (t k + τ )) and col(x(t k ), w d,∆ (k), w d (k)), respectively. Grouping the terms together yieldṡ…”

On the Tightest Interval-Valued State Estimator for Linear Systems

“…Here the signals b w , b v , y and the initial state vector X 0 are viewed as the inputs of system (3). Boundedness is understood in the sense of the infinity norm being finite.…”

“…Assuming these signals take values in known (bounded) time-dependent intervals, the goal is to find an interval containing the state trajectories. In these settings, many contributions have been made for different classes of systems: continuous-time Linear Time Invariant (LTI) [15], [3], [6], [18], discrete-time LTI/LTV systems [7], [17], [16], Linear Parameter Varying (LPV) systems [4], [9], nonlinear systems [22], [19]. For more on the interval observer literature we refer to a recent survey reported in [8].…”

An interval observer for uncertain continuous‐time linear systems

“…[1][2][3][4][5][6][7][8][9] In the case of linear systems, this problem has been tackled in both continuous-time and discrete-time framework. 7,8,[10][11][12] The core insight applied to solve this problem is to construct cooperative dynamics for the estimation errors that satisfy the order-preserving property of monotone systems. 13,14 To achieve that, similarity transformations are proposed in the literature.…”

“…[5][6][7][8]10 For instance, in the works of Raïssi et al 5 and Efimov et al, 11 based on Sylvester equation, an algebraic method to jointly compute an observer gain and a constant linear state coordinate transformation, which guarantee both stability and cooperativity of the dynamics of the estimation error in a new basis of the state variables, is introduced. On the other hand, using the internal positive representation of dynamical systems, a pole placement technique is proposed by Cacace et al 7 to design interval observers. In this approach, a region of the observer eigenvalues, where the stability and the cooperativity of the estimation error are guaranteed by constant linear state transformations, is determined a priori.…”