1 A Novel Approach to Set-Membership Observer for Systems with Unknown Exogenous Inputs

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A Novel Approach to Set-Membership Observer for

Systems with Unknown Exogenous Inputs

Marvin Jesse, Dawei Sun, Student Member, IEEE, and Inseok Hwang, Member, IEEE

Abstract—Motivated by the increasing need to monitor safety-

critical systems subject to uncertainties, a novel set-membership

approach is proposed to estimate the state of a dynamical system

with unknown-but-bounded exogenous inputs. The proposed

method decomposes the system into the strongly observable

and weakly unobservable subsystem in which an unknown

input observer and an ellipsoidal set-membership observer are

designed for each subsystem, respectively. The conditions for the

boundedness of the proposed set estimate are discussed, and

the proposed set-membership observer is also tested numerically

using illustrative examples.

Index Terms—Estimation, linear system observers, linear sys-

tems, set-membership observer.

I. INTRODUCTION

State estimation has been widely used in the control commu-

nity in areas like the secure control of cyber-physical systems

[1] and fault diagnosis [2]. One of the common methods of

state estimation used is the deterministic method, which treats

the noise and disturbance as unknown-but-bounded (UBB) [3].

Related works: One of the most well-established methods

to estimate the system’s state under UBB uncertainty is the

set-membership observer. This method uses geometrical sets,

such as ellipsoids [4], zonotopes [5], or parallelotopes [6], to

enclose all admissible state values. An alternative approach

is the interval observer. This method works by evaluating the

error dynamics generated by the upper and lower bounds of the

estimated states so that the error dynamics are cooperative and

stable [7]. The unknown input observer [8] is a third viable op-

tion that can accurately estimate the system state without much

prior knowledge of the inputs, rendering it more amenable for

use in fault detection schemes. The design of such an observer

does require the system to have strong observability, which

limits its applicability. Strong observability [9] is deﬁned as

a system’s property in which we can infer the true state from

the system’s output for any initial state and unknown input.

Note that each class of approaches has its own advantages

and disadvantages. In particular, the interval observer offers

low computational complexity but more conservative results

than the set-membership observer, and vice versa. In order

to achieve a balance between these two speciﬁc aspects, the

authors in [10] combined both the interval observer and set-

membership observer. Additionally, the idea to decompose the

system into strongly observable and weakly unobservable sub-

systems has been attempted in [11]. In [11], the authors apply

This work is supported in part by NSF CNS-1836952.

The authors are with School of Aeronautics and Astronautics, Purdue

University, West Lafayette, IN 47907 USA (e-mail: jessem@purdue.edu,

sun289@purdue.edu, ihwang@purdue.edu).

a High Order Sliding Mode technique and interval observer in

order to improve the estimation accuracy. Motivated by these

works, we consider a more effective way to use the structure

of the system such that our proposed set-membership observer

has a comparable estimation performance while not requiring

the system to be too restrictive.

Contributions: This paper presents a novel set-membership

approach to state estimation of the linear time-invariant (LTI)

system that integrates the unknown input observer and the

ellipsoidal set-membership observer, which has not been at-

tempted in literature, to the best of our knowledge. Based on

a system decomposition technique, we implement an unknown

input observer and an ellipsoidal set-membership observer for

the strongly observable and weakly unobservable subsystem,

respectively. Compared with existing representative works,

our contributions include relaxing one of the assumptions in

[11], which is the existence of a transformation matrix to

transform the weakly unobservable subsystem into a coop-

erative form, and providing analytical analysis in which the

set estimate computed by our observer is stable, i.e., the set

estimate does not grow to inﬁnity, for which less restrictive

conditions are required compared to [4], [12], [13]. In terms

of estimation accuracy, our proposed observer outperforms

existing ellipsoidal set-membership observers as well, which

is demonstrated thoroughly in the numerical simulations.

The rest of this paper is organized as follows: we ﬁrst

introduce the preliminaries and problem setup in Section II.

The detailed design of the observer is described in Section

III. Section IV discusses properties of the proposed algorithm.

Two numerical simulations are presented in Section V to

illustrate the effectiveness of our proposed algorithm. Finally,

Section VI concludes the paper.

II. PRELIMINARIES AND PROBLEM SETUP

Throughout this paper, we denote an ellipsoid as E`c, K˘ﬁ

txPRx:px´cqTK´1px´cq ď 1u, where cPRxis the center

vector and KPRxˆxis a symmetric positive deﬁnite matrix,

called the shape matrix. The pseudoinverse and transpose of a

matrix Aare denoted as A:and AT, respectively. rankpAq

and trpAqdenotes the rank and trace of A, respectively.

colpa1, a2, . . . , anqdenotes a column vector. || ¨ || denotes

the standard 2-norm. diagpA1, A2, . . . , Anqdenotes a block

diagonal matrix in which the diagonal elements are A1,A2,

. . .,An.max RepλpAqq denotes the largest real part of A’s

eigenvalues. σminpAqdenotes the smallest singular value of

arXiv:2210.10927v1 [eess.SY] 19 Oct 2022

In this paper, we will consider the following continuous

linear time-invariant (LTI) system

Σ : #9xptq “ Axptq ` Bwptq

yptq “ Cxptq ` Dwptq,(1)

where APRnˆn,BPRnˆnw,CPRnyˆn, and DPRnyˆnw

are constant matrices; xptq P Rnis the unknown state to

be estimated; yptq P Rnyis the measurable output; and

wptq P Rnwrepresents the unknown input vector. The fol-

lowing assumptions are commonly made for set-membership

state estimation problems [4], [12], [13].

Assumption 1. The unknown input wptqis bounded, i.e.,

wptq P E`cwptq, Kwptq˘.

Assumption 2. The initial state xp0qis bounded, i.e., xp0q P

E`ˆx0, K0˘.

Given the system Σin (1) with its parametric matrices, the

output yptq, bound on the unknown input Epcwptq, Kwptqq,

and bound on the initial state Epˆx0, K0q, our objectives are to

design a computationally efﬁcient and accurate state estimation

algorithm such that xptkq P Epˆxk,ˆ

Pkqfor all kPNand to

investigate the conditions for which Epˆxk,ˆ

Pkq’s are uniformly

bounded.

III. OBSERVER DESIGN

In this section, we ﬁrst discuss our proposed observer

architecture and system decomposition. Then, we proceed by

designing an observer for each subsystem and fusing these

individual set estimates into a single set estimate. Finally, the

proposed approach is summarized in Algorithm 1.

A. Proposed Observer Architecture and System Decomposi-

tion

To solve the aforementioned problem, we are motivated to

utilize the system’s structure, which can be done by decom-

posing the given system Σinto two subsystems: a strongly

observable subsystem and a weakly unobservable subsystem

such that a different observer can be designed to match each

subsystem. Our proposed scheme, which can facilitate the

characteristics of each subsystem, is illustrated in Figure 1.

To achieve the desired system decomposition, we follow the

methods described in [14], [15] to obtain the transformation

matrix. Next, the following lemma illustrates how to decom-

pose the system Σ.

Lemma 1 (System Decomposition).For the system Σin (1),

there exists a coordinate transformation matrix P1such that

for xp“colpx1, x2q “ P1x, one has

Σp:

’

9xpptq “ „A1A3

A2A4

looooomooooon

xpptq ` „B1

B2

loomoon

wptq

yptq “ “C1C2‰

loooomoooon

xpptq ` Dwptq

,(2)

where Ap“P1AP ´1

1,Bp“P1B,Cp“CP ´1

1,x1PRn1,

and x2PRn2. Additionally, the system Σpsatisﬁes the

property that subsystems Σ1and Σ2, which are deﬁned as

Σ1:#9x1“A1x1`B1

1u1

y“C1x1`D1

1u1

,Σ2:#9

x2“A4x2`B1

2u2

y“C2x2`D1

2u2

,(3)

where B1

1““A3B1‰,D1

1““C2D‰, and u1“

colpx2, wq,B1

2““A2B2‰,D1

2““C1D‰, and u2“

colpx1, wq, are strongly observable and weakly unobservable

[14], respectively.

Proof. See Lemma 3.4 in [14] and Theorem 4 in [15].

B. Set-Membership Observer for Σ1

A set-membership observer based on the unknown input

observer is proposed for subsystem Σ1, and the derivation is

adapted from [16] to be self-contained. The goal is to ﬁnd

a bounding ellipsoid Epˆx1ptq, 2

1ptqIn1qthat contains the true

state x1ptq. First, we are interested in ﬁnding the center of

the ellipsoid ˆx1ptq, which is obtained using an unknown input

observer. The following assumption is needed for the observer:

Assumption 3. The unknown input wptqis a sufﬁciently

smooth function, i.e., wptq P Clfor some l, and the derivatives

of wptqare bounded.

System Proposed Observer

Set-membership Estimate of

Strong

Observable

Subsystem

Weakly

Unobservable

Subsystem

Output

equation of

Set-membership Estimate of

High Gain

Observer

(HGO)

Set-membership

Observer based on

Unknown Input

Observer

Ellipsoidal Set-membership

Observer

Fusion of Set-

membership

Estimates

Estimated output

derivatives

Set-membership Estimate of

Fig. 1: Proposed observer architecture for system Σ

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1ANovelApproachtoSet-MembershipObserverforSystemswithUnknownExogenousInputsMarvinJesse,DaweiSun,StudentMember,IEEE,andInseokHwang,Member,IEEEAbstractMotivatedbytheincreasingneedtomonitorsafety-criticalsystemssubjecttouncertainties,anovelset-membershipapproachisproposedtoestimatethestateofadynamical...

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