Minimal-order Appointed-time Unknown Input Observers Design and Applications Yuezu Lva Zhongkui Lib Zhisheng Duanb

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Minimal-order Appointed-time Unknown Input Observers:

Design and Applications

Yuezu Lv a, Zhongkui Li b, Zhisheng Duan b

aDepartment of Systems Science, School of Mathematics, Southeast University, Nanjing 211189, China

bState Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of

Engineering, Peking University, Beijing 100871, China

Abstract

This paper presents a framework on minimal-order appointed-time unknown input observers for linear systems based on

the pairwise observer structure. A minimal-order appointed-time observer is ﬁrst proposed for the linear system without the

unknown input, which can estimate the state exactly at the preset time by seeking for the unique solution of a system of

linear equations. To further release the computational burden, another form of the appointed-time observer is designed. For

the general linear system with the unknown input acting on both the system dynamics and the measured output, the model

reconﬁguration is made to decouple the eﬀect of the unknown input, and the gap between the existing reduced-order appointed-

time unknown input observer and the possible minimal-order appointed-time observer is revealed. Based on the reconstructed

model, the minimal-order appointed-time unknown input observer is presented to realize state estimation of linear system

with the unknown input at the arbitrarily small preset time. The minimal-order appointed-time unknown input observer is

then applied to the design of fully distributed adaptive output-feedback attack-free consensus protocols for linear multi-agent

systems.

Key words: Appointed-time unknown input observer, minimal-order observer, attack-free protocol, consensus

1 Introduction

The state estimation has been well investigated since

the invention of the well-known Kalman ﬁlter [Kalman

(1960)] and Luenberger observer [Luenberger (1964)] in

1960s, and various observers have been developed for lin-

ear or nonlinear systems [Hou et al. (2002); Deza et al.

(1993); Ding (2012)]. The unknown input observer is a

typical state estimation for the systems with external

unknown inputs, which has attracted widespread atten-

tions due to its resultful applications in the ﬁelds of fault

diagnosis [Gao et al. (2016); Cristofaro et al. (2014)] and

attack detection [Amin et al. (2013); Ameli et al. (2018)].

The generalized dynamic model of linear time-invariant

system with unknown input can be formulated by

˙x=Ax +Bu +Ew,

y=Cx +Du +F w, (1)

Email addresses: yzlv@seu.edu.cn (Yuezu Lv),

zhongkli@pku.edu.cn (Zhongkui Li), duanzs@pku.edu.cn

(Zhisheng Duan).

where x∈Rn,u∈Rp,w∈Rq,y∈Rmare the state,

the control input, the unknown input and the measure-

ment output, respectively. In practice, the unknown in-

put wcan represent the external disturbances, unmod-

eled dynamics or actuator failures.

The unknown input observer has been investigated

from various perspectives. A procedure of the minimal-

order unknown input observer was proposed in Wang

et al. (1975) for the linear system (1) with F= 0,

and the existence conditions were revealed in Kudva

et al. (1980) that the rank of CE equals to that of E

and the triple (A, E, C) has stable or even no invari-

ant zeros. Following the conditions proposed in Kudva

et al. (1980), full-order unknown input observers were

designed in Yang et al. (1988); Darouach et al. (1994),

and the reduced-order observers were presented in Hou

et al. (1992); Syrmos et al. (1993). Syrmos et al. (1983);

Hou et al. (1994) further studied the general model (1),

and presented minimal-order observer design procedure

as well as the existence conditions. Unknown input

observers for discrete-time systems were illustrated in

Syrmos et al. (1999); Sundaram et al. (2007, 2008), and

Preprint submitted to Automatica 7 October 2022

arXiv:2210.02700v1 [eess.SY] 6 Oct 2022

Darouach et al. (1996); Koenig et al. (2002); Zhang et al.

(2020) investigated the unknown input observer design

for descriptor systems. The unknown input functional

observers were designed in Sundaram et al. (2008);

Trinh et al. (2008); Sakhraoui et al. (2020). Unknown

input observers for the switched systems [Bejarano et al.

(2011); Zhang et al. (2020)] and h∞unknown input

observers [Gao et al. (2016)] have also been well studied.

One common feature of the aforementioned unknown

input observers is that the system state is estimated

asymptotically. In practical applications such as the

fault detection, it is desired to realize ﬁnite-time estima-

tion of the state. Among all the categories of ﬁnite-time

convergence, the strictest one is to reach convergence

exactly at the preset time instant, which is named as

appointed-time or speciﬁed-time convergence [Zhao

et al. (2019)]. The appointed-time observer for linear

systems without the unknown input was proposed in

Engel et al. (2002), where a pairwise observer structure

was designed, consisting of two Luenberger observers

and achieving the appointed-time state estimation based

on time-delayed observer information. By introducing a

time-varying coordinate transformation matrix, a novel

observer for linear systems was designed in Pin et al.

(2020), which successfully realized the appointed-time

state estimation with an arbitrarily small predeter-

mined time. Based on the pairwise observer structure,

the appointed-time observers for nonlinear systems were

presented in Kreisselmeier et al. (2003); Menold et al.

(2003), and the appointed-time functional observers for

linear systems were studied in Raﬀ et al. (2005). Li et al.

(2015) further considered the appointed-time state es-

timation of nonlinear systems with measurement noise.

The appointed-time observer for discrete-time systems

was presented in Ao et al. (2018), where the applica-

tions on the attack detection were also investigated.

Following the observer design structure of Engel et al.

(2002), the appointed-time unknown input observer for

linear system (1) with F= 0 was proposed in Raﬀ et al.

(2006). Distributed appointed-time unknown input ob-

servers were further investigated in Lv et al. (2020a),

based on which fully distributed attack-free consensus

protocols were proposed for multi-agent systems.

Notice that the above-mentioned appointed-time ob-

servers based on the pairwise observer structure are ei-

ther of full order 2n, or of reduced order 2(n−rank(E)).

From the point of view of realization, it is favourable to

design minimal-order appointed-time observers, which

is expected to be of order 2(n−rank(C)) when F= 0. In

this paper, we intend to answer whether such minimal-

order appointed-time observer exists and how to design

the observers.

For the linear system without the unknown input, we

ﬁrst give a thorough analysis of the pairwise observer

design structure presented in Engel et al. (2002) to re-

veal how it works on realizing state estimation at the

appointed time. That is, to build a system of 6nlinear

equations in 6nunknowns, and construct the observer

expression based on the unique solution of the system

of linear equations. Following such design methodology,

the pairwise minimal-order observers with diﬀerent poles

are proposed, and a system of (6n−4m) equations in

(6n−4m) unknowns is constructed by adding the 2m

equations of measured output at time instant tas well as

the delayed time instant t−τ. It is demonstrated that the

coeﬃcient matrix is invertible, which gives a unique so-

lution to the system of linear equations. The appointed-

time observer is then designed by taking the portion of

the unique solution. To release the computation burden

caused by calculating the inverse of the high-dimensional

coeﬃcient matrix, another form of the minimal-order

appointed-time observer is formulated, whose structure

is coincident with that of the full-order appointed-time

observer in Engel et al. (2002).

For the linear system with the unknown input, we ﬁrst

reconstruct the model to decouple the eﬀect of the un-

known input, and exhibit both full-order and reduced-

order appointed-time unknown input observers based

on diﬀerent reconstructed models. The gap between the

reduced-order and expected minimal-order appointed-

time unknown input observers is revealed, which moti-

vates us to further decrease the observer order. Follow-

ing the observer design structure of the minimal-order

appointed-time observer for linear systems without the

unknown input, the minimal-order appointed-time un-

known input observer is obtained by designing the ob-

server to estimate the state of the reconstructed model at

the appointed time. The special case that the unknown

input does not act on the measured output, i.e., F= 0, is

also discussed. The proposed minimal-order appointed-

time unknown input observer is then applied into the

consensus problem of linear multi-agent systems, where

distributed minimal-order appointed-time unknown in-

put observer is put forward to estimate the consensus

error by viewing the relative input among neighboring

agents as the unknown input, and the distributed adap-

tive attack-free consensus protocol is presented based on

the consensus error estimation. The proposed protocol

possesses the feature of avoiding information transmis-

sion via communication channel, which takes the advan-

tages of reducing the communication cost and being free

from network attacks.

The rest of this paper is organized as follows. Section

2 presents the design structures of the minimal-order

appointed-time observer for linear system (1) without

unknown input w. Section 3 further studies the minimal-

order appointed-time unknown input observers. Section

4 applies the appointed-time unknown input observer

into the design of fully distributed adaptive attack-free

consensus protocols for linear multi-agent systems, and

gives a simulation example to illustrate the eﬀectiveness

of the proposed methods. Section 5 concludes this paper.

Notations: Let Rn×mbe the set of n×mmatrices. Ip

represents the p-dimensional identity matrix. Symbol

diag(x1,· · · , xn) represents a diagonal matrix with diag-

onal elements being xi. For a matrix A,A+denotes its

generalized inverse with AA+A=Aand A+AA+=A+.

For a square matrix Z,<i{λ(Z)}represents the real part

of the i-th eigenvalue of Z.

2 Minimal-order Appointed-time Observers

We ﬁrst study the minimal-order appointed-time ob-

server for linear systems without the unknown input,

i.e., w= 0, or E= 0, F = 0 in model (1). Without loss

of generality, we assume that Cis of full row rank.

2.1 Problem Analysis

Under the assumption that (A, C) is observable, the full-

order appointed-time observer was given in Engel et al.

(2002) as

¯v1= (A+L1C)v1−L1y+ (B+L1D)u,

¯v2= (A+L2C)v2−L2y+ (B+L2D)u,

¯x(t) = ¯

Dc[¯v(t)−e¯

Acτ¯v(t−τ)],

(2)

where ¯v= [¯vT

1,¯vT

2]T,¯

Ac= diag(A+L1C, A +L2C)

with L1and L2as the gain matrices satisfying <i{λ(A+

L2C)}< σ < <j{λ(A+L1C)}<0,∀i, j = 1,· · · , n,σ

is a negative constant, ¯

Dc=hIn0ih¯

Cce¯

Acτ¯

Cci−1with

Cc="In

In#, and τis a positive constant.

The methodology of observer (2) is to construct a system

of linear equations, which has a unique solution [Engel

et al. (2002)]. Speciﬁcally, deﬁne ˜

¯vi= ¯vi−x, i = 1,2,

and ˜

¯v= [˜

¯vT

1,˜

¯vT

2]T. Then, we have

Ccx(t) + ˜

¯v(t) = ¯v(t),

Ccx(t−τ) + ˜

¯v(t−τ) = ¯v(t−τ),

¯v(t) = e¯

Acτ˜

¯v(t−τ),

(3)

which contains 6nlinear equations in 6nunknowns

x(t), x(t−τ),˜

¯v(t),˜

¯v(t−τ). The unique solution of x(t)

can be calculated as ¯x(t) in (2). Thus, it is concluded

in Engel et al. (2002) that the observer (2) can esti-

mate the exact value of x(t) at appointed time τ, i.e.,

¯x(t)≡x(t),∀t≥τ.

To design minimal-order appointed-time observer, we

can only construct two (n−m)-order observers:

˙v1=M1v1+H1y+ (T1B−H1D)u,

˙v2=M2v2+H2y+ (T2B−H2D)u, (4)

where M1, M2∈R(n−m)×(n−m)are gain matrices sat-

isfying <i{λ(M2)}< σ < <j{λ(M1)}<0,∀i, j =

1,· · · , n −m,H1and H2are matrices such that both

(M1, H1) and (M2, H2) are controllable, T1and T2are

respectively the unique solutions of the Sylvester equa-

tions

TiA−MiTi=HiC, i = 1,2,(5)

such that "Ti

C#are invertible. Let Ui=[Si¯

Si]= "Ti

C#−1

It is well-known that each vican exponentially converge

to Tix. Deﬁne ˜vi=vi−Tix, and we have

Tix(t) + ˜vi(t) = vi(t),

Tix(t−τ) + ˜vi(t−τ) = vi(t−τ),

˜vi(t) = eMiτ˜vi(t−τ), i = 1,2.

(6)

It is clear that (6) contains 6(n−m) equations in 2n+

4(n−m) unknowns x(t), x(t−τ),˜vi(t),˜vi(t−τ). As

a result, x(t) cannot be uniquely determined by the

above system of linear equations. To design minimal-

order appointed-time observer, the main diﬃculty lies in

constructing an appropriate system of linear equations

to calculate a unique solution of x(t).

2.2 Observer Design

To make the system of linear equations (6) have a unique

solution, 2mextra independent equations are required.

It is natural to add the following 2mequations:

Cx(t) = y(t)−Du(t),

Cx(t−τ) = y(t−τ)−Du(t−τ),(7)

and the system of linear equations (6) and (7) can be

written as

A0x0=b0(8)

with x0= [xT(t),˜vT

1(t),˜vT

2(t), xT(t−τ),˜vT

1(t−τ),˜vT

2(t−

τ)]T,b0= [vT

1(t),(y(t)−Du(t))T,0T

n−m, vT

2(t), vT

1(t−

τ),(y(t−τ)−Du(t−τ))T,0T

n−m, vT

2(t−τ)]Tand

A0=







T1In−m0 0 0 0

C0 0 0 0 0

0In−m0 0 −eM1τ0

T20In−m0 0 0

0 0 0 T1In−m0

0 0 0 C0 0

0 0 In−m0 0 −eM2τ

0 0 0 T20In−m







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Minimal-orderAppointed-timeUnknownInputObservers:DesignandApplicationsYuezuLva,ZhongkuiLib,ZhishengDuanbaDepartmentofSystemsScience,SchoolofMathematics,SoutheastUniversity,Nanjing211189,ChinabStateKeyLaboratoryforTurbulenceandComplexSystems,DepartmentofMechanicsandEngineeringScience,CollegeofEnginee...

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