Multi-Objective Robust H∞ Observer Design for One-Sided LipschitzNonlinear Uncertain Systems

Muhammad Abid1, Muhammad Shoaib1, Abdul Qayyum Khan1

1. Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences, Islamabad, PakistanE-mail: mabid@pieas.edu.pk, muhammadshoaib89@gmail.com

Abstract: In this paper, the problem of robust H∞ observer design for one-sided Lipschitz nonlinear systems in

the presence of time-varying parametric uncertainties is addressed. The main idea is to minimize the L2 gain from

the disturbance to the state estimation error and at the same time the admissible one-sided Lipschitz constant is to be

maximized. In addition to asymptotic convergence, the designed observer also ensures robustness against disturbances,

parametric uncertainties and additive one-sided Lipschitz nonlinear uncertainty. The solution is represented in terms

of LMIs which enables us to carry out the design procedure using Matlab LMI toolbox making the nonlinear observer

design an easy task.

Key Words: robust observers, nonlinear uncertain systems, one-sided Lipschitz, multi-objective optimization.

1 Introduction

The state estimation problem has been a subject of great

concern for execution of many control methodologies and

due to its utility in fault diagnosis. State observers serve

the purpose to build the estimate of states. For the past

few decades nonlinear observer design has been a matter of

considerable involvement. Observer design for nonlinear

uncertain systems has been discussed by various authors

proposing numerous techniques [1]. Probably, the work

was initiated by de Souza et. al. in which they proposed

stability conditions based on Riccati equations for the ob-

server designed for Lipschitz nonlinear uncertain systems.

The proposed design ensured disturbance rejection level for

the system with given Lipschitz constant [2, 3]. Since then,

numerous design methods have been proposed for robust

uncertain nonlinear observers [4, 5, 6, 7]. A design method

based on LMIs has been recently suggested in [8], for Lip-

schitz nonlinear uncertain systems in which they have used

convex optimization approach in which Lipschitz constant

is also an optimization variable making the designed ob-

server also robust against Lipschitz nonlinear uncertainty.

A considerable focus has been on Lipschitz nonlinear sys-

tems in all the work discussed so far because of the fact

that this class of nonlinear systems represents many practi-

cal systems. Furthermore, such systems are comparatively

more comfortable to deal with.

Control community has been lately introduced with one-

sided Lipschitz nonlinear systems [9, 10]. An extensive

range of nonlinear systems is encompassed by this class

of nonlinear systems and any Lipschitz nonlinear system

is also a one-sided Lipschitz nonlinear system. Moreover,

the one-sided Lipschitz constant is always smaller as com-

pared to the Lipschitz constant [11], which results into less

conservative results. Recently, there has been some discus-

sion on the observer design for one-sided Lipshitz nonlin-

ear systems. One method of observer design for this class

of nonlinear systems is presented in [11]. The solution ini-

tially obtained consisted of nonlinear matrix inequalities

(NMIs). They converted their NMI based solution into lin-

ear matrix inequalities (LMIs) due to the fact that no effec-

tive method for solving NMIs is available. Further work on

observers for one-sided Lipschitz systems based on LMIs

was discussed in [12, 13].

Design of observers for one-sided Lipschitz systems tak-

ing into account time-varying parametric uncertainties and

disturbances has not been addressed so far. This note for-

mulates a method for the observer design of one-sided Lip-

schitz class of nonlinear systems which leads to H∞ ob-

server with prespecified decay rate which is robust against

parametric uncertainties along with one-sided Lipschitz

nonlinear uncertainties at the same time. The observer

design approach proposed in this paper guarantees distur-

bance attenuation level and minimization of the L2 gain

from the disturbance to the state estimation error. The pro-

posed solution is in the form of LMIs which are both lin-

ear in disturbance attenuation level and the admissible one-

sided Lipschitz constant. Due to this linearity we are able

to optimize both variables using multiobjective optimiza-

tion. The maximization of admissible one-sided Lipschitz

constant allows this observer to be robust against one-sided

Lipschitz nonlinear uncertainty.

The remainder of the note is ordered as follows: Some

preliminary discussion is provided along with the problem

statement in Section II. In Section III robust H∞ observer

design technique is presented for one-sided Lipschitz non-

linear uncertain systems. Section IV addresses robustness

against nonlinear uncertainty. In Section V an example is

simulated to validate the design technique.

1907978-1-4799-3708-0/14/$31.00 c©2014 IEEE

2 Problem Formulation

One-sided Lipschitz nonlinear uncertain systems are repre-

sented as:

x(t) = (A+ΔA)x(t) + Φ(x, u) +Bw(t)

y(t) = (C +ΔC)x(t) +Dw(t)(1)

where x ∈ Rn is the system state vector, y ∈ R

p represents

the output vector and u ∈ Rm is the control vector. We

assume that in a region Ω containing origin, (1) is locally

one-sided Lipschitz with respect to x, uniformly in u, i.e.:

Φ(0, u�) = 0 (2)

〈Φ(x1, u�)− Φ(x2, u

�), x1 − x2〉 ≤ ν ‖x1 − x2‖2 (3)

u� here is any allowable control input. ‖·‖ represents

the Euclidean norm of the argument (·), that is, ‖x‖ =√〈x, x〉. The smallest constant ν > 0 fulfilling condi-

tion in (3) is known as one-sided Lipschitz constant and

(1) defines what is known as one-sided Lipschitz system

locally in Ω. Φ(x, u) is termed globally one-sided Lips-

chitz if the condition (3) is valid everywhere in Rn. The

one-sided Lipschitz constant may be positive, negative or a

zero. w ∈ L2[0,∞) is an unknown exogenous disturbance

input vector, and the uncertain matrices ΔA and ΔC are

described by:

ΔA = U1F (t)W1 (4)

ΔC = U2F (t)W2 (5)

where U1, U2,W1 and W2 are real known constant matri-

ces and F (t) is a time-varying norm-bounded matrix such

that

‖F (t)‖2 ≤ 1 (6)

The parametric uncertainty is considered to be the change

in the operating point. The type of uncertainties mentioned

in (4)-(5) are often used in robust control and can be used

to model numerous practical uncertainties [14, 15].

2.1 Disturbance RejectionTake the observer of the type

˙x(t) = Ax(t) + Φ(x, u) + L(y − Cx),

y(t) = Cx(t).(7)

The error dynamics is given as

e(t) =(A− LC)e(t) + Φ(x, u)− Φ(x, u)

+(B − LD)w(t) + (ΔA− LΔC)x(8)

Assume that z(t) = He(t) represents the controlled output

for the error state vector with H as a known matrix. Our

aim is to design such L (observer gain matrix) so that the

observer error dynamics is asymptotically stable with max-

imal admissible one-sided Lipschitz constant and also the

following upper bound on H∞ norm is concurrently satis-

fied:

‖z‖2 ≤ μ ‖w‖2 (9)

‖·‖2 represents the L2 norm of the argument (·), that is,

‖x‖2 =√∫∞

0(xTx) dt.

2.2 Guaranteed Decay Rate

We want to design observer with a guaranteed decay rate.

Take (1) with w(t) = 0 and ΔC, ΔA = 0. The largest

α > 0 is then known as the decay rate of (8) such that

limt→∞exp(αt) ‖e(t)‖ = 0 (10)

holds for all trajectories e(t). To set up a lower bound

on the decay rate of (8), we use the Lyapunov function

V (e) = eTPe. IfdV (e(t))

dt ≤ −2αV (e(t)) for all trajecto-

ries, then V (e(t)) ≤ exp(−2αt)V (e(0)), hence ‖e(t)‖ ≤exp(−αt)κ(P )

12 ‖e(0)‖ for all trajectories, thus the decay

rate of (8) is at least α, [16]. Decay rate is actually the stan-

dard to assess observer convergence rate. κ(P ) here is the

condition number of matrix P.

3 Robust H∞ Observer Design

This part of the paper proposes an H∞ observer design

method in which the decay rate α is guaranteed. As the

proposed LMIs in the following Theorem are linear in both

disturbance attenuation level and the admissible one-sided

Lipschitz constant, therefore, for a given decay rate, opti-

mization is carried out for maximization of one sided Lips-

chitz constant and minimization of L2 gain simultaneously.

This is known as Pareto multi-objective optimization and it

is useful where optimal decision is a tradeoff among two

or more linearly combined optimization criterion. Design

procedure for such an observer is proposed in Theorem 1

but first a lemma is provided that is to be utilized to prove

Theorem 1.

Lemma 1. [3] For any vectors x, y ∈ Rn, D,S and F are

real matrices of suitable dimensions such that FTF ≤ I ,the following inequality holds for any scalar ε > 0

2xTDFSy ≤ ε−1xTDDTx+ εyTSTSy (11)

Theorem 1. Consider the observer (7) for one-sided Lips-chitz nonlinear system (1). The error dynamics of the ob-server are asymptotically stable with decay rate α and si-multaneously minimized L2 gain μ∗ and maximized admis-sible one-sided Lipschitz constant ν∗, if there exists scalarsζ > 0 and ν > 0, fixed scalars 0 ≤ γ ≤ 1 and α > 0, and matrices G, P1 > 0 and P2 > 0, so that the LMIoptimization problem stated below has a solution.

min [γ(η) + (1− γ)ζ]

s. t.⎡⎣Θ1 0 χ1

Θ2 χ2

−ζI

⎤⎦ < 0 (12)

where

1908 2014 26th Chinese Control and Decision Conference (CCDC)

Θ1 =

⎡⎢⎢⎣HTH −Q I P1U1 Y U2

−0.5Iη 0 0 −I 0 −I

⎤⎥⎥⎦ (13)

Θ2 =

⎡⎣R I P2U1

−0.5Iη 0 −I

⎤⎦ (14)

χ1 =[P1B − Y D 0 0 0

]T(15)

χ2 =[P2B 0 0

]T(16)

After solving the problem

L =P−11 Y (17)

ν∗ � max(ν) =1

η(18)

μ∗ � min(μ) =√ζ (19)

Proof. From (8), we have

e(t) =(A− LC)e(t) + Φ(x, u)− Φ(x, u)

+(B − LD)w(t) + (ΔA− LΔC)x(20)

For easiness, let us have

Φ(x, u) � Φ, Φ(x, u) � Φ. (21)

Take the candidate Lyapunov function as

V (t) = V1 + V2 (22)

where V1 = eTP1e, V2 = xTP2x. Take (1) with w(t) = 0and ΔC, ΔA = 0. We have

V1(t) =eT (t)P1e(t) + eT (t)P1e(t)

=− eTQe+ 2eTP1 (Φ(x, u)− Φ(x, u))T

(23)

For V1 ≤ −2αV1(t), it is sufficient to have (24) < 0 with

ATP1 + P1A− CTY T − Y C + 2αP1 = −Q (24)

with

Y � P1L ⇒ LTP1 = Y T (25)

Now considering the system with (1) parametric uncertain-

ties and disturbance, we get

V1(t) =eT (t)P1e(t) + eT (t)P1e(t)

=− eTQe+ 2eTP1(B − LD)w + 2eTP1(Φ− Φ)

2eTP1U1FW1x− 2eTY U2FW2x

(26)

Using Lemma 1, we get

2eTP1U1FW1x ≤ eTP1U1UT1 P1e+ xTWT

1 W1x (27)

2eTY U2FW2x ≤ eTY U2UT2 Y T e+ xTWT

2 W2x (28)

2xTP2U1FW1x ≤ xTP2U1UT1 P2e+ xTWT

1 W1x (29)

Also from definition of one-sided Lipschitz system, we

have

〈PΦ(x1, u�)− PΦ(x2, u

�), x1 − x2〉 ≤ ν ‖x1 − x2‖2(30)

Therefore, we have

2eTP1(Φ− Φ) ≤2νeT e (31)

2xTP2Φ ≤2νxTx (32)

Using (27), (28) and (31), we have

V1 ≤− eTQe+ 2νeT e+ eTP1U1UT1 P1e

+ xT (WT1 W1 +WT

2 W2)x+ eTY U2UT2 Y TE

+ 2eTP1(B − LD)w

(33)

V2 = xT (ATP2 + P2A)x

+ 2xTP2Φ+ 2xTP2U1FW1x+ 2xTP2Bw(34)

Using (29) and (32), we have

V2 ≤ xT (ATP2 + P2A)x+ 2νxTx+ xTP2U1UT1 P2x

+ xTW1WT1 x+ 2xTP2Bw

(35)

Thus

V ≤ eT[−Q+ P1U1U

T1 P1 + Y U2U

T2 Y T + 2νI

]e

+ xT[ATP2 + P2A+ P2U1U

T1 P2 + 2νI

]x

+ xT (2WT1 W1 +WT

2 W2)x+ 2xTP2Bw

+ 2eTP1(B − LD)w

Now, defining

J =

∫ ∞

0

(zT z − ζwTw)dt

where ζ = μ2. Thus,

J <

∫ ∞

0

(zT z − ζwTw + V )dt (36)

So for J ≤ 0

zT z − ζwTw + V ≤ 0, ∀t ∈ [0,∞)

is the sufficient condition. We get

zT z − ζwTw + V ≤ eT (HTH −Q+ P1U1UT1 P1

+ Y U2UT2 Y T + 2νI)e

+ xT (R+ P2U1UT1 P2 + 2νI)x

+ 2eTP1(B − LD)w + 2xTP2Bw − ζwTw

where

R = ATP2 + P2A+ 2WT1 W1 +WT

2 W2 (37)

2014 26th Chinese Control and Decision Conference (CCDC) 1909

so a sufficient condition for J ≤ 0 is that the above matrix

which by applying Schur complement is same as (12) be

negative definite. J ≤ 0 guarantees that∫ ∞

0

(zT z − ζwTw)dt < 0

which, in turn, means

‖z‖2 ≤√ζ ‖w‖2

.

Remark 1. The LMIs proposed in this section are linear in

both ν and ζ(= μ2). Thus any of these can be a defined

constant or the variable to be optimized. If it is desired to

design an observer for a given system with with a prespec-

ified μ and a known one-sided Lipschitz constant, then the

problem shortens to feasibility problem.

Remark 2. The designed observer is robust against both

time-varying parametric uncertainty in the matrices A, Cand one-sided Lipschitz nonlinear uncertainty while the

disturbance attenuation level is guaranteed at the same

time.

4 Robustness Against Nonlinear Uncertainty

Proposition 1. Let us assume that the actual one-sidedLipschitz constant of the given nonlinear system is ν andthe maximum admissible one-sided Lipschitz constantcalculated by Theorem 1, is ν∗. So, when designedthrough Theorem 1, the observer can tolerate any additiveone-sided Lipschitz nonlinear uncertainty with one-sidedLipschitz constant less than or equal to ν∗ − ν.

Proof. Consider a nonlinear uncertainty of the form

ΦΔ(x, u) = Φ(x, u) + ΔΦ(x, u) (38)

x(t) = (A+ΔA)x(t) + ΦΔ(x, u) (39)

where

〈ΔΦ(x1, u)−ΔΦ(x2, u), x1 − x2〉 ≤ Δν ‖x1 − x2‖2(40)

We have

〈ΦΔ(x1, u)− ΦΔ(x2, u), x1 − x2〉= 〈Φ(x1, u)− Φ(x2, u) + ΔΦ(x1, u)−ΔΦ(x2, u), x1 − x2〉= 〈Φ(x1, u)− Φ(x2, u), x1 − x2〉+ 〈ΔΦ(x1, u)−ΔΦ(x2, u), x1 − x2〉≤ν‖x1 − x2‖2 +Δν‖x1 − x2‖2≤(ν +Δν)‖x1 − x2‖2

From Theorem 1, ΦΔ(x, u) can be any one-sided Lipschitz

nonlinearity with one-sided Lipschitz constant less than or

equal to ν∗,

〈ΦΔ(x1, u)− ΦΔ(x2, u), x1 − x2〉 ≤ ν∗ ‖x1 − x2‖2(41)

0 2 4 6 8 10 12 14 16 18 20

0

1

2

time(Sec)

real value of state x1estimated value of state x1

0 2 4 6 8 10 12 14 16 18 20−0.4−0.2

00.20.4

time(Sec)

real value of state x2estimated value of state x2

0 2 4 6 8 10 12 14 16 18 20−1

0

1

time(Sec)

e1e2

Figure 1: Real states, estimated states and state estimation

errors of the example

therefore,

ν +Δν ≤ ν∗ → Δν ≤ ν∗ − ν (42)

5 Illustrative example

Consider a system in which

A =

[0 1−1 −1

], Φ(x) =

[−x1(x21 + x2

2)−x2(x

21 + x2

2)

]

U1 =

[0.1 0.05−2 0.1

], U2 =

[−0.2 0.8]

C =[1 0

], W1 = W2 =

[0.1 00 0.1

]

H = 0.5I

α = 0.5

γ = 0.5

D = 0.2

B =[1 1

]TWe get

ν∗ = 0.0373

μ∗ = 2.5152

L =[6.8750 6.4313

]TThis system is globally one-sided Lipschitz with one-sided

Lipschitz constant ν = 0. But the admissible one-sided

Lipschitz constant is greater than zero, hence allowing for

the handling of one-sided Lipschitz nonlinear uncertainty

by the designed system. The system state trajectories along

with their estimates are shown in Fig. 1

1910 2014 26th Chinese Control and Decision Conference (CCDC)

6 Conclusion

This paper has presented a procedure in the form of LMI

optimization problem to design a robust observer for one-

sided Lipschitz uncertain nonlinear systems . The admis-

sible one-sided Lipschitz constant and L2 gain from the

disturbance to the state estimation error are the LMI opti-

mization variables. The optimal combined performance of

both variables is desired. The observer designed as a result

guarantees asymptotic convergence and ensures robustness

against disturbances, parametric uncertainties and additive

one-sided Lipschitz nonlinear uncertainty. Simulation re-

sults have verified the validity of the method proposed.

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