274 Int. J. Dynamical Systems and Differential Equations, Vol. 4, No. 3, 2012

Copyright © 2012 Inderscience Enterprises Ltd.

Observer-based robust control of time-delay uncertain systems with application to engine idle speed control

Yong-Hong Lan* Key Laboratory of Intelligent Computing and Information, Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, China Email: lyhcsu@yahoo.com.cn *Corresponding author

Hui-Xian Huang School of Information Engineering, Xiangtan University, Xiangtan, Hunan 411105, China Email: huanghx05@yahoo.com.cn

Abstract: A combined controller-observer design procedure for a class of time-delay uncertain systems is proposed. First, a sufficient condition for robust asymptotical stability of the closed-loop systems is presented based on Lyapunov function theory. Next, by using matrix’s Singular Value Decomposition (SVD) and Linear Matrix Inequality (LMI) techniques, the existence condition and method of designing an observer-based feedback controller is derived. The result is obtained in terms of LMI, which make all the gain matrices can be easily obtained by Matlab’s LMI toolbox. Finally, the developed design techniques are applied to an engine idle speed control problem. The simulation results demonstrate the validity of this approach.

Keywords: time-delay dynamical systems; observer; feedback; singular value decomposition; linear matrix inequality.

Reference to this paper should be made as follows: Lan, Y.H. and Huang, H.X. (2012) ‘Observer-based robust control of time-delay uncertain systems with application to engine idle speed control’, Int. J. Dynamical Systems and Differential Equations, Vol. 4, No. 3, pp.274–285.

Biographical notes: Yong-Hong Lan is currently an Assistant Professor in the Xiangtan University’s School of Information Engineering. He received his MS in Applied Mathematics from Xiangtan University, Xiangtan, China in 200 and PhD in Control Science and Engineering from Central South University, Changsha, China in 2010. His research interests include the application of control theory, repetitive control and process control.

Hui-Xian Huang is currently a Professor in the Xiangtan University’s School of Information Engineering. He received his MS in Power Electronics and Drives from Xi’an University of Technology, Xi’an, China in 1994 and PhD in

Observer-based robust control of time-delay uncertain systems 275

Control Theory and Control Engineering from Northwestern Polytechnical University, Xi’an, China in 2000. His research interests include the application of control theory, intelligent control, complex system modelling and process control.

1 Introduction

Time delay exists commonly in dynamic systems due to measurement, transmission and transport lags, computational delays or unmodelled inertia of system components, which has been generally regarded as a main source of instability and poor performance. Therefore, considerable attention has been devoted to the problem of analysis and synthesis for time-delay uncertain systems (He et al., 2007; Gao et al., 2008; Kacem et al., 2011).

As is well known, excellent closed-loop performance can be achieved using state feedback control (Billy et al., 2011). However, in the many real-world systems, state feedback control will fail to guarantee the stability when some of system states are not measurable. In this case, the output feedback control or observer-based control is often needed (Lien et al., 2007; Chen et al., 2008; Sun et al., 2010; Lan and Zhou, 2011). For linear systems, the design of such a controller with an observer is typically carried out based on eigenvalue assignment. Unlike systems of linear systems, where the methods for eigenvalue assignment are well-developed, the design procedure for linear systems with time-delays in the state variables is not straightforward. Observer-based control for system without uncertainties (nominal systems) has studied by Lien et al. (2007). The problem of the delay-dependent non-fragile H∞ observer-based control for a class of continuous time-delay systems was investigated by Chen et al. (2008). Based on the Lambert W function, Sun et al. (2010) developed an observer-based feedback control approach for time-delay systems via assignment of eigenvalues. But this method can not applied to uncertain systems. Using matrix’s Singular Value Decomposition (SVD) and Linear Matrix Inequality (LMI) techniques, Lan and Zhou (2011) investigated the observer-based control and static output feedback control for fractional-order uncertain linear systems.

This paper focuses on the observer-based robust control of time-delay uncertain linear system. The main contribution is a combined controller-observer design methodology in an LMI framework and using SVD techniques. Specifically, a state-observer-controller problem is formulated with a linear controller using the estimated states to asymptotically stabilise the system. Sufficient condition for robust asymptotical stability of the closed-loop control system is presented. By using matrix’s SVD techniques, the efficient LMI-based approach is also proposed for designing a robust stabilising controller for time-delay uncertain linear systems. Furthermore, the control and observer design techniques are applied to a representative problem of automotive engine idle speed control.

The rest of this paper is organised as follows: In Section 2, the problem formulation and some preliminaries are presented. The observer-based robust control in Section 3. Section 4 applies the techniques to the engine idle speed control problem. Simulation results are also given. Finally, some conclusions are drawn in Section 5.

276 Y-H. Lan and H-X. Huang

Throughout this paper, Rn denotes an n-dimensional Euclidean space, Rn × m is the set

of all n × m real matrices, I means an identity matrix of appropriate order, X > 0(< 0) indicates that the matrix X is positive (negative) definite, XT denotes the transpose of

X and : T

X Y X YZ Y Z

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦.

2 Problem formulation and preliminaries

Consider a time-delay uncertain system described by

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )

0 0 1 1 ,x t A A x t A A x t d B B u t

y t Cx t

= + Δ + + Δ − + + Δ⎧⎪⎨

=⎪⎩ (1)

where x(t) ∈ Rn is the state of the plant, u(t) ∈ Rm and y(t) ∈ Rp are the control input and

output, respectively. The delay d is fixed with at least a known upper bound d . The initial condition is given by x(θ) = x0(θ),θ ∈[–d, 0]. A0 A1, B and C are known constant matrices

with appropriate dimensions, ΔA0 ΔA1 and ΔB represent the following admissible time-variant uncertainties:

0 0 0 0 1 1 1 1 2 2 2, ,A H F E A H F E B H F EΔ = Δ = Δ = (2)

where Hi and Ei, i = 0, 1, 2 are known constant matrices and Fi, i = 0, 1, 2 are unknown matrices with Lebesgue measurable elements satisfying

( ) ( ) , 0,1,2Ti iF t F t I i≤ = (3)

The purpose in this paper is to study the problem of asymptotical stabilisation of the time-delay system (1) under a Luenberger-type linear observer (Lien et al., 2007).

( ) ( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( )

0

0 1

ˆ ˆ

ˆ ˆ ,

ˆ ˆ

x t A x t d Bu t

L y t y t L y t d y t d

y t Cx t

⎧ = − +⎪⎪ + − + − − −⎨⎪

=⎪⎩

(4)

where L0 and L1 are the observer gain matrices to be designed and the control input is

( ) ( ) ( )0 1ˆ ˆu t K x t K x t d= + − (5)

where K0 and K1 are the control gain matrices to be designed. Let ( ) ( ) ( )ˆe t x t x t= − denotes the estimation error signal, under the controller (5),

then the closed-loop system is given by

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0 0 1 1 0 1

0 0 0 1 1

0 0 1 1

ˆ ˆ ˆ ,

ˆ ˆ

x t A BK x t A BK x t d L Ce t L Ce t d

e t A A L C e t A L C e t

A BK x t A BK x t d

⎧ = + + + − + + −⎪⎪ = + Δ + + Δ +⎨⎪ − Δ + Δ − Δ + Δ −⎪⎩

(6)

Observer-based robust control of time-delay uncertain systems 277

Denoting ( ) ( ) ( )ˆTT TX t x t e t⎡ ⎤= ⎣ ⎦ , equation (6) can be rewritten as:

( ) ( ) ( )( ) ( )

0 0

1 1

c c

c c

X t A A X t

A A X t d

= + Δ

+ + Δ − (7)

where

0 0 0 1 1 10 1

0 0 1 1

,0 0c c

A BK L C A BK L CA A

A L C A L C+ +⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥+ +⎣ ⎦⎣ ⎦ (8)

( ) ( )0 10 0 1 1

0 0 0 0,

0 0c cA AA BK A BK

⎡ ⎤ ⎡ ⎤Δ = Δ =⎢ ⎥ ⎢ ⎥− Δ + Δ − Δ + Δ⎣ ⎦ ⎣ ⎦

(9)

Now, the design problem can be transformed to the robust stability problem of uncertain time-delay system (7).

In the following, we will develop an LMI-based approach for designing the controller (5) to asymptotically stabilise the time-delay uncertain system (7). To proof the main results in the next section, we need the following lemmas.

Lemma 1: (Xie (1996). Given real symmetric matrix Q, matrices H and E of appropriate dimensions,

( ) ( ) 0T T TQ HF t E E F t H+ + <

for all F(t) satisfying FT(t)F(t) ≤ I if and only if there exists an∈ > 0 such that 1 0T THH E E−∈ +∈ < .

Lemma 2: (Schur complement, Boyd et al. (1994). For a real matrix Σ = ΣT, the following assertions are equivalent:

1 11 12

22

: 0∑ ∑⎡ ⎤

∑ = >⎢ ⎥∑⎣ ⎦.

2 111 22 12 11 120 and 0T −∑ > ∑ −∑ ∑ ∑ > .

3 122 11 12 22 120 and 0T−∑ > ∑ −∑ ∑ ∑ > .

Recall that for any matrix Π ∈ Rm × n with full row rank (i.e. rank(Π) = m), there exists a

SVD of Π as follows:

[ ]0 TU S V∏ =

where S ∈ Rm × n is a diagonal matrix with positive diagonal elements in decreasing order,

U ∈ Rp × p, V ∈ Rn × n are unitary matrices. In particular, the following lemma holds:

278 Y-H. Lan and H-X. Huang

Lemma 3: (MacDuffee et al. (2004).) Given matrix Π ∈ Rm × n with rank(Π) = m, assume

that X∈ Rm × n is a symmetric matrix, then there exists a matrix m mX ×∈R satisfying

X X∏ = ∏ if and only if X can be expressed as:

11

22

00

TXX V V

X⎡ ⎤

= ⎢ ⎥⎣ ⎦

where X11 ∈ Rm × m, X22 ∈ R(n–m)×(n–m).

3 Observer-based robust control

In this section, a sufficient condition is first derived for robust asymptotical stability for uncertain time-delay system (7). Next, an LMI-based approach is proposed for designing the robust stabilising controller to asymptotically stabilise time-delay system (1).

For time-delay uncertain system (7), we have the following Theorem.

Theorem 1: Given matrices L0, L1 and K0, K1, time-delay uncertain system (7) is asymptotically stable if there exist symmetric positive definite matrices Wk, Zk, k = 1, 2 and real scalars εi > 0, i = 1, 2, such that:

11 0 2 13 1 2 1 0 1 0 2

22 24

1

2 1 1 48

1

1

2

2

0 00 0 0 0 0

0 0 0 0 00 0

00 0 0

0 00

T T T

T

L CW L CW W E W K E

ZZ W E

II

II

εε

εε

⎡ ⎤Θ Θ⎢ ⎥Θ Θ⎢ ⎥⎢ ⎥−⎢ ⎥

− Θ⎢ ⎥ <⎢ ⎥−⎢ ⎥−⎢ ⎥

⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

(10)

where

11 1 0 0 1 1 0 0 1 1T T TW A A W W K B BK W ZΘ = + + + + ,

13 1 1 1 1AW BK WΘ = + ,

22 2 0 0 2 2 0 0 2

2 1 0 0 1 2 2 2 1 1 2 2 2

T T T

T T T T

W A A W W C L L CW

Z H H H H H H H Hε ε ε ε

Θ = + + +

+ + + + +,

24 1 2 1 2AW L CWΘ = + ,

48 1 1 2T TW K EΘ = .

Proof: To prove the stability of the closed-loop system (7), we use a classical Lyapunov function as following:

( )( ) ( ) ( ) ( ) ( ),tT T

t dV X t t X t PX t X s SX s ds

−= + ∫

Observer-based robust control of time-delay uncertain systems 279

where the symmetric positive matrices P = diag{P1, P2}, S = diag{S1, S2}. The time derivative of V(X(t), t) taken along the solution trajectories of (7) is

( )( ) ( )( )

( ) ( )( )

11 1 1,T

c cX t X tP A AV X t t

X t d X t dS⎡ ⎤ ⎡ ⎤∏ + Δ⎡ ⎤

= ⎢ ⎥ ⎢ ⎥⎢ ⎥− −∗ −⎣ ⎦⎣ ⎦ ⎣ ⎦

where ( ) ( )11 0 0 0 0T

c c c cP A A A A P S∏ = + Δ + + Δ + .

Therefore, ( )( ), 0V X t t < , which implies that the time-delay uncertain system (1) is

robustly stable if

( ) ( ) ( )0 0 0 0 1 1 0T

c c c c c cP A A A A P S P A AS

⎡ ⎤+ Δ + + Δ + + Δ <⎢ ⎥∗ −⎢ ⎥⎣ ⎦

(11)

Inequality (11) is equivalent to

0 1 00 0 1

0 0 0 1

1 1

1

1 1

0 0 00 0 0 0 0 0

0 0 0 00 0 0 00 00 0 0 0 0 0 0 0

0 0 0 0 00

0 0 0 0

Tc c cc c c

T T Tc c c c

c c

T Tc

c c

PH F EPA A P S PAS

E F PH PHF E

PHE F

⎡ ⎤+ + ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ + ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ <⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦

(12)

where

{ }

{ }

00 0 0 2 0

0 2 2 0

11 1 1 2 1

1 2 2 1

0 0 0, diag , , ,

0

0 0 0, diag , ,

0

c c c

c c c

EH F F F E

H H E K

EH F F F E

H H E K

⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤

= = =⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

It follows from (3) and Lemma 1 that for any real scalars εi > 0, (i = 1, 2).

0 00 0 11

0 0 1 111 2

12

1 1

0 00 0 0 0

0 0 0 00 0 0 0 0 0 0 0

0 0 0 00

0 0

TTc cc c c

T Tc c c c

T T

c c

PH PHPA A P S PAS

E E PH PH

E E

ε

ε ε

ε

−

−

⎡ ⎤+ + ⎡ ⎤ ⎡ ⎤+⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤+ <⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

(13)

As a result, one has

10 0 1 0 0 0 01 1

11 12 2

1 1

0 00 0 0 0

0 000

00 0

T T Tc c c c c c c

Tc c

Tc c

PA A P S PA PH H P E ES

PH H PE E

ε ε

ε ε

−

−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ ++ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤+ + <⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

(14)

280 Y-H. Lan and H-X. Huang

Now, pre-multiplying and post-multiplying both sides of inequality (14) by diag {P–1, P–1} yields:

1 1 1 1 10 0 1

1 1

1 110 0 0 0

1 1

11 12 2 1 1

1 1

0 00 0 0 0

0 000

00 0

Tc c c

TT Tc c c c

TTc c

Tc c

P A A P P SP A PP SP

H H P E E P

H HP E E P

ε ε

ε ε

− − − − −

− −

− −−

−− −

⎡ ⎤+ +⎢ ⎥−⎣ ⎦

⎡ ⎤ ⎡ ⎤+ +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤+ + <⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

(15)

Noting that P = diag{P1, P2}, S = diag{S1, S2}, defining 1i iW P−= , 1 1

i i i iZ P S P− −= , i = 1, 2, by some calculations and using the Schur complement, one obtains inequality (10). This completes the proof.

Due to the existence of the non-linear terms such as L0CW2, BK0W1, the matrix inequality (10) in Theorem 1 is not an LMI. However, applying Lemma 3, it can be transformed into an LMI, as shown below, which presents an LMI-based design method of observer-based control for time-delay uncertain system (1).

Theorem 2: For time-delay uncertain system (1), assume that the SVD of the output matrix C with full row rank is [ ]0 TC S V= . Then, under the control input (5), the closed-loop system is asymptotically stable if there exist symmetrical matrices W1 > 0, W11 > 0, W22 > 0, Z1 > 0, Z2 > 0 matrices Nk, Mk, k = 0, 1 together with two real scalars ε1 > 0, ε2 > 0 such that:

11 0 2 1 1 1 1 1 0 0 2

22 24

1

2 1 1 1 2

1

1

2

2

0 00 0 0 0 0

0 0 0 0 00 0

00 0 0

0 00

T T T

T T T

M CW AW BN M C W E N E

ZZ W E N E

II

II

εε

εε

⎡ ⎤Λ +⎢ ⎥Λ Λ⎢ ⎥⎢ ⎥−⎢ ⎥

−⎢ ⎥ <⎢ ⎥−⎢ ⎥−⎢ ⎥

⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

(16)

where

11 1 0 0 1 0 0 1T T TW A A W N B BN ZΛ = + + + + ,

22 2 0 0 2 0 0 2

2 1 0 0 1 2 2 2 1 1 2 2 2

T T T

T T T T

W A A W C M M C

Z H H H H H H H Hε ε ε ε

Λ = + + +

+ + + + +,

24 1 2 1AW M CΛ = + ,

112

22

00

TWW V V

W⎡ ⎤

= ⎢ ⎥⎣ ⎦

.

Observer-based robust control of time-delay uncertain systems 281

Moreover, the robustly asymptotically stabilising gain matrices are given by 1 1

0 0 1 1 1 1,K N W K N W− −= = (17)

1 1 1 1 1 10 0 11 1 1 11,L M USW S U L M USW S U− − − − − −= = (18)

Proof: Since 112

22

00

TWW V V

W⎡ ⎤

= ⎢ ⎥⎣ ⎦

, it follows from Lemma 3 that there exists

12 11

TW USW S U−= , such that 2 2CW W C= , where 1 1 1 12 11W USW S U− − − −= . Noting that

2 2CW W C= and setting 0 0 1N K W= , 1 1 1N K W= , 0 0 2M L W= and 1 1 2M LW= , inequality (10) is equivalent to (16). The proof is completed.

Remark 1: Note that the feasible problem in Theorem 2 is affined with all their respective arguments. Hence Theorem 2 provides an LMI-based method of designing a state estimated feedback controller for uncertain fractional-order linear system (1). Furthermore, the stabilising controller gain matrices Ki Li, i = 0, 1, can be directly solved by utilising the powerful Matlab’s LMI Toolbox Boyd et al. (1994).

Remark 2: Although the classical observer estimate is considered in this work, the techniques we used can be extended to a Luenberger-type linear observer as following:

( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( )

0 1

0 1

ˆ ˆˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

x t A x t A x t d Bu t

L y t y t L y t d y t d

y t Cx t

⎧ = + − +⎪⎪ + − + − − −⎨⎪

=⎪⎩

where 0A , 1A , L0 and L1 are to be designed. Such can also be formulated as LMIs.

4 Application to engine idle speed control

In this section, control of a engine idle speed is consider to illustrate the validity of the method proposed in Section 3. That is, an observer-based feedback controller is designed using LMI and SVD techniques.

Some of the challenges associated with idle speed control lie with the engine induction-to-torque delay and the system parameter uncertainties resulting from modelling. For this reason, we apply our observer-based robust control and design techniques to this problem. A highly simplified engine idle speed control model is developed by Hrovat and Sun (1996) and used in this work. An overview of the system dynamics and control scheme is given in Figure 1. The model includes the intake manifold dynamics, an induction-to-power delay, and the engine rotational dynamics, which are described below

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1 ,4

m v dm

m

m f

dp t Vt p t k a t

dt Vd t

J t k p t h k t tdt δ

ηω

πω

ξω δ

⎧+ =⎪

⎪⎨⎪ + = − + +⎪⎩

τ τ

.

282 Y-H. Lan and H-X. Huang

Figure 1 Configuration of system dynamics and control structure

The parameter values are consistent with those in the work of Hrovat and Sun (1996): pm is the intake manifold pressure, ηv is the volumetric efficiency of the pumping process (0.55, dimensionless), Vd is the engine displacement (0.0046 m3), Vm is the manifold volume (0.0029 m3), ω is the engine speed, k1 is a constant relating the idle throttle angle to the air mass flow rate past the throttle plate [110, (kPa/s)deg], J is the engine inertia (0.0843 Nm-s2/rad), ξ is the lumped damping coefficient (0.592 Nm-s), kτ is a coefficient relating engine torque to the intake manifold pressure (0.57143 Nm/kPa), kδ is a coefficient relating engine torque to the spark advance (2, Nm/deg), δ is the spark timing advance, h is an induction-to-torque delay, and τf is the disturbance torque; we set τf = 0.

The time delay h generally varies with engine speed. However, for fixed idle speed ω0 = 63.98rad/s, we assume a constant (upper bounded) time delay 0.1sh = . Further, we add the following uncertainties to some of the parameters to account for modelling and other types of errors: ηv = 0.55 ± 0.025, ξ = 0.592 ± 0.0125 Nm-s, k1 = 110 ± 2.5 (kPa/s)/deg, kτ = 0.57143 ± 0.015 Nm/kPa and kδ = 2 ± 0.05 Nm/deg.

Denoting k2 = ηvVd/4πVm and linearising (19) about the nominal operating point (ω0 = 63.98 rad/s, pm0 = 6.628 kPa), one obtains

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 0 2 0 1m

m m

m

d p tk p t k p t k a t

dtd t

J t k p t h k tdt δ

ω ω

ωξ ω δ

Δ⎧+ Δ = − Δ + Δ⎪⎪

⎨Δ⎪ + Δ = − + Δ⎪⎩

τ

(20)

Let ( ) ( ) ( ),T

mx t t p tω= Δ⎡ ⎤⎣ ⎦ and ( ) ( ) ( ),T

u t a t tδ= Δ Δ⎡ ⎤⎣ ⎦ . We can convert the above

equations to a state space form

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( )

2 0 2 0

0 0 0 01 2 0

1 1 1 1 2 2 2 2

0 00 0

0

,

m

J k Jx t x t x t h

k p k

k Ju t A H F t E x t

k k

A H F t E x t h B H F t E u t

y t C t

δ

ξω

ω

⎡ ⎤ ⎡ ⎤= + −⎢ ⎥ ⎢ ⎥− − ⎣ ⎦⎣ ⎦

⎡ ⎤+ = +⎡ ⎤⎢ ⎥ ⎣ ⎦−⎣ ⎦+ + − + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

=

τ

Observer-based robust control of time-delay uncertain systems 283

where

( ) ( )( )

( ) ( )( )

( ) ( )( )

0 0 0

1 1 1

2 2

0

07.0225 0 0.5 0.5, , ,

00.4601 4.4420 0.5 0.5

00 67.7853 0 0.4269, , ,

00 0 0 0

00 2 0 2.9656, , ,

0110 0 0 0

0.3559 00.3140 0.40

r tA H F t

r t

r tA H F t

r t

r tB H F t

r t

E

⎡ ⎤−⎡ ⎤ ⎡ ⎤= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤= = = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦−

=−

[ ] ( )

1 2

0 0.3821 0 0.2, , ,

38 0 0 2.5 01,0 1 1.

E E

C r t

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎪ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎪

= − ≤ ≤⎪⎩

The initial conditions are set as Δω0(t) = 40.36rad/s (ω0(t) = 1000 PM), Δpm0(µ) = 5 kPa

and ( ) [ ]0ˆ 20 5 Tx θ = − − for θ ∈ [–:1, 0]. Using the Matlab’s LMI toolbox, we find the LMI (16) in Theorem 2 is feasible. The stabilising gain matrices are as follows:

[ ] [ ]

0 1

0 1

0.3282 7.0645 0.0008 0.0614, ,

0.8133 21.2443 9.5908 208.6270

0.2610 0.3807 , 0.0312 0.0023 .

K K

L L

− − − −⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦= − = −

Figure 2 Time response of the system (system states: solid line; observer estimates: dashed line)

284 Y-H. Lan and H-X. Huang

Figure 3 Control inputs

Both the system states and estimated system states are shown in Figure 2 and the control inputs are shown in Figure 3. As we can see from Figures 2 and 3, the controller can bring the engine speed from 1000 RPM to the nominal idle speed 600 RPM in about 0.4 s. The state variables estimated using the observer (dashed line) converge to those of the plant (solid line), which are stabilised by observer-based feedback controller. As shown in Figure 3, the system is guaranteed to be asymptotically stable with respect to the given uncertainties. Since the existences of uncertainties, it cannot, as with previous approaches, locate the dominant subset of eigenvalues to achieve desired eigenvalue assignment (Sun et al., 2010). It need not fix any gain matrices (such as Bengea et al., 2004), all the stabilising gain matrices K0, K1, L0 and L1 can be directly solved by Matlabs LMI toolbox. This can lead to ease of design, analysis and implementation, which is one of the main advantages of the proposed approach.

5 Conclusion

This paper has presented an LMI-based combined controller-observer design method for asymptotically stabilising a time-delay uncertain system. A sufficient condition for robust asymptotical stability of the closed-loop control system was first presented. Then applying matrix’s SVD and LMI techniques, the existence condition and method of designing a robust stabilising feedback controller using the estimated states were derived. The results are obtained in terms of LMIs, which are very convenient to implement in practice. Finally, the techniques developed are applied to an internal combustion engine idle speed control problem. The simulation results show excellent performance of the designed observer and controller.

Observer-based robust control of time-delay uncertain systems 285

Acknowledgements

This work was supported in part by the National Science Foundation of China under grants 61104072. The authors would like to acknowledge the many helpful suggestions of the anonymous reviewers and the Editor of this Journal.

References Bengea, S.C., Li, X.Q. and DeCarlo, R.A. (2004) ‘Combined controller-observer design for

uncertain time delay systems with application to engine idle speed control’, Journal of Dynamic Systems, Measurement, and Control, vol. 126, pp.772–780.

Billy, J.J., John, M.D., Ian, A.G. and Robert, J.M. (2011) ‘Linear state feedback stabilization on time scales, International Journal of Dynamical Systems and Differential Equations, Vol. 3, Nos. 1/2, pp.163–177.

Boyd, S., Ghaoui, L., Feron, E. and Balakrishnan, V. (1994) Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, Philadelphia, Pennsylvania, Vol. 15.

Chen, J.D., Yang, C.D., Lien, C.H. and Horng, J.H. (2008) ‘New delay-dependent non-fragile H∞ observer-based control for continuous time-delay systems’, Information Sciences, Vol. 178, No. 24, pp.4699–4706.

Gao, H.J., Chen, T.W. and Lam, J. (2008) ‘A new delay system approach to network based control’, Automatica, Vol. 44, No. 1, pp.39–52.

He, Y., Wang, Q.G., Lin, C. and Wu, M. (2007) ‘Delay-range-dependent stability for systems with time-varying delay’, Automatica, Vol. 43, pp.371–376.

Hrovat, D. and Sun, J. (1996) ‘Models and control methodologies for ic engine idle speed control design’, Proceedings of 13th IFAC World Congress, 30 June–5 July, San Francisco, CA, USA, pp.243–248.

Kacem, W., Chaabane, M. and Mehdi, D. (2011) ‘On novel exponential stability criterion for systems with time-varying delays’, International Journal of Systems, Control and Communications, Vol. 3, No. 1, pp.95–103.

Lan, Y.H. and Zhou, Y. (2011) ‘LMI-based robust control of fractional-order uncertain linear systems’, Computers and Mathematics with Applications, Vol. 62, No. 3, pp.1460–1471.

Lien, C.H., Cheng, W.C., Tsai, C.H. and Yu, K.W. (2007) ‘Non-fragile observer-based controls of linear system via LMI approach’, Chaos, Solitons and Fractals, Vol. 32, pp.1530–1537.

MacDuffee, C.C. (2004) The Theory of Matrices, Dover Publications, New York. Sun, Y., Ulsoya, A.G. and Patrick W.N. (2010) ‘Design of observer-based feedback control for

time-delay systems with application to automotive powertrain control’, Journal of the Franklin Institute, Vol. 347, No. 1, pp.358–376.

Xie, L. (1996) ‘Output feedback H∞ control of systems with parameter uncertainty’, International Journal of Control, Vol. 63, pp.741–752.