SECOND-ORDER TERMINAL SLIDING MODE CONTROL OF INPUT-DELAY SYSTEMS

12 Asian Journal of Control, Vol. 8, No. 1, pp. 12-20, March 2006

Manuscript received April 15, 2005; accepted November 7, 2005.

Yong Feng and Xuemei Zheng are with Department of Elec-trical Engineering, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected], xuemeizheng_hrb@ sina.com).

Xinghuo Yu is with School of Electrical and Computer En-gineering, Royal Melbourne Institute of Technology, Mel-bourne, VIC 3001, Australia (e-mail: [email protected]).

This paper was supported by the National Natural Science Foundation of China (No.60474016) and the Scientific Re-search Foundation for Returned Overseas Chinese Scholars, State Education Department of China.

SECOND-ORDER TERMINAL SLIDING MODE CONTROL OF INPUT-DELAY SYSTEMS

Yong Feng, Xinghuo Yu, and Xuemei Zheng

ABSTRACT

This paper proposes a second-order terminal sliding mode control for a class of uncertain input-delay systems. The input-delay systems are firstly converted into the input-delay free systems and further converted into the regular forms. A linear sliding mode manifold is predesigned to represent the ideal dynamics of the system. Another terminal sliding mode manifold surface is presented to drive the linear sliding mode to reach zeros in finite time. In order to eliminate the chattering phenomena, a second-order sliding mode method is utilized to filter the high frequency switching control signal. The uncertainties of the systems are analysed in detail to show the effect to the systems. The simulation results validate the method presented in the paper.

KeyWords: Delay systems, time delay, sliding mode control, system design.

I. INTRODUCTION

Time-delay effect is very common in many practical applications such as aircraft, chemical and biological reac-tors, process control systems, rolling mills, automotive engines, manufacturing systems, communication network and economics, etc. Time-delay appears either in the state, or the control input, or the measurements of systems. Time- delay often significantly degrades the control performance of systems or even destabilizes the systems. Therefore, the issues of controlling the time-delay systems are of both theoretical and practical importance.

There are three kinds of time delay systems: the sys-tems with state delay, the systems with input delay, and the systems with simultaneous state and input delays. Although

some engineering systems exhibit delay in the states, delay in the input variables is particularly pervasive in numerous applications, such as chemical processing, for example [2]. This paper is mainly concentrated on the issue of the con-trol of the input-delay systems.

The control of input-delay systems has received much attention in the literature in the last few years. Many re-search results, including both delay-independent and delay- dependent results, have been obtained based on either the Laypunov theory of stability or frequency domain consid-erations [1]. A main method is to use a transformation that converts the original system into a delay-free form, such as a sliding mode control design for a class of uncertain input- delay systems [2], and a sliding mode control scheme to ensure the asymptotic stability of a linear system with de-lay in both the input and state variables [3]. For the robust stabilization of uncertain input-delay systems, a sliding mode control and a sliding mode control with uncertainty adaptation were proposed by using a predictor to compen-sate for the input delay of the system and the adaptation scheme respectively [4,5]. For systems with a single and pure input lag, based on state-space analysis, mixing a finite-dimensional and an abstract evolution model, the standard H∞ problem was resolved [6]. The stability crite-rion of the closed-loop system was derived in terms of lin-ear matrix inequalities (LMIs) [7]. A sub-optimal second order sliding mode control law for a double integrator sys-tem with delayed input was investigated [8]. A disturbance

Y. Feng et al.: Second-Order Terminal Sliding Mode Control of Input-Delay Systems 13

reduction control technique was proposed for systems with input time delays and multiple states to minimize the ad-verse effects of unmeasurable deterministic disturbances of waveforms [9]. A robust control method for some nonlin-ear control problems with an input delay was investigated in [10]. The basic idea is to transform the control problems with input nonlinearity into the robust control problems of linear systems with only structured uncertainty by letting input nonlinearity in the sector bounds as a new diagonal structured uncertainty. For linear systems with a single, constant time-delay input, the design of memoryless state feedback controllers was proposed to guarantee either closed-loop stability or closed-loop stability with an H∞

performance for any delay not larger than a given bound and for all admissible uncertainties [11]. A state feedback controller which maximizes the delay bound of the closed-loop system for a class of linear systems with a con-tinuous bounded time-varying delayed input was presented by transforming the stability problem into a convex opti-mization one [12]. A control design based on input-output feedback linearisation for a class of uncertain nonlinear systems with an input time delay was presented by using the Lyapunov stability theory and Razumikhin’s stability theory [13].

Variable structure systems (VSS) are well known for their robustness to system parameter variations and the external disturbances [14]. Recently, a terminal sliding mode (TSM) controller was developed [15-17]. Compared with the linear hyperplane based sliding mode control, TSM offers some superior properties such as fast, finite time convergence and better static tracking precision. This controller is particularly useful for high precision control as it speeds up the rate of convergence near the equilibrium point. However, common TSM controller design methods have a singularity problem. Based on TSM, some nonsin-gular terminal sliding mode (NTSM) control systems have been presented to avoid the singularity problem in TSM [18,19]. Therefore, NTSM control can be used for time- delay systems. Because of main two properties of finite time convergence and high static tracking precision, NTSM control can guarantee the system to reach the linear sliding mode in finite time, and then force the system to stay on the linear sliding mode manifold surface more perfectly. Compared to the present second-order sliding mode control method, NTSM control can provide a simple and direct way to realize the second-order mode.

This paper proposes a second-order TSM control for a class of uncertain input-delay systems. The input-delay systems are firstly converted into the input-delay free sys-tems using a state transformation and further converted into the regular forms using another state transformation. A linear sliding mode manifold is pre-designed to represent the ideal dynamics of the system. Another TSM manifold surface is proposed to force the linear sliding mode to reach zeros in finite time. In order to eliminate the chattering phenomena in traditional sliding mode control systems, a

second-order sliding mode method is utilized to filter the high frequency switching control signal. The uncertainties in input-delay systems are analysed in detail to show that the uncertainties are distributed into two parts: matched and unmatched. Generally speaking, these two parts always exist in input-delay systems. The simulation results validate the method presented in the paper.

The paper is organized as follows: the problem for-mulation is presented in Section 2. The linear sliding mode design and the second-order nonsingular TSM control are proposed in Section 3 and Section 4 respectively. Section 5 gives the concluding remarks.

II. PROBLEM FORMULATION

Consider the uncertain input-delayed linear system given by

0 1( ) ( ) ( ) ( ) ( , )t t t t h p t= + + − +x A x B u B u f (1)

with

( ) ( ) [ , 0]hτ = τ τ∈ −u ϕ (2)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control, A ∈ Rn×n, B0 ∈ Rn×m, B1 ∈ Rn×m are known constant matri-ces; h is a known constant time delay; ϕ(t) is a continuous vector-valued initial function, and f (p, t) represents model-ing a uncertainty including external disturbance, assume: || f (p, t)|| ≤ ld and || ( , ) || ddp t l≤f , where ld > 0, ldd > 0.

In order to control the uncertain input-delayed linear system (1) with uncertainty f (p, t), a linear transformation can be used to convert the system (1) into a input-delay free system [2-5]. The linear transformation TA depending on a matrix A is defined by

( )1( ) ( ) ( )t t h

t ht t d− −τ−= + τ τ∫ Az x e B u (3)

Differentiating Eq. (3) gives

( )1( ) ( ) ( )t t h

t ht t A d− −τ−= + τ τ∫ Az x e B u

1 1( ) ( )h t t h−+ − −Ae B u B u (4)

By using the coordinate transformation (3), the input- delayed linear system (1) can be transformed to the fol-lowing system:

0 1( ) ( ) ( ) ( ) ( ( ), )ht t t t t−= + + +Az Az B e B u f x (5)

let B ∈ Rn×m be

0 1h−= + AB B e B (6)

then, the system (5) can be rewritten as follows:

( ) ( ) ( ) ( , )t t t p t= + +z A z B u f (7)

Therefore, through the state transformation (3), the input-

14 Asian Journal of Control, Vol. 8, No. 1, March 2006

delay system (1) can be converted into the input-delay free system (7). Without loss of generality, we also assume Rank(B) = m.

The control objective is to force the uncertain input- delayed linear system (1) with uncertainty f ( p, t) to con-verge to the equilibrium points asymptotically or in a finite time from any initial condition x(0) ≠ 0.

III. SLIDING MODE DESIGN

In the previous section, the input-delay system (1) is transformed to the delay free system (7). In this section, the ideal sliding mode dynamics will be designed. Since by assumption Rank(B) = m, there exists an invertible matrix T ∈ Rn×n such that

2

0⎡ ⎤= ⎢ ⎥⎣ ⎦

TBB

where B2 ∈ Rm×m and is non-singular. For system (7), mak-ing the following state transform

( ) ( )t t=z T z (8)

then, system (7) can be written in the following regular form:

1 11 1 12 2 1( ) ( ) ( ) ( , )t t t p t= + +z A z A z T f (9a)

2 21 1 22 2 2 2( ) ( ) ( ) ( ) ( , )t t t t p t= + + +z A z A z B u T f (9b)

where -1

n mR∈z and 2mR∈z are the partitioned states

of z, that is 1 2[ , ]T T T=z z z ; T1 ∈ Rn-m×n and T2 ∈ Rm×n are

the partitioned parts of T, that is 1 2[ , ]T T T=T T T .

Remark 1. In system (9), T1 f ( p, t) and T2 f ( p, t) represent projections of f ( p, t) into the coordinates of the subspace N(S) and R(B) respectively, where S is the hyperplane of the sliding mode.

Remark 2. T1 f (p, t) and T2 f (p, t) represent the unmatched and matched uncertainty components respectively.

Remark 3. For uncertain input-delayed linear system (1), when it is transformed into a delay free system (7), it is difficult to make an assumption for uncertainties to satisfy the matched condition. Suppose that the uncertainties in the input-free linear system (1) (in the case h = 0) satisfy the matched condition, when there exists input-delayed (in the case h ≠ 0), its transformed system (7) cannot guarantee that the matched condition is held.

For the regular form of the system (9), a linear sliding mode is designed as follows:

1 2( ) ( ) ( )t t t= +s M z z (10)

The ideal sliding motion of the system can be de-scribed as follows:

1 11 12 1( ) ( ) ( )t t= −z A A M z (11)

or

1 11 1( ) ( )t t=z A z (12)

where 11 11 12 .= −A A A M That is, the design of the ideal sliding hyperplane (10) is to determine matrix M in Eq. (10) so as (A11 − A12 M) to have stable eigenvalues.

IV. SECOND-ORDER NONSINGULAR TSM CONTROL

The uncertain input-delay system (1) is already in the regular form (9) after the transformation (3) and (8). A second-order nonsingular TSM control strategy is adopted in the paper. The design consists of two steps. The first is to design the second-order nonsingular TSM and ensure that the sliding mode converge to the equilibrium point asymptotically. The second is to design the robust control that ensures the system is robust to the internal parameter uncertainties and the external disturbances.

In order to eliminate the chattering, the paper proposes the following second-order nonsingular TSM manifold utilizing the second-order concept [20]:

1 /( ) ( ) ( )p qt t t−= +l s sγ (13)

where l ∈ Rm; γ = diag(γ1, …, γm), γi > 0 are constant; p, q

are all odds, 1 < p/q < 2; /q ps is denoted as:

/ / /1( ) [ ( ), , ( )]p q p q p q T

mt s t s t=s

From Eq. (10), the derivative of s can be obtained:

1 2( ) ( ) ( )t t t= +s M z z

( )11 1 12 2 1( ) ( ) ( , )t t p t= + +M A z A z T f

21 1 22 2 2 2( ) ( ) ( ) ( , )t t t p t+ + + +A z A z B u T f

11 21 1 12 22 2

1 2 2

( ) ( ) ( ) ( )

( , ) ( ) ( , )

t t

p t t p t

= + + +

+ + +

MA A z MA A z

MT f B u T f

Furthermore, the second-order derivative of s can be ob-tained:

11 21 1 12 22 2

1 2 2

( ) ( ) ( ) ( ) ( )

( , ) ( ) ( , )

t t t

p t t p t

= + + +

+ + +

s MA A z MA A z

MT f B u T f

Y. Feng et al.: Second-Order Terminal Sliding Mode Control of Input-Delay Systems 15

( )11 21 11 1 12 2 1

1 2

12 22 21 1 22 2

2 2 2

( ) ( ) ( ) ( , )

( , ) ( , )

( )( ( ) ( )

( ) ( , ) ( )

t t p t

p t p t

t t

t p t t

= + + +

+ +

+ + +

+ + +

MA A A z A z T f

MT f T f

MA A A z A z

B u T f B u

( )( )( )

11 21 11 12 22 21 1

11 21 12 12 22 22 2

11 21 1 12 22 2

1 2

12 22 2 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( , )

( ) ( , )

( ) ( ) ( )

t

t

p t

p t

t t

= + + +

+ + + +

+ + + +

+ +

+ + +

MA A A MA A A z

MA A A MA A A z

MA A T MA A T f

MT T f

MA A B u B u

that is:

11 1 12 2 1( ) ( ) ( ) ( , )ft t t p t= + +s A z A z B f

2 2 2( , ) ( ) ( )f p t t t+ + +B f B u B u (14)

where 11 11 21 11 12 22 21( ) ( ) ,= + + +A MA A A MA A A

12 =A (MA11 + A21) A12 + (MA12 + A22) A22, Bf 1 = (MA11 + A21) T1 + (MA12 + A22) T2, Bf 2 = MT1 + T2 and 2 =B (MA12 + A22)B2.

The aim of introducing s(t) is to control the system (9) using the sliding mode, while introducing l(t) is to realize the second-order sliding mode control and eliminate the chattering effectively. Therefore, the second-order sliding mode is designed as a nonsingular TSM to guarantee the linear sliding mode of s(t) to reach the equilibrium points in finite time and have no singularity.

When l(t) = 0, ∀t ≥ tr, (tr is the time when l(t) reaches to the nonsingular terminal sliding mode manifold l(t) = 0). Assume s(tr) = [s1(tr), …, sm(tr)]T. Solving the Eq. (13) obtains the time to reach s(t) = 0:

1, ,1, ,

max ( )( ) min

pp q

s r i ri mi

i m

pt t s tp q

−

==

⎛ ⎞⎜ ⎟= +⎜ ⎟− γ ⎝ ⎠

(15)

Thus, through suitable control design, s(t) and ( )ts can be driven to reach the second-order sliding mode l(t) = 0 and then remain on l(t) = 0 to realize the sliding mode motion. Among l(t) = 0, s(t) will reach the equilibrium points in finite time ts (15). After s(t) reaches zero, the sys-tem will remain in the linear sliding mode motion, that is, the dynamic characteristics of system (11) can be deter-mined by the design parameters, and has nothing to do with the system’s parameters. The relevant control methodology is given by the following theorem.

Theorem 1. For the uncertain input-delayed linear system (1) in the regular form (9), if the linear sliding mode and

second-order nonsingular TSM are chosen as (10) and (13), and the control law is designed as follows, then the sliding mode hyperplane s(t) can converge to zeros asymptotically:

2 2( ) ( ) ( )t t t= +v B u B u (16)

where v(t) the input of the low-passed filter, which is designed as follows:

( ) ( ) ( )l nt t t= +v v v (17)

with

2 /11 1 12 2( ) ( ) ( ) ( )p q

lqt t t tp

−= − − − γv A z A z s (18)

1 2( ) ( || || || || ) sgn( ( ))n d f dd ft l l t= − + + ηv B B l (19)

where γ and η are the design parameters, η > 0.

Proof. It is considered the following Lyapunov function:

1( ) ( ) ( )2

TV t t t= l l

Differentiating V(t) with respect to time, it gets:

( )1 / 1

( ) ( ) ( )

( ) diag ( ) ( ) ( )

T

T p q

V t t t

pt t t tq

− −

=

⎛ ⎞= +⎜ ⎟

⎝ ⎠γ

l l

l s s s

( )(

)

1 / 111 1 12 2

1 2 2 2

( ) diag ( ) ( ) ( )

( , ) ( , ) ( ) ( ) ( )

T p q

f f

pt t t tq

p t p t t t t

− −⎛= +⎜

⎝

⎞+ + + + + ⎟⎠

γl s A z A z

B f B f B u B u s

( )(

)

1 / 111 1 12 2

1 2

( ) diag ( ) ( ) ( )

( , ) ( , ) ( ) ( )

T p q

f f

pt t t tq

p t p t t t

− −⎛= +⎜

⎝

⎞+ + + + ⎟⎠

γl s A z A z

B f B f v s

( )( )1 / 11 2( ) diag ( ) ( , ) ( , ) ( )T p q

f f npt t p t p t tq

− −= + +γl s B f B f v

( )(1 / 11 2 2 2( ) diag ( ) ( , ) ( , )T p q

f fpt t f p t f p tq

− −= +γl s B B B B

( ) )1 2 2 2sgn ( ) ( || || || || )d f dd ft l l− + +ηl B B B B

notice γ−1 is a diagonal matrix and ( )/ 1diag ( ) 0p q t− >s for ( ) 0t ≠s , the above express can be written:

( )1 / 1( ) ( ) diag ( ) sgn( ( ))T p qpV t t t tq

− −≤ −η γl s l

( )1 / 1

1, ,min ( ) || ( ) ||p q

i ii m

p s t tq

− −

=≤ −η γ l

16 Asian Journal of Control, Vol. 8, No. 1, March 2006

since p, q are all odds, / 1( ) 0p qis t− > for any ( ) 0is t ≠

and / 1 ( ) 0p qis t− = only for ( ) 0is t = . For the case

( ) 0is t = but ( ) 0is t ≠ , the system will not always stay on the points ( ( ) 0is t = , ( ) 0is t ≠ ) [21]. Therefore, the condition for Lyapunov stability is satisfied. The system states can reach the sliding mode l(t) = 0 within finite time. In the sliding mode l(t) = 0, there is 1 / ( ) ( ) 0p q t t− + =s sγ , or /( ) ( ) 0q pt t+ =s sγ , s will reach the zero in ts (15).

Theorem 2. For the uncertain input-delayed linear system (1) in the regular form (9), if the control methodology is designed using Theorem 1, the states of system (9) will converge to the neighborhood of the equilibrium point zΩ asymptotically:

21

2: || || || || 1 || ||ˆ dl

⎧ ⎫Ω ≤ ≤ +⎨ ⎬µ⎩ ⎭

z z z T M

where 11ˆ 2max[Re ( )]µ ≤ − λ A .

Proof. For system (12), since 11A is stable, given any symmetric positive definite matrix Q1, there exists a unique symmetric positive definite matrix P1 satisfying the fol-lowing Lyapunov equation:

1 11 11 1 1T+ = −P A A P Q

The following Lyapunov function is considered:

1 1 111( )2

TV =z z P z

Differentiating V(t) with respect to time yields:

1 1 1 11( ) T TV = +z z z z z

1 1 11 11 1 1 11( ) 2 ( , )TT T p t= + +z P A A P z z P T f

1 1 11 12 ( , )T T p t= − +z Q z z P T f

2min 1 1 max 1 1 1( ) || || 2 ( ) || || || || dl≤ −λ + λQ z P z T

define µ = λmin(Q1) / λmax(P1) and substitute it into above inequation. It can be obtained:

( )1 max 1 1 1 1( ) ( ) || || || || 2 || || dV l≤ −λ µ −z P z z T

that is:

1 1 12( ) 0 , 0 , || || || || dV t l< ∀ ≥ >µ

z z T

According to the results in [21,22], when Q1 = I, the opti-mal choice of Q1 to maximize µ can obtain a further result:

11max 1

1ˆ max( ) 2 max[Re ( )]( )

µ = µ = ≤ − λλ

AP

When system states z1 satisfies:

1 12|| || || ||ˆ dl>µ

z T

it is obtained:

1 1 12( ) 0 , 0 , || || || ||ˆ dV t l< ∀ ≥ >µ

z z T

in other words, 1( ) 0V <z if 1 1∉Ωz , where

1 1 1 12: || || || ||ˆ dl

⎧ ⎫Ω ≤ ≤⎨ ⎬µ⎩ ⎭

z z T

That is, the states z1 of system (21) will converge to the neighborhood of the equilibrium points asymptotically. During the sliding motion, 2 1= −z M z , it can get:

21|| || 1 || || || ||= +z M z

Therefore, the states of system (1) will converge to the neighborhood of the equilibrium point Ω z asymptotically respectively. Ω z is given by:

2 21 1

2: || || 1 || || || || || || 1 || ||ˆ dl

⎧ ⎫Ω ≤ = + ≤ +⎨ ⎬µ⎩ ⎭

z z z M z T M

This completes the proof.

Theorem 3. For the uncertain input-delayed linear system (1) in the regular form (9), if the control methodology is designed using Theorem 1, the states of system (7) and system (1) will converge to the neighborhood of the equi-librium point Ωz and Ωx asymptotically respectively. Ωz and Ωx are given by:

1 21

2: || ( ) || || || || || 1 || ||ˆz dt l−⎧ ⎫

Ω ≤ ≤ +⎨ ⎬µ⎩ ⎭z z T T M

( )1 1 1 11

21

: || || || || || ( ) ( ) ||

2 || || 1 || ||ˆ

hx

dl

− − − − −⎧Ω ≤ ≤ + −⎨⎩

⎫⋅ + ⎬µ ⎭

Ax x T A e I B CB CAT

T M

where C = [M I] T.

Proof. From Theorem 2 and the state transform (8), it can be obtained:

1 1 21

2|| ( ) || || || || ( ) || || || || || 1 || ||ˆ r dt t l− −≤ ≤ +µ

z T z T T M

From the sliding mode manifold (10), s(t) can be rewritten as follows:

Y. Feng et al.: Second-Order Terminal Sliding Mode Control of Input-Delay Systems 17

1 2( ) ( ) ( ) [ ] ( )

[ ] ( ) ( )

t t t t

t t

= + =

= =

s M z z M I z

M I Tz Cz

for the system (7), on the sliding mode, the equivalent con-trol is ueq(t)

1( ) ( ) ( )eq t t−= −u CB CAz

substitute ueq(t) into the linear transformation (3), it can be obtained:

( )1( ) ( ) ( )t t h

t ht t d− −τ−= − τ τ∫ Ax z e B u

( ) 11( ) ( ) ( )t t h

t ht t d− −τ −−= + τ∫ Az e B CB CAz

The norm of x(t) can be found as:

( ) 11|| ( ) || || ( ) || ( ) ( )t t h

t ht t t d− −τ −−≤ + τ∫ Ax z e B CB CAz

( ) 110|| ( ) || ( ) ( )ht d t− τ −≤ + τ∫ Az e B CB CAz

1 11|| ( ) || || ( ) ( ) ( ) ||ht t− − −≤ + −Az A e I B CB CAz

1 1 1 11|| ( ) || || ( ) ( ) ( ) ||ht t− − − − −≤ + −AT z A e I B CB CAT z

1

1 1 11

|| || || ( ) ||

|| ( ) ( ) || || ( ) ||h

t

t

−

− − − −

≤

+ −A

T z

A e I B CB CAT z

( )1 1 1 11

21

|| || || ( ) ( ) ||

2 || || 1 || ||ˆ

h

dl

− − − − −≤ + −

⋅ +µ

AT A e I B CB CAT

T M

This completes the proof.

Remark 4. If there is no any uncertainty in the input-delay system (1), that is f (p, t) = 0, or the uncertainty in the sys-tem (1) satisfies the matched condition, that is T1 f (p, t) = 0 in system (9), the uncertain input-delay system (1) can converge to the equilibrium points asymptotically, that is lim ( ) 0t

t→∞

=x .

Remark 5. Although there exists a switching function in the control signal v(t) (17), through the low-passed filter (16), the control actually used in the input-delay system, u(t), is filtered without high frequency terms and the sys-tem is chattering free.

Remark 6. In the controller designed in Theorem 1, the derivative of sliding mode manifold, ( )ts , should be used. Since it is used only within the closed loop of the system, it

can be directly obtained using a differentiator. It is similar with the traditional PID controllers.

V. SIMULATIONS

A simulation with a input-delay linear system is per-formed for the purpose of evaluating the performance of the proposed scheme.

Consider the following system:

0 1( ) ( ) ( ) ( ) ( , )t t t t h p t= + + − +x A x B u B u f (20)

with

2 1 00 1.2 0.50 0 0.8

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦

A 0

011

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

B

1

001

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

B

0( , ) 1 0.15sin(2 )

1p t t

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

f

it can be seen ld = 0.15, ldd = 0.3. First, using the state transform (3), the time delay free system (7) can be ob-tained and control matrix B is obtained using Eq. (6) as follows:

0.09720.71211.6703

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

B

Second, the state transform (8) is made, and then the regular form of the system (9) can be obtained. The state transform matrix is given below:

1.0629 0.7939 0.40032.1257 0.1102 0.07674.2514 2.2580 0.1165

− −⎡ ⎤⎢ ⎥= − −⎢ ⎥⎢ ⎥− −⎣ ⎦

T

Third, the sliding mode manifold (10) can be designed. The eigenvalues of the system on the sliding mode are designed – 1 and – 2. Then, the matrix M is designed M = [2 3]. That is, the sliding mode manifold is designed

11 12 2( ) 2 ( ) 3 ( ) ( )t t t t= + +s z z z . The second-order nonsin-gular TSM manifold (13) is designed as 5 3( )l t s s= + , that is, the parameters are γ = 1, p = 5, q = 3. The control law is designed according to Theorem 1.

The simulation results are shown in Figs. 1 ~ 8. Figure 1 depicts the state variables of the input-delay system. Fig-ure 2 shows the state variables of the input-delay free sys-tem through the state transmission (3). Figure 3 presents the state variables of the regular form system through the state transform (8). It can be seen that all the states of the system in three different forms converge to the areas near

18 Asian Journal of Control, Vol. 8, No. 1, March 2006

the equilibrium points. Figure 4 shows the linear sliding mode manifold surface s(t). It can be seen that s(t) reaches zeros within finite time. Figure 5 displays the second-order sliding mode manifold surface l(t). It can be seen that l(t) reaches zero within finite time and stay on the sliding sur-face after reaching the sliding mode. The phase plot of s(t) and its derivative is shown in Fig. 6. It can be observed that the s(t) and its derivative realize the terminal sliding mode motion on l(t) = 0. Figures 7 and 8 shows the second-order sliding mode control and the input-delay system control. It can be seen that although there exists chattering phenom-ena in the second-order sliding mode control v(t), through low pass filter the control signal executed actually to the input-delay system u(t) is chattering free. In addition, ac-cording to Theorems 2 and 3, three neighborhoods of the equilibrium points, which the states of three forms of sys-tem (9), (7), and (1) will converge to asymptotically are calculated respectively as follows: || || ≤z 0.2446, || z || ≤ 1.7496 and || x || ≤ 2.0513. From the simulation results, it can be seen that all the states of the system converge to these neighborhoods of the equilibrium points asymptoti-cally.

Fig. 1. State variables of the input-delay system.

Fig. 2. State variables of the delay free system.

Fig. 3. State variables of the regular form system.

Fig. 4. Linear sliding mode manifold surface.

Fig. 5. Second-order sliding mode manifold surface.

Y. Feng et al.: Second-Order Terminal Sliding Mode Control of Input-Delay Systems 19

Fig. 6. Phase plot of s(t) and its derivative.

Fig. 7. Second-order sliding mode control.

Fig. 8. Input-delay system control.

VI. CONCLUSIONS

This paper is devoted to the control of input-delay systems. A second-order terminal sliding mode control has

been proposed for input-delay systems. The advantages of the method proposed in the paper are described as follows: (1) Second-order sliding mode is utilized to eliminate the chattering. (2) The uncertainties in input-delay systems are analyzed in detail to show that the uncertainties are distrib-uted into two parts: matched and unmatched. Generally speaking, these two parts always exist in input-delay sys-tems. (3) Terminal sliding mode is used to accelerate the system to reach the linear sliding mode so as to increase the response rate of systems.

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Yong Feng received the B.S. degree from the Department of Control Engineering in 1982, and M.S. degree from the Department of Electrical Engineering in 1985 and Ph.D. degree from the Department of Control Engi-neering in 1991, in Harbin Institute of Technology, China, respectively. He has

been with the Department of Electrical Engineering, Harbin Institute of Technology since 1985, and is currently a pro-fessor. He has published over 80 journal and conference papers, and 3 books. He has completed over 15 research projects, including that supported by the National Natural Science Foundation of China and supported by the National “863” Hi-Tech Programme of China. His current research interests are nonlinear control systems, sampled data sys-tems, robot control, digital camera modelling and simula-tion.

Xinghuo Yu received B.S. (EEE) and M.S. (EEE) from the University of Sci-ence and Technology of China in 1982 and 1984, respectively, and Ph.D. de-gree from South-East University, China in 1987. From 1989 to 1991, he was Postdoctoral Fellow with University of Adelaide, Australia. From 1991 to 2002, he was with Central Queensland Uni-

versity, Rockhampton Australia where, before he left, he was Professor of Intelligent Systems and the Associate Dean (Research) of the Faculty of Informatics and Com-munication. Since March 2002, he has been with Royal Melbourne Institute of Technology, Australia, where he is a professor, the Associate Dean of the Faculty of Engineering. His research interests include sliding mode and nonlinear control, chaos and chaos control, soft computing and ap-plications. Professor Yu published over 200 technical pa-pers in journals, books and conference proceedings and co-edited seven research books. He serves as an Associate Editor of IEEE Trans Circuits and Systems Part I. He is a Fellow of IEAust and a Senior Member of IEEE.

Xuemei Zheng received the B.S. degree from the Yan Shan University, China, in 1992, and M.S. degree from the De-partment of Control Engineering in 2000 and Ph.D. degree from the Department of Electrical Engineering in 2004, in Harbin Institute of Technology, China, respec-tively. She has been with the Department

of Electrical Engineering, Harbin Institute of Technology since 2001, and is currently an associate professor. Her current research interests are nonlinear control systems and robot control. She published more than ten papers in jour-nals.