SECOND-ORDER NONSINGULAR TERMINAL SLIDING MODE DECOMPOSED CONTROL OF UNCERTAIN MULTIVARIABLE SYSTEMS

Asian Journal of Control, Vol. 5, No. 4, pp. 505-512, December 2003 505

SECOND-ORDER NONSINGULAR TERMINAL SLIDING MODE DECOMPOSED CONTROL OF UNCERTAIN

MULTIVARIABLE SYSTEMS

Yong Feng, Xuemei Zheng, and Xinghuo Yu

ABSTRACT

This paper proposes a second-order nonsingular terminal sliding mode decomposed control method for multivariable linear systems with internal parameter uncertainties and external disturbances. First, the systems are con-verted into the block controllable form, consisting of an input-output subsys-tem and a stable internal dynamic subsystem. A special second-order non-singular terminal sliding mode is proposed for the input-output subsystem. The control law is designed to drive the states of the input-output subsystem to converge to the equilibrium point asymptotically. Then the states of the stable zero-dynamics of the system converge to the equilibrium point asymp-totically. The method proposed in the paper has advantages for higher-dimensional multivariable systems, in the sense that it simplifies the design and makes it possible to realize a robust decomposed control. Meanwhile, be-cause of the adoption of the second-order sliding mode, the control signal is continuous. Simulation results are presented to validate the design.

KeyWords: Multivariable systems, canonical decomposition, nonsingular

terminal sliding mode control, zero dynamics.

I. INTRODUCTION Research on multivariable linear systems with in-

ternal parameter uncertainties and external disturbances is important for both theoretical and practical reasons. Two methodologies have been commonly used: the state-space feedback control and the optimal control [1]. If a system, however, has a relatively high dimension, these two methods may impose a severe computational demand when applied real time control. One approach proposed in [2] alleviates this problem by transforming a multivariable linear uncertain systems into a block con-

trollable canonical form (BC-form), but it does not seem to exhibit robustness to internal parameter uncertainties and external disturbances.

Variable structure systems (VSS) are well known for their robustness to system parameter variations and external disturbances [3,4]. One aspect of VSS of par-ticular interest is the sliding mode control, which is de-signed to drive and constrain the system states to lie within a neighborhood of prescribed switching manifolds that exhibit the desired dynamics. When the system is in the sliding mode, the closed-loop responses of the sys-tem become totally insensitive to both internal parameter uncertainties and external disturbances. One characteris-tic of the conventional VSS is that the convergence of the system states to equilibrium points usually occurs asymptotically due to the asymptotical convergence of the linear switching manifolds that are commonly chosen. Recently, a terminal sliding mode (TSM) controller was developed [5,6,7]. Compared with the linear hyperplane based sliding mode, TSM offers some superior proper-ties, 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

−Brief Paper−

Manuscript received January 15, 2003, 2003; accepted May5, 2003.

Yong Feng and Xumei Zheng are with Department of Elec-trical Engineering, Harbin Institute of Technology, Harbin,150001, P. R. China.

Xinghuo Yu is with School of Electrical and Computer En-gineering, Royal Melbourne Institute of Technology Univer-sity, Melbourne, VIC 3001, Australia.

This work is supported by the Multidiscipline Scientific Re-search Foundation of Harbin Institute of Technology (HIT MD2001.18).

506 Asian Journal of Control, Vol. 5, No. 4, December 2003

of convergence near the equilibrium point. However, TSM controller design methods suffer from a singularity problem. Based on TSM, some nonsingular terminal sliding mode (NTSM) control systems have been pro-posed to avoid the singularity in TSM [8,9,10].

It is well known that the chattering phenomenon is a major drawback of the sliding mode approach in the practical realization of VSS. The chattering phenomenon, that is, high-frequency finite amplitude oscillations with finite frequencies caused by system imperfections, result from the discontinuity of the sign function on the sliding manifold. Two popular methods for eliminating chatter-ing replace this sign function with either a saturation function or a sigmoid-like function at the cost of slight deterioration in system performance. In addition, the second-order sliding mode control can be used to smooth the control signal. It is a continuous control, robust to parameter uncertainties and disturbances. Meanwhile, the characteristics of the traditional sliding mode ap-proaches are guaranteed in second-order sliding mode control systems [11].

This paper proposes a second-order nonsingular TSM decomposed control method for linear multivari-able systems that is robust to both internal parameter uncertainties and external disturbances. Firstly, we in-troduce the controllability index and the parameter r of the systems then, the systems are converted into the BC-form, and the coupled state variables are eliminated. As a result, the original systems are converted into r block decoupled BC-form systems consisting of an in-put-output subsystem and an internal dynamic subsystem. A special second-order nonsingular TSM is proposed for the input-output subsystem. The control law is designed to drive the state variables of the input-output subsystem to the equilibrium points asymptotically. Then, the state variables of the internal dynamic subsystem converge to the equilibrium points asymptotically. Possessing the advantages of a simple controller design and a hierar-chical controller structure, the proposed method can be easily applied to linear high-dimensional multivariable systems. Moreover, the second-order sliding mode pro-posed in this paper can eliminate chattering in systems. Simulation results are presented to validate the method.

II. DECOMPOSITION OF MULTIVARIABLE

LINEAR UNCERTAIN SYSTEM Consider an uncertain multivariable linear system:

( ) ( ( )) ( ) ( ),pert t t t= + +x A A x Bu (1)

where x(t) ∈ Rn is the state variable vector; u(t) ∈ Rm is the control input vector; A ∈ Rn×n and B ∈ Rn×m are the known parameters matrices of the system (1); Aper(t) ∈ Rn×n represents any uncertainties or nonlinearities. As-

sume that the system (1) satisfies the following assump-tions: (1) A, B are constant matrixes; (2) A, B are controllable; (3) Aper(t) satisfies the following matching condition:

( ) ( ),per t t=A Bd (2)

where d(t) ∈ Rm×1 is a bounded time-varying matrix and satisfies ( ) dt l≤d , 0dl > .

The control objective is to force the system (1) with uncertainty Aper(t) to converge to the equilibrium points asymptotically or in a finite amount of time from any initial condition (0) 0≠x .

In order to convert the system (1) into the BC-form, the following nonsingular transformation is performed:

,′ =x Fx (3)

where nR′∈x and the state transformation matrix F is defined as

1 2 1.r−=F F F F (4)

By means of the transformation (3), the system (1) is converted into:

( ) ( ) ( ) ( ),t t t t′ ′ ′ ′ ′= + +x A x B d B u (5)

where

1 11,0 1,0; 0 ; ;

T n mT ×− ′ ′= = = ∈ A FAF B FB B B R

rankB10 = n1, n1 ≤ m, n1 is the controllable index of the system (1). Therefore, the pseudo-inverse matrix of 1,0

+B exists and is given by:

1

1,0 1,0 10 1,0 .T T −+ = B B B B (6)

Then, the system (5) can be rewritten as

, , 1 1

, 1 1

1 1 1,0 1,01

( ) ( ) ( )

( ) ( ) ( ), 2, 1

( ) ( ) ( ) ( ),

r r r r r r rr

i ij j i i ij i

r

j jj

t t t

t t t i r

t t t t

− −

− −=

=

′ ′ ′= + ′ ′ ′= + = − ′ ′= + +

∑

∑

x A x B x

x A x B x

x A x B d B u

(7)

where 1[ ]T T T Tr′ ′ ′=x x x , ni

i′ ∈x R , i = 1, …, r; Bi, i-1, i = 1, …, r, have full rank, i.e., rank Bi, i-1 = ni, i = 1, …, r;

1

r

ii

n n=

=∑ , (n1, …, nr) is the controllability index vector

of the system (1). 1i−′x is regarded as the virtual control vector of the ith layer of the system (7).

For the convenience of design, the system (7) is

Y. Feng et al.: Second-Order Nonsingular Terminal Sliding Mode Decomposed Control 507

further converted into the decoupled BC-form expres-sion.

The following nonsingular state transformation is performed again:

,′ ′=x F z (8)

where nR∈z and the nonsingular transformation ma-trix F′ is

1,

2, 2, 1

2, 2, 1 2, 2

1, 1, 1 1, 2 1,2

0 0 0 00 0 0

0 0

0

r r

r r r r

r r r

r r r

−

− − −

− −

− −

− − −

′ = − − − − − − −

IK IK K I

F

K K K IK K K K I

(9)

with Ki,α, i = 1, …, r, α = 2, …, r, being the designed parameter, which will be determined by the designed matrix Ni (11), i =1, …, r, which in turn will be deter-mined by the system specifications.

Then, the BC-form system (7) is converted into the following form by using nonsingular transformation (8):

( ) ( ) ( ) ( ),t t t t′′ ′′ ′′= + +z A z B d B u (10)

where

1 110( ) ; ( ) 0 .

TT− − ′′ ′ ′ ′ ′′ ′ ′= = = A F A F B F B B

Eq. (10) can be rewritten as

, 1 1

1 1, 1,0 1,01

( ) ( ) ( ) , 2,

( ) ( ) ( ) ( ),

i i i i i ir

t t t i r

t t t tα αα

− −

=

= + = = + +

∑

z N z B z

z A z B d B u (11)

where

1[ ] , ; , 2, , ,nT T T ir i i i r= ∈ = …z z z z R N

are the matrices designed according to the system speci-fications. After Ni is determined, F′ and 1,αA can also be determined.

III. SECOND-ORDER NONSINGULAR

TSM CONTROL For the convenience of design, we assume that the

uncertain multivariable linear system (1) is already in the canonical form (11) after the transformation (3) and (8). A second-order nonsingular TSM control strategy is then adopted. The design consists of two steps. The first step is to design the second-order nonsingular TSM and en-sure that the sliding mode converges to the equilibrium

point asymptotically. The second step is to design the robust control so as to ensure that the system is robust to internal parameter uncertainties and external distur-bances.

We now proposes the following second-order nonsingular TSM manifold for the system (11)

1 10( ) ,tt dtβ

−= + ∫s z z (12)

where 1n∈s R ; 11

n∈z R ; 1 1diag( , , )nβ β=β , where

0iβ > is a constant; 10

t dt−∫ z are denoted as:

1 11 1 10 0 0, , .

Tt t tnz dt z dt z dt

− − − = ∫ ∫ ∫

In order to eliminate chattering, we propose the following second-order nonsingular TSM manifold util-izing the second-order concept [11]:

1( ) ,p qt γ −= +l s s (13)

where 1; diag( , , )mmγ γ∈ =l R γ , where 0iγ > remain

constant; p and q are both odd, 1 < p/q < 2; q ps is de-noted as

1 , , .Tp q p q p q

ms s = s

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

When ( ) 0, rt t t= ∀ ≥l (tr is the time when l(t) reaches the nonsingular terminal sliding mode manifold l(t) = 0), assume that 1 1

( ) [ ( ), , ( )]Tr n rt s t s t=s . Solving

Eq. (13), we obtain the time needed to reach s(t) = 0:

1, , 11, , 1

max ( ( ) )( ) min

pp q

s r i ri nii n

pt t s tp q γ

−

==

= +−

(14)

Thus, through the suitable control design, s(t) and ( )ts can be driven to reach the second-order sliding

mode l(t) = 0 and then remain at l(t) = 0 to realize sliding mode motion. Among l(t) = 0, s(t) will reach the equilib-rium points in finite time ts (14). After s(t) reaches zero, the system will remain in linear sliding mode motion (12); that is, the dynamic characteristics of the system (11) can be determined by the design parameters γ, β, p and q, and have nothing to do with the system’s parame-ters (13). The relevant control methodology is given in the following theorem.


Theorem 1. For the uncertain multivariable linear sys-tem (1) in the canonical form (13), if the linear sliding mode and second-order nonsingular TSM are chosen as (12) and (13), and if the control law is designed as shown below, then the system (1) is asymptotically sta-ble:

0 1( ) ( ) ( ),t t t= +u u u (15)

where

0 10 1,1

( ) .r

t α αα

+

== − ∑u B A z (16)

Here, u1(t) is obtained using the low-passed filter:

1 1( ) ( ) ( ),t t t= +v u uβ (17)

where β is the design parameter (12); in addition, the input v(t) of the low-passed filter is:

( ) ( ) ( ),eq nt t t= +v v v (18)

210( ) ,p q

eqqtp

+ −= −v B γ s (19)

( )(( ) )

1 11,0

21 11,0

1 11,0

2

( ( ) )( )

( )

( )

1/ 2 ,

T p q T

n T p q

p qd dd

diagt

diag

diag l l

qp

ηηα

− −

− −

− −

= −

+

+

l s Bv

l s B

l B s

l

γ

γ

γ β (20)

where γ , α and η are the design parameters, α > 0, 0 < η < 1.

Proof. Consider the following Lyapunov function:

1( ) ( ) ( ).2

TV t t t= l l

Differentiating V(t) with respect to time, we get:

1 1( ) ( )T T p qpV t diagq

− − = = +

l l l γ s s s

( )( )1 11,0 1,0( ) ( ) ( )T p qp diag t t

q− −

= + + +

l s B v B d d sγ β

( )( )1 11,0 1,0( ) ( ) ( )T p q

np diag t tq

− − = + +

l s B v B d dγ β

( )( )1 11,0( ) ( ) ( )T p q

np diag t tq

− − = + +

l s B v d dγ β

( )

( ) ( )

1 11,0

2 1 11,0

( )

1/2 ( ) ( ) ( )

p qd dd

T p q

p diag l lq

p diag t tq

ηηα

− −

− −

= − +

− − +

l B s

l l s B d d

γ β

γ β

( ) 21/ 2 ( );V tηη ηα α≤ − = −l

that is,

( ) ( ), 0, (0) 0.V t V t t Vηα≤ − ∀ ≥ ≥

Therefore, according to Lemma 2 [12], the condition for Lyapunov stability is satisfied. The system states can reach the sliding mode l(t) = 0 within finite time

( )10( ) (1 )rt V tη α η−= − . In the sliding mode l(t) = 0,

either 1 0p q− + =s sγ , or 0q p+ =s sγ , and s will reach the zero in ts (14). When s = 0, 1 10

d 0t t−

+ =∫z zβ . The

input-output subsystem, z1, is stable asymptotically for t > ts. Furthermore, according to Ni = −λi Ini, i = 2, …, r, −λ2 < −λ3 < … < −λr < 0, the other state variables, z2, …, zr of the system (11) will converge to the equilibrium points asymptotically. Since the state transformations between x and z, (3) and (8), are linear, the variable states x of the original system (1) will converge to the equilibrium points asymptotically. This completes the proof.

IV. SIMULATIONS

A simulation with a seventh-order system was per-

formed for the purpose of evaluating the performance of the proposed scheme.

Consider the following system [13]:

( ) ,t= + +x Ax Bd Bu (21)

where the disturbance is d(t) = [0.1sin(2t) 0.1sin(2t) 0.1sin(2t)]T, and A and B are given by

2 1 0 0 0 0 0 2 1 1

0 2 0 0 0 0 0 2 1 1

0 0 2 0 0 0 0 1 1 1

.0 0 0 2 0 0 0 3 2 1

0 0 0 0 1 1 0 1 0 0

0 0 0 0 0 1 0 1 0 1

0 0 0 0 0 0 1 1 0 0

= = −

A B

Firstly, the system (21) was transformed into the BC-form (7):


[ ]3 3 2

2 3 2 1

1 1 2 3

2 1 0 0 ,

0 2 2 0 2 1 10 0 0 0 1 0 1 ,0 0 4 2 3 2 1

0 0 2 0 2 0 00 1 0 0 0 0 00 0 1 0 0 0 0

1 1 11 0 1 ( )1 0 0

x x

x

x

t

′ ′ ′= +

− ′ ′ ′ ′= + + − −

− ′ ′ ′ ′= + +

+

x

x x x

x x x

d1 1 11 0 1 .1 0 0

+

u

(22)

Then, Eq. (22) was transformed into the following decoupled block controllable canonical form by means of state transformation (8):

[ ]3 3 2

2 2 1

1 1,1 1 1,2 2 1,3 3

0.6 1 0 0 ,

2 1 11 0 1 ,

3 2 1

1 1 1 1 1 11 0 1 ( ) 1 0 1 .1 0 0 1 0 0

z z

t

= − + = − + −

= + + + +

z

z z z

z A z A z A z d u

(23)

In the above transformation, the parameters N2 and N3 were chosen as follows: N2 = diag(−1, −1, −1), and N3 = −0.6. Then, we get:

11 12

7.2 2.6 1.6 1.16 3.5 1.57.4 1.2 8.2 , 35.92 3 9 ,

6.2 2.6 5.6 17.96 1 3

− − = − − − = − − −

A A

13

13.5213.52 .6.76

− = −

A

The state transformation matrix from z to x is:

3 3

2 2

1 1

1 0 0 0 0 0 02.6 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 0

6.76 5.6 0.5 1.5 1 0 013.52 11.2 1.5 4.5 0 1 06.76 5.6 0.5 1.5 0 0 1

x z

− = − − − − −

x zx z

.(24)

The three linear sliding mode manifolds were chosen follows:

1 11 11 2 12 120 0

3 13 130

0.8 ( d ), 0.8 ( d ),

0.8 ( d ).

t t

t

s z z t s z z t

s z z t

− −

−

= + = +

= +

∫ ∫

∫

The three nonsingular terminal sliding mode manifolds were chosen as follows:

5 / 3 5 / 3 5 / 31 1 1 2 2 2 3 3 3, , .l s s l s s l s s= + = + = +

According to Theorem 1, the controller was designed as follows:

0 1,= +u u u (25)

0 1

2 3

0 0 1 7.2 2.6 1.61 1 0 7.4 1.2 8.20 1 1 6.2 2.6 5.6

1.16 3.5 1.5 13.5235.92 3 9 13.52 .17.96 1 3 6.76

= − − − − − −

− − − + − + − − −

u z

z z

According to (17), u1 was designed as follows:

1 1

0.8 0 00 0.8 0 ( ) ( ).0 0 0.8

t t = +

v u u

According to (18), (19) and (20), veq and vn were designed, respectively, as follows:

1/ 31 1

1/ 32 2

1/ 33 3

2 / 31

1 2 / 33 3 2

2 / 33

2 / 31

1 2 / 33 3 2

2 / 33

0 0 1 13 1 1 0 1 ,5

0 1 1 1

0 0 1 1 10 0 1 0 10 0 1 0 0

0 00 00 0

eq

T

T

n

T

s ss s

s s

sI s

s

sI s

s

−×

−×

= − − −

= −

v

l

v

l

( )

2

2 / 31

1 2 / 33 3 2

2 / 33

0.5

,1 1 11 0 11 0 0

1 1 1 0 01 0 1 0 0

.1 0 0 0 0

0.1732 0.8*0.3464 (3/5)(1/ 2) 0.01

ss

s

−×

⋅ + +

l I

l

We assumed that the initial state variables of the system were as follows: x3(0) = 0, x21(0) = 0, x22(0) = 0, x23(0) = 1, x11(0) = 2.36, x12(0) = −3.16 and x13(0) = 2.16.


−0.2 0 0.2 0.4 0.6 0.8 1 1.2

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

the state of s1

the

deriv

ate

of s

tate

s1

Fig.1. Phase plane of s1and its differential.

the state of s2

the

deri

vate

of

stat

e s2

0 0.2 0.4 0.6 0.8 1

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

Fig.2. Phase plane of s2and its differential.

−0.2 1 0.2 0.4 0.6 0.8

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

the state of s3

the

deriv

ate

stat

e of

s3

Fig.3. Phase plane of s3 and its differential.

The simulation results are illustrated in Fig. 1 to

Fig. 10. The phase planes of s1, s2 and s3 and their dif-ferentials are shown in Fig. 1 to Fig. 3 respectively. It is seen that s1, s2 and s3 realized the nonsingular terminal sliding mode. The phase plane of z11, z12 and z13 of the

0 0.2 0.4 0.6 0.8 1

0

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

the integral of z11

syst

em s

tate

z11

Fig.4. Phase plane of z11and its integral.

0 0.2 0.4 0.6 0.8 1

0.2

0.1

0

−0.1

−0.2

−0.3

−0.4

−0.5

the integral of z12

syst

em s

tate

z12

Fig.5. Phase plane of z12 and its integral.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

−0.05

−0.1

−0.15

−0.2

−0.25

−0.3

−0.35

−0.4

the integral of z13

syst

em s

tate

z13

Fig.6. Phase plane of z13 and its integral.

input-output subsystem and their integrals are depicted in Fig. 4 to Fig. 6, respectively. It shows that the in-put-output subsystem realized the linear sliding mode control. The plots of z vs time and x vs time are depicted in Fig. 7 and Fig.8, respectively. They both converge to the equilibrium points asymptotically. Fig. 9 shows the


0 5 10 15

1.5

1

0.5

0

−0.5

−1

−1.5

−2

t/s

the

syst

em s

tate

z

z1

z2

z3

Fig. 7. State variable z.

0 5 10 15−4

−3

−2

−1

0

1

2

3

4

t/s

the

syst

em s

tate

x

x1

x2

x3

Fig. 8. State variable x.

input v of the low-passed filter. The control signals are shown in Fig. 10. It is seen that no chattering phenome-non occurred.

V. CONCLUSIONS

This paper has proposed a special second-order

nonsingular TSM decomposed control method for linear uncertain multivariable systems. Through decoupled state transformation, the systems are converted into the BC–form for the convenience of design. A second-order nonsingular TSM has been proposed for the system, which can drive the state variables of the input-output subsystem to the equilibrium points asymptotically; then, the state variables of the stable zero-dynamic subsystem converge to the equilibrium points asymptotically. The method proposed here simplifies the design of the con-troller and realizes hierarchical control. Moreover, the second-order sliding mode proposed in this paper elimi-nates chattering.

0 2 4 6 8 10−1.5

-1

−0.5

0

0.5

1

1.5

2

t/s

the

cont

rol

sign

al v

v13

v12

v13

Fig.9. The control signal v.

0 5 10 15−30

−20

−10

10

20

30

t/s

the

cont

rol s

igna

l u

0

u2 u1

u3

Fig. 10. The control signal u.

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SECOND-ORDER NONSINGULAR TERMINAL SLIDING MODE DECOMPOSED CONTROL OF UNCERTAIN MULTIVARIABLE SYSTEMS

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