978-1-4244-7815-6/10/$26.00 ©2010 IEEE ICARCV2010

Reaching Law Based Sliding Mode Control for Discrete MIMO Systems

Mija S J Electrical Engineering Department

National Institute of Technology Calicut Kerala, India.

mija@nitc.ac.in

Susy Thomas Electrical Engineering Department

National Institute of Technology Calicut Kerala, India.

susy@nitc.ac.in

Abstract— This paper analyzes the effectiveness of the reaching law approach for sliding mode control of discrete time systems. The method is extended for sliding mode control for multi-input multi-output (MIMO) systems. Modified reaching law for reducing the chattering is proposed. The performance of the proposed controller is analyzed by carrying out simulation studies on MIMO systems. The performance of the controller is also compared with that of the equivalent control.

Keywords— Discrete Time Systems, Multi Input Multi Output Systems, Sliding Mode Control, Reaching Law

I. INTRODUCTION Remarkable attention is currently being devoted by control

systems researchers to variable structure systems (VSS), mainly due to their robustness characteristics. Sliding mode control of discrete time systems has not been studied as much as its continuous counterpart. There are relatively few papers about discrete sliding mode control and most of them discuss Single-Input Single-Output (SISO) discrete-time systems. In control applications, most processes have multiple inputs and multiple outputs. Lacking general-purpose multivariable controllers, a large percentage of multivariable processes are treated as single variable processes resulting in poor control, wasted energy and materials, inconsistent quality, and plant upsets. The discrete version of Variable Structure Control (VSC) is important when the implementation of the control is realized by computers. It should be pointed out that, theoretically, discrete VSCs cannot be obtained from their continuous counterpart by means of simple equivalence.

Dote and Hoft [1] first considered the discrete VSC problem and used an equivalent form of the continuous reaching condition to give a discrete reaching condition. Discrete sliding mode control (DSMC) of discrete-time systems has been considered by Milosavljevic [2] in the context of sampled-data systems and he named the discrete sliding mode the quasi-sliding mode. Sarpturk et al. [3] presented a new sufficient condition for the existence of DSMC and discussed the stability. Reaching law approach in continuous time domain is well studied and implemented. The objective is to find a suitable reaching condition for discrete systems such that, when sampling period tends to zero, the continuous sliding mode reaching condition is satisfied. For SISO systems some authors have used the ideal sliding mode conditions to design control and others have given sliding

mode reaching conditions [4]. In [2] a reaching sliding condition is presented which is only a necessary condition for the existence of the sliding mode. Gao et al. [5] used an equivalent form of the so-called continuous reaching law to give a discrete-time reaching law and defined a quasi-sliding mode band. This paper discusses the reaching law for MIMO systems.

In discrete-time sliding mode systems the control can be chosen as a simple state feedback [6]. This control guarantees the state to converge onto the sliding hyperplane and the state trajectories may not cross the sliding surface. In the discrete sliding mode, the effect of external disturbances on the discrete-time system is reduced, but to eliminate the disturbances completely an additional condition like the invariance condition is to be satisfied. In continuous time domain the system states can be retained exactly on the sliding surface. But for discrete systems, in the presence of disturbances the state may not lie precisely on the sliding lattice hyperplane. However, when certain conditions like the cone inequality for the norm of the state and disturbance input, and the matched uncertainty condition hold, the sliding points may lie on the pre specified sliding lattice [4]. For continuous time systems, control strategies using linear or/and nonlinear surfaces (Terminal Sliding Mode) is found to be effective [7, 8, 9]. The effects of discretization of continuous-time terminal sliding mode have been analyzed in [10]. It is found that the finite-time convergence behaviour observed in continuous-time terminal sliding mode is lost on discretization. So a control strategy based on linear surface is adopted for discrete case.

In this paper the effectiveness of reaching law approach is analyzed and is extended to MIMO systems. The performance enhancement is clearly illustrated using few examples. The performance of the designed controller is compared with that of the equivalent control.

The paper is organized as follows. In Section 2, discrete time sliding mode control and sliding lattice for MIMO is discussed. Problem formulation and stability constraints are discussed in section 3. Section 4 deals with SMC for MIMO discrete time system. Section 5 explains the extended reaching law approach to discrete MIMO systems. Simulation results are discussed in Section 6 and the conclusions are drawn in Section 7.

2010 11th Int. Conf. Control, Automation, Robotics and VisionSingapore, 7-10th December 2010

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II. DISCRETE TIME SLIDING MODE CONTROL AND SLIDING LATTICE FOR MIMO SYSTEMS

In the case of discrete time sliding mode (DSM), control action can only be activated at sampling instants and the control effort is constant over each sampling period. The condition 0ss <� , that assures the sliding motion on the switching manifold for continuous time system, is no longer applicable in DSM. Thus a discrete time sliding mode condition is required.

DSMC of discrete-time systems have been considered by Milosavljevic [2] for SISO systems. Sarpturk et al. [3] presented the reaching condition for SISO systems.

( 1) ( )s k s k+ < (1)

Sira-Ram�´rez proposed [11] the following reaching condition

2[ [ 1] [ ]] [ ]s k s k s k+ < (2)

Furuta used the equivalent form of a Lyapunov-type of continuous reaching condition to give the discrete version [12]

[ 1] [ ] 0V k V k+ − < with 21[ ] [ [ ]]2

V k s k= (3)

Spurgeon, Yu and Potts [4] showed that the condition [ 1] [ ]s k s k+ < is only a sufficient condition for existence

of the discrete sliding mode. Sarpturk et al.and Sira-Ram�´rez presented the necessary condition for the existence of the sliding mode as stated by Milosavljevic´

2[ 1] [ ] [ ]s k s k s k+ < (4)

This yields an unstable sliding mode along the sliding surface s=0. This condition is not a sufficient condition for the existence of the discrete sliding mode and only guarantees that the sliding points approach and/or cross the sliding hyperplane. It is not sufficient for convergence to the sliding lattice wise hyperplane.

Equivalent control in the Furuta approach [12] is obtained by setting

[ 1] [ ]s k s k+ = (5)

But it should be emphasized that, when the sliding mode occurs, s(k)=0 and the equivalent control in the sense of Furuta is the same as in the traditional case. Assume for all

sk k≥ (5) is satisfied. Then, for all sk k≥ , [ ] [ ]ss k s k= . So according to Furuta’s definition, the discrete-time sliding mode occurs if there exists a finite time ks such that after this time the value of the sliding function is constant.

Koshkouei and Zinober [4] have presented a condition, which is weaker than the above conditions. In discrete-time systems instead of having a hyperplane as in the continuous case, a countable set of points is defined comprising a so-called lattice; and the surface on which these sliding points lie is named the lattice wise hyperplane. The sliding lattice is

defined as

[ ] 0s k = (6)

One way to design a sliding lattice hyperplane for MIMO (m-input p-output) systems is to consider the intersection of the m sliding lattice surfaces. Let

1 2[ ]Tms s s s= � (7.a)

The ith sliding lattice is 0is = and

{ }1

[ ] : [ ] 0i m

ii

x k s k=

==�

(7.b)

is a sliding lattice for the system. The ith component of [ ](1 )iu k i m≤ ≤ of the state feedback control vector u(k) is

selected such that the state lies on the ith sliding lattice. The sliding mode of discrete-time systems is completely different from SMC in continuous systems. In continuous systems the sliding variable is a linear transformation

: n ms R R→ with x Cx→ . In contrast, for discrete-time systems the sliding function consists of a sequence, which can be considered as the restriction of the function ( [ ])s x k . When

the sequence { } 0[ ]

ks k ∞

= is a null sequence, i.e.,

lim [ ] 0s k = the convergent sliding mode exists.

III. PROBLEM FORMULATION AND STABILITY CONSTRAINTS

Consider a Multi input Multi output discrete-time plant

[ 1] [ ] [ ]x k Ax k Bu k+ = + (8)

where, again, x is an n-vector, u is a m-vector, and A and B are of appropriate dimensions.

When a discrete VSC is applied to the plant, the state response of the system can also be separated into the reaching, sliding, and steady-state modes. The problem is to design a globally asymptotically stabilizing feedback control law

*[ ] [ [ ]]u k u x k= assuring the sliding motion on S. It is shown that, once the manifold has been reached, the system behaves like an autonomous linear system of reduced order with poles arbitrarily located in the s-plane[3]. In the discrete-time implementation of the sliding mode methodology, the switching elements are replaced by a computing device, which changes the structure of the system at discrete instants. Since the control input is computed at discrete instants and applied to the system during the sampling interval, inevitably, a non-ideal sliding regime will appear. This quasi-sliding regime is inherently different from the quasi-sliding regime, which may appear in continuous time systems due to non-ideal behavior of the analog components and can make the system unstable.

Let the feedback control be of the form

[ ] ( , ) [ ]u k k x s x k= − (9)

where, the feedback gain k is given by

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; [ ] [ ] 0( , )

; [ ] [ ] 0ij i j

ijij i j

k s k x kk x s

k s k x k

+

−

� >�= � <�� (10)

1i m= � 1j n= �

The difference between ijk + and ijk − depends primarily on the amplitude of the disturbing factors and on the sampling period. If the sampling period decreases the lower and upper bounds of the feedback gains become further apart. When the sampling period tends to zero, the convergence limits tend to ±∞ and the feedback gains satisfying the sliding conditions become practically equal to the ones obtained from the continuous design.

{ }0

lim [ ] [ ] ( [ ]) 2 [ ] 0i i i iTs kT T s kT Sign s kT s kT

→+ + = >

(11.a)

{ }0

lim [ ] [ ] ( [ ])

[ ] ( [ ]) 0i i iT

i i

s kT T s kT Sign s kT

Ts kT Sign s kT→

+ −

= <�

(11.b)

For any value of k, the feedback coefficients (11) will be always positive. Since T is a positive value, it has no effect on the sign of equation (11.b). Therefore, in the limit, equation (11.b) will be completely equivalent to 0is s <� . This means that one of the bounds for each feedback gain tends to ±∞ and the other bound tends to the one obtained from the continuous design. Conversely, if the sampling period increases, the upper and lower bounds become closer and for a sampling period larger than a critical value, no interval can be found for the feedback gains. In general, the lower bounds are functions of the parameter variations and input disturbance bounds while the upper limits appear to be mainly functions of the sampling period. The bounds of the discrete time control guaranteeing the stability of the sliding mode system depend also on the distance of the state from the sliding manifold [13].

In case of sliding-mode control based on a linear sliding hyperplane, the asymptotic stability found in continuous-time system is also found in discrete-time sliding-mode control. However, the finite time convergence property assured by the nonlinear sliding manifold, in continuous-time systems, is not preserved when the concept is extended to discrete-time systems[10]. For a second order system, the sliding manifold for terminal sliding mode would be of the form

2 1 1s=x x xγ ρα β+ + (12)

For a sampling time of τ sec, the discrete representation of system (8) and the sliding manifold (12) can be given as

1 1 2

2

[ 1] [ ] [ ][ 1] ( [ ], [ ], )

x k x k x kx k a x k u kτ

ττ

+ = ++ =

(13)

2 1 1[ ] [ ] x [ ] x [ ]s k x k k kγ ρα β= + + (14)

Where 0, 0α β> > , ρ and γ are rational numbers of

the form pq , with both p and q being odd.

A finite time convergence is possible with the nonlinear sliding line approach only if the control is so designed such that the trajectory can exactly reach one of the points ( , )s sx xτ± � . A discretized system would either settle into a periodic orbit or would be unstable, once it is assumed that the system states do not cross the points ( , )s sx xτ± � . This justifies the choice of linear hyperplane.

IV. SMC FOR MIMO DISCRETE TIME SYSTEMS The controller consists of a linear state feedback and a

switching feedback. A class of stabilizing switching gain, G is derived.

Consider the system given by equation (8) We assume that the pair (A,B) is stabilizable, rank(B) = m and sliding function

0s Gx= = where m nG R ×∈ . We assume that the matrix G

is selected such that 0GB ≠ . Then the Control law is given by [14]

{ }1[ ] ( ) ( ) [ ] [ ]u k GB G A I x k v k−= − − + (15)

0: [ ] [ ]

[ ] [ ][ ] : [ ] [ ]

[ ]

s k x kv k x k

s k s k x ks k

δγ δ

�≤�= � >�

�

���� (16)

Here [ ] mv k R∈ is a switching part of the input. A

positive constant Rγ ∈ is a switching gain, and a positive constant Rδ ∈ is a sector width.

Using the control strategy in (15) the discrete sliding mode control exhibits chattering. As the state of the system stays outside of the sector and goes through the sector at each time step, chattering occurs. So a condition of γ under which the state does not go through the sector. If the state of the system does not go opposite side of the switching surface, then it does not go through the sector. Therefore, chattering elimination is achieved in case the sign of each element of [ ]s k coincides with that, of [ 1]s k + .The same sign condition of [ ]s k and

[ 1]s k + is derived as follows[14].

[ ]1 0

[ ]x ks k

γ− ≥ (17)

Using the relation [ ] [ ]s k x kδ> and equation above we have the following result.

The control law (15) stabilizes the system (8) if γ satisfies

12

γ δ< (18)

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V. REACHING LAW APPROACH FOR MIMO SYSTEMS In this section the reaching law approach which has been

successfully used for continuous time systems [15] and later on applied to SISO discrete time systems [5] is extended to MIMO systems. This reaching law directly dictates the dynamics of the switching function. A VSC control law is synthesized from the reaching law in conjunction with a known model of the plant and the known bounds of perturbations. For the VSC of a continuous plant, a convenient reaching law [5] is

( ) ( ( )) ( )s t sign s t qs tε= − −� 0, 0qε > > (19)

where S is the lattice wise sliding hyperplane with 1ms R ×∈ .

For the VSC of a discrete system, an equivalent form of the reaching law is

[ 1] [ ] [ ] ( [ ])s k s k qTs k Tsign s kε+ − = − − (20.a) with, 1 0qT− > (20.b)

where, T > 0 is the sampling period. The following attributes are in order.

Starting from any initial state, the trajectory will move monotonically towards the switching plane and cross it in finite time.

Once the trajectory has crossed the switching plane the first time, it will cross the plane again in every successive sampling period, resulting in a zigzag motion about the switching plane.

The size of each successive zigzagging step is non increasing and the trajectory stays within a specified band.

The inequality for T must hold to guarantee attribute Al, which implies that the choice of T is restricted. The Presence of the signum term guarantees attributes A2 and A3.

Reaching law approach has following merits.

A desirable reaching mode response can be achieved by a judicious choice of parameters k and q.

The width of the quasi-sliding mode band (QSMB) can conveniently be calculated as will be shown later.

The effect of T on the VSC system can be evaluated using (20), since reaching law (20) contains T as a parameter.

The determination of the control law based on (20) is simple. This is important for designing control law for multiple input VSC systems.

The equality form of the reaching law (20) leads to a control law, which is also in equality form.

For discrete time MIMO systems the control law, with reaching law approach with [ ]s Gx k= is obtained by substituting (8) in (20) as follows:

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

T T T

T T

u k G B G Ax k G x kqTG x k Tsign G x kε

−= − −+ + (21.a)

In order to reduce the chattering the control law (21.a) has been modified as follows:

1[ ] ( ) [ [ ] [ ] [ ] ( [ ])]T T T T Tu k G B G Ax k G x k qTG x k Tsat G x kε−= − − + +

(21.b) The width of QSMB is

1TqT

εδ =− (22)

The results show that the modified control law with saturation function, instead of sign gives better performance. The choice of sat or sign function usually involves a trade off between performance and chattering. But the plots show that there is no deterioration of performance with the use of sat function (Fig. 4, 5).

VI. RESULTS AND DISCUSSION Example 1

Consider the system [14] with matrices 0 1 00 0 11 2 3

A� �� = � � �

0 01 00 1

B� �� = � � � (23)

The G matrix for sliding surface, S is obtained as

1 2 00 0 1

G � �= � � (24)

Control law given by equation (15) is applied to the system.

Fig. 1 shows the effect of adding the term v(k) in the control law (15). It is clear from the plots that without the term v(k) the system states are stabilized but with some offset. Fig. 2 is the plots of control inputs. The control effort is much reduced with v(k) added to control law. Hence the addition of the term v(k) in the control law (15) is justified.

Now using the same control law the system performance has been analyzed for different γ values. Plot of the state x1 vs time is shown in Fig. 3. The plot clearly shows that chattering and settling time is reduced considerably with the switching gain,γ and sector width,δ chosen according to the condition (17) and (18).

The performance of the same system with extended reaching law approach has been evaluated with control law (21.a and 21.b). Same G has been used in [ ]s Gx k= to get the sliding function.

Fig. 4 shows the plots of the system states with proposed reaching law approach. By equation (21.a) the system exhibited considerable chattering. So the modified control law (21.b) has been used. It is clear from the plots that the modified control law with sat function successfully removed chattering, without degrading system performance. By this method in addition to chattering reduction settling time has also reduced (<0.5 sec) compared to ideal sliding mode

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method (<10sec), (Fig. 1).

Example 2

Consider the discrete time system [4]

1 1

2 2

3 3

4 4

[ 1] [ ]0 1 0 0 0 0[ 1] [ ]5 6 1 1 1 0

[ ][ 1] [ ]0 0 0 1 0 0[ 1] [ ]0 0 10 9 0 1

x k x kx k x k

u kx k x kx k x k

+� � � �� � � �� � � � + −� � � � = +� � � � +� � � � + � � � �

(25)

Being convinced with improved performance of control

law (21.b) compared to (21.a), the former is used to control the system (25). The G matrix for sliding surface, S is obtained as in [4].

0.3565 3.0000 0.3417 0.21570.0918 0.2157 1.0767 3.0000

G � �= � �

(26)

0 20 40 60-5

0

5

t(sec)

x1

with out v(k)

0 20 40 60-2

0

2

t(sec)

x2

with out v(k)

0 20 40 603

4

5

t(sec)

x3

with out v(k)

0 20 40 60-5

0

5

t(sec)

x1

with v(k)

0 20 40 60-2

0

2

t(sec)

x2

with v(k)

0 20 40 60-2

0

2

4

t(sec)

x3

with v(k)

Fig.

1 Plot of System State, Vs time

0 20 40 60-4.5

-4

-3.5

-3

t(sec)

u1

with out v(k)

0 20 40 60-10

-9.5

-9

-8.5

-8

t(sec)

u2

0 20 40 60-6

-4

-2

0

2

t(sec)

u1

with v(k)

0 20 40 60-15

-10

-5

0

5

t(sec)

u2

Fig.2 Plot of Control Inputs (u) Vs time

0 5 10 15 20 25 30 35 40 45 50-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

gama=0.9

gama=1.8

Fig. 3 Plot of System State, x1 Vs time

0 1 2 3 4 5 6-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t(sec)x1

with sign

with sat

(a)

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

1.5

2

t

x2

with sign

with sat

(b)

0 1 2 3 4 5 6-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

t

x3

with sign

with sat

(c)

Fig. 4 Plot of System States Vs time, (a) x1, (b) x2 and (c) x3 with sign and saturation functions

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0 0.5 1 1.5 2 2.5 3 3.5-6

-4

-2

0

2

t(sec)

u1

with sign

with sat

0 0.5 1 1.5 2 2.5 3 3.5-15

-10

-5

0

5

t(sec)

u2

with sign

with sat

Fig. 5 Plot of Control Input Vs time (a) u1 and (b) u2

0 5 10 15 20-4

-2

0

2

t(sec)

x1

0 5 10 15 20-1

0

1

2

t

x2

0 5 10 15 20-2

0

2

4

t

x3

0 5 10 15 20-2

-1

0

1

t

x4

Fig. 6 Plot of System states Vs time

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

t(sec)

u1

0 2 4 6 8 10 12 14 16 18 20-30

-20

-10

0

10

20

t(sec)

u2

Fig.7 Plots of Control Inputs (u) Vs time

VII. CONCLUSION Since discretization degrades the system stability, due care

must be taken to ensure stability. In this paper the criteria for the choice of feedback gain matrix, k is discussed and the importance of bounds of k is highlighted. The reaching condition for discrete SISO systems has been successfully extended to discrete MIMO systems. The reaching law is modified by replacing the signum function with saturation function to reduce chattering. The results show that the control strategy proposed is effective in terms of performance and chattering reduction.

REFERENCES

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[13] Kotta, U., ‘‘Comments on ‘On the stability of discrete-time sliding mode control systems’,’’ IEEE Trans. Autom. Control, 34, pp. 1021–1022, 1989.

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