a new algorithm for discrete-time sliding mode control using infrequent

310

Asian Journal of Control, Vol. 7, No. 3, pp. 310-318, September 2005

A NEW ALGORITHM FOR DISCRETE-TIME SLIDING MODE CONTROL USING INFREQUENT OUTPUT OBSERVATIONS S. Janardhanan and B. Bandyopadhyay ABSTRACT This paper presents a new approach for sliding-mode control of discrete-time systems using the reaching law approach together with periodic output feedback technique. This method does not need the system states for feedback as it makes use of only the output samples for designing the controller. Thus, this methodology is more practical and easy to implement. A numerical example is presented to illustrate the design technique. KeyWords: Sliding mode control, periodic output feedback, discrete-time systems.

I. INTRODUCTION The sliding mode control theory (SMC) is based on the concept of varying the structure of the controller based on the changing state of the system in order to obtain a desired response [1]. A high speed switching control action is used to switch between different structures and the system state trajectory is forced to move along a chosen manifold in the state space, called the switching manifold. The behavior of the closed loop system is thus determined by the sliding surface [2,3]. In the recent years, considerable efforts have been put in the study of the concepts of Digital Sliding Mode (DSM) controller design [4-6]. In case of the DSM design, the control input is applicable only at certain sampling instants and the control effort is constant over the entire sampling period. Moreover, when the states reach the switching surface, the subsequent control would be unable to keep the states confined to the surface. As a result, DSM can undergo only quasi-sliding mode, i.e., the system states would approach the sliding surface but would generally be unable to stay on it. Gao et al [6] introduced a “reaching law” approach for the design of control for DSM using state feedback. This reaching law ensures that the system trajectory will hit the switching manifold and thereafter undergo a zigzag motion about the switching manifold. The magnitude of each successive zigzagging step decreases so that the trajectory stays within a specified band called the quasi-sliding-mode band. Manuscript received February 10, 2003; revised July 15, 2004; accepted October 13, 2004. The authors are with Indian Institute of Technology Bombay (IITB), India (e-mails: [email protected]; [email protected]. ac.in).

Most of the sliding mode control strategies require full-state feedback. But, in practice, all the states of the system may not be available for measurement. Such situations would demand the use of observers or dynamic compensators, which would add up to the system complexity. Since the output is available for measurement, output feedback can be used for controller design. Few research works are available which deal with SMC design using output feedback [7-9]. The problem of static output feedback has been studied in considerable details. However, no results are available till today which show that a guaranteed closed loop stability can be achieved by using static output feedback [10]. It was shown in [11] that by the use of periodically time-varying piecewise constant output feedback gain, the poles of the discrete-time control system could be assigned arbitrarily (within the natural restriction that they be located symmetrically with respect to real axis). This techniques has the feature of static output feedback, but, unlike static output feedback, it guarantees the closed loop stability of the system. A discrete-time quasi-sliding mode control law using fast output sampling was proposed by Saaj et al. [12] which realizes a DSM control based on state feedback, but using only the outputs of the system. But, as fast output sampling [13] involves sampling of the system output at a rate faster as compared to the input sample rate, there is a possibility that sensor noise, being repeatedly entered in the loop through the control input, deteriorating the performance of the system. This paper proposes an alternative algorithm to realize DSM using output feedback by combining important features of discrete-time SMC [6] and POF [14]. This technique uses infrequent observations of the output to generate a control input profile. Hence, there would be less prob-

S. Janardhanan and B. Bandyopadhyay: A New Algorithm for Discrete-Time Sliding Mode Control

ability of sensor noise entering into the system as compared to FOSSMC [12]. A single-input single-output linear-time invariant discrete-time system representation has been considered. The outline of the paper is as follows. The system definition and problem statement are introduced in Section II. Section III gives some preliminary results which are critical to the discussion on POFSMC. The new SMC methodology is presented in Section IV. An example including computer simulation results is presented in Section V. Finally, the concluding remarks are made in Section VI.

x(k )

x(k  1) y (k )

) W x (k )  * W u ( k ) ,

(1)

Cx(k )

where W is the sampling period, x is an n-dimensional state vector, u is a scalar, y is the output and the matrices )W, *W, C are of appropriate dimensions. For the following discussion, we assume that the pair ()W, *W) is controllable and ()W, C) is observable and a switching surface s(k) = gcx(k) = 0, which is stable and has the desired dynamics has already been designed. Here, the notation gc is used to denote the transpose of g. The design procedure is as follows. 1. The first step is to obtain a suitable transformation matrix T, such that it relates the states of the autonomous systems x(k + 1) = )W x(k) and its adjoint

(5)

The control is of the form

u

 °­u (u y k ) for ®  °¯u (u y k ) for

g cf (u y k )  0 g cf (u y k ) ! 0

Then, this control would cause the trajectory to undergo a zigzag motion about the switching manifold with decreasing amplitude. This guarantees the quasi sliding motion of the system.

II. SYSTEM DEFINITION AND PROBLEM STATEMENT Consider the discrete-time system representation of the SISO system

f (u y k )

311

III. PRELIMINARY RESULTS 3.1 Reaching law approach The reaching law is a differential equation which specifies the dynamics of a switching function s(t). The constant plus proportional rate reaching law [2] for the VSC of a continuous-time plant is given as

s(t )

E sgn( s (t ))  qs (t )

(6)

where E and q are gains such that E > 0, q > 0. Various choices of E and q specify different rates for s(t) and yield different structures in the reaching law. If E is too small, the reaching time will be too long. On the other hand, a too large value of E will cause severe chattering. Also due to the presence of the proportional rate term – qs(t), the state is forced to approach the switching manifold faster when s is large. An equivalent reaching law for the QSMC of a discrete-time system proposed in [6] is



xˆ (k  1)

) cW xˆ (k )

(2)

by the relation

xˆ (k ) Tx(k )

(3)

2. Next is to design a state feedback based DSMC [6] for the adjoint system

xˆ (k  1) yˆ (k )

) cW xˆ (k )  C cuˆ (k )

(4)

*cW xˆ (k )

so that it slides along an equivalent stable sliding manifold gˆ cxˆ (k ) 0 3. The third step is to translate the DSM control input uˆ of the adjoint system into an equivalent periodic output feedback based quasi-sliding mode control law u for which the system in Eq. (1) satisfies the reaching condition for discrete quasi-sliding mode. A relationship expressing the switching function at k-th sampling interval as a linear combination of the past inputs and outputs has also been derived as

s (k  1)  s (k )

 qWs (k )  EW sgn( s (k ))

(7)

where W > 0 is the sampling period, E > 0, q > 0, 1 – qW> 0. This approach can describe how the switching function decreases toward zero. The inequality for W must hold to guarantee that starting from any initial state, the trajectory will move monotonically towards the switching plane and cross it in finite time. This implies that the choice of W is restricted. Also the presence of the signum term guarantees that 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 and that the size of each successive zigzagging step is non-increasing and the trajectory stays within a specified band.

3.2 Periodic output feedback In this technique, an output feedback control law is used to realize a desired closed-loop eigenspectrum by application of a piecewise constant periodic output feedback [14]. Consider the discrete-time LTI system in (1). The following control law is applied to this system. The

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output is measured at the time instants t = kW, k = 0, 1, } using a sampling and hold system. The output sampling interval W, is divided into N subintervals of length ' = W/N, and the hold function is assumed constant on these subintervals. The choice of N is made such that N t v, the controllability index of the system. Thus the control law becomes

K l y (k W)

u (t )

k W  l '  t d k W  (l  1)' 0 1 "  N  1

Kl  l

Kl  N

(8)

Note that a sequence of N gains {K 0  K1  "  K N 1} when substituted in Eq. (8) generates a time-varying, piecewise constant output feedback gain K(t) for k W d t d (k  1)W . Consider the system (), *, C) obtained by sampling the system in Eq. (1) at sampling interval '

x(k  1) y (k )

)x(k )  *u (k )

(9)

Cx (k )

G

(13)

3.3 Design of the switching surface The switching plane for state feedback DSMC is

s (k )

g c x(k )

0

(14)

For designing the vector g, the system (1) is transformed to the normal form using a suitable transformation

z (k )

M x(k )

z (k  1)

K1 " K N 1 ]c

[ K0

*K

where M is the transformation matrix. The transformed system takes the form

Define

K

where U(˜) denotes the spectral radius. Since the system is observable by assumption, one can find a state feedback Gc that would assign the eigenvalues of the dual system ()Wc + CcGc) in arbitrary locations  However, as the eigenvalues ()Wc + CcGc) are equivalent to those of its transpose, one would be able to find an output injection gain G to achieve any desired self-conjugate set of eigenvalues for the closed loop matrix ()N + GC), and from N t v, it follows that one can guarantee the finding of a periodic output feedback gain K, which realizes the output injection gain G by solving

) W z (k )  * W u (k )

(15)

where

u(k W)

K y ( k W)

ª u (k W) º « » « u (k W  ') »  « » # « » «¬u (k W  W  ' ) »¼

)W

)12 º»

»

)22 »¼

* W [0 " 0 1]c 

then a state space representation for the system sampled over W is

x  (k  1)W

ª) « 11 « «) ¬ 21

z (k ) [ z1 (k )

z2 (k )]c 

)22 and z2(k) are scalars. The switching surface is

) N x (k W)  *u(k W)

g c M 1 z (k ) 0 s (k ) [ g c 1] [ z1 (k ) z2 (k )]c s (k )

y (k )

Cx(k )

where

*

[) N 1* " * ]

() N  * KC ) x ( k W )

() W  *KC ) x(k W)

(11)

The problem has now taken the form of static output feedback problem. Equation (11) suggest that an output injection matrix G be found such that

U() N  GC )  1

(16)

From (15) and (16) we get (10)

Applying periodic output feedback for the '  system in (9), i.e., Ky(kW) is substituted for u(kW), the closed loop system becomes

x ( k W  W)

0

(12)

z1 (k  1) z2 ( k )

)11 z1 (k )  )12 z2 (k )

 g c z1 (k )

(17)

Then g is obtained by assigning the eigenvalues of ()11  )12 g c) in a desired location of the unit circle.

IV. PERIODIC OUTPUT FEEDBACK BASED DSMC 4.1 Determination of transformation matrix T The transformation matrix T which relates the

S. Janardhanan and B. Bandyopadhyay: A New Algorithm for Discrete-Time Sliding Mode Control

autonomous state vectors xˆ and x, as described in Eq. (3) can be computed in the following manner. Using the state equation of the autonomous adjoint system in Eq. (2) and Eq. (3), we can obtain the relationship

xˆ (k  1)

) cW xˆ (k )

x(k  1) T 1 ) cW Tx(k )

(18)

Now, comparing Eq. (18) with the state equation of the autonomous system x(k  1) ) W x(k ) T, may therefore be obtained by solving the equation

)W T

1

1

T ) cW

uˆ (k )

F1 xˆ (k )  J1 ˜ sgn( s(k ))

313

(23)

where

F1

( gˆ cC c) 1 ( gˆ c) cW  gˆ cI  qWgˆ c)

J1

( gˆ cC c) 1 eW

(24)

The closed-loop dynamics of the adjoint system would then be governed by the state equation

xˆ (k  1)

) cW xˆ (k )  C c  F1 xˆ (k )  J1 ˜ sgn( s(k ))

xˆ (k  1)

() cW  C cF1 ) xˆ (k )  C c( gˆ c C c) 1 eW ˜ sgn( s(k ))

(19)

(25)

It should be noted that the solution of Eq. (19) is not unique. Therefore, the designer has the freedom to choose T that has additional desirable properties. In order to enable the proposed control law to satisfy the reaching condition, T1 is therefore chosen to be the one that satisfies the equation.

4.3 Translation of adjoint system control input into DSMC for original system

g c( I  ) W )( I  Qc)

g c qW( I  Qc)

(20)

the original system (1). Then, the closed-loop system dynamics may be represented as

) W x(k )  *  Ky  J 2 ˜ sgn( s ( k ))

x(k  1)

where

Q

Assume that a periodic output feedback based control input of the form u Ky (k )  J 2 sgn( s (k )) is applied to

) W x(k )  *KCx(k )

1

C c( g c T 1 C c) g c T 1

 *[1 1 " 1]cJ 2 ˜ sgn( s (k ))

This equivalence is necessary for the proposed control law to satisfy the reaching law condition (7). The necessity of the equality is evident in the proof presented later in the paper.

In order to design the discrete-time SMC for the adjoint system, the sliding manifold s(k) for the original system (1) is converted into an equivalent sliding manifold gˆ c xˆ (k ) for the adjoint system (4) as

s (k )

g c x(k )

gc T

xˆ (k )

(21)

where gˆ c is computed using the relation

gˆ c

1

gc T 

(27)

Making use of Eq. (3), Eq. (25) can be expressed in terms of x(k) as

Tx(k  1)

() cW  C c F1 ) Tx(k )

C c( gˆ c C c) 1 eW ˜ sgn( s(k ))

(28)

Comparing Eqs. (27) and (28) and equating the switching part of the equations, the switching input J2 can be derived in the following manner

T 1 C c( gˆ c C c) 1 eW

*W J 2

g c *W J 2 (22)

Using the reaching law in Eq. (7) proposed in [6], the control function uˆ (k ) for the adjoint system (4) can be obtained as

() W  *KC ) x(k )  * W J 2 ˜ sgn( s(k ))

T 1 C c( gˆ c C c) 1 eW ˜ sgn( s (k ))

Thus, the sliding manifold of the original system (1) is described equivalently in terms of the adjoint system states xˆ as

gˆ c xˆ (k )

i 0

x(k  1) T 1 () cW  C c F1 )Tx(k )

gˆ c xˆ (k )

s (k )

N 1

() W  *KC ) x(k )  J 2 ¦ ) i * ˜ sgn( s(k )) x(k  1)

4.2 DSMC design for the adjoint system

1

(26)

 g c T 1C c( gˆ c C c) 1 eW ( gˆ c C c)( gˆ c C c) 1 eW

J2

( g c * W ) 1 eW

(29)

This switching input enables the states x of the origi-

Asian Journal of Control, Vol. 7, No. 3, September 2005

314

nal system to move towards the desired switching manifold s(k) = 0. Once the system states are within a band around the switching manifold the control input enables it to stay within the band. However, if the system states are exactly on the manifold, the proportional controller should maintain them on the manifold. Thus, the proportional control Ky is calculated from the control input of the adjoint system in a way that the closed-loop eigenvalues of both the systems are identical when there is no switching input. The closed loop eigenvalues in case of the adjoint system are those of ()Wc + CcF1) and thus would be identical to the eigenvalues of ()Wc + CcF1)c = ()W+ F1cC). Thus, the periodic output feedback gain matrix K, which would realize the same closed-loop eigenspectrum, can be calculated by solving

( ) W  * KC ) *K



F1c C

 [( gˆ c C c) 1( gˆ c ) cW  gˆ cI  qWgˆ c)]cC  [C c( gˆ c C c) 1( gˆ c) cW  gˆ cI  qWgˆ c)]c

c  ª¬C c( gˆ c C c) 1  gˆ c() W  I )c  qWgˆ c º¼





 ª C c( gˆ c C c) 1gˆ c  () W  I )c  qW º ¬ ¼ 1 ˆ c c c ˆ c c   () W  I )  qW [C ( g C ) g ] F1c C

c

  () W  I )  qW Qc

(36)

Substituting the value of F1cC from Eq. (36) in Eq. (35), and making use of Eq. (20)

() W  F1c C )

F1c 

(30)

 g c( I  ) W ) x(k )

[( k )  s ( k )

* 1 F1c 

 g c  () W  I )  q W Q c x ( k )  g c( I  ) W )( I  Qc) x(k )  g c qWQ c x(k )  g c qWx(k )

(31)

To summarize, the proposed periodic output feedback based control law has the form

u(k )

(35)

Now, expanding F1cC,

If the number of gain changes N, is chosen to be equal to the order of the system, then K can be computed from Eq. (30) as

K



 g c ( I  ) W )  F1c C x(k )

[( k )  s ( k )

Ky (k )  J 2  sgn( s (k ))

(32)

where J2 and K are obtained from Eqs. (29) and (31). It can be proved that the control law given in Eq. (32) results in DSM in the following manner.

s (k  1)  eW ˜ sgn  s (k )  s (k )

 qWs ( k )

(37)

which is the reaching law to be satisfied for discrete time quasi-sliding mode control. Hence, it is proved that the periodic output feedback based control described earlier results in quasi-sliding motion. „

4.4 The control law Proof. The reaching law for discrete-time quasi-sliding mode control given in Eq. (7) can be written as

s (k  1)  eW ˜ sgn( s (k ))

s (k )  qWs (k )

[(k )

(33)

Consider the LHS of the reaching law in Eq. (33), if the periodic output feedback based input, as described in Eqs. (30)-(32) is applied, then

[( k )

s (k  1)  eW ˜ sgn( s (k )) g c  ) W x(k )  * W u(k )  eW ˜ sgn( s (k ))

In the control law described in Eqs. (30)-(32), the switching input J2 ˜ sgn(s(k)) requires a measure of the state vector x(k). This can be calculated in the following manner. For k t n  1: The dynamics of the actual system gives

y (k  n  1)

Cx(k  n  1)

y (k  n  2)

Cx(k  n  2) C ) W x(k  n  1)

g c() W  *KC ) x(k )

 C *u(k  n  1)

 ¨§ I  ( g c * W )( g c * W ) 1 ¸· eW ˜ sgn( s (k )) ©

¹

(34)

#

Using the relation between K and F1 in Eq. (30) and

y (k )

g c() W  *KC ) x(k ) (34),

[( k )

Cx(k ) C ) nW 1 x(k  n  1)

s (k  1)  eW ˜ sgn( s (k ))

n2

 C ¦ ) Wn  2 i *u(k  n  1  i )

g c( I  ) W  F1c C  I ) x(k )



i 0



g cx(k )  g c ( I  ) W )  F1c C x(k )

Thus,

S. Janardhanan and B. Bandyopadhyay: A New Algorithm for Discrete-Time Sliding Mode Control

Yk

when k – n + 1 + i < 0 to be zero. This, in effect would mean that the system inputs and outputs are assumed to be zero for time instants t < 0. This evaluation, though not exact, would satisfy the relation y(k) = C ˜ f(u, y, k). Thus, the state-evaluation error decreases at every instant until it becomes zero at k = n  1. Thus, the periodic output feedback based discrete-time quasi-sliding mode control law (POFSMC), based on infrequent output observations is of the form

Cx(k  n  1)  G k 

where

Yk

C

Gk

ª y (k  n  1) º « » « y (k  n  2) »  « » # « » y (k ) »¼ «¬ ª C º» « « » « C) » W » « « » « # » « » « n 1 » ¬« C ) W ¼»

u(k )

0 ª º « » C *u(k  n  1) « » « » # « » « n  2 n  2i » «C ¦ ) W * u ( k  n  1  i ) » ¬ i 0 ¼

x(k  n  1)

C1 (Yk  G k ) { x (k  n  1)

The matrix C is invertible because ()W, C) is assumed to be observable. Here, x (k  n  1) is the reassessed approximation of the states at time-instant k – n + 1. Then the approximate state at kth time instant can now be calculated as n2

x(k )

) Wn 1 x (k  n  1)  ¦ ) Wn  2 i *u(k  n  1  i ) i 0

f (u y k )

) Wn 1 C1 (Yk

 Gk )

n2

 ¦ ) Wn  2 i *u(k  n  1  i ) i 0

s (k )

g c ) Wn 1 C1 (Yk  Gk )

gc

315

n2

¦ ) Wn  2i *u(k  n  1  i)

(38)

i 0

In this manner of evaluation, the error in assumed initial state does not propagate as the states are reassessed at every iteration using only the past outputs and inputs of the system. Moreover, since the state evaluation is to be used only to evaluate the signum function, error propagation is reduced further. It should also be noted that this state evaluated is not being used for the proportional part of the control. For k < n  1: During the time-instants when 0 d k < n  1, there is not enough information about past output and past input available for the states to be evaluated exactly. Hence, for these instants, the evaluation is done partially by assuming the control inputs and outputs in Eq. (38) for time-instants i

­ Ky ( k )  J 2 ® ¯Ky ( k )  J 2

for s (k )  0 for s (k ) ! 0

(39)

where s(k) is computed from the past control inputs and system outputs using Eq. (38). It should be noted here that for k < n  1, the state information, though converging to the actual value, is not exact. Hence, the value of the sliding function s(k) is not known exactly. Hence, quasi-sliding mode is not assured for the time instants k < n  1. However, for time instants k t n  1, there is enough output information to compute the system state accurately, thus enabling the accurate calculation of s(k). Therefore, quasi-sliding mode can be assured after k > n  2 by using the proof in 4.3. We can now summarize the design procedure into the following POF SMC design algorithm. Given: )W, *W, C 1. Choose the number of subintervals N as equal to the order of the system and obtain ' = W/N. 2. Construct the system (), *, C) of sampling period ' from the given system and the lifted input matrix * using Eq. (10). 3. Determine the switching surface parameter g using Eq. (17). 4. Compute the transformation T that relates the states x and xˆ by solving Eqs. (19) and (20). 5. From Eq. (24) obtain a state feedback gain F1 and J1 such that closed loop adjoint system ()Wc + Cc F1) does not have any poles at the origin and ensures DSM of the adjoint system states xˆ (k ) about the equivalent switching manifold. 6. Obtain the periodic output feedback gain sequence K using Eq. (31) and the switching input J2 using Eq. (29). 7. Evaluate the switching function as s(k) at each Winterval using Eq. (38). 8. Test the sign of the switching function s(k) and apply the control input profile u(k) according to Eq. (39).

4.5 Effect of sensor noise In the FOSSMC technique proposed in [12], the system output is measured N times during each W- interval. But, in the proposed POFSMC technique, there is only one measurement of the system output during the same period. Thus, in case of sensor noise entering into the system dur-

Asian Journal of Control, Vol. 7, No. 3, September 2005

316

ing one W-interval, the proposed technique uses the corrupted output only once whereas FOSSMC would use the corrupt output N times to generate the control input. Hence, it may be argued that POFSMC is relatively less responsive to sensor noise as compared to FOSSMC.

From Eq. (24) we get

[00048 00048]

F1

Substituting * and F1 in Eq. (31), we get

[20855 19837]c

Ȁ

V. ILLUSTRATIVE EXAMPLE The example cited in [5] is used here to show how this approach results in quasi-sliding mode control. An added disturbance is considered here to analyse and compare the robustness of the proposed algorithm with the fast output sampling sliding mode control algorithm proposed in [12]. Consider the continuous-time system representation.

x

Ax  bu  d 

y

Cx

e

01

e

W

001 J 2

01051

The POF SMC in this case is

u( k )

­ Ky ( k )  J 2 ® ¯ Ky ( k )  J 2

f (u y k )

for g c f (u y k )0 for g c f (u y k )!0

) nW 1 C1 (Yk  G k ) n2

 ¦ )nW  2 i *u(k n1i ) i 0

with

A

ª0 1º « » b ¬ 1 2 ¼

C

[1 0]

ª0º « » ¬1 ¼

For a sampling time W = 0.1 sec, the discrete-time system representation is

x(k  1)

) W x(k )  * W u (k )  d d (k )

Cx(k )

y (k ) with

)W

Let

ª 09953 00905 º « »  *W ¬ 00905 08144 ¼

d d (k )

ª 00047 º « » ¬ 00905 ¼

sin(02k )e 02 k 

Let

q 1 N

2 '

x1 (0) 1 x2 (0)

The simulation results with E = 0.1 and with a sensor bias of 0.01 and an unmodeled disturbance are shown in Fig. (1). The results are satisfactory. Figure 1(a) shows the comparative phase plot of the actual system states between proposed algorithm and FOSSMC. The phase portrait shows that the trajectory has a quasi-sliding mode motion which is better than that obtained from FOSSMC. The control function is plotted in Fig. 1(b). The switching behaviour of the control is evident here. Figure 1(c) shows the response of the states. It can be seen that magnitude of the states are lesser in POFSMC. The plot of the switching function shown in Fig. 1(d) decreases towards zero from the initial value and stays within a band of decreasing amplitude. This results in a quasi-sliding motion. It is evident in the plot of the switching function that on the face of the sensor bias and unmodelled system dynamics, the proposed technique exhibits better quasi-sliding mode behaviour as compared to FOSSMC.

01sec 0

and the switching surface is designed as

s (k )

x1 (k )  x2 (k )

0

Using Eq. (10), the lifted input matrix * is calculated to be

*

ª 00035 00012 º « » ¬00429 00476 ¼

The transformation matrix T is calculated to be

T

ª1 2 º « » ¬2 3¼

Fig. 1. Response of the system with EW = 0.01. (a) Phase plot (b) Control input (c) Plant states (d) Switching surface

S. Janardhanan and B. Bandyopadhyay: A New Algorithm for Discrete-Time Sliding Mode Control

VI. CONCLUSION A new control technique is developed for discretetime quasi-sliding mode control combining the reaching law approach and the periodic output feedback technique. The key advantage of this approach is that it does not require the states of the system for feedback as only the output of the system which is available is used to generate the control action. Another advantage of the proposed algorithm would be its static structure, making it a simpler output feedback quasi-sliding mode controller as compared to dynamic output feedback quasi-sliding mode control techniques. Moreover, since the output observations are infrequent, there is less probability of sensor noise deteriorating the system dynamics. By proper choice of the parameters in the reaching law, the dynamic response of the closed loop system can be improved. Thus by using the reaching law approach, the quasi-sliding motion is guaranteed. The use of output feedback as opposed to state feedback reduces the cost of implementation and makes the design more practical. A strategy for the evaluation of system states using infrequent output observations has also been investigated here. This enables the control algorithm to compute the switching function with information of past outputs and control signal alone. This control technique works for higher order SISO LTI systems. The simulation results show the effectiveness of the proposed method. The extension of this new control methodology to multi-input systems is under investigation.

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8. Diong, B.M., “On the Invariance of Output Feedback VSCS,” Proc. IEEE Conf. Decis. Contr., San Antonio, TX, U.S.A., pp. 412-413 (1993). 9. Bag, S.K., S.K. Spurgeon, and C. Edwards, “Output Feedback Sliding Mode Design for Linear Uncertain Systems,” Proc. IEE Contr. Theory Appl., Vol. 144, No. 3, pp. 209-216 (1997). 10. Syrmos, V.L., C.T. Abdallah, P. Dorato, and K. Grigoriadis, “Static Output Feedback  A Survey,” Automatica, Vol. 33, Issue 2, pp. 125-137 (1997) 11. Chammas, A.B. and C.T. Leondes, “Pole Assignment by Piecewise Constant Output Feedback,” Int. J. Contr., Vol. 29, No. 1, pp. 31-38 (1979). 12. Saaj, M.C., B. Bandyopadhyay, and H. Unbehauen, “A New Algorithm for Discrete-Time Sliding-Mode Control Using Fast Output Sampling Feedback,” IEEE Trans. Ind. Electron., Vol. 49, No. 3, pp. 518-523 (2002). 13. Werner, H., “Multimodel Robust Control by Fast Output Sampling  An LMI Approach,” Automatica, Vol. 34, No. 12, pp. 1625-1630 (1998). 14. Werner, H. and K. Furuta, “Simultaneous Stabilization by Piecewise Constant Periodic Output Feedback,” Contr. Theory Adv. Tech., Part 4, Vol. 10, No. 4, pp. 1763-1775 (1995).

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S. Janardhanan was born in Hyderabad, India in 1978. He obtained his Bachelor of Engineering degree in Instrumentation and Control Engineering from the University of Madras in 2000. Presently he is pursuing his Ph.D. in Systems and Control Engineering in Indian Institute of Technology Bombay after graduating from the same department. S. Janardhanan is a Student Member of the Institute of Electrical and Electronics Engineers (IEEE). His research interests include discrete-time control systems, sliding mode control, multirate sampling and output feedback control.

B. Bandyopadhyay was born in Birbhum village, West Bengal, India, in 1956. He received Bachelor’s degree in Electronics and Communication Engineering from the University of Calcutta, Calcutta, India, and Ph.D. in Electrical Engineering from the Indian Institute of Technology, Delhi, India in 1978 and 1986, respectively. In 1987, he joined the Interdisciplinary Programme in Systems and Control Engineering, Indian Institute of Technology Bombay, India, as a faculty member, where he is currently a Professor and the Convener. He visited the Center for System Engineering and Applied Mechanics, Universite Catholique de Louvain, Louvainla-Neuve, Belgium, in 1993. In 1996, he was with the Lehrstuhl fur Elecktrische Steuerung und Regelung, Ruhr Universitat Bochum, Bochum, Germany, as an Alexander von Humboldt Fellow. He revisited the Control Engineering Laboratory of Ruhr University of Bochum during May-July 2000. He has authored and coauthored more than 150 journal articles and conference papers. His research interests include the areas of large-scale systems, model reduction, reactor control, smart structures, periodic output feedback control, fast output feedback control and sliding mode control. Prof. Bandyopadhyay served as Co-Chairman of the International Organization Committee and as Chairman of the Local Arrangements Committee for the IEEE International Conference in Industrial Technology, held in Goa, India, in January 2000. His biography was published in Marquis’ Who’s Who in the World in 1997. Prof. B. Bandyopadhyay has been nominated as one of the General Chairmen of IEEE ICIT Conference to be held in Mumbai, India in December 2006.