A Variable Structure Approach to the Robust Stabilization of a Class of Discrete-Time Nonlinear Systems

Copyright co IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998

A VARIABLE STRUCTURE APPROACH TO THE ROBUST STABILIZATION OF A CLASS OF DISCRETE-TIME NONLINEAR SYSTEMS. M.L.Corradini *

* Dipartimento di Eleftronica ed Automatica, Universitd di Ancona, Via Brecce

Bianche, 60131 Ancona, Italy

Abstract. Robust stabilization of a class of discrete-time nonlinear systems, in presence of uncertainties and/or external disturbances, is here addressed. A discrete-time Variable Structure approach is proposed, and a design procedure, based on backstepping, is given. A constructive proof is provided, and simulation results are reported. Copyright © 1998 IFAC

Keywords. Discrete-time Variable Structure Control, Uncertain Systems, Discrete-time Nonlinear Systems, Robust Control.

1. INTRODUCTION

gac, 1992): a digital version of the controller is derived in this paper by simple discretization, but the approximation is of course acceptable only for low frequency dynamics.

Variable Structure Control (VSC) is well established as a general design approach for robust control systems (Utkin, 1984) (DeCarlo et al., 1988) (Utkin, 1992) (Hung et al., 1993). This method is particularly attractive for continuoustime nonlinear systems, but its implementation on digital computers presents well known difficulties (Su et al., 1994), which severely limit the practical application of the technique.

In this note, a robust VSC algorithm is proposed for a class of discrete-time nonlinear systems, linear in control, in presence of uncertainties and/or external disturbances. As it will be shown in the following, a constructive design procedure, based on backstepping, can be given to achieve stabilization.

To overcome this problem, the discrete-time counterpart of sliding mode, known as quasisliding regime, has begun to be addressed in recent years. A number of discrete-time VSC approaches for uncertain systems has been recently proposed in literature (Furuta, 1990) (Furuta, 1993) (Wang et al., 1994) (Gao et al., 1995) (Corradini and Orlando, 1995) (Corradini and Orlando, 1997 a), but in all these studies a linear plant model is considered. Sliding more control with perturbation estimation for a wide class of continuous-time nonlinear systems has been considered in (Elmali and 01-

As well known, the combination of sliding mode control and backstepping has been widely studied in the framework of continuous-time systems. Translating the design approach to the discrete-time case, the resulting control law leads to the definition of a switching region (or sector), whose width depends on the variation range of the uncertainty as well as on the state variables. As already proposed in (Corradini and Orlando, 1997b), the approximate imposition of the sliding condition inside the sector has been performed in this paper using the

225

= Tc Z2(k)+JI(z(k),u(k),Tc)

concept of time delay control (TDC) (Su et al., 1993) (Chan, 1994) (Tesfaye and Tomizuka, 1995), under the assumption that uncertainties are slow with respect to system dynamics. It is worth noting that this approximation is anyway performed only inside the switching region.

= TcZ3(k)

+ h(z(k) , u(k), Tc)

zn-l(k + 1) = Tczn(k)+ + fn-l (z(k), u(k) , Tc) zn(k + 1) = Tc!n(z(k), u(k)) which corresponds to (2.1) if the functions Ji(z(k),u(k),Tc), i = l..n - 1 are considered as the uncertain terms di(x(k), u(k) , k), i = l..n - 1 of (2.1).

2. PRELIMINARIES

Remark 3 The class of systems described by (2.1) may include systems not state feedback linearizable.

Let's consider a SISO discrete-time output-affine nonlinear plant described by the following equations:

To prove tliis\assert, let us consider the example by (Cruz aM Nijmeijer, 1996). It is easy to verify that , after rearranging the second and third row, that system corresponds to (2.1) if one sets:

xl(k+l) = x2(k)+d 1 (x(k) , u(k),8, k) x2(k + 1) = x3(k) + d2(X(k) , u(k) , 8, k) xn(k + 1) = f(x(k)) + b(x(k))u(k)+ +dn(x(k) , u(k) , 8, k)

d 1 (x(k) , u(k) , 8, k) = ~alx2(k)X3(k) d 2(x(k),u(k) , 8, k) = xl(k)X3(k) - x2(k)u(k) d3(X(k) , u(k) , 8, k) = ~a2u(k)

(2.1)

where x( k) is the n-dimensional state vector, u(k) E rn. is the system input, f(x(k)) is a given nonlinear function , and the n functions di(x(k) , u(k) , 8, k) ,i = 1, n , 8 being the vector of uncertain parameters, are unknown terms representing possible external disturbances, deviations from the nominal system (e.g. parametric variations), and eventually unmodelled dynamics.

The following Assumption is introduced: Assumption 1 It is assumed that the uncertainties in the plant are considerably slow with respect to the system dynamics, i.e. di(x(k), u(k), 8, k) ~ d i (x(k-l) , u(k-l), 8, k-l)

The Assumption 1 allows to obtain an estimate of the system perturbations at time (k - 1) by delaying the time k by a sampling interval, assumed small w.r .t . the uncertainties dynamics. This approach is known as Time Delay Control (TDC): though widely employed for continuous-time systems, it has been applied to the control of discretized systems in (Su et al., 1993) (Chan, 1994) (Tesfaye and Tomizuka, 1995) (Chang and Lee, 1996).

Remark 1 Each unknown function di(x(k) ,u(k), 8, k) may depend on the whole state vector, and not only on its components up to the i-th. In other words no triangularity assumption is made. Remark 2 A class of continuous-time discretized systems can be described by equation (2.1). In fact, given a system in the canonical Brunowsky form:

3. MAIN RESULT

Zn-l

zn

Theorem 1 Assume that for system (2.1):

= Zn = g(z) + h(z)u

i) Assumption 1 holds; ii) The vectorial function F(x(k),u(k) , k) = col(x"l(k), x3(k), .. ,J(x(k)) + b(x(k))u(k)) is such that F(O, 0, k) = 0; iii) There exist n vectorial functions pi(x(k), u(k), k) such that

its integration by parts from kTc to (k + I)Tc , under the usual hypothesis that the variation of the term h(z)u is neglectable within a sampling period, provides:

226

Idi(x(k), u(k) , 0, k)1 $ pi(x(k), u(k), k)

derived shifting one time instant backward the following expression:

(i = l..n) and

Pi (Xl (k), x2(k) , .. , xn(k) , u(k), k) = 0 if xi(k) = 0 (i = l..n)

thus obtaining

dl(x(k) , u(k) ,O, k) ~ ::= dl(x(k - I), u(k - 1),0, k - 1) = = sl(k) - x2(k - 1)

iv) The positive vectorial functions pi(x(k) , u( k) , k) are bounded on any bounded domain V E IRn , i.e. there exist n bounding functions Mi(X(k), u(k) , k) such that

pi(x(k), u(k) , k) : IRn ~ ~ [0, Mi(X(k) , u(k), k)]

The condition sl(k guaranteed posing:

Note that this approximation is used only inside the sector of width Pl(x(k) , u(k),k). Collecting together (3.5) and (3.8) one gets:

Proof. A backstepping constructive procedure will be given in the following. Introduce the quantities:

(i

= 2.. n)

II(lsl(k)l- Pl(x(k) , u(k), k)) WI (k) _ if IsICk)1 > PI(x(k) , u(k), k) - { -(sl(k) - x2(k -1)) if ISI(k)1 $ PI(x(k) , u(k), k)

(3 .1)

(3.9)

where the n - 1 functions wiCk) are to be determined. Step 1. Suppose that s2(k) = 0, and consider the following Lyapunov function:

1 2 VI = 2Sl(k)

Step 2. In order to guarantee that s2(k) = 0, consider now the following Lyapunov function under the assumption that s3(k) = 0: 1

(3.2)

V2 = 2S2(k)

The input Wl (k) which guarantees the achievement of a quasi-sliding motion on the hyperplane SI (k + 1) = 0 can be determined defining

6.s i (k + 1) = siCk (i = l..n)

wl(k))

2

(3 .10)

+ d2(X(k), u(k), 0, k)-

It can be easily verified that the term wl(k+I), although unknown at the time instant k, satisfies the following inequality both inside and outside the sector of width Pl(x(k + I), u(k + I) , k + 1):

(3 .3)

1

1 = 2(X2(k) -

6.s 2(k + 1) = w2(k) -s2(k) - wl(k + 1)

+ 1) - siCk)

+ 1) < -2(6.s l (k + 1))2

2

In this case one has:

and imposing the condition (Furuta, 1990):

sl(k)6.s l (k

::= 0 is approximately

wl(k) = -dl(x(k -I) , u(k -1) , 0, k - 1) = = -(sl(k) - x2(k - 1)) (3.8)

(i = l..n)

Then there exists a stabilizing discrete-time sliding-mode control law which can be recursively built through a backstepping approach.

sl(k) = xl(k) siCk) = xi(k) - Wi-l(k)

+ 1)

(3.7)

(3.4)

Noting that

IWl(k + 1)1 < 2Ix2(k)1 + Pl(x(k),u(k),k)+ +PI (x(k + I), u(k + I), k + 1) (3.11)

6.s l (k+l) = wl(k)+dl(x(k),u(k),O,k)-Sl(k) inequality (3.4) yields:

Due to the condition iii), one can always find a function pill (x(k), u(k), k), depending on x(k), u(k) bounding Pl(x(k + I), u(k + I), k + 1):

wI(k) = 'Yl(ISl(k)l- Pl(x(k),u(k) , k)) if ISl(k)1 > Pl(x(k),u(k),k) (3 .5)

PI(x(k+I), u(k+I), k+I) $ pil)(x(k), u(k), k)

where 11 E (-1, 1) is a design constant.

(3.12)

Let us consider now the condition ISl(k)1 ~ Pl(x(k), u(k), k) . According to (3.5), the condition ISl(k+l)1 < ISI(k)1 cannot be imposed exactly inside the sector of width PI (x( k), u( k), k). Under the Assumption I, an estimate of the unknown quantity dl(x(k),u(k),O,k) can be

and (3.11) can be further overbounded as follows:

IWl(k + 1)1 < (2Ix2(k)1 +pP) (x(k) , u(k), k))

227

+ Pl(x(k),u(k),k)+ (3.13)

If one defines: 6(x(k), u(k) , k) ~f 2Ix2(k)I+Pl(X(k) , u(k) , k)+ +Al) (x(k) , u(k), k) + P2(x(k) , u(k) , k) (3.18)

the condition 1

s2(k).6.s 2(k + 1) = -2(.6.s 2(k + 1))2 < 0

Note that the approximation (3.14) is used only inside the sector of width 6(x(k), u(k), k).

gives:

Collecting together (3.13) and (3.17) one finally gets:

(w2(k)f + 2W2(k)[d2(X(k), u(k), e, k)Wl (k + 1)] + (d 2(x(k), u(k) , e, k) -Wl (k+ 1) )2_ (s2(k)f < (w2(k))2 +2W2(k)6 (x(k), u(k), k)+ +({l(x(k), u(k), k))2 - (s2(k))2 < 0 (3.14)

w2(k) = 1'2(ls2(k)l- 6(x(k) , u(k) , k)) _ if IS2(k)1 > 6(x(k) , u(k) , k) - { -(s2(k) ~ x3(k -1) + wl(k)) + wl(k) if IS2(k)h,::; 6(x(k), u(k) , k) (3.19)

and the form of the auxiliary control input w2(k) ensuring that a quasi-sliding motion is achieved on the surface s2(k + 1) = 0 is:

w2(k) = 1'2(ls2(k)l- 6(x(k) , u(k) , k)) if IS2(k)1 > 6(x(k), u(k) , k)

The i-th step is omitted for brevity. However, by iterating the above procedure up to the (n1)-th component, the (n - 1) functions wi(k) can be found. The last step provides the expression for the actual control input u(k) appearing in (2.1) . [::,

(3 .15)

where 1'2 E (-1 , 1) is a design constant. The condition IS2(k)1 ::; 6(x(k) , u(k), k) remains to be considered. Also in this case, the condition IS2(k + 1)1 < IS2(k)1 cannot be imposed exactly inside the sector of width {I (x( k) ,u(k),k) . However, the Assumption 1 can be used again to obtain an estimate of the unknown quantity d2(x(k) , u(k) , e, k) through a time shifting:

d2(x(k), u(k) , e, k) ~ ~ d2(x(k - 1) , u(k - 1) , e, k - 1) = s2(k) - x3(k - 1) + wl(k)

4. A SIMULATION EXAMPLE In this section the controller proposed in Theorem 1 has been tested by simulation using the simple example of a pendulum:

[!~] =

= (3 .16)

The discretization of (4.1) with a sampling interval Tc, integrating (4.1) by parts, gives:

An approximation Wl (k) of the unknown term Wl (k + 1) appearing in the expression (3.17) can be computed using the fact that d1(x(k) ,u(k) ,e,k) ~ dl(x(k -1),u(k -1),e,k -1). In fact an estimation Xl (k) of the state variable xl(k + 1) is given by:

xl(k

+ 1)

x2(k

+ 1)

1

- x2(k -1)

= TcX2(k)

+ xl(k) =

= TcX2(k)+d l (x(k) ,u(k) , e,k) + Tc[-n 2sin(Xl(k))-b1X2(k)] + Tcetu(k) =

= x2(k)

= f(x(k))

+ bu(k) (4.2)

T;

has been newhere a term proportional to glected in the first row. If Tc is chosen sufficiently small, the Assumption 1 holds. Moreover the system (4.2) satisfies the requirements of Theorem 1. In fact, the 'disturbance' term d1(x(k),u(k) , e, k) is bounded by :

Therefore an approximation Pl(x(k),u(k),k) of the bounding function Pl(x(k + 1),u(k + 1), k + 1) can be easily obtained, and the term Wl (k + 1) can be approximated as follows :

wl(k

[~] u

where u is the control input, n2 = gfl , et = 1fml 2 , 9 is the gravity acceleration, l and m are the pendulum length and mass respectively, and bl is the friction coefficient.

w2(k) = -d 2(x(k -1) , u(k -1),e, k -1)+ +wl(k + 1) = -(s2(k) - x3(k - 1) + wl(k))+ +wl(k + 1) (3.17)

+ sl(k)

+

(4.1)

Therefore, the form of the auxiliary control input W2 (k) ensuring the approximate condition s2(k + 1) ~ 0 is:

xl(k) = x2(k)

[ -n 2sin(::) - blX2]

+ 1) ~ wl(k) =

228

Idl(x(k), u(k) , B, k)1 < kdxl(k)1 ~f

where:

~f Pl(x(k) , u(k), k) being kl 2: 1, and f(x(k)) is such that f(O) =

Simulation results are shown in Figs.1-3, where the continuous-time state variables Xl, X2, the control input u and the variables sl(k),S2(k) are plotted. The following physical values have been chosen m = 1 kg , l = 1 m, bl = 0.001 S-l while the initial conditions of the system were x(O) = [0.5 of. A sampling interval of O.ls has been used. The following values of the design parameters have been considered: 1'1 -0.8; 1'2 = -0.65.

O. An extra disturbance term d2(t) has been considered to be present in the second row of (4.1), having the following form (disturbance a)): d2(t) = AXl(t)X2(t) With reference to the discretized model (4.2) , a possible bound for d2(x(k), u(k), B, k) is: P2(x(k), u(k) , k) The value A

= ATclxl(k)llx2(k)1

= 10 has been chosen. 5. CONCLUSIONS

The synthesis procedure sketched in the proof of Theorem 1 can be easily applied, due to the low number of state variables. As far as the Lyapunov function VI is concerned, a possible choice of Wl (k) is: (k)T _ Wl

c -

Stabilization of a class of discrete-time nonlinear systems, in presence of uncertainties and/or external disturbances, has been addressed in this paper. A discrete-time Variable Structure Control approach has been proposed, and a constructive design procedure, based on backstepping, has been provided. In the resulting control law some approximation has been found necessary inside the sector, where the imposition of the sliding condition has been performed using the concept of time delay control. Finally, to show the feasibily of the technique, some simulations have been presented with reference to the motion control of a pendulum.

'Yl(ISl(k)l- Pl(x(k),u(k),k)) if ISl(k)1 > kllxl(k)1 -(sl(k) - Tc X 2(k -1)) { if ISl(k)1 ~ kllxl(k)1 (4.3)

Next, if the Lyapunov function V2 is considered, it is useful to define the following quantity: 6(x(k),u(k),k) ~f

~f If(x(k))1 + P2(x(k), u(k), k) + 2Tclx2(k)l+ +Pl (x(k), u(k) , k) +

pil ) (x(k), u(k), k)

(4.4) where of course pP) (x(k), u(k), k) = kl [Tclx2(k)1 +Pl(x(k),u(k) , k)] Finally we get the following form of the control input u:

buCk)

=

1'2(iS2(k)l- ~l(x(k), u(k),k)) if IS2(k)1 > 6(x(k),u(k),k) wl(k + 1) - [s2(k) - f(x(k - 1))-buCk -1) + wl(k) + f(x(k))] if IS2(k)1 ~ ~l(x(k), u(k), k) (4.5)

6. REFERENCES

1

Chan, C.Y. (1994). Robust discrete-time sliding mode controllers. Syst. & Contr.Lett. 23, 371-374. Chang, P.H. and J .W. Lee (1996) . A model reference observer for time-delay control and its application to robot trajectory control. IEEETrans. Contr. Syst. Technol. 4, 2-10. Corradini, M.L. and G. Orlando (1995). Discrete variable structure control for nonlinear systems. Proc. of the European Control Conference ECC95 2, 1465-1470. Corradini, M.L. and G. Orlando (1997a). A discrete adaptive variable structure controller

The term Wl (k + 1) in (4.4) , under the Assumption 1, can be approximated by: wl(k

+ 1) ~ wl(k) =

1'1(f3(k) _ if f3(k) - { -(sl(k) if f3(k)

The simulations reported in Figs.1-3 demonstrate the effectiveness of the controller proposed in Theorem 1. Being 'tunable' by the designer, the control law presented in this paper seems able to provide a satisfactory tradeoff between transient shape and control effort.

pil)(x(k),u(k),k))/Tc

> pP)(x(k),u(k),k) TcX2(k -l))/Tc ~ Pll)(x(k),u(k),k) (4.6)

229

for MIMO systems, and its application to an underwater ROV. IEEE Trans. Contr. Syst. Technol. 5(3), 349-359. Corradini, M.L. and G. Orlando (1997b). Variable structure control of discretized continuous systems. To appear in: IEEE Trans. Autom. Contr. Cruz, C. and H. Nijmeijer (1996). Feedback stabilitazion of perturbed nonlinear discrete-time systems. Proc.13th IFAC World Congress F , 25-30. DeCarlo, R.A., S.H. Zak and G.P. Matthews (1988) . Variable structure control of nonlinear multivariable systems: a tutorial. Proc. of the IEEE 76, 212-232. Elmali, H. and N. Olgac (1992). Sliding mode control with perturbation estimation:a new approach. Int.J.Control 56, 923-941. Furuta, K. (1990) . Sliding mode control of a discrete system. Syst. & Contr.Lett. 14, 145-152. Furuta, K. (1993) . VSS-type self-tuning control. IEEE Trans. Industrial Electronics 40,37-44. Gao, W ., Y. Wang and A. Homaifa (1995) . Discrete-time variable structure control systems. IEEE Trans. Industrial Electronics 42,117-122. Hung, J.y', W. Gao and J .C. Hung (1993) . Variable structure control: a survey. IEEE Trans. Industrial Electr. 40, 2-22. Su, W.C., S.V. Drakunov and U. Ozguner (1993). Sliding mode control in discretetime linear systems. Proc. of the 12th IFAC World Congress pp. 297-300. Su, W.C., S.V. Drakunov and U. Ozguner (1994). Implementation of variable structure control for sampled-data systems. Proc. of the Workshop on Robust Control via Variable Structure and Lyapunov Techniques pp. 166-173. Tesfaye, A. and M. Tomizuka (1995). R0bust control of discretized continuos systems using the theory of sliding modes. Int.J.Control 62, 209-226. Utkin, V.I. (1984) . Variable structure systems: present and future. Automation and Remote Contr. 44, 1105-1120. Utkin, V.I. (1992). Sliding modes in control optimization. Springer Verlag. Berlin. Wang, W.J., G.H. Wu and D.C. Yang (1994). Variable structure control design for uncertain discrete-time systems. IEEE Trans. Aut.Cont. 39, 99-102.

Disturbance a): State variables 0 .8

::

.

~:: ~::::::::r:::::::::::I::: : : : : l

=~ ~:sJ

J

0.: I: :~:::~::::::!"''''''''''I''-'''''''''r'''''''''' -0.2 ... -;:- .... -~ - .......... -0.4

-0.6

-+ . ...-....+-.. . .. -+ .... -----..-

r·'~·-·······;·· · ··· · ······~··········· · +·-·· ··· ···· · ~ ............ .

~:-· .. -.. · .. ·f .. -.. --..

-·+. . . . . . i........-· ..-j...... -.... ..

-0.8 L-._--'-_--'-_ _'---_--'-_--.J o 4 6 10

time[sJ

Fig. 1. Disturbance a) : continuous-time evolution of tee state variables.

Disturbance a): Control input

.2

L-._~_~

o

20

__

40

samples

~_~_--.J

60

80

100

Fig. 2. Disturbance a): Control input u.

Disturbance a): Sliding surfaces

0 .6 r---_:_,----:, ----.,~-_:_,---,

::·,::rLFI---_.: E 0.2

.. -\- .... -~ - ..........

-+ .. -----..... +-...........;.. -.... -.....

;: J'l-~~-tl• • • r• • • • o

20

40

samples

60

80

100

Fig. 3. Disturbance a): variables si(k) and s2(k).

230