Proceedings of the 39'h IEEE Conference on Decision and Control Sydney, Australia December, 2000

Adaptive Backstepping Control of Nonlinear Systems with Unmatched Uncertainty Ali J. Koshkouei and Alan S. I. Zinober

Department of Applied Mathematics The University of Sheffield

Sheffield S10 2TN UK

Abstract

This paper considers control design using an adaptive backstepping algorithm for a class of nonlinear continu- ous uncertain processes with disturbances which can be converted to a parametric semi-strict feedback form. Sliding mode control using a combined adaptive back- stepping sliding mode control (SMC) algorithm is also studied. The algorithm follows a systematic procedure for the design of adaptive control laws for the output of observable minimum phase nonlinear systems with matched and unmatched uncertainty. An existing suf- ficient condition for sliding is not needed by the new algorithm.

1. Introduction

The backstepping procedure is a systematic design tech- nique for globally stable and asymptotically adaptive tracking controllers for a class of nonlinear systems. Adaptive backstepping algorithms have been applied to systems which can be transformed into a triangu- lar form, in particular, the parametric pure feedback (PPF) form and the parametric strict feedback (PSF) form [4]. This method has been studied widely in r e cent years [3], [4], [8]-[ll]. When plants include un- certainty with lack of information about the bounds of unknown parameters, adaptive control is more conveni- ent; whilst, if some information about the uncertainty, e.g. bounds, is available, robust control is usually em- ployed. Sliding mode control (SMC) is a robust control method, and backstepping can be considered to be a method of adaptive control. The combination of these methods yields benefits from both approaches.

A systematic design procedure has been proposed to combine adaptive control and SMC for nonlinear sys- tems with relative degree one [13]. The sliding mode backstepping approach has been extended to some classes of nonlinear systems which need not be in the PPF or PSF forms [8]-[11]. A symbolic algebra toolbox

allows straightforward design [7] of dynamical back- stepping control.

The adaptive sliding backstepping control of semi-strict feedback systems (SSF) has been studied by Koshkouei and Zinober [5]. Their method ensures that the error state trajectories move on a sliding hyperplane.

In this paper we develop the backstepping approach for SSF systems with unmatched uncertainty. We also design a controller based upon sliding backstepping mode techniques so that the state trajectories approach a specified hyperplane. We remove the sufficient con- dition (for the existence of the sliding mode) given by Ros-Bolivar and Zinober [7]-[9]. We consider nonlinear systems which can be converted to a particular form, the so-called parametric semi-strid feedback form. We extend the classical backstepping method to this class of systems in Section 2 to achieve the output tracking of a dynamical reference signal. The sliding mode con- trol design based upon the backstepping techniques is presented in Section 3. We consider an example to il- lustrate the results in Section 4 with some conclusions in Section 5.

2. Adaptive Backstepping Control

Consider the semi-strict feedback form (SSF) [5], [6], P I ii = + 93z1,z2, . . . , z i p + vi(z,w,t) ,

in = f(4 + d Z > U + Co;(z>e + rink, 20, t> (1) l < i < n - 1

y = z1

where z = [XI 2 2 . . . z,IT is the state, y the output, U the scalar control and pi(z1,. . . ,zj) E Rp, i = 1 , . . . , n, are known functions which are assumed to be suffi- ciently smooth. 0 E RP is the vector of constant un- known parameters and qi(z, w, t ) , i = 1 , . . . , n, are un- known nonlinear scalar functions including all the dis- turbances. w is an uncertain timevarying parameter.

0-7803-6638-7/00$10.00 0 2000 IEEE 4765

Assumption 1 The finctions 7)i(x,w, t ) , i = 1,. . . , n are bounded by known positive functions hi(z1, . . . xi) E P, i.e.

(2) Iqi(x, w,t)l < hi(xl,. ..xi), i = 1,. . . , n

The output y should track a specified bounded reference signal yr(t) with bounded derivatives up to n-th order.

If a plant has unmatched uncertainty, the system may be stabilized via state feedback control [l]. Some tech- niques have been proposed for the case of plants con- taining unmatched uncertainty [2]. The plant may con- tain unmodelled terms and unmeasurable external dis- turbances, bounded by known functions.

First, a classical backstepping method will be extended to this class of systems to achieve the output tracking of a dynamical reference signal. The sliding mode control design based upon the backstepping techniques is then presented in Section 3.

2.1. Backstepping Algorithm We first follow a backstepping approach which differs from Koshkouei and Zinober [5], [SI. The functions that compensate the system disturbances, are continuous.

Step 1. Define the error variable z1 = z1 - gr then T

21 = 22 + 91 (z1)e + w (2 ,909 t ) - air

21 = xa + UT8 + 7 ) l ( x , w , t ) - & + UT6

(3)

(4)

with wl(z1) = cpl(zl) and 6 = 8 - e where 8(t) is an estimate of the unknown parameter 8.

From (3)

Consider the stabiliation of the subsystem (3) and the Lyapunov function

(5) 1 2 2

vi ( z l , 8) = -2; + h - 1 6

where I' is a positive definite matrix. The derivative VI is

i.; (21,e) = 81 (zZ + w ~ e + q1 (z, w, t ) - e r )

+8Tr-1 (rul.Zl - e ) (6)

Define 71 = I'wlzl. Let

PI = al(xl,B,t) + e r

with c1, a and E positive numbers. Define the error variable

22 = 22 - a1 ( ~ 1 , 8 , t ) -

Then

Step k (1 < k < n - 1). error variable Zk is

The time derivative of the

where

with ck > 0. variable z k is

Then the time derivative of the error

i k = -2k-1 - C k z k f zk+l. + UT6 + <k - c k z k

The time derivative of v k is

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since and k

t h$Jo wn < Vn(0) + €(n + 1)/(2a) < 00

Step n. Define Since Wn is a uniformly continuous function, according to the Barbalat Lemma, limt,,Wn = 0. This im- plies that limt,w zj = 0, i = 1,2, . . . , n. Particularly, limt+, - 'Up) =

Remark 1 When there is no unknown parameter 8 in the system equation, one can attain the tracking per- formance directly. Then (21) becomes

(n) zn = zn - pn-l = zn - an-1 - yr

with an-l obtained from (12) for IC = n. Then the time derivative of the error variable zn is

n-1 aan-1

in = f(z> + g(z)u + wT(z7 t>d - F z i + 1 i=l

-- aan-l: e - aan-l - + W , T ( z , t ) B + & --U?) (16) Vn < -CV, + ~ ( n + 1)e-"~/2 (23) a t

with 0 < c < g!sm ci and c # a. Then ad

where wn(z,8) is defined in (11) for k = n. Extend the Lyapunov function to be o<vn<---- (n + (e-ct - e-at) + Vn(O)eTCt (24)

limt+,V, = 0 implies that limt,, zi = 0, i = 1,2 , . . . , n. This guarantees limt,, (y - yr) = 0.

2(a - c)

(17) 1 2 v, = Vn-l + -2:

The time derivative

3. Sliding Backstepping Control

Sliding mode techniques, yielding robust control, and adaptive control techniques are both popular when there is uncertainty in the plant. The combination of these methods has been studied in recent years [8]-[ll]. In general, at each step of the backstepping method, the new update tuning function and the defined error

T~ = T ~ - ~ +rw,Tx, (19) variables (and virtual control law) take the system to the equilibrium position. At the final step, the system is stabilized by suitable selection of the control.

The adaptive sliding backstepping control of SSF sys- tems has been studied by Koshkouei and Zinober 151,

aan-1 aan-l aan-l [SI. The controller is based upon sliding backstepping i=l ae at mode techniques so that the state trajectories approach

a specified hyperplane. The sufficient condition (for the existence of the sliding mode) given by Rios-Bolivar and Zinober [7]-[9], is no longer needed.

Vn = vn- l +zn in n-2

< - 2 cjz' + ( n e - ' ' 2 -5 x i=l

(3 - Tn) + 8Tr-l (Tn - 8) (18)

where

if we select the control

1 U = - [-z,-, - cnzn - f(z) - W,TJ

9 b ) n-1

+ 7&"i+l + -773 + -

- (g . z ~ + ~ !$) Fur, - cnzn] (20)

with cn > 0. Selecting 3 = T ~ , 8 is eliminated from the To provide the adaptive backstepping al- right-hand side of (18). Then gorithm can be modified to yield adaptive sliding out-

put tracking controllers. The modification is carried Vn < -w, + e (n + 1) e-at/2 (21) out at the final step of the algorithm by incorporat-

ing the following sliding surface defined in terms of the error coordinates with Wn = $ C y = 1 ciz:. Then

t 4 = klzl + . . . + kn-lzn-l + Z, = 0 (25) vn-vn(0> < - w,+E(n + 1) (1 -e-'') /(2a) (22)

where ki > 0, i = 1 , . . . , n - 1, are real numbers. Addi- Therefore, tionally, the Lyapunov function is modified as follows

n-1 1 1 v, = Z; + --u2 + -(e - BjTr-l (e - 8) (26) 2 2

i= 1

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(& - T ~ ) + eTr-' ( T ~ - &) aai z i + l x (30)

n-2

i=l

and the adaptive sliding mode output tracking control- ler i=2

with

Setting 8 = rn, 8 is eliminated from the right-hand side of (27). n o m (25) we obtain

2, = c - klzl - k 2 ~ 2 - e . . - kn-lZn-1 (29)

Substituting (29) in (27) removes the need for the suf- ficient condition for the existence of the sliding mode

where kn = 1, K > 0 and W 2 0 are arbitrary real numbers and

with Q as

which is a positive definite matrix:.

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which yields

Vn - Vn(O> < - finds + E(n + 1) (1 - e-at) / ( 2 ~ )

(37) I'

Therefore,

0 < / d ' m n d s < vn(o) + E(n + 1) (1 - e-atj /(2a)

and t Wnds < Vn(0) + E(n + 1)/(2a) < 00

n o m the Barbalat Lemma, limt+,Wn = 0. This implies that limt+,zi = 0, i = 1,2,.. . , n and limt+,o = 0. Particularly, limt+, (XI -yT) = 0. Therefore, the stability of the system along the sliding surface U = 0 is guaranteed. Note that the sufficient condition for the existence of the sliding mode in [lo] has been removed by using (25).

However, if E is a sufficiently small positive number and a suitably large, V < 0. Also, the design parameters K , W , ci and ki, i = 1,. . . , n - 1, can be selected to ensure that V e 0.

Remark 2 Alternatively, one can apply a different procedure at the n-th step which yields

r

1 aan - 1 U = s(z> 1 - Zn-1 - f(x) - w,Te + - Tn ae

4. Example

Consider the second-order system in SSF form

21 = 2 2 + + AX: cos(Bxlx2)

x2 = U (40) where A and B are considered unknown but it is known that !AI < 2 and IBI < 3. We have

hi = 22: 21 = 2 1 -Vr

w1 = 21

Then the control law (20) becomes

.. aa1 act1 8x1 ae U = -zl - c2z2 - w,Te + -x2 + -.72

aa1 +- + $1 - ( i 2 2 a t Simulation results showing desirable transient re- sponses are shown in Fig. 1 with yT = 0.4, a = 1,

c1 = c2 = 0.01. E = 0.06, e = 1, r = 1, A = 2, B = 3 and

Alternatively, we can design a sliding mode control- ler for the system. Assume that the sliding surface is U = klsl + 22 = 0 with kl > 0. The adaptive sliding mode control law (32) is

aa1 (2) klh:zl i=2

+-+gr + at hllall + ie-at

n-1 - (. + (ki + I % [ ) 4) sgn(o) (42)

where 72 = ( a w 1 + a(w2 + klwl)). Simulation res- ults showing desirable transient responses are shown in Fig. 2 with yr = 0.05sin(27rt), kl = 1, K = 5, W = 0,

a n d 8 = 1 . (38) a = 1, E = 0.06, r = 1, A = 2, B = 3, cl = 34, c2 = 1

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5. Conclusions

Backstepping is a systematic Lyapunov method to design control algorithms which stabilize nonlinear sys- tems. Sliding mode control is a robust control method design and adaptive backstepping is an adaptive con- trol design method. In this paper the control design has benefited from both design approaches. Backstep- ping control and sliding backstepping control were ori- ginally developed for a class of nonlinear systems which can be converted to the parametric semi-strict feedback form. The plant may have unmodelled or external dis- turbances. The discontinuous control may contain a gain parameter for the designer to select the velocity of the convergence of the state trajectories to the slid- ing hyperplane. We have extended the previous work of Rios-Bolivar and Zinober [7]-[9] and Koshkouei and Zinober [5], and have removed the sufficient existence condition for the sliding mode to guarantee that the state trajectories converge to a given sliding surface.

References

[l] Corless, M., and G. Leitmann, “Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamical systems,” IEEE “bans. Automat. Control, Vol. 26, pp. 1139-1144, 1981. [2] Freeman, R. A., and P. V. KokotoviC, “Tracking con- trollers for systems linear in unmeasured states,” Automat- ica, Vol. 32, pp. 735-746, 1996. [3] Krstit, M., I. Kanellakopoulos, and P. V. Kokotovit, “Adaptive nonlinear control without overparametrization,” Syst. B Control Letters, Vol. 19, pp. 177-185, 1992. [4] Kanellakopoulos, I., P. V. KokotoviC, and A. S. Morse, “Systematic design of adaptive controllers for feed- back linearizable systems,” IEEE “bans. Automat. Con-

[5] Koshkouei, A. J., and A. S. I. Zinober, “Adaptive Sliding Backstepping Control of Nonlinear Semi-strict Feed- back Form Systems,’’ Proceedings of the 7th IEEE Mediter- ranean Control Conference, Haifa, 1999. [6] Koshkouei, A. J., and A. S. I. Zinober, “Adapt- ive Output Tracking Backstepping Sliding Mode Control of Nonlinear Systems,” Proceedings of the 3rd IFAC Sym- posium on Robust Control Design, Prague, 2000. [7] Rios-Bolivar, M., and A. S. I. Zinober, “A symbolic computation toolbox for the design of dynamical adaptive nonlinear control,” Appl. Math. and Comp. Sci., Vol. 8, pp. 73-88, 1998. [8] Rios-Bolivar, M., and A. S. I. Zinober, “Dynamical adaptive sliding mode output tracking control of a class of nonlinear systems,” Int. J . Robust and Nonlinear Control, Vol. 7, pp. 387405, 1997. [9] Rios-Bolivar, M., and A. S. I. Zinober, “Dynamical adaptive backstepping control design via symbolic computa- tion,” Proceedings of the 3rd European Control Conference, Brussels. 1997.

trol, Vol. 36, pp. 1241-1253, 1991.

[lo] Nos-Bolivar, M., A. S. I. Zinober, and H. Sira- Ramfrez, “Dynamical Sliding Mode Control via Adaptive Input-Output Linearization: A Backstepping Approach,” in: Robust Control via Variable Structure and Lyapunov Techniques (F. Garofalo and L. Gllielmo, Eds.), Springer- Verlag, 15-35, 1996. [ll] Rios-Bolivar, M., and A. S. I. Zinober, “Sliding mode control for uncertain linearizable nonlinear systems: A back- stepping approach,” Proceedings of the IEEE Workshop on Robust Control via Variable Structure and Lyapunov Tech- niques, Benevento, Italy, pp. 78-85, 1994. [12] Yao, B., and M. Tomizuka, “.Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form,” Automatica, Vol. 33, 893-900,1997. [13] Yao, B., and M. Tomizuka, ‘43mooth adaptive sliding mode control of robot manipulatoris with guaranteed tran- sient performance,” ASME J. Dyn. Syst. Man Cybernetics, Vol. SMC-8, pp. 101-109, 1994.

0’2

Parameter estimate Control action

-1OL-- 0 0.5 1 1.5 2

t

Figure 1: Responses of example with control (41)

I I 0 5 10

1

Parameter estimate

1

0.51

0

- 0 . 5 L u 10

t

Control &on

40r------- 20

0

Figure 2: Responses of example with sliding control (42)

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