This paper considers control design using an adaptive backstepping algorithm for a class of nonlinear continu- ous uncertain processes with disturbances which ...
Adaptive Backstepping Control of Nonlinear Systems with Unmatched Uncertainty Ali J. Koshkouei and Alan S. I. Zinober Department of Applied Mathematics The University of Sheeld Sheeld S10 2TN UK Abstract This paper considers control design using an adaptive backstepping algorithm for a class of nonlinear continuous uncertain processes with disturbances which can be converted to a parametric semi-strict feedback form. Sliding mode control using a combined adaptive backstepping 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 cient condition for sliding is not needed by the new algorithm.
1. Introduction The backstepping procedure is a systematic design technique 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 triangular form, in particular, the parametric pure feedback (PPF) form and the parametric strict feedback (PSF) form [4]. This method has been studied widely in recent years [3], [4], [8]-[11]. When plants include uncertainty with lack of information about the bounds of unknown parameters, adaptive control is more convenient; whilst, if some information about the uncertainty, e.g. bounds, is available, robust control is usually employed. 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 bene ts from both approaches. A systematic design procedure has been proposed to combine adaptive control and SMC for nonlinear systems 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 backstepping 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 speci ed hyperplane. We remove the sucient condition (for the existence of the sliding mode) given by Rios-Bolvar and Zinober [7]-[9]. We consider nonlinear systems which can be converted to a particular form, the so-called parametric semi-strict 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 control design based upon the backstepping techniques is presented in Section 3. We consider an example to illustrate the results in Section 4 with some conclusions in Section 5.
2. Adaptive Backstepping Control Consider the semi-strict feedback form (SSF) [5], [6], [12] x_ i = xi+1 + 'Ti (x1 ; x2 ; : : : ; xi ) + i (x; w; t); 16i6n;1 T x_ n = f (x) + g(x)u + 'n (x) + n (x; w; t) (1) y = x1 where x = [x1 x2 : : : xn ]T is the state, y the output, u the scalar control and 'i (x1 ; : : : ; xi ) 2 Rp , i = 1; : : : ; n, are known functions which are assumed to be suciently smooth. 2 Rp is the vector of constant unknown parameters and i (x; w; t), i = 1; : : : ; n, are unknown nonlinear scalar functions including all the disturbances. w is an uncertain time-varying parameter.
Assumption 1 The functions i (x; w; t), i = 1; : : : ; n are bounded by known positive functions hi (x ; : : : xi ) 2 Rp ; i.e. ji (x; w; t)j 6 hi (x ; : : : xi ); i = 1; : : : ; n (2) 1
1
The output y should track a speci ed 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 [1]. Some techniques have been proposed for the case of plants containing unmatched uncertainty [2]. The plant may contain unmodelled terms and unmeasurable external disturbances, 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 rst follow a backstepping approach which di ers from Koshkouei and Zinober [5], [6]. The functions that compensate the system disturbances, are continuous.
Then 2 z_1 = ;c1 z1 +z2 +!1T ~+1 (x; w; t); h jz hj +1 z1 e;at (9) 1
1
2
1
1
V_1 (z1 ; ^) 6 ;c1 z12 + z1 z2 + n e;at + ~T ;;1 1 ; ^_
Step k (1 6 k 6 n ; 1). The time derivative of the error variable zk is
where
k
De ne 1 = ;!1 z1. Let 1 = 1 (x1 ; ^; t) + y_r = ;!1T ^ + y_r ; c1 z1 ;
h21 z1 h1 jz1 j + n e;at (7)
with c1 , a and positive numbers. De ne the error variable z2 = x2 ; 1 (x1 ; ^; t) ; y_r 2 1 z1 = x2 + !1T ^ + c1 z1 ; y_r + h jz hj + ;at (8) 1 1 ne
kX ;1
@ k;1 ' (x ; : : : ; x ) i 1 i i=1 @xi kX ;1 @ 2 2 h k ;1 k = h jz j + e;at + @x k k i n i=1 h2i (11) hi @ @x;1 zk + n e;at kX ;1 @ k ;1 i = k ; @x i i=1
!k = 'k (x1 ; : : : ; xk ) ;
k
i
1
Consider the stabilization of the subsystem (3) and the Lyapunov function V1 (z1 ; ^) = 12 z12 + 21 ~T ;;1 ~ (5) where ; is a positive de nite matrix. The derivative V1 is V_1 (z1 ; ^) = z1 x2 + !1T ^ + 1 (x; w; t) ; y_r +~T ;;1 ;!1 z1 ; ^_ (6)
k;1 X
@ k;1 x ; @ k;1 ^_ i+1 @ ^ i=1 @xi (10) +k ; yr(k) (t) + !kT ~ ; @ @tk;1
z_k = xk+1 + !kT ^ ;
1
From (3) z_1 = x2 + !1T ^ + 1 (x; w; t) ; y_r + !1T ~ (4) with !1 (x1 ) = '1 (x1 ) and ~ = ; ^ where ^(t) is an estimate of the unknown parameter .
n
and V_1 is converted to
Step 1. De ne the error variable z = x ; yr then z_ = x + 'T (x ) + (x; w; t) ; y_r (3) 1
1
k
De ne zk+1 = xk+1 ; k = xk+1 ; k ; yr(k) , where k (x1 ; x2 ; : : : ; xk ; ^; t) = ;zk;1 ; ck zk ; !kT ^ +
kX ;1
@ k;1 x + @ k;1 ; z + @ k;1 i+1 k k @t @ ^ k i=1 @xi ! kX ;2 @ i + zi+1 ^ ;wk (12) @ i=1
with ck > 0. Then the time derivative of the error variable zk is z_k = ;zk;1 ; ck zk + zk+1 + !kT ~ + k ; k zk ! kX ;2 @ @ k;1 ^_ i ; ^ ; k + zi+1 ^ ;wk (13) @ @ i=1
The time derivative of Vk is
V_k 6 ;
k X i=1
ci zi2 + zk zk+1 + k(k2+n 1) e;at
+~T ;;1 k ; ^_ +
kX ;2
@ i z ^ i+1 i=1 @
!
k ; ^_
(14)
since
k = k;1 + ;!k zk = ;
Step n. De ne zn = xn ; n;1
k X i=1
and
!i zi
(15)
= xn ; n;1 ; yrn
( )
with n;1 obtained from (12) for k = n. Then the time derivative of the error variable zn is
z_n =
f (x) + g(x)u + !nT (x; t)^ ;
nX ;1
@ n;1 x i+1 i=1 @xi
; @ n^; ^_ ; @ @tn; + !nT (x; t)~ + n ; yrn (16) @ where !n (x; ^) is de ned in (11) for k = n. Extend the 1
1
( )
Lyapunov function to be
Vn = Vn;1 + 12 zn2
(17)
The time derivative V_ n = V_n;1 + zn z_n n nX ;2 @ X i 6 ; ci zi2 + (n +2 1) e;at ; z ^ i+1 i=1 i=1 @ ^_ ; n + ~T ;;1 n ; ^_ (18) where
n = n;1 + ;!nT zn
(19)
if we select the control h u = g(1x) ;zn;1 ; cn zn ; f (x) ; !nT ^ nX ;1
@ n;1 x + @ n;1 + @ n;1 i+1 @t @ ^ n i=1 @xi ! nX ;2 i ; zi+1 @ ^i ;wn +yr(n) ; n zn (20) @ i=1 +
with cn > 0. Selecting ^_ = n , ~ is eliminated from the right-hand side of (18). Then V_n 6 ;Wn + (n + 1) e;at =2 (21) with Wn = 12
Pn
i=1 ci zi . Then Z t ; Vn ;Vn (0) 6 ; Wn +(n + 1) 1 ; e;at =(2a) 2
0
(22)
Therefore, 06
Z t 0
;
Wn 6 Vn (0) + (n + 1) 1 ; e;at =(2a)
lim
t!1
Z t 0
Wn 6 Vn (0) + (n + 1)=(2a) < 1
Since Wn is a uniformly continuous function, according to the Barbalat Lemma, limt!1 Wn = 0. This implies that limt!1 zi = 0, i = 1; 2; : : :; n. Particularly, limt!1 (x1 ; yr ) = 0.
Remark 1 When there is no unknown parameter in
the system equation, one can attain the tracking performance directly. Then (21) becomes V_n 6 ;cVn + (n + 1)e;at =2 (23) with 0 < c 6 1min 6 6 ci and c 6= a. Then i
n
; 0 6 Vn 6 (2(na+;1)c) e;ct ; e;at + Vn (0)e;ct
(24)
limt!1 Vn = 0 implies that limt!1 zi = 0, i = 1; 2; : : :; n. This guarantees limt!1 (y ; yr ) = 0.
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]-[11]. In general, at each step of the backstepping method, the new update tuning function and the de ned error variables (and virtual control law) take the system to the equilibrium position. At the nal step, the system is stabilized by suitable selection of the control. The adaptive sliding backstepping control of SSF systems has been studied by Koshkouei and Zinober [5], [6]. The controller is based upon sliding backstepping mode techniques so that the state trajectories approach a speci ed hyperplane. The sucient condition (for the existence of the sliding mode) given by Rios-Bolvar and Zinober [7]-[9], is no longer needed. To provide robustness, the adaptive backstepping algorithm can be modi ed to yield adaptive sliding output tracking controllers. The modi cation is carried out at the nal step of the algorithm by incorporating the following sliding surface de ned in terms of the error coordinates = k1 z1 + : : : + kn;1 zn;1 + zn = 0 (25) where ki > 0, i = 1; : : : ; n ; 1, are real numbers. Additionally, the Lyapunov function is modi ed as follows
Vn
nX ;1 1 = 2 zi2 + 21 2 + 21 ( ; ^)T ;;1 ( ; ^) (26) i=1
6 ;
nX ;1 i=1
ci zi2 + (n ; 1)e;at=2 + zn;1 zn
+[ f (x) + g(x)u + !nT ^ ;
nX ;1
@ n;1 x i+1 i=1 @xi
; @ n^;1 ^_ ; @ @tn;1 + n ; yr(n) @ 2 h 1 z1 +k1 z2 ; c1 z1 + 1 ; h jz j + e;at +
with
nX ;1 i=2
1
!
i;2 X ; @ i^;1 ^_ ; i +;wi zl+1 @ ^l @ @ l=1 ! !# nX ;2 nX ; 1 @ i ; zi+1 ^ ; !n + ki !i @ i=1 i=1 nX ;2 ; zi+1 @ ^i ^_ ; n + ~T ;;1 n ; ^_ (27) @ i=1
"
= ;
nX ;1 i=1
nX ;1 i=1
ki ! i
!i zi + !n +
!
nX ;1 i=1
ki !i
!#
(28)
Setting ^_ = n , ~ is eliminated from the right-hand side of (27). From (25) we obtain zn = ; k1 z1 ; k2 z2 ; : : : ; kn;1 zn;1 (29) Substituting (29) in (27) removes the need for the suf cient condition for the existence of the sliding mode [8], [9] and
V_n 6 ;
nX ;1 i=1
ci zi ; zn;1 (k1 z1 + k2 z2 + : : : + kn;1 zn;1 )
nX ;1
@ n;1 x ; @ n;1 ^_ ; @ n;1 + i+1 n @t @ ^ i=1 @xi 2 h 1 z1 (n) ; yr + k1 z2 ; c1 z1 ; h jz j + e;at + 1 1 1 n
;
+
nX ;1 i=2
ki ( ; zi;1 ; ci zi + zi+1 + i ; i zi
i;2 X ; @ i^;1 ^_ ; i + ;wi zl+1 @ ^l @ @ l=1
!
nX ;1
;
i=1 nX ;1
i=1
zi !i + !n +
i=1
ki !i ki !i
(31)
and the adaptive sliding mode output tracking controller
u = g(1x) [ ; zn;1 ; f (x) ; !nT ^ + @ n^;1 n @ nX ;1 @ n;1 + xi+1 + yr(n) + @ @tn;1 @x i i=1 2 ;k1 ;c1 z1 + z2 ; h jz hj +1 z1 e;at 1 1 n
;
nX ;1 i=2
ki ( ; zi;1 ; ci zi + zi+1 ; i zi
! ! i;2 X @ @ i;1 l ; ^ ( n ; i ) + zl+1 ^ ;wi @ @ l=1 ! ! nX ;2 nX ;1 @ i + zi+1 ^ ; !n + ki !i @ i=1 i=1
;W ; K +
n X i=1
!
#
ki i sgn()
(32)
where kn = 1, K > 0 and W > 0 are arbitrary real numbers and
i = h
2
+ (n ; 1)e;at =2 + [ zn;1 + f (x) + g(x)u + !nT ^
;
^_ = n = n;1 + ; !n + = ;
n
1
!#
We now obtain the update law
;1 n X
ki ( ; zi;1 ; ci zi + zi+1 + i ; i zi
n = n;1 + ; !n +
!
nX ;1 @ i ; zi+1 ^ ; !n + ki !i ; @ i=1 i=1 nX ;2 (30) zi+1 @ ^i ^_ ; n + ~T ;;1 n ; ^_ @ i=1 nX ;2
The time derivative is V_ n = V_n;1 + _
i;1 X i+ j =1
@ k;1 h ; @xj j
16i6n
(33)
Then substituting (32) in (27) yields V_n = V_n;1 + _ 6 ; [z1 z2 : : : zn;1] Q [z1 z2 : : : zn;1 ]T ; K jj ; W2 + (n ; 1)e;at=2 (34) with Q as
2
c1 0 : : : 0 c2 : : :
0 6 0 Q = 664 .. .. .. .. . . . . k1 k2 : : : kn;1 + cn;1 which is a positive de nite matrix.
3 7 7 7 5
(35)
Let W~ n = [z1 z2 : : : zn;1 ] Q [z1z2 : : : zn;1 ]T + K jj + W2 Then, similarly to (21), we have V_n 6 ;W~ n + (n + 1)e;at=2
with kn = 1, K > 0 and W > 0 arbitrary real numbers and for all i, 1 6 i 6 n i;1 2 X h2 i = h jj +hi e;at + @ @xi;1 h jj +j e;at (39)
i
(36)
n
Z t 0
;
W~ n ds + (n + 1) 1 ; e;at =(2a)
(37)
Therefore, 06
Z t 0
and lim
;
W~ n ds 6 Vn (0) + (n + 1) 1 ; e;at =(2a) Z t
t!1
0
W~ n ds 6 Vn (0) + (n + 1)=(2a) < 1
From the Barbalat Lemma, limt!1 W~ n = 0. This implies that limt!1 zi = 0, i = 1; 2; : : :; n and limt!1 = 0. Particularly, limt!1 (x1 ; yr ) = 0. Therefore, the stability of the system along the sliding surface = 0 is guaranteed. Note that the sucient condition for the existence of the sliding mode in [10] has been removed by using (25). However, if is a suciently 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_ < 0.
Remark 2 Alternatively, one can apply a di erent procedure at the n-th step which yields
u = g(1x) [ ; zn;1 ; f (x) ; !nT ^ + @ n^;1 n @ nX ;1 @ n;1 xi+1 + yr(n) + @ @tn;1 + @x i i=1 2 ;k1 ;c1z1 + z2 ; h jz hj +1 z1 e;at 1 1 n
;
nX ;1 i=2
!
zl @ ^l ;wi ; @ i^; ( n ; i ) + @ @ l ! ! nX ; nX ; @ i + zi ^ ; !n + ki !i @ i i 1
k1 h21 z1 ; W 1 (2) + @ + y + @t r h1 jz1 j + 2 e;at
j (42) ; K + k + j @ @x h sgn() where = ; (z ! + (! + k ! )). Simulation res-
+1
+1
;K sgn() ; W +
n X i=1
2
=1
! #
ki i
1
1
1
=1
Consider the second-order system in SSF form x_ 1 = x2 + x1 + Ax21 cos(Bx1 x2 ) x_ 2 = u (40) where A and B are considered unknown but it is known that jAj 6 2 and jB j 6 3. We have h1 = 2x21 z1 = x1 ; yr 4 z2 = x2 + x1 ^ + c1 z1 + h jz 4jx+1 z1 e;at ; y_r 1 1 2 4 4 x z 1 1 1 = ;x1 ^ ; c1 z1 ; 2x2 jz j + e;at 1 1 2 !1 = x1 1 !2 = ; @ @x1 x1 2 = ; (!1 z1 + !2 z2) 4 @ 1 2 2 = 2 @ 1 4x 1 2x1 @x1 z2 + 2 e;at @x1 Then the control law (20) becomes @ 1 1 u = ;z1 ; c2 z2 ; !2T ^ + @ @x1 x2 + @ ^ 2 1 (2) + @ (41) @t + yr ; 2 z2 Simulation results showing desirable transient responses are shown in Fig. 1 with yr = 0:4; a = 1, = 0:06, = 1, ; = 1, A = 2, B = 3 and c1 = c2 = 0:01.
!
=1
2
n
Alternatively, we can design a sliding mode controller for the system. Assume that the sliding surface is = k1 z1 + z2 = 0 with k1 > 0. The adaptive sliding mode control law (32) is @ 1 1 u = (c1 k1 ; 1) z1 ; k1 z2 ; !2T ^ + @ x 2 + @x1 @ ^ 2
ki ( ; zi;1 ; ci zi + zi+1 ; i zi i;2 X
j
4. Example
which yields
Vn ; Vn (0) 6 ;
j
j =1
(38)
1
1
1
1
2
1
1
ults showing desirable transient responses are shown in Fig. 2 with yr = 0:05 sin(2t); k1 = 1, K = 5, W = 0, a = 1, = 0:06, ; = 1, A = 2, B = 3, c1 = 34, c2 = 1 and = 1.
5. Conclusions Backstepping is a systematic Lyapunov method to design control algorithms which stabilize nonlinear systems. Sliding mode control is a robust control method design and adaptive backstepping is an adaptive control design method. In this paper the control design has bene ted from both design approaches. Backstepping control and sliding backstepping control were originally 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 disturbances. The discontinuous control may contain a gain parameter for the designer to select the velocity of the convergence of the state trajectories to the sliding hyperplane. We have extended the previous work of Rios-Bolvar and Zinober [7]-[9] and Koshkouei and Zinober [5], and have removed the sucient existence condition for the sliding mode to guarantee that the state trajectories converge to a given sliding surface.
[10] Rios-Bolvar, M., A. S. I. Zinober, and H. SiraRamrez, \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. Glielmo, Eds.), SpringerVerlag, 15{35, 1996. [11] Rios-Bolvar, M., and A. S. I. Zinober, \Sliding mode control for uncertain linearizable nonlinear systems: A backstepping approach," Proceedings of the IEEE Workshop on Robust Control via Variable Structure and Lyapunov Techniques, 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, \Smooth adaptive sliding mode control of robot manipulators with guaranteed transient performance," ASME J. Dyn. Syst. Man Cybernetics, Vol. SMC-8, pp. 101{109, 1994. x1(t)
x2(t) 3
0.8 2
[1] Corless, M., and G. Leitmann, \Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamical systems," IEEE Trans. Automat. Control, Vol. 26, pp. 1139{1144, 1981. [2] Freeman, R. A., and P. V. Kokotovic, \Tracking controllers for systems linear in unmeasured states," Automatica, Vol. 32, pp. 735{746, 1996. [3] Krstic, M., I. Kanellakopoulos, and P. V. Kokotovic, \Adaptive nonlinear control without overparametrization," Syst. & Control Letters, Vol. 19, pp. 177{185, 1992. [4] Kanellakopoulos, I., P. V. Kokotovic, and A. S. Morse, \Systematic design of adaptive controllers for feedback linearizable systems," IEEE Trans. Automat. Control, Vol. 36, pp. 1241{1253, 1991. [5] Koshkouei, A. J., and A. S. I. Zinober, \Adaptive Sliding Backstepping Control of Nonlinear Semi-Strict Feedback Form Systems," Proceedings of the 7th IEEE Mediterranean Control Conference, Haifa, 1999. [6] Koshkouei, A. J., and A. S. I. Zinober, \Adaptive Output Tracking Backstepping Sliding Mode Control of Nonlinear Systems," Proceedings of the 3rd IFAC Symposium on Robust Control Design, Prague, 2000. [7] Rios-Bolvar, 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-Bolvar, 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. 387{405, 1997. [9] Rios-Bolvar, M., and A. S. I. Zinober, \Dynamical adaptive backstepping control design via symbolic computation," Proceedings of the 3rd European Control Conference, Brussels, 1997.
x1
References
0.6 1
0.4
0
0.2
−1
0 0
0.5
1 t
1.5
2
−2 0
0.5
Parameter estimate
1 t
1.5
2
1.5
2
Control action
1.5
5
1 0 0.5 −5 0 −0.5 0
0.5
1 t
1.5
2
−10 0
0.5
1 t
Figure 1: Responses of example with control (41)
x1(t)
x2(t) 1
0.1 0.05
0.5
0 0 −0.05 0
5 t
10
−0.5 0
Parameter estimate
5 t
10
Control action
1.5
40 20
1 0 0.5 −20 0 0
5 t
10
−40 0
5 t
10
Figure 2: Responses of example with sliding control (42)
