- Email: [email protected]
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
European Journal of Control (1998)4:132-139 © 1998 EUCA
European Journal of Control
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances V. O. Nikiforov Laboratory of Cybernetics and Control Systems, Institute of Fine Mechanics and Optics, Saint-Petersburg, Russia
We consider a class of single-input, single-output non-linear systems which are transformable into the parametric-strict-feedback canonical form with unknown constant parameters. A state-feedback adaptive controller is proposed which achieves global asymptotic tracking of the reference signal with simultaneous complete compensation of bounded disturbances generated by a linear exosystem of the known order but with unknown parameters. The main contribution of the paper consists of a new parameter-dependent non-linear observer for unknown and inaccessible deterministic disturbances. The proposed control law contains a regulating component designed via the adaptive integrator backstepping technique and the compensating component providing adaptive counteracting of unknown external disturbances and global stabilisation of the closed-loop system.
Keywords: Adaptive non-linear control; Disturbance rejection; Non-linear observer
1. Introduction This paper concerns the problem of complete compensation of deterministic external disturbances affecting the dynamic system to be controlled. Several conceptually different approaches to the problem have been proposed in the literature during the last decades. One of them is based on the famous Correspondence and offprint requests to: V.O. Nikiforov, Laboratory of Cybernetics and Control Systems, Institute of Fine Mechanics and Optics, Sablinskaya 14, Saint-Petersburg, 197 IQ I, Russia. Email: [email protected]
Internal Model Principle (IMP) [3,7,12]. The Principle means that reference signals (or disturbances) produced by some external generator (exosystem) are asymptotically followed (or completely rejected) if the external generator model is suitably reduplicated in the feedback path of the closed-loop control system. A large number of important control problems have been solved with the use of the IMP for the case of linear systems, and now the applicability of the IMP to non-linear control problems is intensively investigated [2, 4, 5, 10, 11, 13, 17]. Desoer and Lin [4] considered the particular case of an exponentially stable non-linear plant and exogenous signals which tend to constant vectors, and demonstrated the applicability of a simple PI controller. Related results were presented by Tseng [17]. The work of Di Benedetto [5] concerns the case of the one-dimensional exosystem. A necessary and sufficient condition for the solvability of the servomechanism problem for a broad class of non-linear systems was established by Isidori and Byrnes [11]. However, they considered the case of small exogenous signals and small initial states and, therefore, their results are local. The barrier of small exogenous signals was overcome by Huang and Rugh [10]. Khalil [13] considered the case of an uncertain non-linear system which can be transformed into a canonical normal form with no zero dynamics. It was assumed that certain functions in the error model obey the known differential equations specifying an internal model which is augmented to the system. The robust servomechanism output-feedback controller designed in Khalil [13] guarantees semiReceived 18 November 1996 Accepted 27 January Recommended by V. Blondel and A. Isidou.
1998
133
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
global tracking of the reference signal. The practical applicability of the non-linear implementation of the IMP to the physical systems like robotic manipulators was demonstrated by Buchner and Hemami [2]. The main shortcoming of the classical IMP lies in the fact that one must know a priori the sufficiently exact models of both the plant to be controlled and disturbance generator. The case when the plant with an uncertain model is affected by a priori unknown external disturbances seems more realistic and has greater practical significance. For such a case, Elliot and Goodwin [6] proposed adaptive implementation of the IMP. In Elliot and Goodwin [6], the unknown deterministic disturbances are treated as the output of a linear exosystem of the known order but with unknown parameters. Thus, in comparison with the classical IMP, the a priori required information about the disturbance waveform is significantly reduced. A few controllers achieving complete adaptive compensation of uncertain external disturbances for both cases of exactly known and uncertain plants have been designed [1,6,8,15,16,18]. However, in all these papers only the case of linear plants was considered and the obtained results cannot be straightforwardly extended to the non-linear ones. In the present paper, the problem of complete adaptive compensation of unknown external disturbances produced by the linear exosystem of the known order but with unknown parameters is solved for a class of non-linear single-input, single-output systems which are transformable into the parametricstrict-feedback canonical form with unknown constant parameters. The case of the matched disturbance is considered; i.e., it is assumed that the unknown disturbance is in the span of the control signal. The proposed approach combines the original parameter-dependent non-linear observer of unknown disturbances, state-feedback adaptive control designed via the adaptive integrator backstepping procedure and a special non-linear damping term providing global stabilisation. The paper is organised as follows: the control problem is posed in Section 2, a non-linear parameter-dependent observer is designed in Section 3, Section 4 presents the new adaptive regulator, while simulation results are demonstrated in Section 5.
2. Problem Statement We consider a class of single-input, single-output non-linear systems, affected both by unknown constant parameters and a time-varying input disturbance, and which are transformable into the para-
metric-strict-feedback form (for details see Krstic [14]):
= xj + 1 + eT
:::; n-l
(I)
with x E Rn being the state, u E R the control, y E R the output variable to be controlled, d E R inaccessible external disturbance, eT = [e 1, • •• em] E Rm the vector of unknown constant parameters, y, f3 and components of
x=rX
(2)
d = IT X
(3)
where X E Rq is the state vector of the exosystem, the constant q x q matrix r has all its eigenvalues on the imaginary axis and I is a constant vector. Without loss of generality the pair (r,l) is assumed to be observable. (ii) The dimension q of the exosystem is known, but parameters of the matrix r and vector I are unknown. (iii) Neither disturbance d nor the state X are accessible to measurements. Assumption 2 (reference signal assumption). The reference signal Yr and its first n derivatives are known and bounded, and y~l) is piecewise continuous. The control problem posed will be solved in two steps. First, a new parameter-dependent non-linear observer for external disturbances will be proposed. Second, a new state-feedback adaptive regulator will be designed. For the sake of notational simplicity, we use below the following compact description of the nonlinear system (I): X = Ex+<\>(x) T 8+h(f3(x)+y(x)u+y(x)d)
V. 0. Nikiforov
134
(la)
y=x l
where x = [XI,. . .,xll]"
E=
oI o0
0
0
1
0
. .
o0 (x) =
l' 0
0
0
[b) . . .,b)]
It is worth noting that in the considered plant model the disturbance d and control u are matched because
they appear in the same equation. However, the matching condition is not crucial and accepted here to clarify the main idea and avoid tedious mathematics. The proposed technique with minor modifications can be extended to the disturbance-strictfeedback case.
3. Non-linear Parameter-Dependent Observer
(6)
Under Assumption l(ii), the vector l/J is unknown because it depends on the unknown matrix r and vector l. Thus, the lemma reformulates the uncertainty of the disturbance d to the parametric uncertainty of a constant vector l/J associated with the known 'regressor' " However, since the disturbance d is not accessible to measurements, the state variable filter (4) is not realisable. We use this filter, however, as an intermediate step towards the appropriate parametrisation of the inaccessible disturbance d. As a second step in this direction, we introduce a parameter-dependent estimate of the regressor (. We set this estimate ( in the form 111
t = TJo + L 8j TJj + v(x)
In order to construct an appropriate parametrisation for the unknown disturbance d, we represent the disturbance in an especially suitable form. This form is established by the following lemma reformulating some results known in the theory of linear observers.
Lemma (virtual observer for the unknown disturbance). Along with (2)-(3) consider the dynamic system (= G( + bd
G( - biI = GOd + (Mr - GM - bF)x = GOd
with Od(O) = MAO) - (0). The latter equation means that Dd decays exponentially. Then substituting X=M- ' (o{r+0 into (3) we obtain (5) with l/JT = FM-I. 0
0 h=
Bd = MTr -
(4)
where 'ERq is the state, and the pair (G,b) is controllable. Then for any q x q Hurwitz matrix G there exists a unique constant vector l/J E Rq such that the disturbance d can be presented in the form (5)
where the exponentially vanishing vector Dd obeys the equation Dd = GOd' Proof Define the vector Dd = Mx -
(, where the q x q matrix M is a solution to the following matrix equation:
Mr- GM= biT Since the matrices rand G have no common eigenvalues, we conclude that the matrix M exists and is unique [19]. Furthermore, since the pair (r,!) is completely observable and the pair (G,b) is completely controllable, the matrix M is nonsingular [9]. Differentiating Dd with respect to time in view of (2)-(4), we obtain
(7)
j=1
where the auxiliary vectors TJj by the filters 1]0 = GTJo
1]j
+ Gv(x) - : :Ex -
= Gr/j -
'P/x),
Rq are generated
E
b[ + ~i~;]
l::5j::5 m
U
(8) (9)
and the vector function v(x) E Rq is a solution to the following equation in partial derivatives: av(x)h = b_l_ ax y(x)
(l0)
In Eq. (9), 'Pj E Rq means the jth column of the matrix product av/ax·qJ(x)T (i.e., av/ax- qJ(x) T = ['PI' . .'Pm])' Then the stability properties of the non-linear observer (7)-(9) are established by the following theorem.
Theorem 1 (parameter-dependent observer for unknown disturbance). If there exists a vector function v(x) satisfying (l0), then in the closed-loop system consisting of the plant (l), uncertain exosystem (2)-(3) and non-linear observer (7)-(9), the inaccessible disturbance d can be presented in the form (l1) where l/J is a constant vector and the exponentially decaying vector 0 obeys the equation
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
(12)
0= Go
with initial state 0(0) = M x(O) - 7]0(0)
-
kJ~1 Oj7]iO) - v(x(O».
Proof Let us introduce the vector 111
j-'
Differentiating the latter equality with respect to time in view of (1)-(3) and (7)-(10) we obtain . av o{ = G~ + bd - G7]o - Gv + - Ex ax av + ax
av
m
h[ yu + {3] -
2: Op7]j + ax (x) TO j=I
av
- -a Ex x
aV aV - (X) TO - - h[ yu + aX aX
I3l -
135
The parametrisation (16) contains the known regressor ~ generated by the physically realisable non-linear filters (8)-(9), unknown constant vector it and exponentially vanishing term l/J TO. In Section 4 Eq. (16) will be appropriately used to design an adaptive regulator.
Remark 3.1. Equation (10) is essentially simplified if the vector b is chosen in the form b = col(O,O, . .., I) (with matrix G being, for example, in the controller canonical form). In this case, the vector-function v(x) can be chosen as v(x) = col(O,O, . . .,vc/x», where the scalar function vq(x) is a solution to the first-order partial differential equation avix) __l_ ax" - y(x)
(17)
bd
4. Adaptive Regulator Design
m
= G(~ - 7]0 -
2: Oj7]j -
v(X» = GO{
(13)
j='
First, we split the total control effort u into two parts:
with 0{(0) = ~(O) - 7]0(0) - kJ~, Oj7]iO) - v(x(O». Then, substituting ~ = , + o{ into (5), we derive d = l/JT' + l/JT(O"
+ o{)
(14)
Defining 0 = 0" + o{ and taking the models (6) and (13) into account, we finally obtain (11)-(12). 0 From Eq. (7) we can see that the unknown parameters OJ are involved in the estimate of the regressor ,. In order to present the uncertain disturbance in the conventional form as a product of the known regressor and unknown constant parameters, let us define an extended vector of unknown parameters it and extended regressor ~ as
itT = [l/J T,O,l/JT ,02l/JT , ~T = [(7]0
+ V)T,7]J,7]J,
,Oml/JT ], ,7],;]
where the term uy (regulating component) is responsible for making Y to follow Ye> and the term u" (compensating component) is responsible for counteracting d. The regulating component uy is proposed to be constructed in accordance with the known design procedure based on integrator backstepping [14]. This procedure is recursive and at its jth step the new regulated variable Zj, the stabilising function cxj and the tuning function 7j are defined by Step 1 Z, = Y - Yr = x, - Yr cx,(x"e,t) = -CIZI - eT
(15)
Then the final form of parametrisation of the external uncertain disturbance d will be defined by the following corollary to Theorem 1.
Corollary (parametrised form of unknown disturbance). The disturbance d can be expressed in the form of the parametric model d=itT~+l/JTo
(18)
Step 2 Z2=X2 A
CX2(X,,x2,O,t) =
-ZI -
CXI
acx, aCX 1 C2 Z2 + ax X2 + ae 72 l
acx l • aCXI.. AT aCXI +-aYr+-a'Yr- O (
(16)
A
where it and ~ are defined by (15), the signals 7]j (0 ::; j ::; m) are generated by filters (8)-(9), and the term l/JTO exponentially decays by virtue of (12). The corollary is proved by straightforward substitution of Eq. (7) into (11) in view of (15).
XI
Step 3
v. o.
136
After analysis of the model (21), the compensating component Ud is chosen as
Zj = Xj - aj_1 a/x!> . . .,xj,e,t) = -Zj_1 - CjZj 'V ( aaj_1 aaj_, (i) ) j-I + LJ a Xi+1 + (i-I)Y,. i=1 Xi ay,. aaj _, (j) aaj_1 + ayy-l) y,. + ae Tj
j-2 + ( kl~Zi+'O"i -
eT
(24)
where it is the vector of additional adjustable parameters generated by the adaptation law (25)
) Wj
+ k.zjwj where Cj > 0 and k I > 0 are the feedback gains and T/X I ,. . .,Xj,e,t) =
Tj _ 1
j-I _ 'V aaj_1 aaj W/XI,· . .,xj,e) - cPj - LJ ~-cPi' O"j = ~~ i=1 aXi ae ~
The actual regulating component uy and actual update law for the estimate are designed at the final nth step as
e
I [
uy = -
-ZII_I - CIIZII -
Y
11-1 (
aa ll _1
f3 + ~ ~Xi+1 i=1
aall_1
I
(i»)
e = TII _I
(kl~ Zi+IO"i -
(19)
eT)wlI ]
+ k,zlIwn
(20)
Z = A(z,e,t)z + W(z,e,t)e + yh(Ud+iY g+l/J TO) e=-kIW(z,e,t)T z
e=
(21)
A=
[~
(22) 0 -C 2
l-k,O",w 3
0
-l+k\O"\w)
-c)
0
k\O",w"
y,.(t) as t -
00
_1 V(z, it,8) = Z T Z
e,
2
+ k/j8 TP8
where lJ = it - it, positive defined matrix P obeys the equation G T P + PG = -1 and a positive gain k/j is chosen larger than the maximum eigenvalue of the matrix l/Jl/JT. Differentiating V with respect to time along the solutions of the system (21)-(25), we obtain if::::; - Col TZ + ZTwe
- ZT hyzn y
e- e, z= [ZI,Z2, ... ,Zn]T and
-I
Theorem 2. For any initial states, all the signals in the closed-loop system consisting of the plant (1), uncertain exosystem (2)-(3), non-linear observer (8)-(9), and adaptive regulator (19)-(20) and (24)(25) are bounded and global asymptotic tracking is achieved
Proof Taking the skew-symmetry of the off-diagonal terms in A(z,e,t) into account, we choose the following Lyapunov function:
where CII > O. In view of Eq. (16), it can be shown that the resulting system closed by the regulating component takes the form
where
Remark 4.1. In the proposed compensating component Ud the term -ZIIY provides nonlinear damping (in the terminology of work [14]) of the exponentially decaying signal l/JT8. The stability properties of the closed-loop system are defined by the following theorem.
yet) -
+ ay,.(i-I)Y,.
+
Nikiforov
0
-z T we -
::::; -ColT Z - (ZnY? = -CoZ TZ -
3
+ ZThygT {j + ZThyl/JT8 YZnC {j - k/jOT 8
+ Znyl/JT 8 - (l/JT 8?
4( l/JT8? -
(Zn Y - !l/JT8)2
-k,o,w" ] l-k';"_2 W " -1+k\0""_2 W "
e
-en
WT wT
w= [
w: w"
(23)
where Co = min(c!> . . .,cn ). This means that z, and it are uniformly bounded. Since Z. = XI - y,., we see that X I is also bounded. Then the boundedness of xj for all 2 ::::; j ::::; n can be shown from the boundedness of aj_1 and the fact that xj = Zj + aj_l. Since X is bounded, all 1)j (1 ::::; j ::::; m) defined by (9) are also bounded. The boundedness of 1)0 is readily
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
obtained by virtue of Theorem 1. Then from (19) and (24) we readily conclude the boundedness of uy and Ud' The boundedness of all the closed-loop signals is proved. Furthermore, taking Assumption 2 into account, we conclude that the right parts of (21), (22) and (25) are locally Lipschitz in z, e and {f uniformly in t. Then the LaSalle-Y oshizawa theorem (Theorem 2.1. in [14]) guarantees the regulation z -+ and, as a result, asymptotic tracking y -+ y,. as t -+ 00. 0
137
Then for the parametrisation (16) we have ~ = ['T/o,,, 'T/O.2
+ V2, 'T/ 1," 'T/ 1,2] T
The control law design procedure gives the following expressions for the regulating component: Step 1
°
Step 2
5. Illustrative Example To demonstrate the efficiency of the proposed adaptive controller, we consider the system X, = X2 + (}cp(x,) X2 = )I(x) (u + d)
y=x,
where cp(x,) = xf, )I(X) = 2 + cos Xl> parameter () = 3 is unknown and the a priori uncertain deterministic disturbance d is assumed to be generated by the exosystem of the second order (i.e., d is presented by a harmonic signal with unknown frequency, amplitude and initial phase). The control objective is to asymptotically track the output of the reference model 25 y,. = (p + 5fr where p = d/dt denotes the differential operator and r = sin t. In this case, Eq. (17) takes the form dV2
1
dX 2
2 + cos X,
and, therefore, the function V2 can be defined as X2 V2(X) = -----"'----
2 + cos
X,
Choosing
G=[ -25° -101],b=[Oj,V(X)=[o] 1
The compensating component takes the form: Ud
=
-C {} -
{} = k 2(2
(2 + cos
X I )Z2
+ cos XI)Z2~
The simulation results are shown in Figs 1 and 2. The results are obtained with d(t) = 11sin1.4t Cl = C 2 = 5, k, = 5, k2 = 100 and zero initial states of the plant, reference model and all additional filters. The plots in Fig. 1 present the processes in the control system when the proposed component of adaptive compensation for unknown external disturbances Ud is turned off (i.e., the control signal contains the regulating component uy only). It is worth noting that in this case the adaptive controller changes into that proposed by Krstic et al. [14]. The non-zero residual tracking error is seen in Fig. 1(a). Figure 2 illustrates the processes in the closed-loop control system with adaptive compensation of external disturbances. It is seen from the presented plots that the proposed regulator ensures asymptotic tracking of the reference model in spite of parameter uncertainties of the plant and presence of an uncertain external disturbance acting on the system.
~
we obtain the following expressions for the observer of the external disturbance: 1Jo., = 'T/O,2 + V2, 1JO,2 = -25'T/o" - 1O'T/o,2 -
1J l,l
V2(10
+
V2
sin Xl) -
U
= 'T/ 1,2.
1J,,2 = -25'T/", - 1O'T/l,2 - (v2xf sin x,)!)I
6. Conclusion The problem of adaptive complete compensation of unknown external disturbances generated by the linear exosystem of the known order but with unknown parameters is solved for a class of nonlinear systems transformable into the parametric-
V. O. Nikiforov
138 Zl
2
u
100
1
50
0
0
-1
-50
-2
0
5
10
15
20
-100
25
0
5
time (sec)
10
15
20
25
time (sec)
Fig. 1. Non-zero residual tracking error due to the presence of external disturbances (adaptive compensation of the external disturbances is turned off: 11 = 11,.).
Zl
2
u
50
1
25
0
0
-1
-25
-2
0
5
10
15
20
25
-50
0
5
10
15
20
25
time (sec)
time (sec)
Fig. 2. Asymptotic tracking with complete compensation of unknown external disturbances (adaptive compensation of the external disturbances is switched on: 11 = 11,. + 11,,).
strict-feedback canonical form with unknown parameters. The main contributions of the paper are the following: (i) non-linear parameter-dependent observer for unknown external disturbances; (ii) nonlinear state-feedback adaptive regulator providing asymptotical tracking of the reference signal for the class of non-linear systems affected both by unknown constant parameters and uncertain external disturbances. It is worth noting that the total dynamic order of the proposed non-linear observer (8)-(9) is q(m + 1) (as we see it depends on both the dynamic order of the exogenous system q and the number of unknown parameters m). As a result, we need to tune q(m + I) adjustable parameters involved in the compensating component Ud of the adaptive regulator. In a certain sense we can talk about overparametrisation, that is, more than one adjustable parameter per unknown parameter. The same situation is faced when we design an adaptive regulator rejecting unknown deterministic disturbances in linear systems [16]. Thus, the overparametrisation is caused by the
complexity of the considered control problem, rather than by non-linearity of the plant model. The problem of removal of the overparametrisation deserves a deeper study and constitutes the perspective of further research.
References I. Bonilla MA, Cheang JA, Lozano R. On the use of adaptive precompensators for rejecting output disturbances in non minimum phase SISO systems. In: Proceedings of 33rd IEEE Conference on Decision and Control vol 2, Florida, 1994. pp 1183-1184 2. Buchner HJ, Hemami H. Servocompensation of disturbance in robotic systems. Int J Control, 1988; 48: 273-288 3. Davison EJ. The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans Autom Control 1976; 21 (I): 25-34 4. Desoer CA, Lin C-A. Tracking and disturbance rejection of MIMO non-linear systems with PI controller. IEEE Trans Autom Control 1985; 30(9): 861-867 5. Di Benedetto MD. Synthesis of an internal model for
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
6.
7. 8. 9. 10. 11. 12.
nonlinear output regulation. Int J Control 1987; 45: 1023-1034 Elliot H, Goodwin Ge. Adaptive implementation of the internal model principle. In: Proceedings of the 23rd IEEE Conference on Decision and Control, USA, 1984. pp 1292-1297 Francis BA, Wonham WM. The internal model principle for linear multivariable regulators. Appl Math Opt 1975; 2: 170-194 Gang Feg and Palaniswami M. Unified treatment of internal model principle based adaptive control algorithms. Int J Contr 1991; 54(4): 883-901 Grigoriev V, Drozdov V, Lavrentyev V, Ushakov A. Computer-aided design of discrete controllers. Mashinostroenie, Leningrad 1983 (in Russian) Huang J, Rugh WJ. Stabilization on zero-error manifolds and the nonlinear servomechanism problem. IEEE Trans Autom. Control 1992; 37(7): 1009-1013 Isidori A, Byrnes Cl. Output regulation of nonlinear systems, IEEE Trans Autom Control 1990; 35(2): 131-140 Johnson CD. Accommodation of external disturbances
13.
14. 15.
16.
17. 18.
19.
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in linear regulator and servomechanism problems. IEEE Trans Autom Control 1971; 16(6): 635-644 Khalil HK. Robust servomechanism output feedback controller for feedback linearizable systems. Automatica 1994; 30(10): 1587-1599 Krstic M, Kanellakopoulos I, Kokotovic PV. Nonlinear and adaptive control design. Wiley, New York 1995 Nikiforov VO. Adaptive servocompensation of input disturbances. 13th IFAC world congress, vol K, USA, 1996. pp 175-180 Nikiforov VO. Adaptive controller rejecting uncertain deterministic disturbances in SISO systems. In: European control conference, Brussels, Belgium, 1997 Tseng He. Error-feedback servomechanisms in nonlinear systems. Int J Control 1992; 55: 1093-1114 Tsypkin YaZ, Nadezhdin PV. Robust nominal continuous-time system design with internal models. Prepr. 12th IFAC world congress, vol 10, Australia, 1993. pp 23-26 Wonham WM. Linear multivariable control: a geometric approach. Springer, Berlin 1979
European Journal of Control
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances V. O. Nikiforov Laboratory of Cybernetics and Control Systems, Institute of Fine Mechanics and Optics, Saint-Petersburg, Russia
We consider a class of single-input, single-output non-linear systems which are transformable into the parametric-strict-feedback canonical form with unknown constant parameters. A state-feedback adaptive controller is proposed which achieves global asymptotic tracking of the reference signal with simultaneous complete compensation of bounded disturbances generated by a linear exosystem of the known order but with unknown parameters. The main contribution of the paper consists of a new parameter-dependent non-linear observer for unknown and inaccessible deterministic disturbances. The proposed control law contains a regulating component designed via the adaptive integrator backstepping technique and the compensating component providing adaptive counteracting of unknown external disturbances and global stabilisation of the closed-loop system.
Keywords: Adaptive non-linear control; Disturbance rejection; Non-linear observer
1. Introduction This paper concerns the problem of complete compensation of deterministic external disturbances affecting the dynamic system to be controlled. Several conceptually different approaches to the problem have been proposed in the literature during the last decades. One of them is based on the famous Correspondence and offprint requests to: V.O. Nikiforov, Laboratory of Cybernetics and Control Systems, Institute of Fine Mechanics and Optics, Sablinskaya 14, Saint-Petersburg, 197 IQ I, Russia. Email: [email protected]
Internal Model Principle (IMP) [3,7,12]. The Principle means that reference signals (or disturbances) produced by some external generator (exosystem) are asymptotically followed (or completely rejected) if the external generator model is suitably reduplicated in the feedback path of the closed-loop control system. A large number of important control problems have been solved with the use of the IMP for the case of linear systems, and now the applicability of the IMP to non-linear control problems is intensively investigated [2, 4, 5, 10, 11, 13, 17]. Desoer and Lin [4] considered the particular case of an exponentially stable non-linear plant and exogenous signals which tend to constant vectors, and demonstrated the applicability of a simple PI controller. Related results were presented by Tseng [17]. The work of Di Benedetto [5] concerns the case of the one-dimensional exosystem. A necessary and sufficient condition for the solvability of the servomechanism problem for a broad class of non-linear systems was established by Isidori and Byrnes [11]. However, they considered the case of small exogenous signals and small initial states and, therefore, their results are local. The barrier of small exogenous signals was overcome by Huang and Rugh [10]. Khalil [13] considered the case of an uncertain non-linear system which can be transformed into a canonical normal form with no zero dynamics. It was assumed that certain functions in the error model obey the known differential equations specifying an internal model which is augmented to the system. The robust servomechanism output-feedback controller designed in Khalil [13] guarantees semiReceived 18 November 1996 Accepted 27 January Recommended by V. Blondel and A. Isidou.
1998
133
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
global tracking of the reference signal. The practical applicability of the non-linear implementation of the IMP to the physical systems like robotic manipulators was demonstrated by Buchner and Hemami [2]. The main shortcoming of the classical IMP lies in the fact that one must know a priori the sufficiently exact models of both the plant to be controlled and disturbance generator. The case when the plant with an uncertain model is affected by a priori unknown external disturbances seems more realistic and has greater practical significance. For such a case, Elliot and Goodwin [6] proposed adaptive implementation of the IMP. In Elliot and Goodwin [6], the unknown deterministic disturbances are treated as the output of a linear exosystem of the known order but with unknown parameters. Thus, in comparison with the classical IMP, the a priori required information about the disturbance waveform is significantly reduced. A few controllers achieving complete adaptive compensation of uncertain external disturbances for both cases of exactly known and uncertain plants have been designed [1,6,8,15,16,18]. However, in all these papers only the case of linear plants was considered and the obtained results cannot be straightforwardly extended to the non-linear ones. In the present paper, the problem of complete adaptive compensation of unknown external disturbances produced by the linear exosystem of the known order but with unknown parameters is solved for a class of non-linear single-input, single-output systems which are transformable into the parametricstrict-feedback canonical form with unknown constant parameters. The case of the matched disturbance is considered; i.e., it is assumed that the unknown disturbance is in the span of the control signal. The proposed approach combines the original parameter-dependent non-linear observer of unknown disturbances, state-feedback adaptive control designed via the adaptive integrator backstepping procedure and a special non-linear damping term providing global stabilisation. The paper is organised as follows: the control problem is posed in Section 2, a non-linear parameter-dependent observer is designed in Section 3, Section 4 presents the new adaptive regulator, while simulation results are demonstrated in Section 5.
2. Problem Statement We consider a class of single-input, single-output non-linear systems, affected both by unknown constant parameters and a time-varying input disturbance, and which are transformable into the para-
metric-strict-feedback form (for details see Krstic [14]):
= xj + 1 + eT
:::; n-l
(I)
with x E Rn being the state, u E R the control, y E R the output variable to be controlled, d E R inaccessible external disturbance, eT = [e 1, • •• em] E Rm the vector of unknown constant parameters, y, f3 and components of
x=rX
(2)
d = IT X
(3)
where X E Rq is the state vector of the exosystem, the constant q x q matrix r has all its eigenvalues on the imaginary axis and I is a constant vector. Without loss of generality the pair (r,l) is assumed to be observable. (ii) The dimension q of the exosystem is known, but parameters of the matrix r and vector I are unknown. (iii) Neither disturbance d nor the state X are accessible to measurements. Assumption 2 (reference signal assumption). The reference signal Yr and its first n derivatives are known and bounded, and y~l) is piecewise continuous. The control problem posed will be solved in two steps. First, a new parameter-dependent non-linear observer for external disturbances will be proposed. Second, a new state-feedback adaptive regulator will be designed. For the sake of notational simplicity, we use below the following compact description of the nonlinear system (I): X = Ex+<\>(x) T 8+h(f3(x)+y(x)u+y(x)d)
V. 0. Nikiforov
134
(la)
y=x l
where x = [XI,. . .,xll]"
E=
oI o0
0
0
1
0
. .
o0 (x) =
l' 0
0
0
[
It is worth noting that in the considered plant model the disturbance d and control u are matched because
they appear in the same equation. However, the matching condition is not crucial and accepted here to clarify the main idea and avoid tedious mathematics. The proposed technique with minor modifications can be extended to the disturbance-strictfeedback case.
3. Non-linear Parameter-Dependent Observer
(6)
Under Assumption l(ii), the vector l/J is unknown because it depends on the unknown matrix r and vector l. Thus, the lemma reformulates the uncertainty of the disturbance d to the parametric uncertainty of a constant vector l/J associated with the known 'regressor' " However, since the disturbance d is not accessible to measurements, the state variable filter (4) is not realisable. We use this filter, however, as an intermediate step towards the appropriate parametrisation of the inaccessible disturbance d. As a second step in this direction, we introduce a parameter-dependent estimate of the regressor (. We set this estimate ( in the form 111
t = TJo + L 8j TJj + v(x)
In order to construct an appropriate parametrisation for the unknown disturbance d, we represent the disturbance in an especially suitable form. This form is established by the following lemma reformulating some results known in the theory of linear observers.
Lemma (virtual observer for the unknown disturbance). Along with (2)-(3) consider the dynamic system (= G( + bd
G( - biI = GOd + (Mr - GM - bF)x = GOd
with Od(O) = MAO) - (0). The latter equation means that Dd decays exponentially. Then substituting X=M- ' (o{r+0 into (3) we obtain (5) with l/JT = FM-I. 0
0 h=
Bd = MTr -
(4)
where 'ERq is the state, and the pair (G,b) is controllable. Then for any q x q Hurwitz matrix G there exists a unique constant vector l/J E Rq such that the disturbance d can be presented in the form (5)
where the exponentially vanishing vector Dd obeys the equation Dd = GOd' Proof Define the vector Dd = Mx -
(, where the q x q matrix M is a solution to the following matrix equation:
Mr- GM= biT Since the matrices rand G have no common eigenvalues, we conclude that the matrix M exists and is unique [19]. Furthermore, since the pair (r,!) is completely observable and the pair (G,b) is completely controllable, the matrix M is nonsingular [9]. Differentiating Dd with respect to time in view of (2)-(4), we obtain
(7)
j=1
where the auxiliary vectors TJj by the filters 1]0 = GTJo
1]j
+ Gv(x) - : :Ex -
= Gr/j -
'P/x),
Rq are generated
E
b[ + ~i~;]
l::5j::5 m
U
(8) (9)
and the vector function v(x) E Rq is a solution to the following equation in partial derivatives: av(x)h = b_l_ ax y(x)
(l0)
In Eq. (9), 'Pj E Rq means the jth column of the matrix product av/ax·qJ(x)T (i.e., av/ax- qJ(x) T = ['PI' . .'Pm])' Then the stability properties of the non-linear observer (7)-(9) are established by the following theorem.
Theorem 1 (parameter-dependent observer for unknown disturbance). If there exists a vector function v(x) satisfying (l0), then in the closed-loop system consisting of the plant (l), uncertain exosystem (2)-(3) and non-linear observer (7)-(9), the inaccessible disturbance d can be presented in the form (l1) where l/J is a constant vector and the exponentially decaying vector 0 obeys the equation
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
(12)
0= Go
with initial state 0(0) = M x(O) - 7]0(0)
-
kJ~1 Oj7]iO) - v(x(O».
Proof Let us introduce the vector 111
j-'
Differentiating the latter equality with respect to time in view of (1)-(3) and (7)-(10) we obtain . av o{ = G~ + bd - G7]o - Gv + - Ex ax av + ax
av
m
h[ yu + {3] -
2: Op7]j + ax (x) TO j=I
av
- -a Ex x
aV aV - (X) TO - - h[ yu + aX aX
I3l -
135
The parametrisation (16) contains the known regressor ~ generated by the physically realisable non-linear filters (8)-(9), unknown constant vector it and exponentially vanishing term l/J TO. In Section 4 Eq. (16) will be appropriately used to design an adaptive regulator.
Remark 3.1. Equation (10) is essentially simplified if the vector b is chosen in the form b = col(O,O, . .., I) (with matrix G being, for example, in the controller canonical form). In this case, the vector-function v(x) can be chosen as v(x) = col(O,O, . . .,vc/x», where the scalar function vq(x) is a solution to the first-order partial differential equation avix) __l_ ax" - y(x)
(17)
bd
4. Adaptive Regulator Design
m
= G(~ - 7]0 -
2: Oj7]j -
v(X» = GO{
(13)
j='
First, we split the total control effort u into two parts:
with 0{(0) = ~(O) - 7]0(0) - kJ~, Oj7]iO) - v(x(O». Then, substituting ~ = , + o{ into (5), we derive d = l/JT' + l/JT(O"
+ o{)
(14)
Defining 0 = 0" + o{ and taking the models (6) and (13) into account, we finally obtain (11)-(12). 0 From Eq. (7) we can see that the unknown parameters OJ are involved in the estimate of the regressor ,. In order to present the uncertain disturbance in the conventional form as a product of the known regressor and unknown constant parameters, let us define an extended vector of unknown parameters it and extended regressor ~ as
itT = [l/J T,O,l/JT ,02l/JT , ~T = [(7]0
+ V)T,7]J,7]J,
,Oml/JT ], ,7],;]
where the term uy (regulating component) is responsible for making Y to follow Ye> and the term u" (compensating component) is responsible for counteracting d. The regulating component uy is proposed to be constructed in accordance with the known design procedure based on integrator backstepping [14]. This procedure is recursive and at its jth step the new regulated variable Zj, the stabilising function cxj and the tuning function 7j are defined by Step 1 Z, = Y - Yr = x, - Yr cx,(x"e,t) = -CIZI - eT
(15)
Then the final form of parametrisation of the external uncertain disturbance d will be defined by the following corollary to Theorem 1.
Corollary (parametrised form of unknown disturbance). The disturbance d can be expressed in the form of the parametric model d=itT~+l/JTo
(18)
Step 2 Z2=X2 A
CX2(X,,x2,O,t) =
-ZI -
CXI
acx, aCX 1 C2 Z2 + ax X2 + ae 72 l
acx l • aCXI.. AT aCXI +-aYr+-a'Yr- O (
(16)
A
where it and ~ are defined by (15), the signals 7]j (0 ::; j ::; m) are generated by filters (8)-(9), and the term l/JTO exponentially decays by virtue of (12). The corollary is proved by straightforward substitution of Eq. (7) into (11) in view of (15).
XI
Step 3
v. o.
136
After analysis of the model (21), the compensating component Ud is chosen as
Zj = Xj - aj_1 a/x!> . . .,xj,e,t) = -Zj_1 - CjZj 'V ( aaj_1 aaj_, (i) ) j-I + LJ a Xi+1 + (i-I)Y,. i=1 Xi ay,. aaj _, (j) aaj_1 + ayy-l) y,. + ae Tj
j-2 + ( kl~Zi+'O"i -
eT
(24)
where it is the vector of additional adjustable parameters generated by the adaptation law (25)
) Wj
+ k.zjwj where Cj > 0 and k I > 0 are the feedback gains and T/X I ,. . .,Xj,e,t) =
Tj _ 1
j-I _ 'V aaj_1 aaj W/XI,· . .,xj,e) - cPj - LJ ~-cPi' O"j = ~~ i=1 aXi ae ~
The actual regulating component uy and actual update law for the estimate are designed at the final nth step as
e
I [
uy = -
-ZII_I - CIIZII -
Y
11-1 (
aa ll _1
f3 + ~ ~Xi+1 i=1
aall_1
I
(i»)
e = TII _I
(kl~ Zi+IO"i -
(19)
eT)wlI ]
+ k,zlIwn
(20)
Z = A(z,e,t)z + W(z,e,t)e + yh(Ud+iY g+l/J TO) e=-kIW(z,e,t)T z
e=
(21)
A=
[~
(22) 0 -C 2
l-k,O",w 3
0
-l+k\O"\w)
-c)
0
k\O",w"
y,.(t) as t -
00
_1 V(z, it,8) = Z T Z
e,
2
+ k/j8 TP8
where lJ = it - it, positive defined matrix P obeys the equation G T P + PG = -1 and a positive gain k/j is chosen larger than the maximum eigenvalue of the matrix l/Jl/JT. Differentiating V with respect to time along the solutions of the system (21)-(25), we obtain if::::; - Col TZ + ZTwe
- ZT hyzn y
e- e, z= [ZI,Z2, ... ,Zn]T and
-I
Theorem 2. For any initial states, all the signals in the closed-loop system consisting of the plant (1), uncertain exosystem (2)-(3), non-linear observer (8)-(9), and adaptive regulator (19)-(20) and (24)(25) are bounded and global asymptotic tracking is achieved
Proof Taking the skew-symmetry of the off-diagonal terms in A(z,e,t) into account, we choose the following Lyapunov function:
where CII > O. In view of Eq. (16), it can be shown that the resulting system closed by the regulating component takes the form
where
Remark 4.1. In the proposed compensating component Ud the term -ZIIY provides nonlinear damping (in the terminology of work [14]) of the exponentially decaying signal l/JT8. The stability properties of the closed-loop system are defined by the following theorem.
yet) -
+ ay,.(i-I)Y,.
+
Nikiforov
0
-z T we -
::::; -ColT Z - (ZnY? = -CoZ TZ -
3
+ ZThygT {j + ZThyl/JT8 YZnC {j - k/jOT 8
+ Znyl/JT 8 - (l/JT 8?
4( l/JT8? -
(Zn Y - !l/JT8)2
-k,o,w" ] l-k';"_2 W " -1+k\0""_2 W "
e
-en
WT wT
w= [
w: w"
(23)
where Co = min(c!> . . .,cn ). This means that z, and it are uniformly bounded. Since Z. = XI - y,., we see that X I is also bounded. Then the boundedness of xj for all 2 ::::; j ::::; n can be shown from the boundedness of aj_1 and the fact that xj = Zj + aj_l. Since X is bounded, all 1)j (1 ::::; j ::::; m) defined by (9) are also bounded. The boundedness of 1)0 is readily
Adaptive Non-linear Tracking with Complete Compensation of Unknown Disturbances
obtained by virtue of Theorem 1. Then from (19) and (24) we readily conclude the boundedness of uy and Ud' The boundedness of all the closed-loop signals is proved. Furthermore, taking Assumption 2 into account, we conclude that the right parts of (21), (22) and (25) are locally Lipschitz in z, e and {f uniformly in t. Then the LaSalle-Y oshizawa theorem (Theorem 2.1. in [14]) guarantees the regulation z -+ and, as a result, asymptotic tracking y -+ y,. as t -+ 00. 0
137
Then for the parametrisation (16) we have ~ = ['T/o,,, 'T/O.2
+ V2, 'T/ 1," 'T/ 1,2] T
The control law design procedure gives the following expressions for the regulating component: Step 1
°
Step 2
5. Illustrative Example To demonstrate the efficiency of the proposed adaptive controller, we consider the system X, = X2 + (}cp(x,) X2 = )I(x) (u + d)
y=x,
where cp(x,) = xf, )I(X) = 2 + cos Xl> parameter () = 3 is unknown and the a priori uncertain deterministic disturbance d is assumed to be generated by the exosystem of the second order (i.e., d is presented by a harmonic signal with unknown frequency, amplitude and initial phase). The control objective is to asymptotically track the output of the reference model 25 y,. = (p + 5fr where p = d/dt denotes the differential operator and r = sin t. In this case, Eq. (17) takes the form dV2
1
dX 2
2 + cos X,
and, therefore, the function V2 can be defined as X2 V2(X) = -----"'----
2 + cos
X,
Choosing
G=[ -25° -101],b=[Oj,V(X)=[o] 1
The compensating component takes the form: Ud
=
-C {} -
{} = k 2(2
(2 + cos
X I )Z2
+ cos XI)Z2~
The simulation results are shown in Figs 1 and 2. The results are obtained with d(t) = 11sin1.4t Cl = C 2 = 5, k, = 5, k2 = 100 and zero initial states of the plant, reference model and all additional filters. The plots in Fig. 1 present the processes in the control system when the proposed component of adaptive compensation for unknown external disturbances Ud is turned off (i.e., the control signal contains the regulating component uy only). It is worth noting that in this case the adaptive controller changes into that proposed by Krstic et al. [14]. The non-zero residual tracking error is seen in Fig. 1(a). Figure 2 illustrates the processes in the closed-loop control system with adaptive compensation of external disturbances. It is seen from the presented plots that the proposed regulator ensures asymptotic tracking of the reference model in spite of parameter uncertainties of the plant and presence of an uncertain external disturbance acting on the system.
~
we obtain the following expressions for the observer of the external disturbance: 1Jo., = 'T/O,2 + V2, 1JO,2 = -25'T/o" - 1O'T/o,2 -
1J l,l
V2(10
+
V2
sin Xl) -
U
= 'T/ 1,2.
1J,,2 = -25'T/", - 1O'T/l,2 - (v2xf sin x,)!)I
6. Conclusion The problem of adaptive complete compensation of unknown external disturbances generated by the linear exosystem of the known order but with unknown parameters is solved for a class of nonlinear systems transformable into the parametric-
V. O. Nikiforov
138 Zl
2
u
100
1
50
0
0
-1
-50
-2
0
5
10
15
20
-100
25
0
5
time (sec)
10
15
20
25
time (sec)
Fig. 1. Non-zero residual tracking error due to the presence of external disturbances (adaptive compensation of the external disturbances is turned off: 11 = 11,.).
Zl
2
u
50
1
25
0
0
-1
-25
-2
0
5
10
15
20
25
-50
0
5
10
15
20
25
time (sec)
time (sec)
Fig. 2. Asymptotic tracking with complete compensation of unknown external disturbances (adaptive compensation of the external disturbances is switched on: 11 = 11,. + 11,,).
strict-feedback canonical form with unknown parameters. The main contributions of the paper are the following: (i) non-linear parameter-dependent observer for unknown external disturbances; (ii) nonlinear state-feedback adaptive regulator providing asymptotical tracking of the reference signal for the class of non-linear systems affected both by unknown constant parameters and uncertain external disturbances. It is worth noting that the total dynamic order of the proposed non-linear observer (8)-(9) is q(m + 1) (as we see it depends on both the dynamic order of the exogenous system q and the number of unknown parameters m). As a result, we need to tune q(m + I) adjustable parameters involved in the compensating component Ud of the adaptive regulator. In a certain sense we can talk about overparametrisation, that is, more than one adjustable parameter per unknown parameter. The same situation is faced when we design an adaptive regulator rejecting unknown deterministic disturbances in linear systems [16]. Thus, the overparametrisation is caused by the
complexity of the considered control problem, rather than by non-linearity of the plant model. The problem of removal of the overparametrisation deserves a deeper study and constitutes the perspective of further research.
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