- Email: [email protected]
Backstepping adaptive control of time-varying plants
Systems & Control Letters 36 (1999) 245–252
Backstepping adaptive control of time-varying plants F. Giri ∗ , A. Rabeh, F. Ikhouane Department of Electrical Engineering, Laboratoire d’Automatique, Ecole Mohammadia d’IngÃenieurs, B.P. 765 Agdal, Rabat, Morocco Received 25 March 1997; received in revised form 19 January 1998
Abstract In this paper, the problem of controlling linear plant whose parameters are unknown and time varying is dealt with using – for the ÿrst time – a backstepping adaptive controller. The parameter time variations are accounted for by including a switching - modiÿcation in the parameter update law. The resulting closed-loop system is shown to be locally stable and a bound on the asymptotic performance is established. The slower the plant variation, the larger the region of attraction and c 1999 Elsevier Science B.V. All rights reserved. the best the asymptotic performance. Keywords: Adaptive backstepping; Time-varying linear systems; -modiÿcation
1. Introduction The ultimate motivation of adaptive control theory is to provide solutions to the problem of controlling plants whose dynamics are unknown and time varying. However, most works reported in the literature have been devoted to the time-invariant case, see e.g. [14] and a reference list therein. The time-varying case has been addressed in the 1980s using certainty equivalence adaptive control schemes [1, 4, 11, 13]. Di erent classes of time-varying plants have been distinguished: stochastic parameters with constant expected value [1], exponentially fast converging parameters [4], ÿnite number parameter jumps [11], periodic parameters [13], general time variation parameters [2, 3, 8, 12, 15]. In general time-variation case, it is accounted in particular for continuous but slow time-variation as well as for infrequent jumps. The problem is dealt with using robustiÿed versions of the standard adaptive control schemes. Such a robustiÿcation is mainly achieved by introducing dead-zones, parameter projections, ∗
Corresponding author.
or parameter leakage in the parameter adaptation laws. In this paper, we consider the problem of adaptively controlling slowly varying linear plants, i.e. the plant model parameters are functions of time with bounded and continuous derivatives. The problem is dealt with using – for the ÿrst time – the backstepping design approach [10] along with switching -modiÿcation in the parameter update laws [7]. It is shown that if the plant parameter variation is suciently slow, then all the closed loop signals are uniformly bounded and the region of attraction is inversely proportional to the parameter variation rate. This means that the boundedness becomes global in the case of time-invariant parameters. Furthermore, the output tracking error is shown to be asymptotically proportional to the parameter variation rate. The paper is organized as follows: the control problem of concern is formulated in Section 2. An adaptive controller that meets the control objectives is designed in Section 3. The performance aspect is discussed in Section 4. Some concluding remarks and a reference list end the paper.
c 1999 Elsevier Science B.V. All rights reserved. 0167-6911/99/$ – see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 9 8 ) 0 0 0 7 3 - 5
246
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
2. Control problem statement We are interested in controlling linear time-varying systems that can be represented by the canonical observer form x˙ (t) = A0 x(t) + (k − a(t))y(t) + b(t)u(t); y = x1 with
(1)
Assumption 1 (1) The plant order n and its relative degree are known. (2) The sign of the plant high-frequency gain bm (t) is known and there exist a constant ¿0 such that |bm (t)|¿;
−k1 In−1 .. A0 = . ; −kn 0 : : : 0
Given the above notations, the controlled system description is completed by the following assumptions:
(4)
(3) The polynomial Bt (s) is uniformly Hurwitz, i.e. all zeros zi (t) (i = 1; : : : ; m) of Bt (s) satisfy the inequality: Re(zi (t))6−0 , for some constant 0 ¿0.
T
k = (k1 ; : : : ; kn ) ;
a(t) = (an−1 (t); : : : ; a0 (t))T ; b(t) = (0(−1)×1 ; bm (t); : : : ; b0 (t))T
∀t¿0:
(2)
where • n, an integer, is the plant order, its relative degree, and m = n − p, • u(t); y(t) and x(t) are the plant input, output and state, respectively, • ai (t); bi (t), are the unknown time-varying plant parameters, • k is real vector that can be freely assigned. It is chosen such that A0 is a stability matrix. Note that a large family of linear time-varying systems can be transformed to the canonical form (1) by a Lyapunov transformation [5]. Such a form was proposed in [9] for observable SISO linear time-invariant systems (a(t) and b(t) constants). To complete the plant description, let us introduce the following notations: At (s) = s n + an−1 (t)s n−1 + · · · + a1 (t)s + a0 (t); Bt (s) = bm (t)s m + · · · + b1 (t)s + b0 (t);
d ˙ )) = ;
(At (A0 )) = (At (A 0 1
dt ∞ ∞
d ˙ ))
(Bt (A0 )) = = 2 ; (3)
(Bt (A 0
dt ∞ ∞ where the real parameters 1 and 2 turn out to be together measures of the plant time-variation rate. It can be checked that in the case of time-invariant plants one has Bt (s) = B(s) and At (s) = A(s), for some constant polynomials B(s); A(s). In such a case, the plant (1) has a transfer function equal to B(s)=A(s). With this interpretation in mind, the reals 1 ; 2 can be viewed as the variation rate of the plant poles and zeros, respectively.
Assumption 2 (1) The time vector functions a(t) and b(t) are differentiable and bounded. Their derivatives are both piecewise continuous and bounded. (2) Upper bounds M and M% of k(t)k and k%(t)k = k1=bm (t)k are known, where (t) = (bm (t); : : : ; b0 (t); an−1 (t); : : : ; a0 (t))T is the unknown parameter vector. The controlled system (1) subject to assumptions A1–A2 will be referred to as the plant. Now let yr (t) be a reference signal such that • yr (t) and its ÿrst derivatives are known and bounded, • yr() is piecewise continuous and bounded. Our purpose is to design a controller such that if the parameter vector (t) is slowly varying then one has 1. All the closed-loop signals should remain bounded. 2. The output tracking error y(t) − yr (t) should be suciently small asymptotically. 3. Control law design First, the following notations will be used throughout the paper, unless otherwise stated: • c: a generic positive constant independent of the parameter variation rates (1 ; 2 ) and the initial conditions, • h: a generic constant scalar, vector or matrix constant independent of the initial conditions and uniformly bounded with respect to the parameter variation rates (1 ; 2 ). • : a generic bounded function of time, independent of the initial conditions and uniformly bounded with respect to the parameter variations rates (1 ; 2 ). Following closely the backstepping design procedure [10], the input and output signals are ÿrst ÿltered as
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
follows:
Table 1 Tuning functions design
˙ = A0 + en y; ˙ = A0 + en u;
(5)
ei is the ith basis vector in the R n . The signals and obtained yield the state estimate deÿned by x(t) ˆ = Bt (A0 ) − At (A0 );
(6)
where Bt (X ) and At (X ) are as deÿned in Eq. (3). It can easily be checked that the resulting state estimation error = x − xˆ undergoes the di erential equation: ˙ = A0 + D(t)
z1 = y − yr (i−1) zi = vm; i − %y ˆ r − i−1 ;
d d (At (A0 )) − (Bt (A0 )): dt dt Let us introduce the vectors [11; p: 421]:
(8)
vj = =
−[A0n−1 ; : : : ; ];
i = −zi−1 − ci + di + i + i =
06j6m;
! = [vm; 2 ; : : : ; v0; 2 ; (2) −
i P
+
@ i−1 (j−1)
P
m+i−1 j=1
ye1T ]T ;
2
@ i−1 @y
2
+
@ i−1 @%ˆ
i = i−1 −
where %(t) = 1=bm (t) and, ¿0 and = ¿0, are the adaptation gains. The switching -modiÿcation used to deal with the plant parameter time variation and to insure perfect tracking in the absence of timevarying e ect is deÿned as ˆ 0 if kk6M ; ˆ if k k¿2M s ; (11) = smooth connecting function otherwise 0 if k%k6M ˆ %; ˆ s% if k%k¿2M %; (12) % = smooth connecting function otherwise; for some design positive constants s and s% .
@ i−1 zj @y
%ˆ˙
where vi; 2 (i = 0; : : : ; m) is the second component of the vector vi deÿned in Eq. (9). Then, the control law and the parameter update laws, designed in steps are summarized in Table 1, where the parameter update laws contains a switching -modiÿcation [6, 7], i.e:
(10)
zi
+ ki vm; 1
1 = (! − %( ˆ 1 + y˙r )e1 )z1 − ˆ
T
@ 1 2 @ˆ
@ i−1 (−kj 1 + j+1 ) @j
(i−1)
+ yr
(j)
yr
! = [0; vm−1; 2 ; : : : ; v0; 2 ; (2) − ye1T ]T ;
ˆ˙ = ; ˆ %ˆ˙ = − sgn(bm )( 1 + y˙ r )z1 − % %;
z2 + 2 +
i−1 P @ i−1 @ i−1 i − ˆ @ˆ j=2 @
@yr
j=1
(9)
@ 1 @y
@ i−1 ˆ + @ i−1 (A0 + en y) (2 + !T ) @y @ +
= −A0n = [1 ; : : : ; n ]T ;
2 = −bˆm z1 − c2 + d2
(7)
D(t) =
i = 2; : : : ;
1 = %ˆ 1 ; 1 = −(c1 + d1 )z1 − 2 − ! T ˆ
with
A0j ;
247
@ i−1 !zi ; @y
i = 2; : : : ;
Adaptive control law : () ˆ r u = − vm; +1 + %y
4. Closed-loop stability and performance analysis In this section, we will quantify the e ect of the plant time variation. We will show that the closed loop is locally uniformly stable and we give an estimate of the region of attraction. Then, we will derive an upper bound on the asymptotic performance of the adaptive closed-loop system. The stability analysis is carried out by considering the similarity transformation T (t), [10; p: 343]: T (t) = [Ab (t)e1 ; : : : ; Ab (t)e1 ; Im ]; −bm−1 (t)=bm (t) IM −1 .. Ab (t) = : . 0:::0 −b0 (t)=bm (t)
(13)
248
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
that for • 1 61∗ and 2 62∗
Plant (1) can then be transformed to the form x˙ 1 = x2 − an−1 (t)x1 ;
• kyr k∞ + ky˙ r k∞ + · · · + kyr() k∞ 6c=21=q
x˙ 2 = x3 − an−2 (t)x1 ; .. .
• k (0)k2 6c=21=q we have: 1. All the signals of the closed loop are bounded. 2. The output tracking error is proportional to the parameter variation rates:
x˙ −1 = x − am+1 (t)x1 ; x˙ = cbT x − am (t)x1 + bm (t)u; ˙ = Ab (t) + bb (t)x1 + T˙ (t)x; y = x1 ;
(14)
where x = (x1 ; : : : ; x ; T )T ;
t→∞
√ c(12 + + % + 22 ) + 2 2 : 6
Proof. Just as in [11] let us consider the following Lyapunov function candidate: X 1 T 1 2 z + P V = 0 2 j dj
= T (t)x; bb (t) = T (t)A b(t) − a(t); 0 In−1 A = ... ; 0
lim sup(y(t) − yr (t))2
j=1
:::0
+
n
cb ∈ R : The term T˙ (t)x does not appear in [10; p: 436] because T is time invariant in [10]. For the stability analysis, we are interested in the deviation ˜ = − r which is governed by ˜˙ = Ab (t)˜ + bb (t)z1 + T˙ (t)x;
(15)
r (0) = (0):
(16)
(0) is any initial condition for (t) ((0) need not be known). For the variable, we deÿne analogously ˜ = − r which is governed by ˜˙ = A0 ˜ + en z1 ˙r = A0 r + en yr ;
r (0) = (0):
(17)
The closed-loop state vector is deÿned as follows: T T ˜ T: = (z T ; T ; ˜T ; ˜ ; ˜ ; %)
(18)
We are now ready to present the main result of this paper. Theorem 1. Consider plant (1) subject to assumptions A1–A2 in closed loop with the controller of Table 1. There exist positive constants ; c; 1∗ and 2∗ independent of 1 ; 2 and the initial conditions such
−1
ˆ ( − ):
(19)
Noting that d T ( P0 ) = − T + 2T P0 D(t) (20) dt and using Eq. (10) and Table 1, it follows that V˙ 6 −
X j=1
where r is deÿned as ˙r = Ab (t)r + bb (t)yr ;
|bm | ˆT (% − %) ˆ 2 + 12 ( − ) 2
cj zj2 −
X 1
() dj j=1
T
ˆ + ˜ −1 (˙ + ) 2 |bm | ˙ ) %( ˜ %˙ + % %) ˆ + T P0 At (A + 0
d0 2 ˙ ); − T P0 Bt (A 0 d0
(21)
where ˙ and %˙ denote time derivative of and %, re˜ %˜ = % − %ˆ and spectively, ˜ = − ; −1 1 1 + ··· + ; d0 = d1 d
() = 34 (12 + · · · + n2 ):
(22)
Eq. (21) di ers from its counterpart in [10; p: 431] mainly by the presence of the perturbation terms con˙ ). These terms are not ˙ ), and B (A ˙ At (A taining , 0 t 0 bounded a priori (because of the presence of the partial state ; ; ) and the subsequent analysis is devoted to the study of their e ect on the closed-loop stability.
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
Lyapunov function (19) does not contain the whole state vector , thus we need to augment it as follows (see [6, 10]): V = V + where Pb (t) =
Z
1 T 1 T ˜ ˜ P0 ˜ + ˜ Pb (t); k k +∞
0
T
eAb (t) eAb (t) d
(23)
(24)
Pb (t) is such that Pb (t)Ab (t) + ATb (t)Pb (t) = − Im ; and according to the assumptions A1(3) and A2(1), 1=∗ ¿ min (Pb (t))¿∗ for some positive constant ∗ (this property will be used in Eq. (30)). Note that V is a quadratic function V = T P of the vector , for some time varying matrix P : On the other hand, using Eqs. (21) and (23) we get c1 V˙ 6 − z12 − 2
X
cj zj2 −
j=2
X 1
() dj
2 1 kk ˜ 2 + ˜T P0 en z1 k k 1 ˜ 2 2 T − kk + ˜ Pb (t)bb (t)z1 k k 2 T 1 T + ˜ Pb (t)T˙ (t)x + ˜ P˙b (t)˜ k k
−
c1 V˙ 6 − z12 − 2
j=2
cj zj2
1 kk ˜ 2+ 4k 1 − kk ˜ 2+ 4k
2 T c1 ˜ P0 en z1 − z12 k 4 2 T ˙ )˜ − 1 () P0 At (A 0 d0 4d0
4kPb bb k2 c1
(29)
V˙ 6− V + + c22 kk2 + c22 kxk2 ;
(30)
where ( (25)
= inf
c1 ; 2c2 ; : : : ; 2c ; 4 2 s% ; |bm |
(26)
−1 (P0 ) min ; 4 )
2s −1 min ( −1 )
= c(12 + 22 + s + s% ):
−
=−
(28)
which yields
1 ˜ 2 1 kk ˜ 2− kk − 2k 2k
where
For 1 , suciently small (1 6((3=210 ) (d0 =kP0 k) × (1=kP0 en k2 ))1=2 ), there exist a positive constant k such that
k ¿
1 −
() 2d0
˜ 2 s kk %˜2 − |bm (t)|s% ; 2 2 +
(27)
We also choose k such that
which implies X
1 2 ˙ )
() + T P0 At (A 0 r 8d0 d0 1 2 ˙ ) −
() − T P0 Bt (A 0 8d0 d0 2 T c1 1 ˜ 2 kk + ˜ Pb bb z1 − z12 − 4k k 4 1 T 1 ˜ 2 2 T − kk + ˜ Pb T˙ x + ˜ P˙b ˜ 4k k k ˜ 2 T kk ˆ + ˜ −1 (˙ + ) +s 2 |bm (t)| %˜2 %( ˜ %˙ + % %): ˆ +|bm (t)|s% + 2
−
3 d0 16kP0 en k2 6k 6 : c1 64 kP0 k12
j=1
T ˆ +˜ −1 (˙ + ) |bm | %( ˜ %˙ + % %) ˆ +
2 ˙ ) − 2 T P B (A ˙ ); + T P0 At (A 0 0 0 t d0 d0
249
1 ; 2∗
; (31)
The perturbation terms c22 kk2 and c22 kxk2 are not bounded a priori. They are not even upper bounded by some simple function of the state vector (Recall V = k k2P ): To see that these perturbations terms are bounded by a C 1 function of the state vector , a more elaborate analysis is needed, this is what the following is about. We concentrate on the e ect of the residual term c2 kk2 + c2 kxk2 in Eq. (30). This term does not depend explicitly on the state vector . Thus, we need to bound it by a function of . To this end, we deÿne
250
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
Yr and m as Yr = kyr k∞ + ky˙ r k∞ + · · · + kyr() k∞ ; m = (1 ; : : : ; m )T :
(32)
It was shown in [10, p. 345] that if Bt (s) and K(s) = sn + k1 sn−1 + · · · + kn are uniformly coprime then m = h + h˜ + h˜ + ; m+1 = h + :
(33)
Note that the coprimeness condition is realized with probability one. We may force K(s) and Bt (s) to be coprime by knowing an upper bound on the norm of b(t), see [6]. On the other hand, we have from Table 1: ˆ r(i−1) : i−1 = (Am 0 )i − zi − %y From Eq. (2) it follows T (Am 0 )i = m+i + gm; i
1 .. .
(34)
(35)
m+i−1
Fig. 1.
bounded by a simple function of the state , namely Vfq+1 . In the following analysis, we will show that Vf is uniformly bounded within a region of attraction proportional to 1=2 : To this end, we deÿne D1 = Vf
and
C1 = c2 Vfq+1 + ∗ :
Since i−1 is a multilinear polynomial function of state variables, of y; yr and its ÿrst derivatives, it follows from Eqs. (36) and (33) that
Fig. 1 shows the curves D1 and C1 as function of Vf . It is readily seen that for small values of 2 the curves D and C have two intersection points, whose Vf -coordinates are vf1 and vf2 , respectively. If Vf (0) ∈ [0; vf2 ] then lim supt→∞ Vf (t) will be smaller than vf1 . This means that the region of attraction for the adaptive closed loop system is deÿned by vf2 , the value of vf2 can be bounded from below by vf3 , the Vf -coordinate of the intersection point between C1 -curve and its D1 -parallel tangent. The value of vf3 can simply be obtained by solving the equation dC1 =dVf = which implies
kk2 6c(V + Yr )q+1
vf3 =
for some vector gm; i ∈ Rm+i−1 . Combining Eqs. (34) and (35), we obtain 1 .. + (y; y ; y˙ ; : : : ; y(i−1) ; T m+i = − gm; i−1 r i r r . m+i−1
ˆ ; 1 ; : : : ; m+i−1 ) + zi + %y ˆ r(i−1) : %; ˆ ;
(36)
(37)
for some positive integer q. It is shown in [10, p. 345] that x is a linear function of and , which yields kxk2 6c(V + Yr )q+1 :
(38)
Combining Eqs. (37), (38) and (30) we get V˙f 6− Vf + ∗ + c2 Vfq+1
(39)
with ∗ = + Yr ; Vf = V + Yr :
(40)
Up till now, we have shown that the perturbations terms 2 kk2 and 2 kxk2 in Eq. (30) can be uniformly
c 21=q
:
(41)
This shows that Vf is uniformly bounded, provided Vf (0) ∈ [0; vf2 ] and 1 ; 2 are suciently small. It then follows that the state vectors ; and x are uniformly bounded. Hence, all signals of the closed-loop system are bounded and the region of attraction is proportional to 1=21=q . It is worth noting that the size of the region of attraction is limited by the variation of the plant zeros 2 . Analytical expressions of vf1 and vf2 cannot be found as we know from ring theory that analytical solutions for the roots of a polynomial whose degree is higher than 4 cannot be obtained. Hence, we need to ÿnd bounds on these values. The following analysis gives a bound on vf1 :
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
˜ k− ˜ Using the fact that % |%|(| ˜ %|−|%|)¿0 ˜ and kk(k kk)¿0 yields
We notice the following from Eq. (39) √ ∗ + 2 ⇒
Vf =
√ V˙f 6 − 2 + c2
V˙ 6 −
√ ∗ + 2 q+1 60
t→∞
√ c(12 + +% +22 ) + 2 2 2 : = lim sup z1 6 t→∞
It is worth noting that, by choosing appropriately the design parameters, the tracking error z1 can be made only dependent of the plant time variation velocities 1 and 2 . Consequently, the slower the plant time-variation the better the asymptotic performances. Moreover, it follows from Eq. (41) that the reference signal must verify c : 1=q
In the particular case where the plant parameters are time-invariant, the previous stability results become global and the tracking error converges asymptotically to zero. This is stated in the following corollary. Corollary 2. Consider the plant (1) subject to assumptions A1–A2, along with the controller of Table 1; if the parameters (t) = and %(t) = % are time invariant, then the closed-loop system is uniformly globally stable and limt→∞ (y(t)−yr (t)) = 0. Proof. Starting from Eq. (21) and putting ˙ = 0 and %˙ = 0 we get X
cj zj2 −
j=1
˜ % %: ˆ +|bm |%
cj zj2 −
X 1
(): dj
(43)
j=1
The rest of proof is standard (see [10]).
lim sup(y(t) − yr (t))2
V˙ 6 −
X j=1
for suciently small 2 ( ∗ and do not depend on 2 ). This clearly implies that vf1 is upper bounded by the above value of Vf . We now focus on the asymptotic performance of the closed loop adaptive system. Since lim supt→∞ Vf (t)6vf1 ; the asymptotic performance is deÿned by vf1 : Hence, one has in particular
Yr 6
251
X T 1
() + ˜ ˆ dj j=1
(42)
5. Concluding remarks We have considered the problem of adaptively controlling linear plants whose parameters are unknown and time-varying. The closed-loop analysis di ers from the one in [10, Ch. 10] by the presence of perturbation terms that are not bounded a priori as they are function of the state vector. These perturbation terms do not appear in [10] where the linear system to be controlled is time-invariant. We have showed in this paper that the perturbation terms resulting for the time-variation of the parameters can be dealt with using a switching -modiÿcation (this modiÿcation does not appear in [10] and is used in a backstepping context for the ÿrst time in [6]). The resulting closed loop is shown to be locally uniformly stable with a region of attraction which is inversely proportional to the rate of the parameters time-variation. On the other hand, the smaller the variation rates, 1 and 2 , the better the asymptotic performance. Loss of globality, mostly due to the strong nonlinear feature of the proposed controller, is a topic for further research. References [1] H.F. Chen, P.E. Caines, On the adaptive control of stochastic systems with random parameters, Proc. 23rd Conf. Decision Control, 1984, p. 39. [2] Ph. De Larminat, H.F. Raynaud, A robust solution to the stabilizability problem in indirect passive adaptive control, Proc. 25th Conf. Decision Control, 1986, pp. 462– 467. [3] F. Giri, M. M’saad, L. Dugard, J.M. Dion, Pole placement direct adaptive control for time varying ill-modeled plants, IEEE Trans. Automat. Control, 35 (1990) 723–726. [4] G.C. Goodwin, D.J. Hill, X. Xianya, Stochastic adaptive control for exponentially convergent time-varying systems, Proc. 23rd Conf. Decision Control, 1984, p. 39. [5] M. Gevers, I.M.Y. Mareels, G. Bastin, Robustness of adaptive observers for time varying systems, Proc. IFAC Workshop on Robust Adaptive Control, Newcastle, 22–24 August, 1988. [6] F. Ikhouane, M. Krstic, Robustness of the tuning functions adaptive backstepping design for linear systems, Proc. Conf. Decision Control, New Orleans, December, 1995. [7] P.A. Ioannou, J. Sun, Stable and Robust Adaptive Control, Prentice-Hall, Englewood Cli s, NJ, 1995.
252
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
[8] G. Kreisselmeier, Adaptive control of a class of time varying plants, University of Kassel, Rep. 1986. [9] G. Kreisselmeier, Adaptive observers with exponential rate of convergence, IEEE Trans Automat. Control 22 (1977) 2–8. [10] M. Krstic, I. Kanellakopoulos, P.V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. [11] J.M. Martin-Sanchez, Adaptive control of time-variant processes, Int. J. Control 44 (1) (1986). [12] R.H. Middleton, G.C. Goodwin, Adaptive control of timevarying linear systems, IEEE Trans. Aut. Control 33 (2) (1988) 150–155.
[13] F. Okhawa, Model reference adaptive control systems for linear time varying systems with periodically varying parameters and time delays, Int. J. Control 33 (2) (1988). [14] R. Ortega, Y. Tang, Robustness of adaptive controllers – A survey, Automatica 25 (5) (1989) 651– 677. [15] K. Tsakalis, P.A. Ioannou, Adaptive control of linear timevarying plants, Proc. IFAC Workshop on Adaptive Syst. Contr. Sign. Processing, Lund, Sweden, 1986.
Backstepping adaptive control of time-varying plants F. Giri ∗ , A. Rabeh, F. Ikhouane Department of Electrical Engineering, Laboratoire d’Automatique, Ecole Mohammadia d’IngÃenieurs, B.P. 765 Agdal, Rabat, Morocco Received 25 March 1997; received in revised form 19 January 1998
Abstract In this paper, the problem of controlling linear plant whose parameters are unknown and time varying is dealt with using – for the ÿrst time – a backstepping adaptive controller. The parameter time variations are accounted for by including a switching - modiÿcation in the parameter update law. The resulting closed-loop system is shown to be locally stable and a bound on the asymptotic performance is established. The slower the plant variation, the larger the region of attraction and c 1999 Elsevier Science B.V. All rights reserved. the best the asymptotic performance. Keywords: Adaptive backstepping; Time-varying linear systems; -modiÿcation
1. Introduction The ultimate motivation of adaptive control theory is to provide solutions to the problem of controlling plants whose dynamics are unknown and time varying. However, most works reported in the literature have been devoted to the time-invariant case, see e.g. [14] and a reference list therein. The time-varying case has been addressed in the 1980s using certainty equivalence adaptive control schemes [1, 4, 11, 13]. Di erent classes of time-varying plants have been distinguished: stochastic parameters with constant expected value [1], exponentially fast converging parameters [4], ÿnite number parameter jumps [11], periodic parameters [13], general time variation parameters [2, 3, 8, 12, 15]. In general time-variation case, it is accounted in particular for continuous but slow time-variation as well as for infrequent jumps. The problem is dealt with using robustiÿed versions of the standard adaptive control schemes. Such a robustiÿcation is mainly achieved by introducing dead-zones, parameter projections, ∗
Corresponding author.
or parameter leakage in the parameter adaptation laws. In this paper, we consider the problem of adaptively controlling slowly varying linear plants, i.e. the plant model parameters are functions of time with bounded and continuous derivatives. The problem is dealt with using – for the ÿrst time – the backstepping design approach [10] along with switching -modiÿcation in the parameter update laws [7]. It is shown that if the plant parameter variation is suciently slow, then all the closed loop signals are uniformly bounded and the region of attraction is inversely proportional to the parameter variation rate. This means that the boundedness becomes global in the case of time-invariant parameters. Furthermore, the output tracking error is shown to be asymptotically proportional to the parameter variation rate. The paper is organized as follows: the control problem of concern is formulated in Section 2. An adaptive controller that meets the control objectives is designed in Section 3. The performance aspect is discussed in Section 4. Some concluding remarks and a reference list end the paper.
c 1999 Elsevier Science B.V. All rights reserved. 0167-6911/99/$ – see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 9 8 ) 0 0 0 7 3 - 5
246
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
2. Control problem statement We are interested in controlling linear time-varying systems that can be represented by the canonical observer form x˙ (t) = A0 x(t) + (k − a(t))y(t) + b(t)u(t); y = x1 with
(1)
Assumption 1 (1) The plant order n and its relative degree are known. (2) The sign of the plant high-frequency gain bm (t) is known and there exist a constant ¿0 such that |bm (t)|¿;
−k1 In−1 .. A0 = . ; −kn 0 : : : 0
Given the above notations, the controlled system description is completed by the following assumptions:
(4)
(3) The polynomial Bt (s) is uniformly Hurwitz, i.e. all zeros zi (t) (i = 1; : : : ; m) of Bt (s) satisfy the inequality: Re(zi (t))6−0 , for some constant 0 ¿0.
T
k = (k1 ; : : : ; kn ) ;
a(t) = (an−1 (t); : : : ; a0 (t))T ; b(t) = (0(−1)×1 ; bm (t); : : : ; b0 (t))T
∀t¿0:
(2)
where • n, an integer, is the plant order, its relative degree, and m = n − p, • u(t); y(t) and x(t) are the plant input, output and state, respectively, • ai (t); bi (t), are the unknown time-varying plant parameters, • k is real vector that can be freely assigned. It is chosen such that A0 is a stability matrix. Note that a large family of linear time-varying systems can be transformed to the canonical form (1) by a Lyapunov transformation [5]. Such a form was proposed in [9] for observable SISO linear time-invariant systems (a(t) and b(t) constants). To complete the plant description, let us introduce the following notations: At (s) = s n + an−1 (t)s n−1 + · · · + a1 (t)s + a0 (t); Bt (s) = bm (t)s m + · · · + b1 (t)s + b0 (t);
d ˙ )) = ;
(At (A0 )) = (At (A 0 1
dt ∞ ∞
d ˙ ))
(Bt (A0 )) = = 2 ; (3)
(Bt (A 0
dt ∞ ∞ where the real parameters 1 and 2 turn out to be together measures of the plant time-variation rate. It can be checked that in the case of time-invariant plants one has Bt (s) = B(s) and At (s) = A(s), for some constant polynomials B(s); A(s). In such a case, the plant (1) has a transfer function equal to B(s)=A(s). With this interpretation in mind, the reals 1 ; 2 can be viewed as the variation rate of the plant poles and zeros, respectively.
Assumption 2 (1) The time vector functions a(t) and b(t) are differentiable and bounded. Their derivatives are both piecewise continuous and bounded. (2) Upper bounds M and M% of k(t)k and k%(t)k = k1=bm (t)k are known, where (t) = (bm (t); : : : ; b0 (t); an−1 (t); : : : ; a0 (t))T is the unknown parameter vector. The controlled system (1) subject to assumptions A1–A2 will be referred to as the plant. Now let yr (t) be a reference signal such that • yr (t) and its ÿrst derivatives are known and bounded, • yr() is piecewise continuous and bounded. Our purpose is to design a controller such that if the parameter vector (t) is slowly varying then one has 1. All the closed-loop signals should remain bounded. 2. The output tracking error y(t) − yr (t) should be suciently small asymptotically. 3. Control law design First, the following notations will be used throughout the paper, unless otherwise stated: • c: a generic positive constant independent of the parameter variation rates (1 ; 2 ) and the initial conditions, • h: a generic constant scalar, vector or matrix constant independent of the initial conditions and uniformly bounded with respect to the parameter variation rates (1 ; 2 ). • : a generic bounded function of time, independent of the initial conditions and uniformly bounded with respect to the parameter variations rates (1 ; 2 ). Following closely the backstepping design procedure [10], the input and output signals are ÿrst ÿltered as
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
follows:
Table 1 Tuning functions design
˙ = A0 + en y; ˙ = A0 + en u;
(5)
ei is the ith basis vector in the R n . The signals and obtained yield the state estimate deÿned by x(t) ˆ = Bt (A0 ) − At (A0 );
(6)
where Bt (X ) and At (X ) are as deÿned in Eq. (3). It can easily be checked that the resulting state estimation error = x − xˆ undergoes the di erential equation: ˙ = A0 + D(t)
z1 = y − yr (i−1) zi = vm; i − %y ˆ r − i−1 ;
d d (At (A0 )) − (Bt (A0 )): dt dt Let us introduce the vectors [11; p: 421]:
(8)
vj = =
−[A0n−1 ; : : : ; ];
i = −zi−1 − ci + di + i + i =
06j6m;
! = [vm; 2 ; : : : ; v0; 2 ; (2) −
i P
+
@ i−1 (j−1)
P
m+i−1 j=1
ye1T ]T ;
2
@ i−1 @y
2
+
@ i−1 @%ˆ
i = i−1 −
where %(t) = 1=bm (t) and, ¿0 and = ¿0, are the adaptation gains. The switching -modiÿcation used to deal with the plant parameter time variation and to insure perfect tracking in the absence of timevarying e ect is deÿned as ˆ 0 if kk6M ; ˆ if k k¿2M s ; (11) = smooth connecting function otherwise 0 if k%k6M ˆ %; ˆ s% if k%k¿2M %; (12) % = smooth connecting function otherwise; for some design positive constants s and s% .
@ i−1 zj @y
%ˆ˙
where vi; 2 (i = 0; : : : ; m) is the second component of the vector vi deÿned in Eq. (9). Then, the control law and the parameter update laws, designed in steps are summarized in Table 1, where the parameter update laws contains a switching -modiÿcation [6, 7], i.e:
(10)
zi
+ ki vm; 1
1 = (! − %( ˆ 1 + y˙r )e1 )z1 − ˆ
T
@ 1 2 @ˆ
@ i−1 (−kj 1 + j+1 ) @j
(i−1)
+ yr
(j)
yr
! = [0; vm−1; 2 ; : : : ; v0; 2 ; (2) − ye1T ]T ;
ˆ˙ = ; ˆ %ˆ˙ = − sgn(bm )( 1 + y˙ r )z1 − % %;
z2 + 2 +
i−1 P @ i−1 @ i−1 i − ˆ @ˆ j=2 @
@yr
j=1
(9)
@ 1 @y
@ i−1 ˆ + @ i−1 (A0 + en y) (2 + !T ) @y @ +
= −A0n = [1 ; : : : ; n ]T ;
2 = −bˆm z1 − c2 + d2
(7)
D(t) =
i = 2; : : : ;
1 = %ˆ 1 ; 1 = −(c1 + d1 )z1 − 2 − ! T ˆ
with
A0j ;
247
@ i−1 !zi ; @y
i = 2; : : : ;
Adaptive control law : () ˆ r u = − vm; +1 + %y
4. Closed-loop stability and performance analysis In this section, we will quantify the e ect of the plant time variation. We will show that the closed loop is locally uniformly stable and we give an estimate of the region of attraction. Then, we will derive an upper bound on the asymptotic performance of the adaptive closed-loop system. The stability analysis is carried out by considering the similarity transformation T (t), [10; p: 343]: T (t) = [Ab (t)e1 ; : : : ; Ab (t)e1 ; Im ]; −bm−1 (t)=bm (t) IM −1 .. Ab (t) = : . 0:::0 −b0 (t)=bm (t)
(13)
248
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
that for • 1 61∗ and 2 62∗
Plant (1) can then be transformed to the form x˙ 1 = x2 − an−1 (t)x1 ;
• kyr k∞ + ky˙ r k∞ + · · · + kyr() k∞ 6c=21=q
x˙ 2 = x3 − an−2 (t)x1 ; .. .
• k (0)k2 6c=21=q we have: 1. All the signals of the closed loop are bounded. 2. The output tracking error is proportional to the parameter variation rates:
x˙ −1 = x − am+1 (t)x1 ; x˙ = cbT x − am (t)x1 + bm (t)u; ˙ = Ab (t) + bb (t)x1 + T˙ (t)x; y = x1 ;
(14)
where x = (x1 ; : : : ; x ; T )T ;
t→∞
√ c(12 + + % + 22 ) + 2 2 : 6
Proof. Just as in [11] let us consider the following Lyapunov function candidate: X 1 T 1 2 z + P V = 0 2 j dj
= T (t)x; bb (t) = T (t)A b(t) − a(t); 0 In−1 A = ... ; 0
lim sup(y(t) − yr (t))2
j=1
:::0
+
n
cb ∈ R : The term T˙ (t)x does not appear in [10; p: 436] because T is time invariant in [10]. For the stability analysis, we are interested in the deviation ˜ = − r which is governed by ˜˙ = Ab (t)˜ + bb (t)z1 + T˙ (t)x;
(15)
r (0) = (0):
(16)
(0) is any initial condition for (t) ((0) need not be known). For the variable, we deÿne analogously ˜ = − r which is governed by ˜˙ = A0 ˜ + en z1 ˙r = A0 r + en yr ;
r (0) = (0):
(17)
The closed-loop state vector is deÿned as follows: T T ˜ T: = (z T ; T ; ˜T ; ˜ ; ˜ ; %)
(18)
We are now ready to present the main result of this paper. Theorem 1. Consider plant (1) subject to assumptions A1–A2 in closed loop with the controller of Table 1. There exist positive constants ; c; 1∗ and 2∗ independent of 1 ; 2 and the initial conditions such
−1
ˆ ( − ):
(19)
Noting that d T ( P0 ) = − T + 2T P0 D(t) (20) dt and using Eq. (10) and Table 1, it follows that V˙ 6 −
X j=1
where r is deÿned as ˙r = Ab (t)r + bb (t)yr ;
|bm | ˆT (% − %) ˆ 2 + 12 ( − ) 2
cj zj2 −
X 1
() dj j=1
T
ˆ + ˜ −1 (˙ + ) 2 |bm | ˙ ) %( ˜ %˙ + % %) ˆ + T P0 At (A + 0
d0 2 ˙ ); − T P0 Bt (A 0 d0
(21)
where ˙ and %˙ denote time derivative of and %, re˜ %˜ = % − %ˆ and spectively, ˜ = − ; −1 1 1 + ··· + ; d0 = d1 d
() = 34 (12 + · · · + n2 ):
(22)
Eq. (21) di ers from its counterpart in [10; p: 431] mainly by the presence of the perturbation terms con˙ ). These terms are not ˙ ), and B (A ˙ At (A taining , 0 t 0 bounded a priori (because of the presence of the partial state ; ; ) and the subsequent analysis is devoted to the study of their e ect on the closed-loop stability.
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
Lyapunov function (19) does not contain the whole state vector , thus we need to augment it as follows (see [6, 10]): V = V + where Pb (t) =
Z
1 T 1 T ˜ ˜ P0 ˜ + ˜ Pb (t); k k +∞
0
T
eAb (t) eAb (t) d
(23)
(24)
Pb (t) is such that Pb (t)Ab (t) + ATb (t)Pb (t) = − Im ; and according to the assumptions A1(3) and A2(1), 1=∗ ¿ min (Pb (t))¿∗ for some positive constant ∗ (this property will be used in Eq. (30)). Note that V is a quadratic function V = T P of the vector , for some time varying matrix P : On the other hand, using Eqs. (21) and (23) we get c1 V˙ 6 − z12 − 2
X
cj zj2 −
j=2
X 1
() dj
2 1 kk ˜ 2 + ˜T P0 en z1 k k 1 ˜ 2 2 T − kk + ˜ Pb (t)bb (t)z1 k k 2 T 1 T + ˜ Pb (t)T˙ (t)x + ˜ P˙b (t)˜ k k
−
c1 V˙ 6 − z12 − 2
j=2
cj zj2
1 kk ˜ 2+ 4k 1 − kk ˜ 2+ 4k
2 T c1 ˜ P0 en z1 − z12 k 4 2 T ˙ )˜ − 1 () P0 At (A 0 d0 4d0
4kPb bb k2 c1
(29)
V˙ 6− V + + c22 kk2 + c22 kxk2 ;
(30)
where ( (25)
= inf
c1 ; 2c2 ; : : : ; 2c ; 4 2 s% ; |bm |
(26)
−1 (P0 ) min ; 4 )
2s −1 min ( −1 )
= c(12 + 22 + s + s% ):
−
=−
(28)
which yields
1 ˜ 2 1 kk ˜ 2− kk − 2k 2k
where
For 1 , suciently small (1 6((3=210 ) (d0 =kP0 k) × (1=kP0 en k2 ))1=2 ), there exist a positive constant k such that
k ¿
1 −
() 2d0
˜ 2 s kk %˜2 − |bm (t)|s% ; 2 2 +
(27)
We also choose k such that
which implies X
1 2 ˙ )
() + T P0 At (A 0 r 8d0 d0 1 2 ˙ ) −
() − T P0 Bt (A 0 8d0 d0 2 T c1 1 ˜ 2 kk + ˜ Pb bb z1 − z12 − 4k k 4 1 T 1 ˜ 2 2 T − kk + ˜ Pb T˙ x + ˜ P˙b ˜ 4k k k ˜ 2 T kk ˆ + ˜ −1 (˙ + ) +s 2 |bm (t)| %˜2 %( ˜ %˙ + % %): ˆ +|bm (t)|s% + 2
−
3 d0 16kP0 en k2 6k 6 : c1 64 kP0 k12
j=1
T ˆ +˜ −1 (˙ + ) |bm | %( ˜ %˙ + % %) ˆ +
2 ˙ ) − 2 T P B (A ˙ ); + T P0 At (A 0 0 0 t d0 d0
249
1 ; 2∗
; (31)
The perturbation terms c22 kk2 and c22 kxk2 are not bounded a priori. They are not even upper bounded by some simple function of the state vector (Recall V = k k2P ): To see that these perturbations terms are bounded by a C 1 function of the state vector , a more elaborate analysis is needed, this is what the following is about. We concentrate on the e ect of the residual term c2 kk2 + c2 kxk2 in Eq. (30). This term does not depend explicitly on the state vector . Thus, we need to bound it by a function of . To this end, we deÿne
250
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
Yr and m as Yr = kyr k∞ + ky˙ r k∞ + · · · + kyr() k∞ ; m = (1 ; : : : ; m )T :
(32)
It was shown in [10, p. 345] that if Bt (s) and K(s) = sn + k1 sn−1 + · · · + kn are uniformly coprime then m = h + h˜ + h˜ + ; m+1 = h + :
(33)
Note that the coprimeness condition is realized with probability one. We may force K(s) and Bt (s) to be coprime by knowing an upper bound on the norm of b(t), see [6]. On the other hand, we have from Table 1: ˆ r(i−1) : i−1 = (Am 0 )i − zi − %y From Eq. (2) it follows T (Am 0 )i = m+i + gm; i
1 .. .
(34)
(35)
m+i−1
Fig. 1.
bounded by a simple function of the state , namely Vfq+1 . In the following analysis, we will show that Vf is uniformly bounded within a region of attraction proportional to 1=2 : To this end, we deÿne D1 = Vf
and
C1 = c2 Vfq+1 + ∗ :
Since i−1 is a multilinear polynomial function of state variables, of y; yr and its ÿrst derivatives, it follows from Eqs. (36) and (33) that
Fig. 1 shows the curves D1 and C1 as function of Vf . It is readily seen that for small values of 2 the curves D and C have two intersection points, whose Vf -coordinates are vf1 and vf2 , respectively. If Vf (0) ∈ [0; vf2 ] then lim supt→∞ Vf (t) will be smaller than vf1 . This means that the region of attraction for the adaptive closed loop system is deÿned by vf2 , the value of vf2 can be bounded from below by vf3 , the Vf -coordinate of the intersection point between C1 -curve and its D1 -parallel tangent. The value of vf3 can simply be obtained by solving the equation dC1 =dVf = which implies
kk2 6c(V + Yr )q+1
vf3 =
for some vector gm; i ∈ Rm+i−1 . Combining Eqs. (34) and (35), we obtain 1 .. + (y; y ; y˙ ; : : : ; y(i−1) ; T m+i = − gm; i−1 r i r r . m+i−1
ˆ ; 1 ; : : : ; m+i−1 ) + zi + %y ˆ r(i−1) : %; ˆ ;
(36)
(37)
for some positive integer q. It is shown in [10, p. 345] that x is a linear function of and , which yields kxk2 6c(V + Yr )q+1 :
(38)
Combining Eqs. (37), (38) and (30) we get V˙f 6− Vf + ∗ + c2 Vfq+1
(39)
with ∗ = + Yr ; Vf = V + Yr :
(40)
Up till now, we have shown that the perturbations terms 2 kk2 and 2 kxk2 in Eq. (30) can be uniformly
c 21=q
:
(41)
This shows that Vf is uniformly bounded, provided Vf (0) ∈ [0; vf2 ] and 1 ; 2 are suciently small. It then follows that the state vectors ; and x are uniformly bounded. Hence, all signals of the closed-loop system are bounded and the region of attraction is proportional to 1=21=q . It is worth noting that the size of the region of attraction is limited by the variation of the plant zeros 2 . Analytical expressions of vf1 and vf2 cannot be found as we know from ring theory that analytical solutions for the roots of a polynomial whose degree is higher than 4 cannot be obtained. Hence, we need to ÿnd bounds on these values. The following analysis gives a bound on vf1 :
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
˜ k− ˜ Using the fact that % |%|(| ˜ %|−|%|)¿0 ˜ and kk(k kk)¿0 yields
We notice the following from Eq. (39) √ ∗ + 2 ⇒
Vf =
√ V˙f 6 − 2 + c2
V˙ 6 −
√ ∗ + 2 q+1 60
t→∞
√ c(12 + +% +22 ) + 2 2 2 : = lim sup z1 6 t→∞
It is worth noting that, by choosing appropriately the design parameters, the tracking error z1 can be made only dependent of the plant time variation velocities 1 and 2 . Consequently, the slower the plant time-variation the better the asymptotic performances. Moreover, it follows from Eq. (41) that the reference signal must verify c : 1=q
In the particular case where the plant parameters are time-invariant, the previous stability results become global and the tracking error converges asymptotically to zero. This is stated in the following corollary. Corollary 2. Consider the plant (1) subject to assumptions A1–A2, along with the controller of Table 1; if the parameters (t) = and %(t) = % are time invariant, then the closed-loop system is uniformly globally stable and limt→∞ (y(t)−yr (t)) = 0. Proof. Starting from Eq. (21) and putting ˙ = 0 and %˙ = 0 we get X
cj zj2 −
j=1
˜ % %: ˆ +|bm |%
cj zj2 −
X 1
(): dj
(43)
j=1
The rest of proof is standard (see [10]).
lim sup(y(t) − yr (t))2
V˙ 6 −
X j=1
for suciently small 2 ( ∗ and do not depend on 2 ). This clearly implies that vf1 is upper bounded by the above value of Vf . We now focus on the asymptotic performance of the closed loop adaptive system. Since lim supt→∞ Vf (t)6vf1 ; the asymptotic performance is deÿned by vf1 : Hence, one has in particular
Yr 6
251
X T 1
() + ˜ ˆ dj j=1
(42)
5. Concluding remarks We have considered the problem of adaptively controlling linear plants whose parameters are unknown and time-varying. The closed-loop analysis di ers from the one in [10, Ch. 10] by the presence of perturbation terms that are not bounded a priori as they are function of the state vector. These perturbation terms do not appear in [10] where the linear system to be controlled is time-invariant. We have showed in this paper that the perturbation terms resulting for the time-variation of the parameters can be dealt with using a switching -modiÿcation (this modiÿcation does not appear in [10] and is used in a backstepping context for the ÿrst time in [6]). The resulting closed loop is shown to be locally uniformly stable with a region of attraction which is inversely proportional to the rate of the parameters time-variation. On the other hand, the smaller the variation rates, 1 and 2 , the better the asymptotic performance. Loss of globality, mostly due to the strong nonlinear feature of the proposed controller, is a topic for further research. References [1] H.F. Chen, P.E. Caines, On the adaptive control of stochastic systems with random parameters, Proc. 23rd Conf. Decision Control, 1984, p. 39. [2] Ph. De Larminat, H.F. Raynaud, A robust solution to the stabilizability problem in indirect passive adaptive control, Proc. 25th Conf. Decision Control, 1986, pp. 462– 467. [3] F. Giri, M. M’saad, L. Dugard, J.M. Dion, Pole placement direct adaptive control for time varying ill-modeled plants, IEEE Trans. Automat. Control, 35 (1990) 723–726. [4] G.C. Goodwin, D.J. Hill, X. Xianya, Stochastic adaptive control for exponentially convergent time-varying systems, Proc. 23rd Conf. Decision Control, 1984, p. 39. [5] M. Gevers, I.M.Y. Mareels, G. Bastin, Robustness of adaptive observers for time varying systems, Proc. IFAC Workshop on Robust Adaptive Control, Newcastle, 22–24 August, 1988. [6] F. Ikhouane, M. Krstic, Robustness of the tuning functions adaptive backstepping design for linear systems, Proc. Conf. Decision Control, New Orleans, December, 1995. [7] P.A. Ioannou, J. Sun, Stable and Robust Adaptive Control, Prentice-Hall, Englewood Cli s, NJ, 1995.
252
F. Giri et al. / Systems & Control Letters 36 (1999) 245–252
[8] G. Kreisselmeier, Adaptive control of a class of time varying plants, University of Kassel, Rep. 1986. [9] G. Kreisselmeier, Adaptive observers with exponential rate of convergence, IEEE Trans Automat. Control 22 (1977) 2–8. [10] M. Krstic, I. Kanellakopoulos, P.V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. [11] J.M. Martin-Sanchez, Adaptive control of time-variant processes, Int. J. Control 44 (1) (1986). [12] R.H. Middleton, G.C. Goodwin, Adaptive control of timevarying linear systems, IEEE Trans. Aut. Control 33 (2) (1988) 150–155.
[13] F. Okhawa, Model reference adaptive control systems for linear time varying systems with periodically varying parameters and time delays, Int. J. Control 33 (2) (1988). [14] R. Ortega, Y. Tang, Robustness of adaptive controllers – A survey, Automatica 25 (5) (1989) 651– 677. [15] K. Tsakalis, P.A. Ioannou, Adaptive control of linear timevarying plants, Proc. IFAC Workshop on Adaptive Syst. Contr. Sign. Processing, Lund, Sweden, 1986.