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An algorithm for robust adaptive control with less prior knowledge
2
An algorithm for robust adaptive control with less prior knowledge G. Feng, Y. A. J iang and R. Zmood
Abstract A new robust discrete-time singularity free direct adaptive control scheme is proposed with respect to a class of modelling uncertainties in this chapter. Two key features of this scheme are that a relative dead zone is used but no knowledge of the parameters of the upper bounding function on the class of modelling uncertainties is required, and no knowledge of the lower bound on the leading coefficient of the parameter vector is required to ensure the control law singularity free. Global stability and convergence results of the scheme are provided.
2.1
Introduction
Since it was shown (e.g. [1], [2]) that unmodeUed dynamics or even a small bounded disturbance could cause most of the adaptive control algorithms to go unstable, much effort has been devoted to developing robust adaptive control algorithms to account for the system uncertainties. As a consequence, a number of adaptive control algorithms have been developed, for example, see [3] and references therein. Among those algorithms are simple projection (e.g. [4], [5]), normalization (e.g. [6], [7]), dead zone (e.g. [8-12]), adaptive law modification (e.g. [13], [14]), tT-modification (e.g. [15], [16]), as well as persistent excitation (e.g. [17], [18]). In the case of the dead zone based methods, a fixed dead zone can be used [6-8] in the presence of only bounded disturbance, which turns off the algorithm when the identification error is smaller than a certain threshold. In
24 An algorithm for robust adaptive control with less prior knowledge
order to choose an appropriate size of the dead zone, an upper bound on the disturbance must be known. When unmodelled dynamics are present, a relative dead zone modification should be employed [11], [12]. Here the knowledge of the parameters of bounding function on the unmodelled dynamics and bounded disturbances is required. However, such knowledge, especially knowledge of the nonconservative upper bound or the parameters of the upper bounding function, can be hardly obtained in practice. Therefore, the robust adaptive control algorithm which does not rely on such knowledge is in demand but remains absent in the literature. One may argue that the robustness of the adaptive control algorithms can be achieved with only simple projection techniques in parameter estimation [4], [5]. However, it should be noted that using the robust adaptive control algorithms such as the dead zone, the robustness of the resulting adaptive control systems will be improved in the sense that the tolerable unmodelled dynamics can be enlarged [19]. Therefore, discussion of the robust adaptive control approaches such as those based on the dead zone technique is still of interest and the topic of this chapter. Another potential problem associated with adaptive control is its control law singularity. The estimated plant model could be in such a form that the polezero cancellations occur or the leading coefficient of the estimated parameter vector is zero. In such cases, the control law becomes singular and thus cannot be implemented. In order to secure the adaptive control law singularity free, various approaches have been developed. These approaches can be classified into two categories. One reties on persistent excitation. The other depends on modifications of the parameter estimation schemes. In the latter case, the most popular method is to hypothesize the existence of a known convex region in which no pole-zero cancellations occur and then to develop a convergent adaptive control scheme by constraining the parameter estimates inside this region (e.g. [11], [20-22]) for pole placement design; or to hypothesize the existence of a known lower bound on the leading coefficient of the parameter vector and then to use an ad hoc projection procedure to secure the estimated leading coefficient bounded away from zero and thus achieve the convergence and stability of the direct adaptive control system. However, such methods suffer the problem of requirement for significant a priori knowledge about the plant. Recently, another approach has been developed which also modifies the parameter estimation algorithm. This approach is to re-express the plant model in a special input-output representation and then use a correction procedure in the estimation algorithm to secure the controllability and observability of the estimated model of the system [23-24]. They also addressed the robustness problem of such algorithms with respect to bounded disturbance [25] using the dead zone technique. They did not address the robustness problem with respect
Adaptive Control Systems 25 to unmodelled dynamics. Moreover, those algorithms also suffer the same problem as the usual dead zone based robust adaptive control algorithms. That is, they still require the knowledge of the upper bound on the disturbance or the parameters of the upper bounding function on the unmodelled dynamics and disturbances. In this chapter, a new robust direct adaptive control algorithm will be proposed which does use dead zone but does not require the knowledge of the parameters of the upper bounding functions on the unmodelled dynamics and the disturbance. It has also been shown that our algorithm can be combined with the parameter estimate correction procedure, which was originated in [24] to ensure the control law singularity free, so that the least a priori information is required on the plant. The chapter is organized as follows. The problem is formulated in Section 2.2. Ordinary discrete time direct adaptive control algorithm with dead zone is reviewed in Section 2.3. Our main results, a new robust direct adaptive control algorithm and its improved version with control law singularity free are presented in Section 2.4 and Section 2.5 respectively. Section 2.6 presents one simulation examples to illustrate the proposed adaptive control algorithms, which is followed by some concluding remarks in Section 2.7.
2.2
Problem formulation
Consider a discrete time single input single output plant
U(t) + V(t) y(t) = z-dB(z-I) L A(z_il)l
(2.1)
i
where y(t) and u(t) are plant output and input respectively, v(t) represents the class of unmodelled dynamics and bounded disturbances and d is the time delay. A(z -1) and B(z -l) are polynomials in z -1, written as
A(z -l) = 1 + a l z -1 + . . . + a n z - n B(z -1) = blz -1 + b2z -2 + . . . + bmz -m Specify a reference model as
E ( z - l ) y , (t) = z-dR(z-l)r(t) where E(z -l) is a strictly stable monie polynomial written as E(z -1)
=
I + elz-l + ... + e~z -~
(2.2)
26 An algorithm for robust adaptive control with less prior knowledge
Then, there exist unique polynomials F(z -l) and G(z -l) written as F(z -t) = 1 4- f l z -l 4-... 4-fd_l z-d+l G(z -l) = go + glz -l + . . . + gn-lz -"+l
such that
E(z -1) = F ( z - l ) A ( z -1) 4- z-dG(z -I)
(2.3)
Using equation (2.3), it can be shown that the plant equation (2.1) can be rewritten as y(t + d) = a(z-l)y(t) +/~(z-l)u(t) = orq~(t) 4- rl(t 4- d)
(2.4)
where y(t 4- d) = E ( z - l ) y ( t 4- d) rl(t -t- d) = F(z-1)A(z-1)v(t 4- d) ok(t) r = [u(t),u(t- 1),... , u ( t - m - d 4- 1 ) , y ( t ) , y ( t - 1),... ,y(t - n + 1)]
:= [u(t), r
r]
Or= [0~,... q,+m+d] := [0~, 0 'r] ~(Z -~) = G(Z -~)
~(z -~) = e(z-~)a(z -~) We make the following standard assumptions [26], [18]. (A1) The time delay d and the plant order n are known. (A2) The plant is minimum phase. For the modelling uncertainties, we assume only: (A3) There exists a function [11] 7(0 such that Ir/(t) 12 < 3'(t) where 7(0 satisfies
~(t) ~ ~ ~up IJx(~)ll 2 + ~2 O__r_~t
for some unknown constants el > O, e2 > O, and x(t) is defined as x(t) = [ y ( t - 1),... , y ( t - n ) , u ( t - 1),... , u ( t - m - d)] T
For the usual direct adaptive control, in order to facilitate the implementation of projection procedure to secure the control law singularity free, the following assumption is required.
Adaptive Control Systems 27
(A4) There is a known constant 0~ satisfying
10~1 ___10~1
and
0~01 > 0
For the usual relative dead zone based direct adaptive control algorithm, another assumption is needed as follows: (A5) The constants eland e2 in (A3) are known a priori. Remark 2.1
It should be noted that the assumptions (A4) and (A5) will not be required in our new adaptive control algorithm to be developed in the next few sections. It is believed that the elimination of assumptions (A4) and (A5) will improve the applicability of the adaptive control systems.
2.3
Ordinary direct adaptive control with dead zone
Let 0(t) denote the estimate of the unknown parameter 0 for the plant model (2.4). Defining the estimation error as
e(t) = .P(t) - q~r(t- d)O(t- 1)
(2.5)
and a dead zone function as
f(g,e) =
e-g
ife>g
0 e+ g
if lel _< g if e < - g
0 < g < oo
(2.6)
then the following least squares algorithm with a relative dead zone can be used for parameter estimation /~(t) = proj{/~(t- 1) + a(t) P(t) = P(t -
1) - a(t) P ( t
P ( t - 1)~b(t- d) e(t)} 1 + ~ ( t - d)Tp(t- 1)~b(t- d) 1)r
-
d)cb(t - d)re(t
1 + cb(t - d ) r e ( t
P(-1) = kol,
-
-
I)
1)cb(t - d )
ko > 0
(2.7)
where the term a(t) is a dead zone, which is defined as follows: 0 a(t)= with0
af(61/27(t)l/2,e(t))/e(t)
if le(t)l 2 _< ~7(t) otherwise
(2.8)
~ = . l . _~j0 a , ~0 > 1, and proj is the projection operator [26]
28 An algorithm for robust adaptive control with less prior knowledge
such that
{~1 (t) if 01(t)sgn (0~) >_ [O~l #l(t) =
0~
(2.9)
otherwise
It has been shown that the above parameter estimation algorithm has the following properties: (i) 0(t) is bounded
1 (ii) [01(t)>_ 10~ml and ~i(0 f (~l/27(t) 1/2, e(t) )2 (iii) i + ~b(t- d)rP(i
(iv)
=
1
I~i(t 01 )
01
i)~b(t- d) E 12
f (~l/27(t) 1/2, e(t) )2
1 + ~b(t- d) Tp(t -- 1)O(t-- d) E/2 .
.
.
.
.
where
e(t) = y(t) - 0(t - d) r ~b(t - d)
(2.10)
The direct adaptive control law can be written as
u(t) where
rf(t) =
- O'(t)r~b'(t)
0'(t)
rf(t) = R(z-l)r(t) and R(z -l) is specified
(2.11) (2.12)
with r(t) a reference input by the reference model equation (2.2). Then with the parameter estimation properties (i)-(iv), the global stability and convergence results of the adaptive control systems can be established as in [26], [11], which are summarized in the following theorem.
Theorem 3.1
The direct adaptive control system satisfying assumptions (A1)(A5) with adaptive controller described in equations (2.7)-(2.9) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded. However, as discussed in the first section, the requirement for knowledge of the parameters of the upper bounding function on the unmodelled dynamics and bounded disturbances is very restrictive. In the next section, we are attempting to propose a new approach to get rid of such a requirement.
2.4
New robust direct adaptive control
Here we develop a new robust adaptive control algorithm which does not need such knowledge. That is, we drop the assumption (A5).
Adaptive Control Systems 29
The key idea is to use an adaptation law to update those parameters. The new parameter estimation algorithm is the same as equations (2.7-2.9) but with a different dead zone a(t), where
a(t) = {
0 r f (~l/2('~(t) 4" Q(t))l/2, e(t))/e(t)
if le(t)[ 2 < ~(-~(t) + otherwise
. cO(l+#5(t . .
d)rP(t .
1)~b(t
(2.13)
[supx,2]supxT,2]
and
Q(t) = 4(1
Q(t))
d)o<_r<_, 1
LO
(2.14) And ,~(t)is calculated by
[
~(t) = C(t)r O
C(t) = (~(t - 1) 4-
(2.15) 1
2(1 - a)(1
a(t)/3 + ~(t- d)rP(t-
sup llx(r)ll2]
O_<'r_
1)~b(t- d))
1
/3 > 0 (2.16) where C(t) r =
e2]
[~l
with zero initial condition. It should be noted that ~l and ~2 will be always positive and non decreasing. As shown in [23], the projection operation does not alter the convergence properties of the original parameter estimation algorithms. Therefore, in the following analysis, the projection operation will be neglected. The properties of the above modified least squares parameter estimator with a relative dead zone are summarized in the following lemma. Lemma 4.1 The least squares algorithm with equations (2.7), (2.9), (2.13)(2.16) applied to any system has the following properties irrespective of the control law: (i) ~(t) is bounded (ii) C(t) is bounded and non-decreasing, and thus ~1 convvrges to a constant, say el (iii) ]0l(t)> 10~l -
and ~11 -[ 0~Tt)] 1
(iv) lid(t)- 0 ( t - 1)[ I ~/2
(0
-
0-r
30 An algorithm for robust adaptive control with less prior knowledge
f(t) 2 (v) f(t) 2 :='1 q- ~b(t-d)T'P(t 1)~b(t- d)
:=
f(~l/2(~/(t) + Q(t)) 1/2 e(t)) 2 ' 1+ ~(t - d)Tp(t - 1)~b(t -d)
El2
Proof Define a Lyapunov function candidate V(t + 1) = 89
1)-10(t) + C'(t + 1)rfl-lC'(t + 1))
(2.17)
where 0(t) = 0(t) - 0", C'(t + 1) = C(t + 1) - [e~ e2]. Then, its difference becomes V(t + 1) - V(t) =
a(t)
l)~bit - d)
1+ r X
[
.
1 + 4)(t- a ) r p ( t - 1)~b(t- d) ] a(t))~(t- d ) r P ( t - 1)~b(t- d) r/(t)2 - e(t)2 .
.
.
.
.
.
.
.
.
.
1 + (1 -
a(t)C(t) r + (1 - c~)(1 + dp(t- d ) T e ( t - l)~b(t- d)) +
X
[]o_<~_<, supIlxff)'12 1
a(t)2fl 4(1 - a)2(1 + ~ ( t - d)rP(t - 1)~b(t - d)) 2 O
O
1
1
a(t) - 1 + ~ ( t - d ) r P ( t - 1)~b(t- d) X
1 + qb(t - d)Tp(t - 1 ) ~ ( t - d) r/(t) 2 -e(t) 2] 1 + (1 - a(t))dp(t- d ) r p ( t - 1)~b(t- d)
a(t) + (1 = a)(1 + ~ ( t - ' d ) T p ( t - l i ' ~ b ( t - d))(C(t)T
sup Ilxff)ll2] + Q(t))
O
1
<
a(t) [1 ] 1 + ~ ( t - d ) r P ( t - 1)~b(t- d) (1 - ai r/(t)2 - e(t)2 +
a(t) (~'(t) - 9'(t) + Q(t)) (1- c~)(l q- qb(t - d)Tp(t - 1)~b(t - d))
Adaptive Control Systems 31 <
a(t)
- 1 + cb(t - d) r e ( t - 1 ) r
x
(1
- d)
a) r/(t)2 - e(t)2 + (i
< - 1 + ~(t-
'"~)('~(t) - 7(t) -t- Q(t))
[ - e(t) 2 + 1
a(t) d)rP(t-
1)~b(t- d)
< a(t) - 1 + ~b(t- d ) r e ( t -
1)~b(t- d)
< - 1 + ~(t-
< _ ~o - 1 -
~o
< _~o-
~o
1
a(t) d)rp(t-
(i
(-~(t) + Q(t))
_ e(t)2 + _
(1
]
a) e(t)2
~)
[ _ e(t) 2 + 1e(t)2 ] 1)~b(t- d)
~0
a(t) e(t) 2 1 + qb(t - d) 7"p(t - 1)~(t - d)
(2.18)
f(t) 2 1 + 4~(t- d)re(t-
1)r
d)
where the fact that a ( t ) e ( t ) 2 = f ( t ) e ( t ) > f ( t ) 2 has been used. Therefore, following the same arguments in [20], [21], [23], the results in Lemma 4.1 are thus proved. If using the same adaptive control law as in equation (2.11), then with the parameter estimation properties (i)-(v) in Lemma 4.1, the global stability and convergence results of the new adaptive control system can be established as in [26], [11] as long as the estimated ~l is small enough, which are summarized in the following theorem. Theorem 4.1 The direct adaptive control system satisfying assumptions (A1)(A4) with the adaptive controller described in equations (2.7), (2.9), (2.13)(2.16) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded. In this approach, we have eliminated the requirement for the knowledge of the parameters of the upper bounding function on the modelling uncertainties. But the requirement for the knowledge of the lower bound on the leading coefficient of the parameter vector, i.e. assumption (A4) is still there. In the next section, the technique of the parameter correction procedure will be combined with the algorithm developed in this section to ensure the least prior knowledge on the plant. That is, only assumptions (A1)--(A3) are needed.
32 An algorithm for robust adaptive control with less prior knowledge
2.5 Robust adaptive control with least prior knowledge The following modified least squares algorithm will be used for robust parameter estimation: O(t) = O ( t - 1) + a(t)
e ( t - 1)~b(t- d)
~,(t)
1 + ~(t - d) T e ( t -- 1)~(t -- d)
P(t) = P(t -
1) - a(t) P ( t - 1)r 1+ r
P(-l) = kd,
d)cb(t- d ) r e ( t - d) r e ( t - 1 ) r
1)
(2.19)
- d)
go > 0
and the parameter estimate is then corrected [24] as
where
O-(t) = O(t) + P(t)fl(t)
(2.20)
~(t) = .~(t) - O(t - 1) T~b(t - d)
(2.21)
the vector ~(t) is described in Figure 2.1
~(t)
p(t)
/~(t) ---IIP(t)ll /3(0 = 0
-611p(t)ll
elO'(t)l
lip(Oil- 21t~l(t)l
Figure 2.1 Parameter correction vector where p(t) is the first column of the covariancc matrix P(t), and the term a(t) is now defined as follows:
a(t)
{
o
if ~(t)l 2 < ~('~(t) + Q(t)) otherwise
with 0 < a < 1, ( = 12(o - a , (o > 1, and
Q ( t ) - [ B ( t - 1)rP(t - 1 ) ~ ( t - d ) ] 2 +
~ [ sup IIx(r)ll2]r[ suP IIx(r)lf2] 2(1-- ot)(l+ ~b(t-- d ) T p ( t -- 1)~b(t-d) o
Adaptive Control Systems 33
And ~,(t)is calculated by sup
IIxO-)ll:]
= ~(t)r o<~_<,
?(t)
(2.22)
1
(~(t) = (~(t- 1) +
(1 - cz)(l
[
~up I1~(~)112]
a(t)A
+ ~(t- d)rp(t- 1)$(t- d)
O<'r
1
> 0 (2.23) where with zero initial condition. It should be noted that ~l and ~2will be always positive and non-decreasing. Remark 5.1
The prediction error ~(t) is used in the modified least squares algorithm to ensure that the estimator property (iii) in the following lemma can be established. The properties of the above modified least squares parameter estimator are summarized in the following lemma. Lemma 5.1 If the plant satisfies the assumptions (A1)-(A3), the least squares algorithm (2.19)--(2.23) has the following properties: (i) 0(t) is bounded, and ]10(t)- 0 ( t - 1)1[ E/2. (ii) C(t) is bounded and non-decreasing, thus converges
f(~l/2(~/(t) + Q(t))lD' ~(t))2
(iii)
1 + ~(t - d) r p ( t - 1)~(t - d) ~
12
(iv) Ilp(t)ll + IOl(t)l > b ~ where bmin =
Ioll max(l, [I/~*ll)
with/3* defined such that O" = O(t) = P(t)/~*
1-e (v) l~ (0l > 3 +-e bmm (vi) 0(t)is bounded, and ll0(t)- 0(t - l)ll~ 6 Proof
Define a Lyapunov function candidate
v(t + l) = 89(~i(0rP(0-1~i(0+ (~(t+ l)r~-~t~(t+ l))
(2.24)
34 An algorithm for robust adaptive control with less prior knowledge
where 0(t) = 0(t) - 0", C(t 4" 1) = C(t 4" 1) - [el e21r. Noting that ~'(t) = ,~(t) - 0(t - 1 ) T~b(t - d) = .~(t) -/~(t-
1)~b(t- d) - fl(t - l ) r p ( t -
1)~b(t- d)
= e(t) - 13(t- 1 ) T p ( t - 1)~b(t- d)
(2.25)
Then, the difference of the Lyapunov function candidate becomes
v(t + l)- v(t)=
a(t) 1 + Jp(t- d ) r P ( t -
l)~b(t- d)
1 4" dp(t- d ) T p ( t - 1)~b(t- d)
•
1 + (1 - a ( t ) ) ~ ( t - d) r p ( t - 1)~(t - d)
x (rl(t)- ~ ( t - 1)Te(t - 1)~b(t- d)) 2 - ~'(t)2]
+
+
[ 0
a(t)~(t)r
(1 -- a)(1 + t k ( t - d ) r P ( t - 1)O(t- d))
Iix(~.)I121
a(t)2A (1
-- ~ ) 2 ( 1 4" dp(t- d)Tp(t -
1)~b(t- d)) 2
a(t) - 1 + ~(t - d ) r p ( t - l)~b(t - d)
,
]
X [] _ ~ (rl(t) -- iO(t- 1 ) T p ( t - 1)~b(t- d)) 2 - ~'(t)2 +
2a(t) (q(t) - '7(t)) (1 - cO(1 + dp(t- d ) r p ( t -
1)~b(t- d))
a(t)2A +
(~ - ~)' (i
+ o(t - d) ~ eft - ~i~(t - di):
sup[Jx(r)ll2]r [O
O
1
1
Adaptive Control Systems 35
< a(t) - 1 + qb(t- d ) T p ( t - l)~b(t- d) ,7(t) 2 - ~ 12-
+
2 (t)
(/3(t- 1 ) r p ( t - 1)~b(t- d)) 2 - ~(t) 2] - 7(0)
(1 - a)(1 + qb(t- d)rp(t - 1)~b(t- d)) a(t) 2A
[sup[Ix(r)[[2]T[ O<_r
+ (1-- a)2(l + #b(t- d l r P ( t - l)~b(t- d)) 2
-
I +
a(t) qb(t- d ) r p ( t - 1)~b(t- d)
x
2 1-c~
1
1
77(t)2 - c'(0 2 + i 2 c~('~(t) - q(t) + Q(t))]
< a(t) - 1 + ~b(t- d ) r P ( t - 1)~b(t- d)
[ - ~,(t)2 + 2
]
~ - a ( ~ ( t ) + Q(t))
a(t) < -g'(t)2+i" 2 cr~1 (t)2] - 1 + ~b(t- d ) r P ( t - 1)~b(t- d) < a(t) [ _ g,(t)2 + l~,(t)2] - 1 + qb(t- d ) r p ( t - 1)~b(t- d) ~o J ~0- 1
1+ < -
a(t)~(t) 2 d ) r P ( t - 1) b(t- d)
~o - 1 f({l/2(~/(t) + Q(t))l/2,~(t)) 2 ~o 1 + qb(t- d ) r P ( t - 1)~b(t- d)
(2.26)
where the fact that a(t)e(t)2=f(t)e(t)>f(t) 2 has been used. Therefore, following the same arguments in [26], [11], [21], the results (i)-(iii) in Lemma 5.1 are thus proved. The properties (iv)-(vi) in the lemma can also be obtained directly from the results in [23]. If using the same adaptive control law as in equation (2.11), then with the parameter estimation properties (i)-(vi), the global stability and convergence results of the new adaptive control system can be established as in [26], [11] as long as the estimated s is small enough, which are summarized in the following theorem.
36 An algorithm for robust adaptive control with less prior knowledge
Theorem 5.1
The direct adaptive control system satisfying assumptions (A1)(A3) with the adaptive controller described in equations (2.19)-(2.23) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded.
2.6 Simulation example In this section, one numerical example is presented to demonstrate the performance of the proposed algorithm. A fourth order plant is given by the transfer function as =
a,(s)
with
5(s + 2) s(s+ 1)
=
as a nominal part, and
G,(s) =
229 s2 + 30s + 229
as the unmodelled dynamics. With the sampling period T = 0.1 second, we have the following corresponding discrete-time model
G(q-l) =
1
0.09784q -l + 0.1206q -2 - 0.1414q -3 - 0.01878q -4 2.3422q 2t + 1.0788q - 2 - 0.4906q -3 + ().04505qL4
The reference model is chosen as Cm(S) =
0.2s+ 1
whose corresponding discrete-time model is
Gm(q-~) =
0.3935q -l 1 - 0.6065q -1
We have chosen k0 = 1, and 0(0) = [0.6 0 0 0] r. If no dead zone is used, the simulation results are divergent. If using the algorithm developed in this chapter with ,~ = l0 -s, the simulation results are shown as in Figure 2.2, where (a) represents the system output y(t) and reference model output y*(t), (b) is the control signal u(t), (c) is the estimated parameter ~l, and (d) denotes the estimated bounding parameters ~l and g2. In order to demonstrate the effect of the update rate parameter ,~, the following simulation with A = 1.4 • 10-5 was also conducted. The result is shown in Figure 2.3. The steady state values of the several important parameters and the tracking error in both eases are summarized in Table 2.1.
Adaptive Control Systems 37 4
(a)
G) 0 r
(b)
0 ,..2 O
2 O l_
~o
,-'0 O
m
e~. 2
-2
O
40
50 100 150 Time in seconds
2
.C
....
200
50 100 150 Time in seconds I~ 2 x10"3
u
(c)
E
(d)
E 1 E.5
1.5
epsl
r O.
"D
" ~E
200
_7
ca 1
1
r
"10 t--
,,~ 0.5 - ' n
. . . .
.~0.5
eps2 ._.._J
50
I00 150" Time in s e c o n d s
Figure 2,2
2(10
=o o
Robust adaptive control with A = 10-
50 100 150 Time in seconds
200
$
4
(a)
0 rG)
(b)
"-- 2
2
(I)
"~
-6
i
l,. ,e.a
o
c
u0
0
e~ -2 O d 0
s'o i~o
1~0
200
50 100 150 Time in seconds
Time in seconds 2 O
(c) m 1.5
2
(d) J
D. O~
c
~ "O r
E
epsl
1 eps2
~0.5
,,~ 0.5
Figure 2.3
xl0 ~
~1.5 (u
0o
200
IJJ
50 100 150 Time in seconds
200
Robust adaptive control with
00
50 100 150 Time in seconds
A = 1.4 x 10 -5
200
38 An algorithm for robust adaptive control with less prior knowledge Table 2.1
Steady state values ,.,.
,
,,
..
A = 10 -5 ,,
,
.
,,
.
,
.
.
.
A = 1.4 x 10 -5 ,
1.1898 x 10-3 0.3509 x 10-3 0.5649 0.0179
~2
ly-y'l
,
.
_
,
1.4682 x 10-3 0.4467 x 10-3 0.5833 0.07587
,.
It can be observed from the above simulation results that the algorithm developed in this chapter can guarantee the stability of the adaptive system in the presence of the modelling uncertainties, and the smaller tracking error could be achieved with smaller update rate parameter A. Most importantly, the knowledge of the parameters 6~ and e2 of the upper bounding function and the knowledge of the leading coefficient of the parameter vector 0 ~ are not required a priori.
2.7
Conclusions
In this chapter, a new robust discrete-time direct adaptive control algorithm is proposed with respect to a class of unmodelled dynamics and bounded disturbances. Dead zone is indeed used but no knowledge of the parameters of the upper bounding function on the unmodelled dynamics and disturbances is required a priori. Another feature of the algorithm is that a correction procedure is employed in the least squares estimation algorithm so that no knowledge of the lower bound on the leading coefficient of the plant numerator polynomial is required to achieve the singularity free adaptive control law. The global stability and convergence results of the algorithm are established.
References [1] Rohrs, C., Valavani, L., Athans, M. and Stein, G. (1985). 'Robustness of Adaptive Control Algorithms in the Presence of Unmodelled Dynamics', IEEE Trans. Automat. Contr., Vol. AC-30, 881-889. [2] Egardt, B. (1979). Stability of Adaptive Controllers, Lecture Notes in Control and Information Sciences, New York, Springer Verlag. [3] Ortega, R. and Tang, Y. (1989). 'Robustness of Adaptive Controllers - A Survey', Automatica, Vol. 25, 651-677. [4] Ydstie, B. E. (1989). 'Stability of Discrete MRAC-revisited', Systems and Control Letters, Vol. 13, 429-438.
Adaptive Control Systems 39 [5] Naik, S. M., Kumar, P. R., Ydstie, B. E. (1992). 'Robust Continuous-time Adaptive Control by Parameter Projection', IEEE Trans. Automat. Contr., Vol. AC-37, No. 2, 182-197. [6] Praly, L. (1983). 'Robustness of Model Reference Adaptive Control', Proc. 3rd Yale Workshop on Application of Adaptive System Theory, New Haven, Connecticut. [7] Praly, L. (1987). 'Unmodelled Dynamics and Robustness of Adaptive Controllers', presented at the Workshop on Linear Robust and Adaptive Control, Oaxaca, Mexico. [8] Petersen, B. B. and Narendra, K. S. (1982). 'Bounded Error Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-27, 1161-1168. [9] Samson, C. (1983). 'Stability Analysis of Adaptively Controlled System Subject to Bounded Disturbances', Automatica, Vol. 19, 81-86. [10] Egardt, B. (1980). 'Global Stability of Adaptive Control Systems with Disturbances', Proc. JACC, San Francisco, CA. [11] Middleton, R. H., Goodwin, G. C., Hill, D. J. and Mayne, D. Q. (1988). 'Design Issues in Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-33, 50-58. [12] Kreisselmeier, G. and Anderson, B. D. O. (1986). 'Robust Model Reference Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-31, 127-133. [13] Kreisselmeier, G. and Narendra, K. S. (1982). 'Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances', IEEE Trans. Automat. Contr., Vol. AC-27, 1169-1175. [14] Iounnou, P. A. (1984). 'Robust Adaptive Control', Proc. Amer. Contr. Conf., San Diego, CA. [15] Ioannou, P. and Kokotovic, P. V. (1984). 'Robust Redesign of Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-29, 202-211. [16] Iounnou, P. A. (1986). 'Robust Adaptive Controller with Zero Residual Tracking Error', IEEE Trans. Automat. Contr., Vol. AC-31,773-776. [17] Anderson, B. D. O. (1981). 'Exponential Convergence and Persistent Excitation', Proc. 20th IEEE Conf. Decision Contr., San Diego, CA. [18] Narendra, K. S. and Annaswamy, A. M. (1989). Stable Adaptive Systems, PrenticeHall, NJ. [19] Feng, G. and Palaniswami, M. (1994). 'Robust Direct Adaptive Controllers with a New Normalization Technique', IEEE Trans. Automat. Contr., Vol. 39, 2330-2334. [20] Goodwin, G. C. and Sin, K. S. (1981) 'Adaptive Control of Nonminimum Phase Systems', IEEE Trans. Automat. Contr., Vol. AC-26, 478-483. [21] Feng, G. and Palaniswami, M. (1992). 'A Stable Implementation of the Internal Model Principle', IEEE Trans. Automat. Contr., Vol. AC-37, 1220-1225. [22] Feng, G., Palaniswami, M. and Zhu, Y. (1992). 'Stability of Rate Constrained Robust Pole Placement Adaptive Control Systems', Systems and Control Letters, Vol. 18, 99-107. [23] Lazono-Lcal, R. and Goodwin, G. C. (1985). 'A Globally Convergent Adaptive Pole Placement Algorithm without a Persistency of Excitation Requirement', IEEE Trans. Automat. Contr., Vol. AC-30, 795-799.
40 An algorithm for robust adaptive control with less prior knowledge
[24] Lazono-Leal, R., Dion, J. and Dugard, L. (1993). 'Singularity Free Adaptive Pole Placement Using Periodic Controllers', IEEE Trans. Automat. Contr., Vol. AC-38, 104-108. [25] Lazono-Leal, R. and Collado, J. (1989). 'Adaptive Control for Systems with Bounded Disturbances', IEEE Trans. Automat. Contr., Vol. AC-34, 225-228. [26] Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering, Prediction and Control, Prentice-Hall, NJ. [27] Lazono-Leal, R. (1989). 'Robust Adaptive Regulation without Persistent Excitation', IEEE Trans. Automat. Contr., Vol. AC-34, 1260-1267.
An algorithm for robust adaptive control with less prior knowledge G. Feng, Y. A. J iang and R. Zmood
Abstract A new robust discrete-time singularity free direct adaptive control scheme is proposed with respect to a class of modelling uncertainties in this chapter. Two key features of this scheme are that a relative dead zone is used but no knowledge of the parameters of the upper bounding function on the class of modelling uncertainties is required, and no knowledge of the lower bound on the leading coefficient of the parameter vector is required to ensure the control law singularity free. Global stability and convergence results of the scheme are provided.
2.1
Introduction
Since it was shown (e.g. [1], [2]) that unmodeUed dynamics or even a small bounded disturbance could cause most of the adaptive control algorithms to go unstable, much effort has been devoted to developing robust adaptive control algorithms to account for the system uncertainties. As a consequence, a number of adaptive control algorithms have been developed, for example, see [3] and references therein. Among those algorithms are simple projection (e.g. [4], [5]), normalization (e.g. [6], [7]), dead zone (e.g. [8-12]), adaptive law modification (e.g. [13], [14]), tT-modification (e.g. [15], [16]), as well as persistent excitation (e.g. [17], [18]). In the case of the dead zone based methods, a fixed dead zone can be used [6-8] in the presence of only bounded disturbance, which turns off the algorithm when the identification error is smaller than a certain threshold. In
24 An algorithm for robust adaptive control with less prior knowledge
order to choose an appropriate size of the dead zone, an upper bound on the disturbance must be known. When unmodelled dynamics are present, a relative dead zone modification should be employed [11], [12]. Here the knowledge of the parameters of bounding function on the unmodelled dynamics and bounded disturbances is required. However, such knowledge, especially knowledge of the nonconservative upper bound or the parameters of the upper bounding function, can be hardly obtained in practice. Therefore, the robust adaptive control algorithm which does not rely on such knowledge is in demand but remains absent in the literature. One may argue that the robustness of the adaptive control algorithms can be achieved with only simple projection techniques in parameter estimation [4], [5]. However, it should be noted that using the robust adaptive control algorithms such as the dead zone, the robustness of the resulting adaptive control systems will be improved in the sense that the tolerable unmodelled dynamics can be enlarged [19]. Therefore, discussion of the robust adaptive control approaches such as those based on the dead zone technique is still of interest and the topic of this chapter. Another potential problem associated with adaptive control is its control law singularity. The estimated plant model could be in such a form that the polezero cancellations occur or the leading coefficient of the estimated parameter vector is zero. In such cases, the control law becomes singular and thus cannot be implemented. In order to secure the adaptive control law singularity free, various approaches have been developed. These approaches can be classified into two categories. One reties on persistent excitation. The other depends on modifications of the parameter estimation schemes. In the latter case, the most popular method is to hypothesize the existence of a known convex region in which no pole-zero cancellations occur and then to develop a convergent adaptive control scheme by constraining the parameter estimates inside this region (e.g. [11], [20-22]) for pole placement design; or to hypothesize the existence of a known lower bound on the leading coefficient of the parameter vector and then to use an ad hoc projection procedure to secure the estimated leading coefficient bounded away from zero and thus achieve the convergence and stability of the direct adaptive control system. However, such methods suffer the problem of requirement for significant a priori knowledge about the plant. Recently, another approach has been developed which also modifies the parameter estimation algorithm. This approach is to re-express the plant model in a special input-output representation and then use a correction procedure in the estimation algorithm to secure the controllability and observability of the estimated model of the system [23-24]. They also addressed the robustness problem of such algorithms with respect to bounded disturbance [25] using the dead zone technique. They did not address the robustness problem with respect
Adaptive Control Systems 25 to unmodelled dynamics. Moreover, those algorithms also suffer the same problem as the usual dead zone based robust adaptive control algorithms. That is, they still require the knowledge of the upper bound on the disturbance or the parameters of the upper bounding function on the unmodelled dynamics and disturbances. In this chapter, a new robust direct adaptive control algorithm will be proposed which does use dead zone but does not require the knowledge of the parameters of the upper bounding functions on the unmodelled dynamics and the disturbance. It has also been shown that our algorithm can be combined with the parameter estimate correction procedure, which was originated in [24] to ensure the control law singularity free, so that the least a priori information is required on the plant. The chapter is organized as follows. The problem is formulated in Section 2.2. Ordinary discrete time direct adaptive control algorithm with dead zone is reviewed in Section 2.3. Our main results, a new robust direct adaptive control algorithm and its improved version with control law singularity free are presented in Section 2.4 and Section 2.5 respectively. Section 2.6 presents one simulation examples to illustrate the proposed adaptive control algorithms, which is followed by some concluding remarks in Section 2.7.
2.2
Problem formulation
Consider a discrete time single input single output plant
U(t) + V(t) y(t) = z-dB(z-I) L A(z_il)l
(2.1)
i
where y(t) and u(t) are plant output and input respectively, v(t) represents the class of unmodelled dynamics and bounded disturbances and d is the time delay. A(z -1) and B(z -l) are polynomials in z -1, written as
A(z -l) = 1 + a l z -1 + . . . + a n z - n B(z -1) = blz -1 + b2z -2 + . . . + bmz -m Specify a reference model as
E ( z - l ) y , (t) = z-dR(z-l)r(t) where E(z -l) is a strictly stable monie polynomial written as E(z -1)
=
I + elz-l + ... + e~z -~
(2.2)
26 An algorithm for robust adaptive control with less prior knowledge
Then, there exist unique polynomials F(z -l) and G(z -l) written as F(z -t) = 1 4- f l z -l 4-... 4-fd_l z-d+l G(z -l) = go + glz -l + . . . + gn-lz -"+l
such that
E(z -1) = F ( z - l ) A ( z -1) 4- z-dG(z -I)
(2.3)
Using equation (2.3), it can be shown that the plant equation (2.1) can be rewritten as y(t + d) = a(z-l)y(t) +/~(z-l)u(t) = orq~(t) 4- rl(t 4- d)
(2.4)
where y(t 4- d) = E ( z - l ) y ( t 4- d) rl(t -t- d) = F(z-1)A(z-1)v(t 4- d) ok(t) r = [u(t),u(t- 1),... , u ( t - m - d 4- 1 ) , y ( t ) , y ( t - 1),... ,y(t - n + 1)]
:= [u(t), r
r]
Or= [0~,... q,+m+d] := [0~, 0 'r] ~(Z -~) = G(Z -~)
~(z -~) = e(z-~)a(z -~) We make the following standard assumptions [26], [18]. (A1) The time delay d and the plant order n are known. (A2) The plant is minimum phase. For the modelling uncertainties, we assume only: (A3) There exists a function [11] 7(0 such that Ir/(t) 12 < 3'(t) where 7(0 satisfies
~(t) ~ ~ ~up IJx(~)ll 2 + ~2 O__r_~t
for some unknown constants el > O, e2 > O, and x(t) is defined as x(t) = [ y ( t - 1),... , y ( t - n ) , u ( t - 1),... , u ( t - m - d)] T
For the usual direct adaptive control, in order to facilitate the implementation of projection procedure to secure the control law singularity free, the following assumption is required.
Adaptive Control Systems 27
(A4) There is a known constant 0~ satisfying
10~1 ___10~1
and
0~01 > 0
For the usual relative dead zone based direct adaptive control algorithm, another assumption is needed as follows: (A5) The constants eland e2 in (A3) are known a priori. Remark 2.1
It should be noted that the assumptions (A4) and (A5) will not be required in our new adaptive control algorithm to be developed in the next few sections. It is believed that the elimination of assumptions (A4) and (A5) will improve the applicability of the adaptive control systems.
2.3
Ordinary direct adaptive control with dead zone
Let 0(t) denote the estimate of the unknown parameter 0 for the plant model (2.4). Defining the estimation error as
e(t) = .P(t) - q~r(t- d)O(t- 1)
(2.5)
and a dead zone function as
f(g,e) =
e-g
ife>g
0 e+ g
if lel _< g if e < - g
0 < g < oo
(2.6)
then the following least squares algorithm with a relative dead zone can be used for parameter estimation /~(t) = proj{/~(t- 1) + a(t) P(t) = P(t -
1) - a(t) P ( t
P ( t - 1)~b(t- d) e(t)} 1 + ~ ( t - d)Tp(t- 1)~b(t- d) 1)r
-
d)cb(t - d)re(t
1 + cb(t - d ) r e ( t
P(-1) = kol,
-
-
I)
1)cb(t - d )
ko > 0
(2.7)
where the term a(t) is a dead zone, which is defined as follows: 0 a(t)= with0
af(61/27(t)l/2,e(t))/e(t)
if le(t)l 2 _< ~7(t) otherwise
(2.8)
~ = . l . _~j0 a , ~0 > 1, and proj is the projection operator [26]
28 An algorithm for robust adaptive control with less prior knowledge
such that
{~1 (t) if 01(t)sgn (0~) >_ [O~l #l(t) =
0~
(2.9)
otherwise
It has been shown that the above parameter estimation algorithm has the following properties: (i) 0(t) is bounded
1 (ii) [01(t)>_ 10~ml and ~i(0 f (~l/27(t) 1/2, e(t) )2 (iii) i + ~b(t- d)rP(i
(iv)
=
1
I~i(t 01 )
01
i)~b(t- d) E 12
f (~l/27(t) 1/2, e(t) )2
1 + ~b(t- d) Tp(t -- 1)O(t-- d) E/2 .
.
.
.
.
where
e(t) = y(t) - 0(t - d) r ~b(t - d)
(2.10)
The direct adaptive control law can be written as
u(t) where
rf(t) =
- O'(t)r~b'(t)
0'(t)
rf(t) = R(z-l)r(t) and R(z -l) is specified
(2.11) (2.12)
with r(t) a reference input by the reference model equation (2.2). Then with the parameter estimation properties (i)-(iv), the global stability and convergence results of the adaptive control systems can be established as in [26], [11], which are summarized in the following theorem.
Theorem 3.1
The direct adaptive control system satisfying assumptions (A1)(A5) with adaptive controller described in equations (2.7)-(2.9) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded. However, as discussed in the first section, the requirement for knowledge of the parameters of the upper bounding function on the unmodelled dynamics and bounded disturbances is very restrictive. In the next section, we are attempting to propose a new approach to get rid of such a requirement.
2.4
New robust direct adaptive control
Here we develop a new robust adaptive control algorithm which does not need such knowledge. That is, we drop the assumption (A5).
Adaptive Control Systems 29
The key idea is to use an adaptation law to update those parameters. The new parameter estimation algorithm is the same as equations (2.7-2.9) but with a different dead zone a(t), where
a(t) = {
0 r f (~l/2('~(t) 4" Q(t))l/2, e(t))/e(t)
if le(t)[ 2 < ~(-~(t) + otherwise
. cO(l+#5(t . .
d)rP(t .
1)~b(t
(2.13)
[supx,2]supxT,2]
and
Q(t) = 4(1
Q(t))
d)o<_r<_, 1
LO
(2.14) And ,~(t)is calculated by
[
~(t) = C(t)r O
C(t) = (~(t - 1) 4-
(2.15) 1
2(1 - a)(1
a(t)/3 + ~(t- d)rP(t-
sup llx(r)ll2]
O_<'r_
1)~b(t- d))
1
/3 > 0 (2.16) where C(t) r =
e2]
[~l
with zero initial condition. It should be noted that ~l and ~2 will be always positive and non decreasing. As shown in [23], the projection operation does not alter the convergence properties of the original parameter estimation algorithms. Therefore, in the following analysis, the projection operation will be neglected. The properties of the above modified least squares parameter estimator with a relative dead zone are summarized in the following lemma. Lemma 4.1 The least squares algorithm with equations (2.7), (2.9), (2.13)(2.16) applied to any system has the following properties irrespective of the control law: (i) ~(t) is bounded (ii) C(t) is bounded and non-decreasing, and thus ~1 convvrges to a constant, say el (iii) ]0l(t)> 10~l -
and ~11 -[ 0~Tt)] 1
(iv) lid(t)- 0 ( t - 1)[ I ~/2
(0
-
0-r
30 An algorithm for robust adaptive control with less prior knowledge
f(t) 2 (v) f(t) 2 :='1 q- ~b(t-d)T'P(t 1)~b(t- d)
:=
f(~l/2(~/(t) + Q(t)) 1/2 e(t)) 2 ' 1+ ~(t - d)Tp(t - 1)~b(t -d)
El2
Proof Define a Lyapunov function candidate V(t + 1) = 89
1)-10(t) + C'(t + 1)rfl-lC'(t + 1))
(2.17)
where 0(t) = 0(t) - 0", C'(t + 1) = C(t + 1) - [e~ e2]. Then, its difference becomes V(t + 1) - V(t) =
a(t)
l)~bit - d)
1+ r X
[
.
1 + 4)(t- a ) r p ( t - 1)~b(t- d) ] a(t))~(t- d ) r P ( t - 1)~b(t- d) r/(t)2 - e(t)2 .
.
.
.
.
.
.
.
.
.
1 + (1 -
a(t)C(t) r + (1 - c~)(1 + dp(t- d ) T e ( t - l)~b(t- d)) +
X
[]o_<~_<, supIlxff)'12 1
a(t)2fl 4(1 - a)2(1 + ~ ( t - d)rP(t - 1)~b(t - d)) 2 O
O
1
1
a(t) - 1 + ~ ( t - d ) r P ( t - 1)~b(t- d) X
1 + qb(t - d)Tp(t - 1 ) ~ ( t - d) r/(t) 2 -e(t) 2] 1 + (1 - a(t))dp(t- d ) r p ( t - 1)~b(t- d)
a(t) + (1 = a)(1 + ~ ( t - ' d ) T p ( t - l i ' ~ b ( t - d))(C(t)T
sup Ilxff)ll2] + Q(t))
O
1
<
a(t) [1 ] 1 + ~ ( t - d ) r P ( t - 1)~b(t- d) (1 - ai r/(t)2 - e(t)2 +
a(t) (~'(t) - 9'(t) + Q(t)) (1- c~)(l q- qb(t - d)Tp(t - 1)~b(t - d))
Adaptive Control Systems 31 <
a(t)
- 1 + cb(t - d) r e ( t - 1 ) r
x
(1
- d)
a) r/(t)2 - e(t)2 + (i
< - 1 + ~(t-
'"~)('~(t) - 7(t) -t- Q(t))
[ - e(t) 2 + 1
a(t) d)rP(t-
1)~b(t- d)
< a(t) - 1 + ~b(t- d ) r e ( t -
1)~b(t- d)
< - 1 + ~(t-
< _ ~o - 1 -
~o
< _~o-
~o
1
a(t) d)rp(t-
(i
(-~(t) + Q(t))
_ e(t)2 + _
(1
]
a) e(t)2
~)
[ _ e(t) 2 + 1e(t)2 ] 1)~b(t- d)
~0
a(t) e(t) 2 1 + qb(t - d) 7"p(t - 1)~(t - d)
(2.18)
f(t) 2 1 + 4~(t- d)re(t-
1)r
d)
where the fact that a ( t ) e ( t ) 2 = f ( t ) e ( t ) > f ( t ) 2 has been used. Therefore, following the same arguments in [20], [21], [23], the results in Lemma 4.1 are thus proved. If using the same adaptive control law as in equation (2.11), then with the parameter estimation properties (i)-(v) in Lemma 4.1, the global stability and convergence results of the new adaptive control system can be established as in [26], [11] as long as the estimated ~l is small enough, which are summarized in the following theorem. Theorem 4.1 The direct adaptive control system satisfying assumptions (A1)(A4) with the adaptive controller described in equations (2.7), (2.9), (2.13)(2.16) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded. In this approach, we have eliminated the requirement for the knowledge of the parameters of the upper bounding function on the modelling uncertainties. But the requirement for the knowledge of the lower bound on the leading coefficient of the parameter vector, i.e. assumption (A4) is still there. In the next section, the technique of the parameter correction procedure will be combined with the algorithm developed in this section to ensure the least prior knowledge on the plant. That is, only assumptions (A1)--(A3) are needed.
32 An algorithm for robust adaptive control with less prior knowledge
2.5 Robust adaptive control with least prior knowledge The following modified least squares algorithm will be used for robust parameter estimation: O(t) = O ( t - 1) + a(t)
e ( t - 1)~b(t- d)
~,(t)
1 + ~(t - d) T e ( t -- 1)~(t -- d)
P(t) = P(t -
1) - a(t) P ( t - 1)r 1+ r
P(-l) = kd,
d)cb(t- d ) r e ( t - d) r e ( t - 1 ) r
1)
(2.19)
- d)
go > 0
and the parameter estimate is then corrected [24] as
where
O-(t) = O(t) + P(t)fl(t)
(2.20)
~(t) = .~(t) - O(t - 1) T~b(t - d)
(2.21)
the vector ~(t) is described in Figure 2.1
~(t)
p(t)
/~(t) ---IIP(t)ll /3(0 = 0
-611p(t)ll
elO'(t)l
lip(Oil- 21t~l(t)l
Figure 2.1 Parameter correction vector where p(t) is the first column of the covariancc matrix P(t), and the term a(t) is now defined as follows:
a(t)
{
o
if ~(t)l 2 < ~('~(t) + Q(t)) otherwise
with 0 < a < 1, ( = 12(o - a , (o > 1, and
Q ( t ) - [ B ( t - 1)rP(t - 1 ) ~ ( t - d ) ] 2 +
~ [ sup IIx(r)ll2]r[ suP IIx(r)lf2] 2(1-- ot)(l+ ~b(t-- d ) T p ( t -- 1)~b(t-d) o
Adaptive Control Systems 33
And ~,(t)is calculated by sup
IIxO-)ll:]
= ~(t)r o<~_<,
?(t)
(2.22)
1
(~(t) = (~(t- 1) +
(1 - cz)(l
[
~up I1~(~)112]
a(t)A
+ ~(t- d)rp(t- 1)$(t- d)
O<'r
1
> 0 (2.23) where with zero initial condition. It should be noted that ~l and ~2will be always positive and non-decreasing. Remark 5.1
The prediction error ~(t) is used in the modified least squares algorithm to ensure that the estimator property (iii) in the following lemma can be established. The properties of the above modified least squares parameter estimator are summarized in the following lemma. Lemma 5.1 If the plant satisfies the assumptions (A1)-(A3), the least squares algorithm (2.19)--(2.23) has the following properties: (i) 0(t) is bounded, and ]10(t)- 0 ( t - 1)1[ E/2. (ii) C(t) is bounded and non-decreasing, thus converges
f(~l/2(~/(t) + Q(t))lD' ~(t))2
(iii)
1 + ~(t - d) r p ( t - 1)~(t - d) ~
12
(iv) Ilp(t)ll + IOl(t)l > b ~ where bmin =
Ioll max(l, [I/~*ll)
with/3* defined such that O" = O(t) = P(t)/~*
1-e (v) l~ (0l > 3 +-e bmm (vi) 0(t)is bounded, and ll0(t)- 0(t - l)ll~ 6 Proof
Define a Lyapunov function candidate
v(t + l) = 89(~i(0rP(0-1~i(0+ (~(t+ l)r~-~t~(t+ l))
(2.24)
34 An algorithm for robust adaptive control with less prior knowledge
where 0(t) = 0(t) - 0", C(t 4" 1) = C(t 4" 1) - [el e21r. Noting that ~'(t) = ,~(t) - 0(t - 1 ) T~b(t - d) = .~(t) -/~(t-
1)~b(t- d) - fl(t - l ) r p ( t -
1)~b(t- d)
= e(t) - 13(t- 1 ) T p ( t - 1)~b(t- d)
(2.25)
Then, the difference of the Lyapunov function candidate becomes
v(t + l)- v(t)=
a(t) 1 + Jp(t- d ) r P ( t -
l)~b(t- d)
1 4" dp(t- d ) T p ( t - 1)~b(t- d)
•
1 + (1 - a ( t ) ) ~ ( t - d) r p ( t - 1)~(t - d)
x (rl(t)- ~ ( t - 1)Te(t - 1)~b(t- d)) 2 - ~'(t)2]
+
+
[ 0
a(t)~(t)r
(1 -- a)(1 + t k ( t - d ) r P ( t - 1)O(t- d))
Iix(~.)I121
a(t)2A (1
-- ~ ) 2 ( 1 4" dp(t- d)Tp(t -
1)~b(t- d)) 2
a(t) - 1 + ~(t - d ) r p ( t - l)~b(t - d)
,
]
X [] _ ~ (rl(t) -- iO(t- 1 ) T p ( t - 1)~b(t- d)) 2 - ~'(t)2 +
2a(t) (q(t) - '7(t)) (1 - cO(1 + dp(t- d ) r p ( t -
1)~b(t- d))
a(t)2A +
(~ - ~)' (i
+ o(t - d) ~ eft - ~i~(t - di):
sup[Jx(r)ll2]r [O
O
1
1
Adaptive Control Systems 35
< a(t) - 1 + qb(t- d ) T p ( t - l)~b(t- d) ,7(t) 2 - ~ 12-
+
2 (t)
(/3(t- 1 ) r p ( t - 1)~b(t- d)) 2 - ~(t) 2] - 7(0)
(1 - a)(1 + qb(t- d)rp(t - 1)~b(t- d)) a(t) 2A
[sup[Ix(r)[[2]T[ O<_r
+ (1-- a)2(l + #b(t- d l r P ( t - l)~b(t- d)) 2
-
I +
a(t) qb(t- d ) r p ( t - 1)~b(t- d)
x
2 1-c~
1
1
77(t)2 - c'(0 2 + i 2 c~('~(t) - q(t) + Q(t))]
< a(t) - 1 + ~b(t- d ) r P ( t - 1)~b(t- d)
[ - ~,(t)2 + 2
]
~ - a ( ~ ( t ) + Q(t))
a(t) < -g'(t)2+i" 2 cr~1 (t)2] - 1 + ~b(t- d ) r P ( t - 1)~b(t- d) < a(t) [ _ g,(t)2 + l~,(t)2] - 1 + qb(t- d ) r p ( t - 1)~b(t- d) ~o J ~0- 1
1+ < -
a(t)~(t) 2 d ) r P ( t - 1) b(t- d)
~o - 1 f({l/2(~/(t) + Q(t))l/2,~(t)) 2 ~o 1 + qb(t- d ) r P ( t - 1)~b(t- d)
(2.26)
where the fact that a(t)e(t)2=f(t)e(t)>f(t) 2 has been used. Therefore, following the same arguments in [26], [11], [21], the results (i)-(iii) in Lemma 5.1 are thus proved. The properties (iv)-(vi) in the lemma can also be obtained directly from the results in [23]. If using the same adaptive control law as in equation (2.11), then with the parameter estimation properties (i)-(vi), the global stability and convergence results of the new adaptive control system can be established as in [26], [11] as long as the estimated s is small enough, which are summarized in the following theorem.
36 An algorithm for robust adaptive control with less prior knowledge
Theorem 5.1
The direct adaptive control system satisfying assumptions (A1)(A3) with the adaptive controller described in equations (2.19)-(2.23) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded.
2.6 Simulation example In this section, one numerical example is presented to demonstrate the performance of the proposed algorithm. A fourth order plant is given by the transfer function as =
a,(s)
with
5(s + 2) s(s+ 1)
=
as a nominal part, and
G,(s) =
229 s2 + 30s + 229
as the unmodelled dynamics. With the sampling period T = 0.1 second, we have the following corresponding discrete-time model
G(q-l) =
1
0.09784q -l + 0.1206q -2 - 0.1414q -3 - 0.01878q -4 2.3422q 2t + 1.0788q - 2 - 0.4906q -3 + ().04505qL4
The reference model is chosen as Cm(S) =
0.2s+ 1
whose corresponding discrete-time model is
Gm(q-~) =
0.3935q -l 1 - 0.6065q -1
We have chosen k0 = 1, and 0(0) = [0.6 0 0 0] r. If no dead zone is used, the simulation results are divergent. If using the algorithm developed in this chapter with ,~ = l0 -s, the simulation results are shown as in Figure 2.2, where (a) represents the system output y(t) and reference model output y*(t), (b) is the control signal u(t), (c) is the estimated parameter ~l, and (d) denotes the estimated bounding parameters ~l and g2. In order to demonstrate the effect of the update rate parameter ,~, the following simulation with A = 1.4 • 10-5 was also conducted. The result is shown in Figure 2.3. The steady state values of the several important parameters and the tracking error in both eases are summarized in Table 2.1.
Adaptive Control Systems 37 4
(a)
G) 0 r
(b)
0 ,..2 O
2 O l_
~o
,-'0 O
m
e~. 2
-2
O
40
50 100 150 Time in seconds
2
.C
....
200
50 100 150 Time in seconds I~ 2 x10"3
u
(c)
E
(d)
E 1 E.5
1.5
epsl
r O.
"D
" ~E
200
_7
ca 1
1
r
"10 t--
,,~ 0.5 - ' n
. . . .
.~0.5
eps2 ._.._J
50
I00 150" Time in s e c o n d s
Figure 2,2
2(10
=o o
Robust adaptive control with A = 10-
50 100 150 Time in seconds
200
$
4
(a)
0 rG)
(b)
"-- 2
2
(I)
"~
-6
i
l,. ,e.a
o
c
u0
0
e~ -2 O d 0
s'o i~o
1~0
200
50 100 150 Time in seconds
Time in seconds 2 O
(c) m 1.5
2
(d) J
D. O~
c
~ "O r
E
epsl
1 eps2
~0.5
,,~ 0.5
Figure 2.3
xl0 ~
~1.5 (u
0o
200
IJJ
50 100 150 Time in seconds
200
Robust adaptive control with
00
50 100 150 Time in seconds
A = 1.4 x 10 -5
200
38 An algorithm for robust adaptive control with less prior knowledge Table 2.1
Steady state values ,.,.
,
,,
..
A = 10 -5 ,,
,
.
,,
.
,
.
.
.
A = 1.4 x 10 -5 ,
1.1898 x 10-3 0.3509 x 10-3 0.5649 0.0179
~2
ly-y'l
,
.
_
,
1.4682 x 10-3 0.4467 x 10-3 0.5833 0.07587
,.
It can be observed from the above simulation results that the algorithm developed in this chapter can guarantee the stability of the adaptive system in the presence of the modelling uncertainties, and the smaller tracking error could be achieved with smaller update rate parameter A. Most importantly, the knowledge of the parameters 6~ and e2 of the upper bounding function and the knowledge of the leading coefficient of the parameter vector 0 ~ are not required a priori.
2.7
Conclusions
In this chapter, a new robust discrete-time direct adaptive control algorithm is proposed with respect to a class of unmodelled dynamics and bounded disturbances. Dead zone is indeed used but no knowledge of the parameters of the upper bounding function on the unmodelled dynamics and disturbances is required a priori. Another feature of the algorithm is that a correction procedure is employed in the least squares estimation algorithm so that no knowledge of the lower bound on the leading coefficient of the plant numerator polynomial is required to achieve the singularity free adaptive control law. The global stability and convergence results of the algorithm are established.
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Adaptive Control Systems 39 [5] Naik, S. M., Kumar, P. R., Ydstie, B. E. (1992). 'Robust Continuous-time Adaptive Control by Parameter Projection', IEEE Trans. Automat. Contr., Vol. AC-37, No. 2, 182-197. [6] Praly, L. (1983). 'Robustness of Model Reference Adaptive Control', Proc. 3rd Yale Workshop on Application of Adaptive System Theory, New Haven, Connecticut. [7] Praly, L. (1987). 'Unmodelled Dynamics and Robustness of Adaptive Controllers', presented at the Workshop on Linear Robust and Adaptive Control, Oaxaca, Mexico. [8] Petersen, B. B. and Narendra, K. S. (1982). 'Bounded Error Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-27, 1161-1168. [9] Samson, C. (1983). 'Stability Analysis of Adaptively Controlled System Subject to Bounded Disturbances', Automatica, Vol. 19, 81-86. [10] Egardt, B. (1980). 'Global Stability of Adaptive Control Systems with Disturbances', Proc. JACC, San Francisco, CA. [11] Middleton, R. H., Goodwin, G. C., Hill, D. J. and Mayne, D. Q. (1988). 'Design Issues in Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-33, 50-58. [12] Kreisselmeier, G. and Anderson, B. D. O. (1986). 'Robust Model Reference Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-31, 127-133. [13] Kreisselmeier, G. and Narendra, K. S. (1982). 'Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances', IEEE Trans. Automat. Contr., Vol. AC-27, 1169-1175. [14] Iounnou, P. A. (1984). 'Robust Adaptive Control', Proc. Amer. Contr. Conf., San Diego, CA. [15] Ioannou, P. and Kokotovic, P. V. (1984). 'Robust Redesign of Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-29, 202-211. [16] Iounnou, P. A. (1986). 'Robust Adaptive Controller with Zero Residual Tracking Error', IEEE Trans. Automat. Contr., Vol. AC-31,773-776. [17] Anderson, B. D. O. (1981). 'Exponential Convergence and Persistent Excitation', Proc. 20th IEEE Conf. Decision Contr., San Diego, CA. [18] Narendra, K. S. and Annaswamy, A. M. (1989). Stable Adaptive Systems, PrenticeHall, NJ. [19] Feng, G. and Palaniswami, M. (1994). 'Robust Direct Adaptive Controllers with a New Normalization Technique', IEEE Trans. Automat. Contr., Vol. 39, 2330-2334. [20] Goodwin, G. C. and Sin, K. S. (1981) 'Adaptive Control of Nonminimum Phase Systems', IEEE Trans. Automat. Contr., Vol. AC-26, 478-483. [21] Feng, G. and Palaniswami, M. (1992). 'A Stable Implementation of the Internal Model Principle', IEEE Trans. Automat. Contr., Vol. AC-37, 1220-1225. [22] Feng, G., Palaniswami, M. and Zhu, Y. (1992). 'Stability of Rate Constrained Robust Pole Placement Adaptive Control Systems', Systems and Control Letters, Vol. 18, 99-107. [23] Lazono-Lcal, R. and Goodwin, G. C. (1985). 'A Globally Convergent Adaptive Pole Placement Algorithm without a Persistency of Excitation Requirement', IEEE Trans. Automat. Contr., Vol. AC-30, 795-799.
40 An algorithm for robust adaptive control with less prior knowledge
[24] Lazono-Leal, R., Dion, J. and Dugard, L. (1993). 'Singularity Free Adaptive Pole Placement Using Periodic Controllers', IEEE Trans. Automat. Contr., Vol. AC-38, 104-108. [25] Lazono-Leal, R. and Collado, J. (1989). 'Adaptive Control for Systems with Bounded Disturbances', IEEE Trans. Automat. Contr., Vol. AC-34, 225-228. [26] Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering, Prediction and Control, Prentice-Hall, NJ. [27] Lazono-Leal, R. (1989). 'Robust Adaptive Regulation without Persistent Excitation', IEEE Trans. Automat. Contr., Vol. AC-34, 1260-1267.