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Discrete model reference adaptive control with an augmented error signal
Automatica. Vol. 13. pp. 507 517.
Pergamon Press, 1977. Printed in Great Britain
Discrete Model Reference Adaptive Control with an Augmented Error Signal* T U D O R IONESCUt:~ and RICHARD M O N O P O L I §
Using a Lyapunov based design for discrete single-input single-output plants, stable model reference adaptive control systems avoid the need for anticipative values of the plant output by using the augmented error concept and only plant input and output signals. Key Word Index--Adaptive control; (augmented error signal); discrete systems; Lyapunov methods; (model reference systems); nonlinear control systems; stability; system theory; time-varying systems. Summary--A method is developed for designing discrete model reference adaptive control systems when one has access to only the plant's input and output signals. Controllers for single-input, single-output, nonlinear, nonautonomous plants are developed via Lyapunov's second method. The augmented error signal method is employed to ensure that the normally used true error signal approaches zero asymptotically without requiring anticipative values of the plant output signal. Such anticipative signals are replaced by others easily obtained from low pass digital filters operating on the plant's input and output signals.
x(k +n)+alX(k + n - 1)+... +a,x(k)--bou(k +m) +... + b,,u(k) + cf(X(k), k)
(1.1)
where u(k),x(k) are the plant input and output respectively, X(k) is the set of x ( k + n - i ) for i= 1, 2.... and f (X ( k ), k) is a nonlinear time varying function of known form. It is assumed that coefficients ai, bi and c are unknown and constant or slowly varying. The term cf(X(k), k) may be replaced by a sum of terms of the same type. There is no loss of generality in carrying only one term of this kind. It is also assumed that: (a) The function f ( . . . ) satisfies the conditions necessary for solutions of (1.1) to exist and be unique, (b) all roots of the polynomial bo +blz -1 +...b,,z m, are in the unit circle, (c)+oo > b0M > bo > bo,, > 0, and boM and bo,, are known, (d) m < n - 1 and must be known. Assumption (b) is required to ensure a bounded control input by not requiring the adaptive system to act to cancel non minimum phase plant zeroes. For a similar reason u is not included i n f In general, it is not known what restrictions to impose on these functions to ensure that u is bounded. The design objective is to have the plant output follow the output of a model reference defined by the equation:
INTRODUCTION
THE AUGMENTED error signal method for continuous model reference adaptive control systems introduced in [1] is used to solve the discrete model reference adaptive control problem. Extension of results presented in [2], and [-3] are given. Further detail on the results of this paper and related problems can be found in [4]. This paper is divided into three parts. In section I, the problem statement and the notation are given. Section II gives the main result. Simulation results are presented in section III. Proofs of lemmas and peripheral results are confined to the appendices. I. NOTATION AND PROBLEM STATEMENT
A discrete time dynamic system (plant) can be described by the non-linear, nonautonomous difference equation *Received 18 August 1975; revised 8 September 1976; revised 13 December 1976; revised 3 March 1977. The original version of this paper was presented at the 6th IFAC Congress on Control Technology in the Service of Man which was held in Boston, Cambridge, MA 02138, U.S.A. during August 1975. The published Proceedings of this IFAC Meeting may be ordered from: IAS, 400 Stanwix Street, Pittsburgh, PA 15222 or John Wiley, Baffins Lane, Chichester, Sussex, PO 19 IUD, UK. This paper was recommended for publication in revised form by associate editor I. Landau. tlntermetrics, 701 Concord Avenue, Cambridge, MA 02138, U.S.A. l-ormerly with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A. :Wormerly with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, Massachusetts. §Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A.
xM(k +n)+adlxM(k +n - 1)+...+ad,xM(k) = Kor(k)+g(XM(k),r(k),k)
(1.2)
where the nth-degree polynomial in z with coefficients ad,, i = 1..... n has roots inside the unit circle, xM(k) is the model output, r is the reference input, Ko is a known scalar X~(k) is the set of xM(k + n -- i) for i = 1, 2 ..... and g(,, ) is a nonlinear time varying function with the properties required for the existence and uniqueness of solutions to (1.2). The design problem is to synthesize a parameter adaptive control system for (1.1) which will cause 507
508
TtJDOR IONESCU and RICHARD MONOPOLI
the error e ( k ) = x M ( k ) - x ( k } , between the plant and model outputs to a p p r o a c h zero. As previously mentioned the special feature of this work is that anticipative values of the plant output x do not a p p e a r in the design, i.e. if the a d a p t a t i o n takes place at time k, then x ( k + j ) , j = 1,2 .... do not appear in the algorithms for the adaptive gains or the controls. It is convenient at this stage to introduce the delay o p e r a t o r z m, and the following polynomial delay operators:
Zp(n)= 1 +
aiz ' i
1
Zu(rtl):bo+ ~ bi2-i
(*)
i-I
ZM(n)= 1 + ~ ady i i-1
Using (*) in (1.l) and (1.2) one has
Next (1.4) is added to (I .5) to obtain
ZM(n)rl(k + n ) = K o r ( k l + g ( k ) - Z , ( m ) u ( k ~-m) + Z a ( n - 1)x(k + n - 1 ) - q l ( k ) + Z w i n - 1)(w(k + n - l ) + q ( k - + n ) ) ~1.6) Equation (1.6) is the starting point for the synthesis. The design p r o b l e m is to generate u(k), w(k) and q(k), all independent of x ( k + j ) , j = 1,2 ..... and in such a way that e(k)~O as k--, oc. State variable filters must be introduced to avoid anticipative values of x. These play the role of an observer of sorts. However, knowledge of plant parameters is not required to design the state variable filters as it would be to design an observer. Also, although an estimate of the state is not obtained, it is not required. Part of the required filtering is defined in terms of filter output signals
x i, i = O, 1..... 4 as follows Zp(n)x(k + n ) = Z , ( m ) u ( k + m ) + c f (k) (1.1") Z~(n- l)x°(k+n -1)=u(k+m) ZM(n)xM(k+n)=Kor(k)+g(k)
(1.2") Z~,ln - 1 )x I (k + n -
1 ) = r(k)
Z,~(n-t)x2(k+n
1)=x(k-
The concept of a u g m e n t e d error signal detailed in [1] must be introduced to achieve the design objective. Let t/(k) be the augmented error signal:
Zw(n - 1 )x 3 (k + n - 1 ) = f ( k )
tl(k)=e(k)+ ),(k)
(1.3)
Zw(n - 1 )x4(k + n - 11 = g(k)
where the signal y(k) is the output of the error a u g m e n t a t i o n filter defined by:
Using (1.7). one can write (1.6) as
Z w(n)y(k+n)=Z,,.(n- 1)(w(k+n-
1 )+q(k+n))
- q f (k)+ Z a ( n - 1)x(k + n - 1) (1.5)
(1.7t
- Z,(m)x°(k + n - 1 ) + Z A ( n - 1)x2(k + 2n - 1) - c x 3 ( k + n - 1)+ x4(k +n - 1) +w(k+n-1)+q(k+n)]
(t.8)
The additional filtering required is defined in terms of signals x s and u s which satisfy
Zf(n-mZM (n)e(k + n)=Kor(k) + g ( k ) - Z,(m)u(k + m)
1)
ZM(nkl(k + n ) = Z w ( n - 1)EKoxt(k +n - 1)
(1.4) where Zw(n - 1 ) = ~ ' £ ~ ciz -i and w(k ) and q(k ) are auxiliary system inputs to be determined along with the control input u(k) as part of the design. The coefficients c / m u s t be chosen in a special way, the details of which will be explained later in the development. It is noted here that the roots of Z,,. (n - 1 ) will be inside the unit circle
-
1)xS(k +n - m - 1)
=(l+FlZ-l+F2z
2+...
+F,_ m_lz -(n " - l ) ) x f ( k + n - m - l ) = x ( k ) Z s ( n - m - 1 ) u f (k + n - m - 1 )
=u(k) (1.9)
where n
ZA(n--1)~
~ Aaiz (i-1),Aai=ai-adl i-1
for i = 1 ..... n.
where constants F i are such that the polynomial Z s ( n - m - 1 ) has all its roots inside the unit circle. N o t e that this filtering is not required if m = n - 1. This is a special case for which Zr = 1, xS(k)=x(k), and uS(k) = u(k ).
Discrete model reference adaptive control with an augmented error signal The following lemmas, proved in Appendix A, are used to prove the main result.
509
(9i(k + n - 1 ) = z - ~ i - 2~Z~ X( n - m - 1)x° ( k + n - 1); for i = n + 2 .... 2n+l,fli=Bi_l,+2 ,
Lemma 1.1 There exist constants A o through A,_ 2 such that
and Oi(k+n - 1 ) = z -~i m-a)zf 1(n - m -
Zu (m)x ° ( k + n - 1 ) = (bo/co)u f ( k + n - 1 ) n-2 + ~, A i z - ~ i + l ) Z f l ( n - m - 1 ) x ° ( k + n - 1 )
for i = 2 n + 2 .... 3 n - m ,
x Z£ l ( n - 1 ) x ( k + n - 1);
fli=U(n,m)cDi_t2,+ 2),
and
(9i(k-4- n - 1) = U(n, m)z -¢i- 2n- l ) z f
i=0
(1.10)
Lemma 1.2 There exists constants Bo through B,-1, Co through C._ 2, and Do through D,-m-2 such that
1)
1
× ( n - m - 1)xa(k+n-1). The equation to be used later for deriving u(k) and w(k) is u f ( k + n - 1)+w l ( k + n - 1) N-1
n-1
= ~ Ki(k+n-1)(9i(k+n-1)
Z a ( n - 1 ) x 2 ( k + 2 n - 1 ) = Z Biz-~"-'-l+i)z~ 1 i=0
× (n-m-1)x2(k+2n -l)4-U(n,m) -n-2
X
E Ciz-(i+l)zf I(H i=o -m-l)
×x°(k+n-l) n-m-2 4- E Di 2-(i+1)
+n-l)]
where U(n, m ) = 1 for m =
+w(k+n-1) + N-1 ~ fli(9i(k+n-1) + q ( k + n ) 1 i=0
flo =Ko, f l l
=
(1.12)
1,
- - C,
fie = 1, (9°(k + n - 1) = x l ( k + n - 1),
(9l(k + n - 1)=x3(k +n - 1), (92(k+n-1)=x4(k+n-1); fli=U(n,m)Ci_3-Ai_3
and
where w(k)=KN(k)wl(k) and Ki(k) for i = 0 to N are the adaptive gains. Substituting (1.13) into (1.12) yields
ZM(n)rl(k + n) = Zw(n -- 1 ) (i=~° b i ( k + n - 1 ) ( g i ( k + n - l ) + q ( k + n ) ) (1.14) where
6i(k)= -(bo/co)Ki(k)+fii for i = 0 ..... N - 1 , 6N(k)
i=0
× Z -f l ( n - m - 1)cx3(k
where N = 3 n - m +
(1.13)
i-0
for i = 3 , 4 .... n + l ,
=KN ( k ) 4- bo/ co, and (9N(k ) = w 1( k ). Before developing the complete solution to the nonlinear problem, it is useful to consider the application of the results developed thus far to the design for a linear time invariant system with known coefficients. This gives an appreciation of the basic structure of the adaptive system, and some insight into the role of the state variable filters. The linear, time invariant, known parameter problem is specified by letting g = 0 in (1.2), c = 0 in (1.1) and assuming that coefficients ai and bl in (1.1) are known. For this problem, there is no need to adjust the gains Ki(k), therefore they are taken to be constants. Also, the auxiliary signals w(k) and q(k) may be taken to be zero in this case. Then, for these conditions, one has to show that constant gains K ~ can be found such that for u(k) given by 2n+l
u(k)= ~ K i Z I ( n - m - 1 ) ( 9 i ( k + n - m - 1 ) i=0
i~1,2
(1.15)
the plant transfer function X {Z }/R {Z) is equal to the model transfer function xM{Z)/R(Z). Equation (1.15} is derived by applying the operator Z f to both sides of(1.13) and using (1.9). The proof of this result is given in Appendix B. Figure 1 shows the resulti,ag system for this linear, time invariant known parameter case.
510
TUDOR I()NES('ti and RICHARD MON()I>OLI
R(z): _ ~
U(z)
D-
~. 2o
] ]
x
)
n¢L ,=3
z,,
Zn*l
~. K ; Z I n + 2 ,;n+2
;I
Zw
F~c;1. Systemconfigurationfor specialcase. 11. THE MAIN RESULT
Proof
Returning to the main problem, we put (1.14) in the matrix vector form
The proof requires the following two lemmas which are proven in Appendix C. Let
(k)
q(k+l)=Aq(k)+d
i
l+...+a,
~(z)=z"+alz"
6'(k)O;(k)+q(k+l)
(2.1)
(2.4t
be a polynomial with real coefficients. Define the scalars'
where
I
-aa~
1
a, 0 ..... n - - l ' a o = l ( 2 . 5 ) ak k-]:k= ~
~k=det[L,
ql ( k ) = q ( k ) = e ( k ) + y ( k ) . and 0 ... 0-~
Let
A-~-l--(.gd2__adn 00 1 ...... 0C :IdT0= [CO'
7z'(z)=':%z"-l+:q
..... Cn-l]
z"-2 + . . . + e , - I
(2.6)
L e m m a 2.1
The main result is Theorem 1
There exists scalars c/, i = 0,..., n - 1 such that the following conditions are satisfied : (i) Zw(n - 1 ) has all its roots inside the unit circle. (ii) For each Kq>O and for )~imin>I(2Kq) -1, the adaptive laws Ki(k)-Ki(k-
If ~z(z) has all its roots inside the unit circle then r((z) also has its roots inside the unit circle. The next lemma deals with a particular solution of the Lyapunov matrix equation for discrete systems. Let A and Ac denote
A=
Co
1 ) = 2 ~ ° t/1 ( k ) ~ i ( k - 1) i=O .... N - 1
KS(k)-KN(k-
-a•
1 0 ... 01
-a2
0
k-a.
(22)
1
1)= - - ~ q l (k )OS( k - 1 )
AN
where
1 ... 0
0
,
... 0 I
0
1
0
...
0
0
l
0
0 ..
0
A, =
(2.71 0
"~imin= min (20 for i=0,
1.... N and the auxiliary control: --
~ln
--
an-
1
. . . . .
--
(all
_
N q ( k ) = -Kqr/(k) ~-~ (qSi(k- 1)) 2
i=0
yield
lim tt(k)=O k~y
(2.3)
L e m m a 2.2
If A has all its eigenvalues in the unit circle then there exist positive semi-definite matrices R T = R, QT = Q and a positive definite matrix Pov = Po = [Pij]
Discrete model reference adaptive control with an augmented error signal such that (i)
-dTpo~d AV~PoA~-Po = - R
511 6i(k-1)c~i(k-1)
i
(2.8)
+q(k)) 2
and the polynomials N
+ ~ ,~,i(6i(k)-fi(k - 1))(hi(k)
A(z) = p..z ~- 1 +... + p2,z + Pl,
i=O
+hi(k-l))
A'(z) = p l l z " - 1 +... +Pl(n- l)z+Pl, From (2.2) we get
have all their roots inside the unit circle. (ii)
ATPo 1A
-
-
1
Po 1 = _ Q
6'(k)-6'(k-1)=-~th(k)dpi(k-1);i=O
We now turn to the proof of the main result. First note that A in (2.1) is of the form in (2.7) and is stable by assumption. Now choose the scalars czi= 0,..., n - 1, such that tiT=[1
0 ...
0] Po
(2.12)
(2.9)
where Po is given by lemma 2.2. This choice of c, implies that Z w ( n - 1 ) has its roots inside the unit circle. Next consider the following Lyapunov function candidate
.... ,N (2.13)
If 261(k-1) is added to both sides of (2.13) one obtains 6'(k) + 6~(k- 1) = 2 6 i ( k - 1 ) - ~ rh ( k ) O i ( k - 1);
i = 0 ..... N
(2.14)
Substituting (2.13) and (2.14) into (2.12) one has V ( k + 1 ) - V ( k ) ~ -aOmmll~(k)ll2-d T eo Xd
V (k)= Ir/x (k)-dT(i~=° fi~(k- 1 )q~(k-1 ) + q ( k ) ) l
xPo ~ t/(k)-d
N
2
N 1 2
6~(k- 1)~b'(k- 1)
y~ 7q~, (k- ll+2,h(k)q(kl
i
(2.15)
i=0 ~i
+ ~ X:~;~(k-1)
+q(k)
(2.10t
i=0
Then V(k + 1 ) - V(k) = ~tT(k)(ATp o XA - Pot )rl (k) N
+2dTPoXq(k) ~ 6 ' ( k - 1 ) # ( k - 1) i=0
+ 2d~ Po ~ ~l(k)q(k) - d T p o Xd(~ b/(k- 1 ) ¢ ' ( k - 1) + q(k)) 2 N
+ 2 2i((~i(k)-6i(k- 1))((~i(k)+6i(k- 1))
(2.11)
From (3.2) and the assumptions it follows that V(k + 1 ) - V (k) < 0 for q (k) @0. Therefore, since V (k) is positive definite, we have by the Lyapunov theorem that limk, ~ q(k)=0. This ends the proof of the theorem. Q.E.D. Remarks (a) Theorem 1 represents the discrete analog of the results in [1]. (b) Lemmas 2.1 and 2.2 are of independent interest and might be regarded as corollaries to the well known Jury stability criteria. (c) The apparent loop implied by (1.3), (1.4) and (2.3) may be resolved as follows: Substitute (2.3) into (1.4) and use (1.3) to get
i=0
From lemma 2.2 and (2.9) it follows that
ZM(n)y(k + n) = Zw(n - 1 )(w(k + n - 1 ) - ( e ( k + n ) + y ( k + n ) ) m E ( k + n - 1))
V(k + 1 ) - V(k) = - n T ( k ) Q q ( k ) + 2th (k)
(R-l) N
x ~ bi(k-1)ch~(k-1) i=0
+2qs(k)q(k)
where N
m2(k+n - 1 ) = K ~ Z (q~i(k+ n - 1)) a. 0
512
Tt:D()R ]()N['S('t' and RICHARD M()NOPOII
Let
are not required to generate either u(k or w(k) Now, using (l.9)in (2.17) one has ZM(n)=I+z
*Z~(n--l). 2 Ki(k)(//(k) o
wl(k)=
where
Z):l(n-m-I.,
~, adi2
Z~ln-1)=
u (kt
li 11
t2.17')
i=1
Substituting (2.16)into (2,17't gives
and
Z.,(n- 1 )=co
+ z - xZ.,(n-
N
1
}{'l(k)~--- ~, (Ki(k)~i(k)--Z.f l(H--nl--
2),
IlK i
0
where
w n
× (k-n+m)Zr(n-m-
1)~ i ( k ) )
l
Z w ( n - 2 ) = Y~ ciz
(i
11
(2.17"}
i=l
Since rl(k)--+O assures (from (2.2)), then
Then (R-I)may be written as
Ki(k)-Ki(k-
(i +Co m 2 ( k + n - 1 ) ) y ( k + n ) = - - ( Z M ( n - - 1 ) + Zw(n-2)mZ(k +n -1))y(k +n+ Z,~, (n - 1 ) ( w ( k
+ n -
1)
1 ) - m 2 (k + n -
1 )e(k + n))
(R-2) Using (R-2). the algorithm for y(k) becomes y(k) = (1 ÷ com2(k - 1 ))- l( _ (ZM( n _ 1 ) + Zw(n - 2)m 2 (k - l ))y(k - i )
+ Z w ( n - 1 ) ( w ( k - 1 ) - m2(k - 1 )e(k))) (R-3) Thus, for implementation, (R-3) is used to generate y ( k ) and (1.3) to generate ~/(k). Since y(k) and r/(k) depend directly on e(k), it must be assumed that the total time taken for the required multiplication in (R-3) and the required addition in (1.3) is negligible relative to the time interval between k and k + 1. In order to complete the design, the control signal u (k) and the auxiliary control w (k) must be specified in a way which insures that ~/(k)--+0 implies that e( k )-+O. Let Kqk) and q(k) be given by Theorem 1. Then e(k)+0if N-
Ki(k)
approaches constant
1 ) ~ 0 as k--, ~.
Hence, from (2.17"), w l ( k ) (and w(k) also) approaches zero as k-+~:. Since q(k) and w(k) approach zero as q(k) approaches zero, then so does y ( k ) , a n d a l s o e ( k ) . Q.E.D. 1 his prool relies oil the assumption that signats (hi(k) are bounded for all i and k. Such an assumption is also required in Ell, but is not explicitly stated there. This assumption was made in all simulations run by the authors, and no dilficulties were encountered, i.e. these signals did remain bounded yielding the desired result, e(k)---,0. One such simulation is given in the next section. An algorithm that does not require the boundedness assumption for q~q,k)is given next. It is somewhat more complicated than that above and has not been tested by simulation. Basically, it requires a decreasing gain in the adaptive algorithms (2.2) as 3,
( ( h i ( k - 1 )) 2 i=:l)
increases. For this case, u{k) and w 1(k) are still as given by (2. l 6) and {2.17) respectively. However, (2.2) and (2.3) must be modified as follows
1
u(k)= ~, Ki(k - 1 ) Z r ( n - m -
1)qSi(k+n-m - 1)
i=0
Kqk)-Kqk-
l)=fS' U,,{kl~?,(k)eqk-l)
(2.16) and
lor i = 0 ..... N - I N-l wl(k)= 2 Ki(k)(bi(k)-uf(k) i=o
(2.17)
Proof First note that (2.16) and (2.17) satisfy (1.13). Furthermore, anticipative values of the plant output
Khr(k)-K:V{k
- 1)= ---
1
Kl,.(k)~ h (k)qSi(k - 1 )
where Kt,-(k)=K k K k + ~ (4)i(k - 1 ))2 i=0
{2.2')
Discrete model reference adaptive control with an augmented error signal
5l 3
Filtered variables required are defined by
and Kk is a positive constant
q( k ) = - K o ( k )tlx (k )
(2.3')
Zwx°(k + 1) =cox°(k + 1 ) + q x ° ( k ) = u(k)
where
Zwx 1(k + 1) = r(k) K¢2(k)=Kq( Kk +i~o (~bi(k- 1 ))2)
ZwX2(k+ 1 ) = x ( k - 1)
(3.4)
ZyxY(k + 1)= (1 + F l z - 1 )Xf( k _~_1 ) = x ( k ) and K. is a positive constant. It is shown in Appendix D that the modified algorithm as given by (2.2') and {2.3') does insure that e(k)--+O without the a priori assumption that ~bi(k) are bounded for all i and k. Conditions forKq a n d K k are also given there.
Z ruY(k + 1) = u(k) where
F1 = 0 . 5 .
The gain a d j u s t m e n t a l g o r i t h m s for this e x a m p l e are
K ° (k) = K ° (k - 1 ) + A ° t l ( k ) O ° (k - 1 ) K 3 (k) = K 3 (k - 1 ) + A3r/(k)q~3 (k - 1 )
111. S I M U L A T I O N RESULTS
Digital computer simulations were performed for an inherently discrete second order plant described by
g 5 (k) = K 5 (k - 1 ) + Ast/(k)q~5 (k - 1 )
x(k + 2)+alX(k + l )+a2x(k)=bou(k ) (3.1)
K 7 (k) = K 7 (k - 1 ) + Avt/ (kkb 7 (k - 1 )
3--
-3
--
K*(k)=K4(k-1)+A4tl(k)O4(k-1)
•
^
•
(3.4)
M
-'__Z ;,';'
•
F1G. 2. Simulation results.
The model used was x M(k + 2) + adlxM(k + 1) + ad2X(k) =Kor(k) (3.2) where
ad, = 0.5, an2 = 0.6 and Ko - 2.0. The augmenting variable v is defined by
y(k + 2) + ady(k + 1 ) + ad2y(k) = CoW(k + 1) + clw(k) + coq(k + 2) + caq(k + 1) where Co =0.64 and cl =0.2.
(3.3)
where A i =Co/(2ibo) and q9 are as defined in (1.12}(1.14). The control input u(k) and the auxiliary input w(k) were generated according to (2.16) and (2.17) and the definition below (1.13). The second auxiliary input q(k) was generated according to Theorem 1 with Kq = 1/2~ml,. In Fig. 2, xU(k) and e(k) are plotted for k = 0 to 36. For this experiment r ( k ) = +1 for 0 < k < 1 0 , r(k) = - 1 for 11 _
514
TIID()R
IONESCII
and
adaptive gains K " , K ~ and K: converged to their correct values but not K '~ a n d K s . This is unimportant, however, since control is the objective. If a suitably exciting input were used, these gains would converge to correct values. Attempts to increase the speed of convergence by increasing the gains A ~did not work well. With A~= 10 rather than one, the error response up to k = 15 was essentially the same except that the error peak at k = 12 was reduced t o - 0 . 6 5 . After k =20, these higher gains did keep the error approximately an order of magnitude smaller than the lower gains. By that time, however, the error is already quite small. Experimentation with different values of F~ led to no conclusive results. Different weightings of e(k)in the definition of r/(k) were also tried, again with inconclusive results.
RICHARI)
Mt)NOPOI.I APPENDIX A
Proofs of lemmas 1.1 and 1.2 are contained in this Appendix
Proo/ ol lemma 1.1 One has Io determine A~, i = 0 ..... n
ZiZ=b(~Z~r+[Aoz
2 from the identity
*+AIz 2 + . . . + A n 2z
In
I)](A_I)
CO
This is possible since both sides are polynomials of the same order in z ~ and the free term in both is bo. Now multiplying both sides of (A-1) by Z} 1 applying both sides to x ° (k + n - 1), and replacing Z w ( n - 1)x°(k + n - 1) by u(k+ml, onegets(l.lO). Q.E.D.
Prool'qlLemma 1.2 Clearly, from (1.7)
x2{k+2n - 1 ) = Z , , I [n - l }x(k + n - 1) x°(k + n - 1 )= Z~I t (n - 1)u(k+m) x3(k + n - I )=Z~ ~(n- I ~(k)
(A-2)
IV. C O N C L U S I O N
A new method for designing discrete model reference adaptive control systems has been presented. Through the use of auxiliary inputs for control purposes, the gains in the adaptive loops are not constrained as in other methods for discrete designs. Simulation results show the method to work quite well. The advantages of the method are that no anticipative values of the plant output are required, and that convergence to zero of the error between plant and model is assured. Further research is needed with this method in order to minimize the convergence time and to determine its immunity to measurement noise.
Thus (1.11 ) is equivalent to n
Za(n_l)Z~tx(k+n_l)=
I
y~ Bi.z (n- m l+i)zflz~ 0
×x(k+n-1) + Uln, m)
(;
Ciz ii+llZflZ~.lu(k+m)
,i=0
+ . ~,.-2 Diz ti+l'ZllcZ~,, lj,(k)' ) i = 0
Let Co ..... C, 2 be such that (A-4) is satisfied for m
Co+Clz ~+...+
Cn
-2-.
tn
2)
=[Do+DIz l+...+D" m 2z I.... 21]Z,,(m } Acknowledgements--This research was supported partially by grant number N G R 22010018 from the National Aeronautics and Space Administration, and by grant number NSF E N G 74 03399 from the National Science Foundation. The helpful comments of the reviewers are gratefully acknowledged, as is the assistance of Mr. Mario Troiani in performing the simulations, and carefully reading the manuscript.
(A-3)
/
(A-4)
These constants are undefined and not used for m > n - 2. It follows from (A-4) that (A-3) is equivalent to n
I
Z A ( n _ l l x ( k + n _ l ) = ~ Biz-(n ,,-i ~itZ~lx{k+n_l} i=0 ¢i
m - 2
+U{n,m) ~
Diz-lZ~lZ,,x(k+n - 1 )
(A-5)
i=0
REFERENCES [1] R. V. MONOPOLI: Model reference adaptive control with augmented error signal. IEEE-TAC AC-19, (5) 474-484 Oct. (1974). [2] T. IONESCU and R. V. MONOPOH: Discrete model reference adaptive control for single input-single output plants.
Milwaukee Symposium on Control and Computer Engineering, April (1975). [3] T. IONESCU and R. V. MONOPOH: Discrete model reference adaptive control with an augmented error signal. Preprints IF AC 6th World Congress, Part 1D, August (1975). [4] T. 1ONESCU: Linear and nonlinear adaptive discrete systems. Ph.D. Dissertation, Electrical and Computer Engineering Department, University of Massachusetts, May (1976). [5] B.C. K u o : Analysis and Synthesis of Sampled Data Control Systems, pp. 156-- 157. Prentice-Hall, New York (19631. [6] I. G. SARMAand M. A. PAl: A note on the Lyapunov matrix equation for discrete systems. IEEE-TAC AC-13, (1) 119 t21, Feb. (1968).
Let B o..... B, and D o..... D,_,, 2 be such that (A-6) is satisfied : n
l
i = 0 n
m
+U(n, nl) ~
2
Diz-iZp(n)
(A-61
~=0
Then relationships (A-5) and (A-3) are verified and
( 1.1 l ) follows. Q.E.D.
APPENDIX B A SPECIAL CASE, LINEAR T I M E I N V A R I A N T K N O W N SYSTEMS In this appendix, it is shown that constant gains K ~ can be found such that for u(k) given by (1.15), the plant transfer
Discrete model reference adaptive control with an augmented function X ( z ) / R ( z ) is equal to the model transfer function X~(z)/R(z). With the definitions provided in Section I, (1.15) may be written as
error signal
51 5
.+1
G1 = ~ K iz-ti-2) i=3
n+l
=cobo
u(k) = K ° Z f Z ~ lz " r ( k ) + ~ Kiz-ti-2)Z~ lu(k)
U(n,m)Ciz -"+11- }~ Aiz li+ll i
i=0
i=3
Note that
2n+1
+ y~ K~z~"+2)-~Zwlx(k)
(B-I)
i=n+2
n
2
Aiz-,+ 11 = Z I Z " _boco IZw i=0
Define 2 . and Zp by and Z . = (bo + bl z- l +... + b,,z-")z" = Z.z" 2p = (1 + alz- 1 +... +a,z-")z" =Zpz"
n-2 Z
(B-2)
Ci Z - ( i + l ) =
i=0
n-m-2 ~J~ Di z-(i+ l)Zu = Z o Z u i=o
form
(B-3)
Thus: G1 =cob ° ll- (U (n, m)Z D - Z y )Z. + boco 1Zw]
Let G~ (z~ ~) and G2(z- 11 be defined as Similarly
n÷l
G 1 = ~ Kiz -tl-2t
2n+l
G2 = ~ 2.+1 G2 =
(B-9)
2n+!
Kizt.+2-i~=cobol
n+2
Kiz(n+2
~
i)
~
fllzt,+2 0
n+2 2n+l
=cobo I ~
i=n+2
Bti-.-2)z ~+2-I~
(B-10)
n+2
Then the closed loop plant transfer function X (z)/R (z) becomes But from lemma 1.2 it follows that
K°Zl.2.z -" Gp - (Z~ - Gl )2~-- G22 ~
(B-4) Biz -i z - ~ n - m l = z - l Z a Z y _ /
One now has to show that Gp as given in (B-4) is equal to GM(z ) =XM(z)/R(z) =KoZ~Xz -"
~
U(n,m)Diz-~i+l)Zp
o
(B-11)
~(Zf-U(n,m)ZD)Zp-ZMZ f (B-5) Using (B-11) in (B-10), it follows that
Here we are dealing with the case where 61(k)=0 for all i, i.e. G2 =cobo az~"-")[(Zf - U(n, m)Zo)Z p - Z M Z f ]
(B-12)
K ~= cobo lfll, i = 0 ..... 2n + 1. Plasing G1 from (B-9) and G 2 from (B-12) into (B-8) one has
Now: Gp - GM = 0 is equivalent to K ° 2 . Z f Z M z " - " =Ko[(Z~ - G 1)2p - G2 "2,]
(B-6) Z~u , 2 - (n - m) =
Zp
which in turn is equivalent to ;2.(K°ZyZMZ"-" +KoG2)=Ko22p(Z~-GI)
(B-7)
From (B-2) and (B-7) it follows that
Ko [ Zw - cobo 1(U (n, m)Zo - Z I ) z . - Zw] K ° Z IZM zl" - "t) + Cobo i zt. -,,)Ko (Z f - U (n, m )Z o )Zp - c o b o l zt"-")KoZMZ r (B-13)
~ 2p
Z. Zp
~" ~
K°(Z~-G1) K°ZyZMz ~- " +KoG 2
(B-8)
Claim. There exist A~, B~, C~, D~ such that lemmas 1.1 and 1.2 and equation (B-8) are satisfied.
For 6 ° =0, K ° =Kocob o 1. Hence (B-13) becomes Z, Zp
_ _ . Z-(n-m) --
Kobo IZ, (Zy - U(n, m)Zo) Z . Z -(n-m) z~.-,.~Kobo l ( Z y - U(n, m)Zo)Z p Z~ (B-14)
Proof By definition which shows that Gp - GM = O.
n+l
G 1 = ~ Kiz -(i-2}
Q.E.D.
3
where K i = cobo lfl~ = cob~ 1(U(n, m)C i_ 3 -- A i - 3), i=3,...,n-- 1 Using lemmas 1.1 and 1.2 one has
APPENDIX C P R O O F S O F LEMMAS 2.1 AND 2.2 Proof for lemmas 2.1 and 2.2 are contained in this appendix.
516
TUDOR
IONESC/I
and
f'roo/'[i~r & , m m , 2 1
RICHARD
MONOPOLI
where:
Applying Jury's stability test[5], to rclzj one has
ROW
Z0
Z'
~n" k
.n
1
~k an 2n
J, L
fin 2 k
flk an
an
2
1
aa
3
bo
bl
4
b.._
5
C1
l
Cn
1
bn-
1
2
C1 2
('n
1
, , .
a k
. , .
a i
1 an
an k
...
an- l
...
bk
...
h,_
"'"
bn
"'"
bn
- k
. . . . . .
Cn
. . . . . .
C(t
k = 0 . . . . . t]
2
k
,
k = O ..... n - 3
Vo= det [ t/° kq3
(C-5)
1
F r o m the a s s u m p t i o n and (C-4) it follows that
2
ek = - - b k
k = 0 ..... n - 1
flk = - - G
k = 0 ..... n - 2
2,,_ 5
Po
Pl
P2
z, 4
P3
p2
P~
qk =Pk
k=0,...,2
z,_3
qo
qt
q2
Vk = q k
k = 0 ..... 2
(C-6)
"fk = d k :
Using (C-3) and (C-6) one has where ao = l,
]~.-,I =
b,, t
bk
>01 = 1 -
a," = a o (since la,l < 1)
I%1-Iqol>lq21=lv21
qodeP°3 q2 = d e t ~Po LP3
, I<
I~ol = ICor> It._21 =1~.-21 11'01=Idol >ld.-3l =1>'.-31
, ck = det a,_kJ
lb.
Therefore the inequalities (C-4) are verified. It remains to show that (C-2a) and (C-2b) imply
Pl] P2
(C-l)
>0
n - 1 even
<0
n - I odd
n ' ( 1 ) > 0 and r t ' ( - 1 )
Since n ( z ) has no roots on or outside the unit circle the following inequalities are satisfied:
n(1)=a,+a,_l
rr(- 1)=a"-a"-
+-.-+ 1>0
1+"'+
>0 ( - 1)" < 0
(C-2a) neven nodd
Indeed: n ' { 1 ) = l - a , + (2l - a . ) ( l + a l + . . . + a ,
l
but from (C-2a) l+a l+a2+...+a,,
i>-[l+a,,)
(C-2b) which replaced above gives ~z'( 1 ) > 0 If n - 1 is odd (n even)
la,] < 1 7~( - 1)= 1 - a 1 +a2 - . . . -
Ibol>lb°-,I I%1>1c°-~1
r((-1)=(1-a,)(a,-
IPo] >IP31 Iqol>lq2[
(C-3)
a -G
a,- 1+a.>0
2+...+al)-(1-a,
(C-7) 2)
(C-8)
Using (C-7) in (C-8) one obtains ~'( - 1) < 0 . In a similar fashion it can be verified that g'( - 1) < 0 if n - 1 is even. Q.E.D. P r o o J f o r l e m m a 2.2
Applying the same test to g'(z) one has to show first that
[0(n
Notice first that A c is a stable matrix since it has the same eigenvalues as A. Moreover, for a stable c o m p a n i o n matrix, from equations (2), (6) and (9) in [6] it follows that there exists a particular solution for (2.8), Po = [Pu], having the last column entries given by
iI<~0
Pi, = a , _ ~- a.ai, i = 1..... n ao = 1
l,lol>l.~l (C-4)
(C-9)
Po is not only symmetric about the main diagonal but it is also symmetric about the cross diagonal, [6]. But from the definition of ~k one has Pi. = % - 1 , i = t ..... n and by assumption z" + al z"- 1 + . . . + a, is a stable polynomial. Thus from lemma 2.1 A(z)_= rr'(z) has roots inside the unit circle. Since
Discrete model reference adaptive control with an augmented error signal Po is symmetric about the cross diagonal as well, A(z)-=A'(z) thus concluding the proof of part (i). Now (2.8) can be written as
,4~Pop~IpoA¢-Po = -R
e>O is a col+tst;.ii+t. ~nt+,.l Xl ( k J n,, the m+ttl tX
Mlk)=
. 1 + 1-K~
or equivalently (C.IO)
and D = dTP, 1 d > 0. It can be shown that
DKQ
+
DK~
M(k) is a positive definite matrix if 1
I
K~K~>I ( ~ n +-D)"
However, by direct computation using (2.7) and equation (6) in [6], one can verify that ~ P o = Po A~. Therefore (ii) is satisfied for
Q=PoIRPo 1.
K2A ~ (q~i(k_l))2; + i
1 + . 2 . . . . ~.= 0. . . . .
KQ
pol/t~Po.PolPoA+Pot-Pol= -polRPo I
517
(D-2)
Using (2.2') and (2.Y). A 1,~(k) = l/~ (/,-+ 1 ) - 1:~(k) satisfies Q.E.D.
AI',(k)~-~Qm+nltl(k)i 2-(D-~:)Ox(k)M(k)O{k).
(D-3)
Thus ife < D, AVI (k) is negative definite in r/(k), APPENDIX D P R O O F O F C O N V E R G E N C E FOR MODIFIED ALGORITHM and q (k). Consequently,
Let lq (k) be given by
ii~l,i~O, qqh~-*O
lq (k)= I~ (k)+e¢~C(k- 1)M(k- 1)~O(k- 1)
(D-l)
and
~/Si(k)(o'th)---,O
~T(k)=[(i=~oi3i(k-1)dpi(k-
as k---, ~.
i=0
where 1~tk) is as in (2.10)
1 )),q(k)~,
These conclusions can be used to sho~ that ytkj--*O, and consequently that e(k)--*O.It is the authors' contention that this modified algorithm insures that ~bi(k) are bounded for all i and k, but a detailed proof for this is not available at this time.
Pergamon Press, 1977. Printed in Great Britain
Discrete Model Reference Adaptive Control with an Augmented Error Signal* T U D O R IONESCUt:~ and RICHARD M O N O P O L I §
Using a Lyapunov based design for discrete single-input single-output plants, stable model reference adaptive control systems avoid the need for anticipative values of the plant output by using the augmented error concept and only plant input and output signals. Key Word Index--Adaptive control; (augmented error signal); discrete systems; Lyapunov methods; (model reference systems); nonlinear control systems; stability; system theory; time-varying systems. Summary--A method is developed for designing discrete model reference adaptive control systems when one has access to only the plant's input and output signals. Controllers for single-input, single-output, nonlinear, nonautonomous plants are developed via Lyapunov's second method. The augmented error signal method is employed to ensure that the normally used true error signal approaches zero asymptotically without requiring anticipative values of the plant output signal. Such anticipative signals are replaced by others easily obtained from low pass digital filters operating on the plant's input and output signals.
x(k +n)+alX(k + n - 1)+... +a,x(k)--bou(k +m) +... + b,,u(k) + cf(X(k), k)
(1.1)
where u(k),x(k) are the plant input and output respectively, X(k) is the set of x ( k + n - i ) for i= 1, 2.... and f (X ( k ), k) is a nonlinear time varying function of known form. It is assumed that coefficients ai, bi and c are unknown and constant or slowly varying. The term cf(X(k), k) may be replaced by a sum of terms of the same type. There is no loss of generality in carrying only one term of this kind. It is also assumed that: (a) The function f ( . . . ) satisfies the conditions necessary for solutions of (1.1) to exist and be unique, (b) all roots of the polynomial bo +blz -1 +...b,,z m, are in the unit circle, (c)+oo > b0M > bo > bo,, > 0, and boM and bo,, are known, (d) m < n - 1 and must be known. Assumption (b) is required to ensure a bounded control input by not requiring the adaptive system to act to cancel non minimum phase plant zeroes. For a similar reason u is not included i n f In general, it is not known what restrictions to impose on these functions to ensure that u is bounded. The design objective is to have the plant output follow the output of a model reference defined by the equation:
INTRODUCTION
THE AUGMENTED error signal method for continuous model reference adaptive control systems introduced in [1] is used to solve the discrete model reference adaptive control problem. Extension of results presented in [2], and [-3] are given. Further detail on the results of this paper and related problems can be found in [4]. This paper is divided into three parts. In section I, the problem statement and the notation are given. Section II gives the main result. Simulation results are presented in section III. Proofs of lemmas and peripheral results are confined to the appendices. I. NOTATION AND PROBLEM STATEMENT
A discrete time dynamic system (plant) can be described by the non-linear, nonautonomous difference equation *Received 18 August 1975; revised 8 September 1976; revised 13 December 1976; revised 3 March 1977. The original version of this paper was presented at the 6th IFAC Congress on Control Technology in the Service of Man which was held in Boston, Cambridge, MA 02138, U.S.A. during August 1975. The published Proceedings of this IFAC Meeting may be ordered from: IAS, 400 Stanwix Street, Pittsburgh, PA 15222 or John Wiley, Baffins Lane, Chichester, Sussex, PO 19 IUD, UK. This paper was recommended for publication in revised form by associate editor I. Landau. tlntermetrics, 701 Concord Avenue, Cambridge, MA 02138, U.S.A. l-ormerly with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A. :Wormerly with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, Massachusetts. §Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A.
xM(k +n)+adlxM(k +n - 1)+...+ad,xM(k) = Kor(k)+g(XM(k),r(k),k)
(1.2)
where the nth-degree polynomial in z with coefficients ad,, i = 1..... n has roots inside the unit circle, xM(k) is the model output, r is the reference input, Ko is a known scalar X~(k) is the set of xM(k + n -- i) for i = 1, 2 ..... and g(,, ) is a nonlinear time varying function with the properties required for the existence and uniqueness of solutions to (1.2). The design problem is to synthesize a parameter adaptive control system for (1.1) which will cause 507
508
TtJDOR IONESCU and RICHARD MONOPOLI
the error e ( k ) = x M ( k ) - x ( k } , between the plant and model outputs to a p p r o a c h zero. As previously mentioned the special feature of this work is that anticipative values of the plant output x do not a p p e a r in the design, i.e. if the a d a p t a t i o n takes place at time k, then x ( k + j ) , j = 1,2 .... do not appear in the algorithms for the adaptive gains or the controls. It is convenient at this stage to introduce the delay o p e r a t o r z m, and the following polynomial delay operators:
Zp(n)= 1 +
aiz ' i
1
Zu(rtl):bo+ ~ bi2-i
(*)
i-I
ZM(n)= 1 + ~ ady i i-1
Using (*) in (1.l) and (1.2) one has
Next (1.4) is added to (I .5) to obtain
ZM(n)rl(k + n ) = K o r ( k l + g ( k ) - Z , ( m ) u ( k ~-m) + Z a ( n - 1)x(k + n - 1 ) - q l ( k ) + Z w i n - 1)(w(k + n - l ) + q ( k - + n ) ) ~1.6) Equation (1.6) is the starting point for the synthesis. The design p r o b l e m is to generate u(k), w(k) and q(k), all independent of x ( k + j ) , j = 1,2 ..... and in such a way that e(k)~O as k--, oc. State variable filters must be introduced to avoid anticipative values of x. These play the role of an observer of sorts. However, knowledge of plant parameters is not required to design the state variable filters as it would be to design an observer. Also, although an estimate of the state is not obtained, it is not required. Part of the required filtering is defined in terms of filter output signals
x i, i = O, 1..... 4 as follows Zp(n)x(k + n ) = Z , ( m ) u ( k + m ) + c f (k) (1.1") Z~(n- l)x°(k+n -1)=u(k+m) ZM(n)xM(k+n)=Kor(k)+g(k)
(1.2") Z~,ln - 1 )x I (k + n -
1 ) = r(k)
Z,~(n-t)x2(k+n
1)=x(k-
The concept of a u g m e n t e d error signal detailed in [1] must be introduced to achieve the design objective. Let t/(k) be the augmented error signal:
Zw(n - 1 )x 3 (k + n - 1 ) = f ( k )
tl(k)=e(k)+ ),(k)
(1.3)
Zw(n - 1 )x4(k + n - 11 = g(k)
where the signal y(k) is the output of the error a u g m e n t a t i o n filter defined by:
Using (1.7). one can write (1.6) as
Z w(n)y(k+n)=Z,,.(n- 1)(w(k+n-
1 )+q(k+n))
- q f (k)+ Z a ( n - 1)x(k + n - 1) (1.5)
(1.7t
- Z,(m)x°(k + n - 1 ) + Z A ( n - 1)x2(k + 2n - 1) - c x 3 ( k + n - 1)+ x4(k +n - 1) +w(k+n-1)+q(k+n)]
(t.8)
The additional filtering required is defined in terms of signals x s and u s which satisfy
Zf(n-mZM (n)e(k + n)=Kor(k) + g ( k ) - Z,(m)u(k + m)
1)
ZM(nkl(k + n ) = Z w ( n - 1)EKoxt(k +n - 1)
(1.4) where Zw(n - 1 ) = ~ ' £ ~ ciz -i and w(k ) and q(k ) are auxiliary system inputs to be determined along with the control input u(k) as part of the design. The coefficients c / m u s t be chosen in a special way, the details of which will be explained later in the development. It is noted here that the roots of Z,,. (n - 1 ) will be inside the unit circle
-
1)xS(k +n - m - 1)
=(l+FlZ-l+F2z
2+...
+F,_ m_lz -(n " - l ) ) x f ( k + n - m - l ) = x ( k ) Z s ( n - m - 1 ) u f (k + n - m - 1 )
=u(k) (1.9)
where n
ZA(n--1)~
~ Aaiz (i-1),Aai=ai-adl i-1
for i = 1 ..... n.
where constants F i are such that the polynomial Z s ( n - m - 1 ) has all its roots inside the unit circle. N o t e that this filtering is not required if m = n - 1. This is a special case for which Zr = 1, xS(k)=x(k), and uS(k) = u(k ).
Discrete model reference adaptive control with an augmented error signal The following lemmas, proved in Appendix A, are used to prove the main result.
509
(9i(k + n - 1 ) = z - ~ i - 2~Z~ X( n - m - 1)x° ( k + n - 1); for i = n + 2 .... 2n+l,fli=Bi_l,+2 ,
Lemma 1.1 There exist constants A o through A,_ 2 such that
and Oi(k+n - 1 ) = z -~i m-a)zf 1(n - m -
Zu (m)x ° ( k + n - 1 ) = (bo/co)u f ( k + n - 1 ) n-2 + ~, A i z - ~ i + l ) Z f l ( n - m - 1 ) x ° ( k + n - 1 )
for i = 2 n + 2 .... 3 n - m ,
x Z£ l ( n - 1 ) x ( k + n - 1);
fli=U(n,m)cDi_t2,+ 2),
and
(9i(k-4- n - 1) = U(n, m)z -¢i- 2n- l ) z f
i=0
(1.10)
Lemma 1.2 There exists constants Bo through B,-1, Co through C._ 2, and Do through D,-m-2 such that
1)
1
× ( n - m - 1)xa(k+n-1). The equation to be used later for deriving u(k) and w(k) is u f ( k + n - 1)+w l ( k + n - 1) N-1
n-1
= ~ Ki(k+n-1)(9i(k+n-1)
Z a ( n - 1 ) x 2 ( k + 2 n - 1 ) = Z Biz-~"-'-l+i)z~ 1 i=0
× (n-m-1)x2(k+2n -l)4-U(n,m) -n-2
X
E Ciz-(i+l)zf I(H i=o -m-l)
×x°(k+n-l) n-m-2 4- E Di 2-(i+1)
+n-l)]
where U(n, m ) = 1 for m =
+w(k+n-1) + N-1 ~ fli(9i(k+n-1) + q ( k + n ) 1 i=0
flo =Ko, f l l
=
(1.12)
1,
- - C,
fie = 1, (9°(k + n - 1) = x l ( k + n - 1),
(9l(k + n - 1)=x3(k +n - 1), (92(k+n-1)=x4(k+n-1); fli=U(n,m)Ci_3-Ai_3
and
where w(k)=KN(k)wl(k) and Ki(k) for i = 0 to N are the adaptive gains. Substituting (1.13) into (1.12) yields
ZM(n)rl(k + n) = Zw(n -- 1 ) (i=~° b i ( k + n - 1 ) ( g i ( k + n - l ) + q ( k + n ) ) (1.14) where
6i(k)= -(bo/co)Ki(k)+fii for i = 0 ..... N - 1 , 6N(k)
i=0
× Z -f l ( n - m - 1)cx3(k
where N = 3 n - m +
(1.13)
i-0
for i = 3 , 4 .... n + l ,
=KN ( k ) 4- bo/ co, and (9N(k ) = w 1( k ). Before developing the complete solution to the nonlinear problem, it is useful to consider the application of the results developed thus far to the design for a linear time invariant system with known coefficients. This gives an appreciation of the basic structure of the adaptive system, and some insight into the role of the state variable filters. The linear, time invariant, known parameter problem is specified by letting g = 0 in (1.2), c = 0 in (1.1) and assuming that coefficients ai and bl in (1.1) are known. For this problem, there is no need to adjust the gains Ki(k), therefore they are taken to be constants. Also, the auxiliary signals w(k) and q(k) may be taken to be zero in this case. Then, for these conditions, one has to show that constant gains K ~ can be found such that for u(k) given by 2n+l
u(k)= ~ K i Z I ( n - m - 1 ) ( 9 i ( k + n - m - 1 ) i=0
i~1,2
(1.15)
the plant transfer function X {Z }/R {Z) is equal to the model transfer function xM{Z)/R(Z). Equation (1.15} is derived by applying the operator Z f to both sides of(1.13) and using (1.9). The proof of this result is given in Appendix B. Figure 1 shows the resulti,ag system for this linear, time invariant known parameter case.
510
TUDOR I()NES('ti and RICHARD MON()I>OLI
R(z): _ ~
U(z)
D-
~. 2o
] ]
x
)
n¢L ,=3
z,,
Zn*l
~. K ; Z I n + 2 ,;n+2
;I
Zw
F~c;1. Systemconfigurationfor specialcase. 11. THE MAIN RESULT
Proof
Returning to the main problem, we put (1.14) in the matrix vector form
The proof requires the following two lemmas which are proven in Appendix C. Let
(k)
q(k+l)=Aq(k)+d
i
l+...+a,
~(z)=z"+alz"
6'(k)O;(k)+q(k+l)
(2.1)
(2.4t
be a polynomial with real coefficients. Define the scalars'
where
I
-aa~
1
a, 0 ..... n - - l ' a o = l ( 2 . 5 ) ak k-]:k= ~
~k=det[L,
ql ( k ) = q ( k ) = e ( k ) + y ( k ) . and 0 ... 0-~
Let
A-~-l--(.gd2__adn 00 1 ...... 0C :IdT0= [CO'
7z'(z)=':%z"-l+:q
..... Cn-l]
z"-2 + . . . + e , - I
(2.6)
L e m m a 2.1
The main result is Theorem 1
There exists scalars c/, i = 0,..., n - 1 such that the following conditions are satisfied : (i) Zw(n - 1 ) has all its roots inside the unit circle. (ii) For each Kq>O and for )~imin>I(2Kq) -1, the adaptive laws Ki(k)-Ki(k-
If ~z(z) has all its roots inside the unit circle then r((z) also has its roots inside the unit circle. The next lemma deals with a particular solution of the Lyapunov matrix equation for discrete systems. Let A and Ac denote
A=
Co
1 ) = 2 ~ ° t/1 ( k ) ~ i ( k - 1) i=O .... N - 1
KS(k)-KN(k-
-a•
1 0 ... 01
-a2
0
k-a.
(22)
1
1)= - - ~ q l (k )OS( k - 1 )
AN
where
1 ... 0
0
,
... 0 I
0
1
0
...
0
0
l
0
0 ..
0
A, =
(2.71 0
"~imin= min (20 for i=0,
1.... N and the auxiliary control: --
~ln
--
an-
1
. . . . .
--
(all
_
N q ( k ) = -Kqr/(k) ~-~ (qSi(k- 1)) 2
i=0
yield
lim tt(k)=O k~y
(2.3)
L e m m a 2.2
If A has all its eigenvalues in the unit circle then there exist positive semi-definite matrices R T = R, QT = Q and a positive definite matrix Pov = Po = [Pij]
Discrete model reference adaptive control with an augmented error signal such that (i)
-dTpo~d AV~PoA~-Po = - R
511 6i(k-1)c~i(k-1)
i
(2.8)
+q(k)) 2
and the polynomials N
+ ~ ,~,i(6i(k)-fi(k - 1))(hi(k)
A(z) = p..z ~- 1 +... + p2,z + Pl,
i=O
+hi(k-l))
A'(z) = p l l z " - 1 +... +Pl(n- l)z+Pl, From (2.2) we get
have all their roots inside the unit circle. (ii)
ATPo 1A
-
-
1
Po 1 = _ Q
6'(k)-6'(k-1)=-~th(k)dpi(k-1);i=O
We now turn to the proof of the main result. First note that A in (2.1) is of the form in (2.7) and is stable by assumption. Now choose the scalars czi= 0,..., n - 1, such that tiT=[1
0 ...
0] Po
(2.12)
(2.9)
where Po is given by lemma 2.2. This choice of c, implies that Z w ( n - 1 ) has its roots inside the unit circle. Next consider the following Lyapunov function candidate
.... ,N (2.13)
If 261(k-1) is added to both sides of (2.13) one obtains 6'(k) + 6~(k- 1) = 2 6 i ( k - 1 ) - ~ rh ( k ) O i ( k - 1);
i = 0 ..... N
(2.14)
Substituting (2.13) and (2.14) into (2.12) one has V ( k + 1 ) - V ( k ) ~ -aOmmll~(k)ll2-d T eo Xd
V (k)= Ir/x (k)-dT(i~=° fi~(k- 1 )q~(k-1 ) + q ( k ) ) l
xPo ~ t/(k)-d
N
2
N 1 2
6~(k- 1)~b'(k- 1)
y~ 7q~, (k- ll+2,h(k)q(kl
i
(2.15)
i=0 ~i
+ ~ X:~;~(k-1)
+q(k)
(2.10t
i=0
Then V(k + 1 ) - V(k) = ~tT(k)(ATp o XA - Pot )rl (k) N
+2dTPoXq(k) ~ 6 ' ( k - 1 ) # ( k - 1) i=0
+ 2d~ Po ~ ~l(k)q(k) - d T p o Xd(~ b/(k- 1 ) ¢ ' ( k - 1) + q(k)) 2 N
+ 2 2i((~i(k)-6i(k- 1))((~i(k)+6i(k- 1))
(2.11)
From (3.2) and the assumptions it follows that V(k + 1 ) - V (k) < 0 for q (k) @0. Therefore, since V (k) is positive definite, we have by the Lyapunov theorem that limk, ~ q(k)=0. This ends the proof of the theorem. Q.E.D. Remarks (a) Theorem 1 represents the discrete analog of the results in [1]. (b) Lemmas 2.1 and 2.2 are of independent interest and might be regarded as corollaries to the well known Jury stability criteria. (c) The apparent loop implied by (1.3), (1.4) and (2.3) may be resolved as follows: Substitute (2.3) into (1.4) and use (1.3) to get
i=0
From lemma 2.2 and (2.9) it follows that
ZM(n)y(k + n) = Zw(n - 1 )(w(k + n - 1 ) - ( e ( k + n ) + y ( k + n ) ) m E ( k + n - 1))
V(k + 1 ) - V(k) = - n T ( k ) Q q ( k ) + 2th (k)
(R-l) N
x ~ bi(k-1)ch~(k-1) i=0
+2qs(k)q(k)
where N
m2(k+n - 1 ) = K ~ Z (q~i(k+ n - 1)) a. 0
512
Tt:D()R ]()N['S('t' and RICHARD M()NOPOII
Let
are not required to generate either u(k or w(k) Now, using (l.9)in (2.17) one has ZM(n)=I+z
*Z~(n--l). 2 Ki(k)(//(k) o
wl(k)=
where
Z):l(n-m-I.,
~, adi2
Z~ln-1)=
u (kt
li 11
t2.17')
i=1
Substituting (2.16)into (2,17't gives
and
Z.,(n- 1 )=co
+ z - xZ.,(n-
N
1
}{'l(k)~--- ~, (Ki(k)~i(k)--Z.f l(H--nl--
2),
IlK i
0
where
w n
× (k-n+m)Zr(n-m-
1)~ i ( k ) )
l
Z w ( n - 2 ) = Y~ ciz
(i
11
(2.17"}
i=l
Since rl(k)--+O assures (from (2.2)), then
Then (R-I)may be written as
Ki(k)-Ki(k-
(i +Co m 2 ( k + n - 1 ) ) y ( k + n ) = - - ( Z M ( n - - 1 ) + Zw(n-2)mZ(k +n -1))y(k +n+ Z,~, (n - 1 ) ( w ( k
+ n -
1)
1 ) - m 2 (k + n -
1 )e(k + n))
(R-2) Using (R-2). the algorithm for y(k) becomes y(k) = (1 ÷ com2(k - 1 ))- l( _ (ZM( n _ 1 ) + Zw(n - 2)m 2 (k - l ))y(k - i )
+ Z w ( n - 1 ) ( w ( k - 1 ) - m2(k - 1 )e(k))) (R-3) Thus, for implementation, (R-3) is used to generate y ( k ) and (1.3) to generate ~/(k). Since y(k) and r/(k) depend directly on e(k), it must be assumed that the total time taken for the required multiplication in (R-3) and the required addition in (1.3) is negligible relative to the time interval between k and k + 1. In order to complete the design, the control signal u (k) and the auxiliary control w (k) must be specified in a way which insures that ~/(k)--+0 implies that e( k )-+O. Let Kqk) and q(k) be given by Theorem 1. Then e(k)+0if N-
Ki(k)
approaches constant
1 ) ~ 0 as k--, ~.
Hence, from (2.17"), w l ( k ) (and w(k) also) approaches zero as k-+~:. Since q(k) and w(k) approach zero as q(k) approaches zero, then so does y ( k ) , a n d a l s o e ( k ) . Q.E.D. 1 his prool relies oil the assumption that signats (hi(k) are bounded for all i and k. Such an assumption is also required in Ell, but is not explicitly stated there. This assumption was made in all simulations run by the authors, and no dilficulties were encountered, i.e. these signals did remain bounded yielding the desired result, e(k)---,0. One such simulation is given in the next section. An algorithm that does not require the boundedness assumption for q~q,k)is given next. It is somewhat more complicated than that above and has not been tested by simulation. Basically, it requires a decreasing gain in the adaptive algorithms (2.2) as 3,
( ( h i ( k - 1 )) 2 i=:l)
increases. For this case, u{k) and w 1(k) are still as given by (2. l 6) and {2.17) respectively. However, (2.2) and (2.3) must be modified as follows
1
u(k)= ~, Ki(k - 1 ) Z r ( n - m -
1)qSi(k+n-m - 1)
i=0
Kqk)-Kqk-
l)=fS' U,,{kl~?,(k)eqk-l)
(2.16) and
lor i = 0 ..... N - I N-l wl(k)= 2 Ki(k)(bi(k)-uf(k) i=o
(2.17)
Proof First note that (2.16) and (2.17) satisfy (1.13). Furthermore, anticipative values of the plant output
Khr(k)-K:V{k
- 1)= ---
1
Kl,.(k)~ h (k)qSi(k - 1 )
where Kt,-(k)=K k K k + ~ (4)i(k - 1 ))2 i=0
{2.2')
Discrete model reference adaptive control with an augmented error signal
5l 3
Filtered variables required are defined by
and Kk is a positive constant
q( k ) = - K o ( k )tlx (k )
(2.3')
Zwx°(k + 1) =cox°(k + 1 ) + q x ° ( k ) = u(k)
where
Zwx 1(k + 1) = r(k) K¢2(k)=Kq( Kk +i~o (~bi(k- 1 ))2)
ZwX2(k+ 1 ) = x ( k - 1)
(3.4)
ZyxY(k + 1)= (1 + F l z - 1 )Xf( k _~_1 ) = x ( k ) and K. is a positive constant. It is shown in Appendix D that the modified algorithm as given by (2.2') and {2.3') does insure that e(k)--+O without the a priori assumption that ~bi(k) are bounded for all i and k. Conditions forKq a n d K k are also given there.
Z ruY(k + 1) = u(k) where
F1 = 0 . 5 .
The gain a d j u s t m e n t a l g o r i t h m s for this e x a m p l e are
K ° (k) = K ° (k - 1 ) + A ° t l ( k ) O ° (k - 1 ) K 3 (k) = K 3 (k - 1 ) + A3r/(k)q~3 (k - 1 )
111. S I M U L A T I O N RESULTS
Digital computer simulations were performed for an inherently discrete second order plant described by
g 5 (k) = K 5 (k - 1 ) + Ast/(k)q~5 (k - 1 )
x(k + 2)+alX(k + l )+a2x(k)=bou(k ) (3.1)
K 7 (k) = K 7 (k - 1 ) + Avt/ (kkb 7 (k - 1 )
3--
-3
--
K*(k)=K4(k-1)+A4tl(k)O4(k-1)
•
^
•
(3.4)
M
-'__Z ;,';'
•
F1G. 2. Simulation results.
The model used was x M(k + 2) + adlxM(k + 1) + ad2X(k) =Kor(k) (3.2) where
ad, = 0.5, an2 = 0.6 and Ko - 2.0. The augmenting variable v is defined by
y(k + 2) + ady(k + 1 ) + ad2y(k) = CoW(k + 1) + clw(k) + coq(k + 2) + caq(k + 1) where Co =0.64 and cl =0.2.
(3.3)
where A i =Co/(2ibo) and q9 are as defined in (1.12}(1.14). The control input u(k) and the auxiliary input w(k) were generated according to (2.16) and (2.17) and the definition below (1.13). The second auxiliary input q(k) was generated according to Theorem 1 with Kq = 1/2~ml,. In Fig. 2, xU(k) and e(k) are plotted for k = 0 to 36. For this experiment r ( k ) = +1 for 0 < k < 1 0 , r(k) = - 1 for 11 _
514
TIID()R
IONESCII
and
adaptive gains K " , K ~ and K: converged to their correct values but not K '~ a n d K s . This is unimportant, however, since control is the objective. If a suitably exciting input were used, these gains would converge to correct values. Attempts to increase the speed of convergence by increasing the gains A ~did not work well. With A~= 10 rather than one, the error response up to k = 15 was essentially the same except that the error peak at k = 12 was reduced t o - 0 . 6 5 . After k =20, these higher gains did keep the error approximately an order of magnitude smaller than the lower gains. By that time, however, the error is already quite small. Experimentation with different values of F~ led to no conclusive results. Different weightings of e(k)in the definition of r/(k) were also tried, again with inconclusive results.
RICHARI)
Mt)NOPOI.I APPENDIX A
Proofs of lemmas 1.1 and 1.2 are contained in this Appendix
Proo/ ol lemma 1.1 One has Io determine A~, i = 0 ..... n
ZiZ=b(~Z~r+[Aoz
2 from the identity
*+AIz 2 + . . . + A n 2z
In
I)](A_I)
CO
This is possible since both sides are polynomials of the same order in z ~ and the free term in both is bo. Now multiplying both sides of (A-1) by Z} 1 applying both sides to x ° (k + n - 1), and replacing Z w ( n - 1)x°(k + n - 1) by u(k+ml, onegets(l.lO). Q.E.D.
Prool'qlLemma 1.2 Clearly, from (1.7)
x2{k+2n - 1 ) = Z , , I [n - l }x(k + n - 1) x°(k + n - 1 )= Z~I t (n - 1)u(k+m) x3(k + n - I )=Z~ ~(n- I ~(k)
(A-2)
IV. C O N C L U S I O N
A new method for designing discrete model reference adaptive control systems has been presented. Through the use of auxiliary inputs for control purposes, the gains in the adaptive loops are not constrained as in other methods for discrete designs. Simulation results show the method to work quite well. The advantages of the method are that no anticipative values of the plant output are required, and that convergence to zero of the error between plant and model is assured. Further research is needed with this method in order to minimize the convergence time and to determine its immunity to measurement noise.
Thus (1.11 ) is equivalent to n
Za(n_l)Z~tx(k+n_l)=
I
y~ Bi.z (n- m l+i)zflz~ 0
×x(k+n-1) + Uln, m)
(;
Ciz ii+llZflZ~.lu(k+m)
,i=0
+ . ~,.-2 Diz ti+l'ZllcZ~,, lj,(k)' ) i = 0
Let Co ..... C, 2 be such that (A-4) is satisfied for m
Co+Clz ~+...+
Cn
-2-.
tn
2)
=[Do+DIz l+...+D" m 2z I.... 21]Z,,(m } Acknowledgements--This research was supported partially by grant number N G R 22010018 from the National Aeronautics and Space Administration, and by grant number NSF E N G 74 03399 from the National Science Foundation. The helpful comments of the reviewers are gratefully acknowledged, as is the assistance of Mr. Mario Troiani in performing the simulations, and carefully reading the manuscript.
(A-3)
/
(A-4)
These constants are undefined and not used for m > n - 2. It follows from (A-4) that (A-3) is equivalent to n
I
Z A ( n _ l l x ( k + n _ l ) = ~ Biz-(n ,,-i ~itZ~lx{k+n_l} i=0 ¢i
m - 2
+U{n,m) ~
Diz-lZ~lZ,,x(k+n - 1 )
(A-5)
i=0
REFERENCES [1] R. V. MONOPOLI: Model reference adaptive control with augmented error signal. IEEE-TAC AC-19, (5) 474-484 Oct. (1974). [2] T. IONESCU and R. V. MONOPOH: Discrete model reference adaptive control for single input-single output plants.
Milwaukee Symposium on Control and Computer Engineering, April (1975). [3] T. IONESCU and R. V. MONOPOH: Discrete model reference adaptive control with an augmented error signal. Preprints IF AC 6th World Congress, Part 1D, August (1975). [4] T. 1ONESCU: Linear and nonlinear adaptive discrete systems. Ph.D. Dissertation, Electrical and Computer Engineering Department, University of Massachusetts, May (1976). [5] B.C. K u o : Analysis and Synthesis of Sampled Data Control Systems, pp. 156-- 157. Prentice-Hall, New York (19631. [6] I. G. SARMAand M. A. PAl: A note on the Lyapunov matrix equation for discrete systems. IEEE-TAC AC-13, (1) 119 t21, Feb. (1968).
Let B o..... B, and D o..... D,_,, 2 be such that (A-6) is satisfied : n
l
i = 0 n
m
+U(n, nl) ~
2
Diz-iZp(n)
(A-61
~=0
Then relationships (A-5) and (A-3) are verified and
( 1.1 l ) follows. Q.E.D.
APPENDIX B A SPECIAL CASE, LINEAR T I M E I N V A R I A N T K N O W N SYSTEMS In this appendix, it is shown that constant gains K ~ can be found such that for u(k) given by (1.15), the plant transfer
Discrete model reference adaptive control with an augmented function X ( z ) / R ( z ) is equal to the model transfer function X~(z)/R(z). With the definitions provided in Section I, (1.15) may be written as
error signal
51 5
.+1
G1 = ~ K iz-ti-2) i=3
n+l
=cobo
u(k) = K ° Z f Z ~ lz " r ( k ) + ~ Kiz-ti-2)Z~ lu(k)
U(n,m)Ciz -"+11- }~ Aiz li+ll i
i=0
i=3
Note that
2n+1
+ y~ K~z~"+2)-~Zwlx(k)
(B-I)
i=n+2
n
2
Aiz-,+ 11 = Z I Z " _boco IZw i=0
Define 2 . and Zp by and Z . = (bo + bl z- l +... + b,,z-")z" = Z.z" 2p = (1 + alz- 1 +... +a,z-")z" =Zpz"
n-2 Z
(B-2)
Ci Z - ( i + l ) =
i=0
n-m-2 ~J~ Di z-(i+ l)Zu = Z o Z u i=o
form
(B-3)
Thus: G1 =cob ° ll- (U (n, m)Z D - Z y )Z. + boco 1Zw]
Let G~ (z~ ~) and G2(z- 11 be defined as Similarly
n÷l
G 1 = ~ Kiz -tl-2t
2n+l
G2 = ~ 2.+1 G2 =
(B-9)
2n+!
Kizt.+2-i~=cobol
n+2
Kiz(n+2
~
i)
~
fllzt,+2 0
n+2 2n+l
=cobo I ~
i=n+2
Bti-.-2)z ~+2-I~
(B-10)
n+2
Then the closed loop plant transfer function X (z)/R (z) becomes But from lemma 1.2 it follows that
K°Zl.2.z -" Gp - (Z~ - Gl )2~-- G22 ~
(B-4) Biz -i z - ~ n - m l = z - l Z a Z y _ /
One now has to show that Gp as given in (B-4) is equal to GM(z ) =XM(z)/R(z) =KoZ~Xz -"
~
U(n,m)Diz-~i+l)Zp
o
(B-11)
~(Zf-U(n,m)ZD)Zp-ZMZ f (B-5) Using (B-11) in (B-10), it follows that
Here we are dealing with the case where 61(k)=0 for all i, i.e. G2 =cobo az~"-")[(Zf - U(n, m)Zo)Z p - Z M Z f ]
(B-12)
K ~= cobo lfll, i = 0 ..... 2n + 1. Plasing G1 from (B-9) and G 2 from (B-12) into (B-8) one has
Now: Gp - GM = 0 is equivalent to K ° 2 . Z f Z M z " - " =Ko[(Z~ - G 1)2p - G2 "2,]
(B-6) Z~u , 2 - (n - m) =
Zp
which in turn is equivalent to ;2.(K°ZyZMZ"-" +KoG2)=Ko22p(Z~-GI)
(B-7)
From (B-2) and (B-7) it follows that
Ko [ Zw - cobo 1(U (n, m)Zo - Z I ) z . - Zw] K ° Z IZM zl" - "t) + Cobo i zt. -,,)Ko (Z f - U (n, m )Z o )Zp - c o b o l zt"-")KoZMZ r (B-13)
~ 2p
Z. Zp
~" ~
K°(Z~-G1) K°ZyZMz ~- " +KoG 2
(B-8)
Claim. There exist A~, B~, C~, D~ such that lemmas 1.1 and 1.2 and equation (B-8) are satisfied.
For 6 ° =0, K ° =Kocob o 1. Hence (B-13) becomes Z, Zp
_ _ . Z-(n-m) --
Kobo IZ, (Zy - U(n, m)Zo) Z . Z -(n-m) z~.-,.~Kobo l ( Z y - U(n, m)Zo)Z p Z~ (B-14)
Proof By definition which shows that Gp - GM = O.
n+l
G 1 = ~ Kiz -(i-2}
Q.E.D.
3
where K i = cobo lfl~ = cob~ 1(U(n, m)C i_ 3 -- A i - 3), i=3,...,n-- 1 Using lemmas 1.1 and 1.2 one has
APPENDIX C P R O O F S O F LEMMAS 2.1 AND 2.2 Proof for lemmas 2.1 and 2.2 are contained in this appendix.
516
TUDOR
IONESC/I
and
f'roo/'[i~r & , m m , 2 1
RICHARD
MONOPOLI
where:
Applying Jury's stability test[5], to rclzj one has
ROW
Z0
Z'
~n" k
.n
1
~k an 2n
J, L
fin 2 k
flk an
an
2
1
aa
3
bo
bl
4
b.._
5
C1
l
Cn
1
bn-
1
2
C1 2
('n
1
, , .
a k
. , .
a i
1 an
an k
...
an- l
...
bk
...
h,_
"'"
bn
"'"
bn
- k
. . . . . .
Cn
. . . . . .
C(t
k = 0 . . . . . t]
2
k
,
k = O ..... n - 3
Vo= det [ t/° kq3
(C-5)
1
F r o m the a s s u m p t i o n and (C-4) it follows that
2
ek = - - b k
k = 0 ..... n - 1
flk = - - G
k = 0 ..... n - 2
2,,_ 5
Po
Pl
P2
z, 4
P3
p2
P~
qk =Pk
k=0,...,2
z,_3
qo
qt
q2
Vk = q k
k = 0 ..... 2
(C-6)
"fk = d k :
Using (C-3) and (C-6) one has where ao = l,
]~.-,I =
b,, t
bk
>01 = 1 -
a," = a o (since la,l < 1)
I%1-Iqol>lq21=lv21
qodeP°3 q2 = d e t ~Po LP3
, I<
I~ol = ICor> It._21 =1~.-21 11'01=Idol >ld.-3l =1>'.-31
, ck = det a,_kJ
lb.
Therefore the inequalities (C-4) are verified. It remains to show that (C-2a) and (C-2b) imply
Pl] P2
(C-l)
>0
n - 1 even
<0
n - I odd
n ' ( 1 ) > 0 and r t ' ( - 1 )
Since n ( z ) has no roots on or outside the unit circle the following inequalities are satisfied:
n(1)=a,+a,_l
rr(- 1)=a"-a"-
+-.-+ 1>0
1+"'+
>0 ( - 1)" < 0
(C-2a) neven nodd
Indeed: n ' { 1 ) = l - a , + (2l - a . ) ( l + a l + . . . + a ,
l
but from (C-2a) l+a l+a2+...+a,,
i>-[l+a,,)
(C-2b) which replaced above gives ~z'( 1 ) > 0 If n - 1 is odd (n even)
la,] < 1 7~( - 1)= 1 - a 1 +a2 - . . . -
Ibol>lb°-,I I%1>1c°-~1
r((-1)=(1-a,)(a,-
IPo] >IP31 Iqol>lq2[
(C-3)
a -G
a,- 1+a.>0
2+...+al)-(1-a,
(C-7) 2)
(C-8)
Using (C-7) in (C-8) one obtains ~'( - 1) < 0 . In a similar fashion it can be verified that g'( - 1) < 0 if n - 1 is even. Q.E.D. P r o o J f o r l e m m a 2.2
Applying the same test to g'(z) one has to show first that
[0(n
Notice first that A c is a stable matrix since it has the same eigenvalues as A. Moreover, for a stable c o m p a n i o n matrix, from equations (2), (6) and (9) in [6] it follows that there exists a particular solution for (2.8), Po = [Pu], having the last column entries given by
iI<~0
Pi, = a , _ ~- a.ai, i = 1..... n ao = 1
l,lol>l.~l (C-4)
(C-9)
Po is not only symmetric about the main diagonal but it is also symmetric about the cross diagonal, [6]. But from the definition of ~k one has Pi. = % - 1 , i = t ..... n and by assumption z" + al z"- 1 + . . . + a, is a stable polynomial. Thus from lemma 2.1 A(z)_= rr'(z) has roots inside the unit circle. Since
Discrete model reference adaptive control with an augmented error signal Po is symmetric about the cross diagonal as well, A(z)-=A'(z) thus concluding the proof of part (i). Now (2.8) can be written as
,4~Pop~IpoA¢-Po = -R
e>O is a col+tst;.ii+t. ~nt+,.l Xl ( k J n,, the m+ttl tX
Mlk)=
. 1 + 1-K~
or equivalently (C.IO)
and D = dTP, 1 d > 0. It can be shown that
DKQ
+
DK~
M(k) is a positive definite matrix if 1
I
K~K~>I ( ~ n +-D)"
However, by direct computation using (2.7) and equation (6) in [6], one can verify that ~ P o = Po A~. Therefore (ii) is satisfied for
Q=PoIRPo 1.
K2A ~ (q~i(k_l))2; + i
1 + . 2 . . . . ~.= 0. . . . .
KQ
pol/t~Po.PolPoA+Pot-Pol= -polRPo I
517
(D-2)
Using (2.2') and (2.Y). A 1,~(k) = l/~ (/,-+ 1 ) - 1:~(k) satisfies Q.E.D.
AI',(k)~-~Qm+nltl(k)i 2-(D-~:)Ox(k)M(k)O{k).
(D-3)
Thus ife < D, AVI (k) is negative definite in r/(k), APPENDIX D P R O O F O F C O N V E R G E N C E FOR MODIFIED ALGORITHM and q (k). Consequently,
Let lq (k) be given by
ii~l,i~O, qqh~-*O
lq (k)= I~ (k)+e¢~C(k- 1)M(k- 1)~O(k- 1)
(D-l)
and
~/Si(k)(o'th)---,O
~T(k)=[(i=~oi3i(k-1)dpi(k-
as k---, ~.
i=0
where 1~tk) is as in (2.10)
1 )),q(k)~,
These conclusions can be used to sho~ that ytkj--*O, and consequently that e(k)--*O.It is the authors' contention that this modified algorithm insures that ~bi(k) are bounded for all i and k, but a detailed proof for this is not available at this time.