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Limit Behaviour of a Multivariable Adaptive Stabilization System with Identification with an Unidentifiable Model
Copyright © I FAC haluation of Adaptive Control Strategies, Tbilisi, USSR , 1989
LIMIT BEHAVIOUR OF A MULTIVARIABLE ADAPTIVE ST ABILIZATION SYSTEM WITH IDENTIFICATION WITH AN UNIDENTIFIABLE MODEL B. G. Vorchik and O. A. Gaysin Central R esearch IlI stitllte of COlllprehensh'f Automatioll, Alascoli', USSR
Abstract. Influence of unidentifiability of a model upon limit (with the increase of time) properties of the multiveriable adaptive stabilization system with identification is analyzed. A procedure of carrying out an a-priori analysis of limit properties of the system is described. For procedure to be executed, one only requires the knowledge of the object structure and control algorithm structure. The identification algorithm is not concretized when an a-priori analysis is carried out. But it is assumed that the estimates of unknown parameters of the identification model converge in a certain probability sense with the increase of time: possibilities of carrying out an a-priori analysis of limit properties of the system are illustrated with an example of an adaptive stabilization system with identification and a locally optimal control strategy. As a working apparatus used are the equations of identifiability (Vorchik, 198 5), which make it possible to reduce solving the problem to solving simple algebraic operations over the sy s tem operators. Key words. Adaptive stabilization system with identification, limit properties; multi variable system; identifiability equations. I NTRODUCTION
with a concrete adaptation algorithm. It seems that solving this pro blem with the help of an a-posteriori analysis may be found difficult for the following reasons.
A procedure of examining limit properties of adaptive stabilization system with identification is offered hereby (Vorchik, 1988). The attention shall be drawn to a practically unstudied aspect of the problem, i.e. limit properties of the adaptive system are being analysed, with the object being unidentifiable.
Firstly, the analysis of the system behaviour with a concrete algorithm does not give answer to this question. The received result is determined by general properties of the adaptive system or particular properties of the concrete algorithm of adaptation are inherited by it; or, moreover, the superposed conditions have determined by the ap proved proving circuit;
Persistent excitation of a desired reference trajectory in the adaptive control s ystem is one of the basic sufficient con ditions of identifiability of an object. 'l 'herefore, a class of adaptive control systems with identification, for which the analysis of limit properties, with the model being unidentifiable, is of interest, i.e. the systems with a constant desired trajectore, and preCisely this class of adaptive sys t ems is dealt with in this paper.
secondly, and this especially re nders the a-posteriori analysis of common properties of the adaptive control sy s tem difficult, and even in case of a linear stationary object, such an anal ysis is not a trivial problem.
When limit properties of adaptive control systems with identification are analy zed, as a rule one concretizes the algorithm of calculating the estimates of unknown parameters of the object (further referred to as adaptation algorithm). Let's call such a traditional analysis of limit properties as an a-posteriori analysis.
To take account of the influe n ce of the changes posted to the structure of the object, controller, to the control criteria etc. upon the limit properties of the adaptive system during a-posteriori analysis, serious analy tical efforts are required. Therefore, an a-posteriori approach is difficult to be used as a working apparatus of synthesis, of deSigning adaptive stabilization systems with identification; unidentifiability of the object renders the a-posteriori analysis more difficult and as a result of it,the latter difficulty in works on adaptive control with identification is simply excluded from the conSideration; when boundary properties of the adaptive cont-
An a-posteriori analysis makes it possible
to analyze the efficiency of concrete adaptation algorithms for an adaptive control system of a desired structure, but nonetheless it is of importance as well to establish general properties of the adaptive control system with identification, that are not interconnected
177
178
B. G. Vorchik and O. A. Gaysin
rol system with identification are analyzed, the assumed unidentifiability is in advance eliminated by this or that method used, most often by introducing a desired trajectory of the output variable or by feeding excitation to the controller, often without due substantiation. Evidently due to these circumstances no reply to the main problem - in which conditions does the unidentifiability of the object influence upon the optimality of the adaptive system controller; no link between the limit properties of the system and the structure of the object and controller as well as a priori knowledge of a number of parameters of the object. The paper is devoted to a description of the procedure of analyzing the limit properties of adaptive stabilization s ystems with identification, with the object being unidentifiable. To implement t he offered analysis procedure, one uses only the knowledge of the structure of the object and control algorithm structure. The adaptation algorithm is not concretized, but at the same time it is assumed that the estimates of unknown parameters of the object do converge in a certain probability sense with the increase of time (not obligatorily to true values). The accepted assumption, when analyzing identifiability of the limit adaptive control systems makes it possible to consider the m as stationary parametric stochastic systems. To describe parametriC set of limit points of successive estimates of unknown parameters of the object, to be called furtheron, when the object is not identifiable, as set of unidentifiability. In this case rather developed methods of the theory of unidentifiability of stochastic stationary parametriC systems may be used. I dentifiability (Vorchik, 1985) equation apparatus, that makes it possible to reduce the solution of the problem to simple algebraic operations over the system operators is used in this paper. Let's call t he offered procedure of analyzing the limit properties of the adaptive stabilization systems with identification an a-priori examination. The a-priori examination procedure does not depend upon a concrete type of the object equation,upon the accepted control strate gy, but the concretizing substantially simplifies the statement and at the same time the concretizing of the object and control strategy incidentally allow to get interesting results. Possibilities of an a-priori analysis of limit properties of adaptive stabilization systems with identification are illustrated in this paper with an example of adaptive systems with the simplest version of widely-spread locally-optimal strategies of control, without any constraints to the control signal and to the output variable. This paper is an extension of the results (Vorchik, 1988) on multivariable adaptive stabilization systems with identification. STATEMENT OF THE PROBLEM Let the behaviour of the multivariable control object be described by the following equations:
A(q)y t = qkB(q)ut+G(q)
Et
(1)
In this case Y € Rn, u E. Rn - the observed vector output vari~ble and control Signal t: t E..Rn - unobserved stationary (in a wide sense) white noise; E E t =0, E Et = diag ( 6} , ... , 0;-),0< ()/ < 00
E;
i=1,n; E - symbol of mathematical expec- f tation; t - discrete time; q - backward shift operator ( qY t=Y t - 1 );
A(q)=IIAij(q)lI~ , B(q)= /lBij(q)lI~ ,
UG ij (q)11 ~ tors;
- matrice polynomial opera-
Aij(q)=~ij+ ~ij
aij(l)ql,Bij(q)=
1=T:" . lJ
B .. ~J
b . . (1) ql
lJ
1=0
G"
+g~L(e) lj, tOff,; }
t
Gij(q)=
~ij+
.
)
dij - Kroneker's symbol; aij(l), 1= r-ij , Aij; bij(l), 1=0, Bij ; gij(l), l=~i~ Gij , 1 ~ i, j ~ n - unknown parameters of the object; Aij , Bij , k, 1:ij~1'fij ~ 1, 1 ~ i, j ~ n - known orders and delays of the object. The paper as well uses the description of the object operators in the form of : A(q)=I n +A1 q+ ••• + An q B(q)=Bo+B UG
f
nA
;
A nB
q+ ... +B nB q
; G(q)=In+G f q+ •••
-
+ Gn q ; Ai' i = 1, n A; Bi, i=O, nB; G __ Gi , i=1, nG - numerical real matrices of size n; In - single matrix of size n n A, nB' nG - known orders of the object (nA= max (A ij ) nB=max(B ij ), nG=max(G ij ).
Let's designate the vector of unknown parameters of the object as Le. the vector of unknown coefficient of polynomials of the object matrix operators A(q), B(q), G(q), the element of a limit parametriC set
e ,
o-f..".
Let the control strategy in the adaptive system be described by equation as follows: (2) n
H
where Ht (q)=I n +H 1t q+ ••• +H q nHt nL Lt(q)=Lot+L1t q + ••• +Ln t q ; Hit' L
i= I,nH; Lit' i=O, n L - numerical real matrices of size n, the elements of which are interconnected with functional dependences with et, calculated at the moment of time t by a certain algorithm of adaptation and estimation vector of unknown parameters of the object (I) :
Multivariable Adaptive Stabilization Sys tem
~,nL
- known orders of the controller.
One may always admit that the controller parametrization in the adaptation control system wi til identification coincides with the object parametrization. Then et - vector of the controller (2) parameters at the moment of time t. Similarly (Vorchik, 1988), limit properties of the adaptive control system with identification shall be considered without the adaptation algorithm being concretized, i.e. without the calculation algorithm being concretized at each tact of vector et. It is assumed that met is only the condition: A) Controller parameter vector of the limit (with the increase of time) adaptive control system with identification on the set of full probabilistic measure is an element of some parametric set V'(,"'" (set 0(,. -I< is entered and discussed in (Vorchik,1988). In (Vorchik, 1985) suggests an a-priori before the adaptation algorithm was concretized - analysis of limit properties of the adaptive control system with identification. A control object with single input - single output was considered. This paper is meant to sum up the results of (Vorchik, 1988) in case there is a multivariable control object. A number of conditions that will be formulated in the next section, when controller operators are concretized are supposed to be met alingside with condition A). OBJE CT IDENTIFIABILITY EQUATIONS IN THE LIMIT ADAPTIVE CONTROL SYSTEM WITH IDENTIFICATION The procedure of describing set with the help of identifiability equations does not depend upon a concrete control algorithm (2), but algorithm concretization simplifies the procedure of describing. Therefore furtheron, as in (Vorchik, 1988), a widely spread class of locally optimal control strategies without any limitations of control was considered as well. Let for the control problem be chosen the following criterium: J= E (Yt+k Q Yt+k+Ut Ru t l1;)
(3)
where Q= diag (q1 , ••• qn»O' R= diag (r 1 ,... r ) ~ 0 - diagonal numerin ccal real matrices of size n; :F t - 6'"-algebra generated by co' ••• , Et
allows to put down: A(q) (Yt+k -F (q) ct+k=B(q)ut+C(q)Ct (4 )
Reversibility of the matrix A(q) q=o =I n involves reversibility of the operator A(q), therefore (4) may be expressed as Yt+k=F(q) et+k+A-" (q)(B(q)ut+C(q) Et)
(5) Let alongside with the condition A) the operators of object (1) meet the conditions: B) operators - Bo ' Bo+(BoQ)-1 R, C(q) are not degenerate; C) matrix operators G(q), B(q), A(q) are pairwise simple; D) matrix Gk _ -1 Ak _i-A k _ II F 1 - A-1 Fk_:t. has no null elements. Conditions C) , D) exclude the set of the null parametric measure from the consideration; condition D) involves equality de g Fij (q)=k - 1, where F (q)=/lFij(q)\I
1
Furtheron the limit adaptive system is considered on {)(,If- • Matrices and parameters of the controller on ~ -I' are marked with upper index
*.
Minimization of (3) with respect to Ut' wit h (5) taken into account, makes it possible to get on : 1{ k ~ Yt - F ( q) t - q S Ut =0 (B ~ (q)+A
F(q)=In+F -1 q+ ••• +F k _1 qk-1, C(q)=C o + + C-1 q+ ••• +C nc qnC, satisfying the polynomial equation C(q)=A(q) F(q)+ qkC (q). Substitution of this equation into (1)
e * (q)S * )ut+C ¥ (q)
Ct=O,
S -ll=(B* Q)-1 R o
(6)
Here reversibility of matrix Bo follows from the condition B). Thus, we 'll get equation of the controller on 0( ~ • Following (Keviczky, Kumar~ 1981)~let's input polynomial matrices u (q), F (q), 'ir (q), satisfying the equation "',..,
k
G(q) =F(q) A(q)+ q C(q) with F (q) C(q)= =C (q) F(q) det F(q) = det F(q) (7) Having excluded £ from (6), with (7) taken into consid~ration we'll get the equation of the controller on ¥
_{
~¥
If
-¥
*
(F (q) B (q)+G (q) S )u t + ~* +S '*" ) - 4 C ( q) Yt =0 (8)
(Bo+S) + (B 0
The design of the control minimizing criterion (3) for object (1), is sufficiently formalized (Astrom, 1970). Following (Astrom, 1970), let's introduce matrix polynomial operators
179
Reversibility of the matrix B; + S~ in (8) results from condition B). Equation (1) and (8) make it possible, with condition A) in force, to analyze the parametric identifiability of object (1) in the limit adaptive system of control with identification. Identifiability equations (Vorchik,1985) are the formal apparatus of describing the set CJ-{" Qf unidentifiability • To simplify the
ISO
B. G. Vorchik and O. A. Gaysin
the equations of identifiability, let's exclude the vector variable Yt from (1) and (8). With (6) taken into consideration, the equations (1) and (8) may be put down as k
w
are independent as well.
~
q (B(q)+A(q)S )ut+(G(q)-A(q)F (q»Ct=O
(9)
q k¥(B (q)+A ¥ (q)S -tf )ut+(G-I'- (q)-A * (q) F¥(q») (10) ~C t =0
x
Let's scrutinize the parametric identifiability of the object (9) in the limit system (9), (10). Let ~t be vector of coefficient of polynomials - elements of matrix operators B (q)+ A(q) S·, G(q) - A(q) F*(q) of object (9). I t follows (Vorchik, 1985) that parametrization of object (9) by vector 9 1 is not the uniqueness iff the variables of the object, with the parameters - coordinates of vector 9., being unknown, are linearly dependant. The said linear dependence of object (9) variables may be caused by the object structure independently of the controller structure (10). The linear dependence of the object variables may be as well caused by the controller (10).
The above stated is informal explanation of the equation of object (9) identifiability in the limit system (9), (10), which is as a matter of fact another form of putting down the equations (11), (12) as follows: r;k lq (B¥(q)+A,,(q)S ~\:G,,(q)-A,,(q) F • (q) ] = =[N (q);M 1 (q)]x x q k (B(q)+A(q)S ¥ IG (q)-A(q) F (q)
Lk(B~(q)+A"*(q)S\
*]'
G(q)_A-l«q) F""(q) (1.3)
where by the lower index ~ marked are the operators of object (9) at the vector of parameters Oi. ~ (set 0(..* is put in and discussed in (Vorchik,1988». Heplacing vector f)~ with vector e I( € 0< * makes it possible to receive the description of the set ()( *" • Let's put down (1.3) with e~=9"'as follows: (I -M 1 ( q» (B~\q)+Av-( q)S"")=N( qXB( q) + n -I' +A(q)S) (14) (In-M~(q»(G
~
~
¥
(q)-A (q) F (q»=N(q)(G(q)-
- A(q) F*(q»
The parametrization of the object of known structure, being not the only one with any structure of the controller, is called incorrect. Let the parametrization of object (9) be incorrect. Then its behaviour will be identically described by the equation: k ~ w N(q)(q (B(q)+A(q)S )ut+(G(q)F (q»€t)=O ( 11)
where N(q) is the matrix of size n, the elements of which are the polynomials of q with coefficients-coordinates of arbitrary vector n. Matrix N(q) meets the following conditions: N(q)$O; N(q)=1, Hf the object is correctly parametrized; multiplying by N(q) from the left won't change the structure of object (9) operators. Let's further assume, that the said linear dependence of object variables is caused by the controller (10). Then the behaviour of object (9) will be identically described by the equation k ~ f( q (B(q)+A(q)S )ut+(G(q)-A(q)F (q)St+ + M (q)(q k (B If (q)+A~ (q)S * JUt +(G ~ (q)1 - A* (q) F*(q) Ct)=O (12) where M1 (q) is a polynomial matrix of size n, the elements of which are polynomials of q with coefficients-coordinates of the arbitrary vector m1 • Matrix M,,(q) meets the following conditions; multiplying by M~ (q) from the left results in no change of the structure of object (9) operators. Note. Vectors i.e. there is between their properties of
(15) Ko
Definition
n and m. are independent, no functional dependence coordinates, and hence the operators N( q) and M 1 (q)
f
Describing 0{ as (14), (15) may be simplified. Let N(q)=N +N 1 q + ••• + nN 0 nM +N q M-f(q)=MfO+M H q+M 1n q nN M Ni' i=O,nN, Mi' i=O, n M are numer~cal real matrices. Let's prove that F (q)= .B' (q). From (14) with M1 (q)=0 we get: B'I + S~=N (B +S*). AS per condition B) o 0 0 ~ metrices Bt + S*, Bo + S are reversible, which involves reversibility of matrix No with Mt(q)=O. Due to independence of vectors m1 and n on matrix No is reversible on (J( ~ with any operator M1 (q). As ~x~Q{~ matrix No is reversible on ot X • From (15) we get equality (I -M 1 (q» qkC(q)_N(q) qkC(q)= n ~ = N(q)A(q)(F(q)- F (q}), hence No(F:- Ff )=0, which due to non-degenerait tion of No leads to F.= Ff • Thus, we have No(F: - F~)=O, which re~ults in F; = F~ etc. ThUS, we have F (q)=F(q) with any k (with k=I we'll have F~(q)= =F(q)=I), Le. set Q{ ~ is described with equations (H), (15) with F*(q)= = F(q). With F~q)=F(q) taken into consideration we'll have (14), (15) as: 11... ,.,. ¥ B (q)+ A (q) S = M(q)(B(q)+A(q) S) (16)
G~(q)- A~(q) F(q)=M(q) (G(q)-A(q)F(q» (17) -1
where M(q)=(In-Mt(q» N(q). The reversibility of the operator (I-Mi(q) )
jVlulli\'ariable Adapli\'e Slabilizalion Syslem
follows from the reversibility of N(q), conditioned by reversibility of No' as well as reversibility due to condition B), operators Bo + S ,B + S~ • o ¥ ~ From (16), (17) we get M(q)=(B (q)+A (q)~ 'S") (B(q)+A(q) S,,)-1 =(G¥(q)-A·(q) F(q» 1\ -'f ~(G(q)-A(q) F(q». Owing to condition C) and dependence of nulls of the operator ~ B(q) + A(q) S upon vectors m( , n , operaif tors B(q)+A(q) S • G(q)-A(q) F(q) are re~. ~ ciprocally simple on subset 0{ C al. Since matrix operator M(q) poles coincide with common nulls of operators (B(q) + ~ + A(q) S ), (G(q)-A(q) x F(q». hence it appears that operator M(q) is a matrix polynominal operator on the set ()-(. ~ of a full parametric measure. Thus, with the accuracy of the null parametric measure the set 0<.. is described by equations (16), (17), where M(q) is a polynomial matrix with the coefficients of polynomials-coordinates of the arbitrary vector m1 •
.,
'"
LIMIT PROPERTIES OF THE ADAPTIVE SYSTEM ON 0<.. if'The description of the set ot. -11- in the form of (14), (15) or in an equivalent form in the shape of (16), (17), cited in above allows getting object (1) identifia bility conditions in the limit system (1), (6) as well as studying the controller invariance and invariant stability of the limit system on set O£~. The notion of the controller invariance on (X ¥. set. corresponding to the notion of strong adaptivity. and the notion of an invariant stability of the limit system on O(~ were put in and analyzed in (Vorchi k , 1988) for the object with single output and single control input. Thus, we'll get sufficient conditions of the parametric identifiability of object (1) in the limit adaptive system (1),(8), i.e. the sufficient cond itions of soleliness of solving the identifiability equations (16), (17). Proposition In the limit adaptive control system wi t h identification (1),(8) meeting the requirements of the conditions A)-D) and if the inequality k >I + max deg G.. (q)=n G + I (1 8 ) ~J
is true, then [A'" ( q ) : B" ( q) : Gj< ( q )] =1,1( q ) [A ( q ) ,: B( q ) ~ G( q) ]
•
(19) where M(q) is a matrix polynomial operator. meeting the requirements of (16).(17). Equation (19) is an equation of identifia bility of a multivariable open-looped object (1) and in general has not a unique solution.
18 1
Let's consider the conditions of the uniqueness solution of equation (19), i.e. the conditions of the equation M( q) I . Let's take advantage of the notion of n object (1) element. put in (Vorchik.1985). Definition A part of the system of equations (1),the behaviour of which is described by its i-m equation (1 in), is called i-m element of the system. Proposition 2 The first element of the open-looped object (1) is parametrically identifiable iff for all S=2, n executed are the inequations min (A ij (n-S)-A s /n-S),j=1:'S; B1j(n-S ) Bi . (n-S ). j = 1. n; G-1 • (n-S ) -G . (n-S) , J_ ~ J sJ j=1,n) - , (n-S) ('0 t"(n-S)=max deg(A .(n-S q). j=;:S ; -2J _ BSj(n-S . q), j=1,n ; GSj (n-S,q),j=1,n) Aij(n-S )=de g Aij(n-s .q), ••• ,Gij(n-S)= =deg Gij(n-S. q ) Aij(l,q) = =A ij (1-1,q) An-l+1.n-l+1 (1-1,q- Ai ,n_l+1(1-1,q) x An _ l + 1 ,j (1-1.q). i=1, (n-l+1), j=1. In-l+1) Bij (1,q)=B ij (1-1.q) An _ l + 1 ,n_l+1(1-1. q ) - Ai , n-l+1 (1-1,q) x x Bn _ l + 1 ,j(1-1. q ), j= 1;n Gij (1.q)=G ij (1-1.1) An _ l + 1;n_l+1(1-1. q ) - A.~.n- 1+1 (1-1,q) xG n- 1+1 ,J. (1-1,q),j=1,n. Aij(o.q)=Aij(q). Bij(o.q)=Bij(q), Gij(o.q) = Gij(q), 1~i, j~n. No argument to prove proposition 2 is g iven. Since rearrangement of columns and rows of operators A(q), B(q) a nd G(q) may put any element of the object to be the first, the inequation (20) solves the problem of uniqueness of equation (19) solution. We'll furnish sufficient conditions of uniqueness of equation (19) solution: a) deg Aij(q)=nA for any couple i.j b) deg Gij(q)=n G for any couple i,j ; c) deg Bij(q)=nB for any couple i.j ; d) object (1) is unconditionally parametrically indentifiable. The condition d) is met, for example, when at least one operator equality is justified: A(q)=diag (A 11 (q), ••• ,A nn (q). B(q)=diag ( B 1~(q) •••• , Bnn(q), G(q)=diag (G 11 (q) •••• , Gnn(q) ). Proposition 3 Object (1) in the limit adaptive control
IS2
B. G. Vorchik and O. A. Gaysin
system with identification (1), (8), meeting the conditions A)-D), is parametrically identifiable, if alongside with inequation (18), at least one of the conditions a), b), c) or d) is met. Then, invariance of the controller of the limit adaptive control system with identification on set ~~ is considered. Let's put down the controller equation (8) as follows: ~ -(F(q) BIf (q)+(F(q) A-)( (q)+(q k C (q) S '* )u t + """"K ~ ... oK. +C (q)Yt=F(q)(B (q)+A (q)S »u t + ~* k If + C (q) (q S Ut +y t ) =0 ( 21 ) With (16) taken into account, we'll get F(q) (B"(q)+A*(q) Sit)=F(q) M(q) (B (q) + +A(q) S*). The equallty F(q)C"(q) = ,...,v
,.....~
=C (q) F(q), C (q)=M(q) C(q) and owing to condition B) of the operator C(q) reversibility allows us to put it down as follows: ,.., F(q) M(q) =C * (q) F(q)C - 4 (q). -(
Th~
receixed equation, with F(q) C (q)= = C- f (q) F(q) taken into account, makes it possible to put (21) down as follows: -4 _( ,.., it(C (q)C (q)F(q)( B(q) +A(q) S »u t + -* (q) (q k" C SUt +y t) =0 (22) N
k
-
Owing to equality G(q)=F(q)A(q)+q C(q) we'll get from (22) controller equation (8) on OC", as follows: "" '" (B +S 11 )- 1 (F(q)B(q)+G(q) SIf )u t + + &- (q) Yt)= 0 (23) Hence it becomes clear from (23) that at R=O or when there is a-priori knowledge of B , the behaviour of controller (8) on 0(- 0 shall be determined by the vector of true parameters of the object.
equation (23) makes it possible to put down a characteristic polynomial of the system (1), (8) as follows "" ",,-It P(q)=det A(q) det (F(q)B(q)+G(q)S 1 _ qk U(q)A- (q)B(q» x (det«B +S-k»-t ~ 0 As at R=O we have S· =0, hence from (24) results a proposition. Proposition 6 The limit adaptive system (1), (8) of a given structure at conditions A)-D) and R=O is invariantly stable on O<~ Proposition 7 The limit adaptive system (1), (8) of a given structure at conditions A)-D) ~nd RtO, with the control object being unidentifiable, is unstable on the set of nonnull parametric measure _)I'" C O-ll( • The availability of unstability set JY is not interconnected with the ideal system stability, i.e. at the conditions of proposition 7 the limit adaptive system won't possess the~roperty of invariant stabili ty on 0< • CONCLUSION The a-priori-prior-to-concritizing-adaptation-algorithm approach to analysis of limit properties of the adaptive control system with identification, offered in (Vorchik,1988) has been summed up in this paper for a multivariable case. Multivariability object has not influenced upon the basic properties of a limit adaptive control s ys tem with identification given in (Vorchik, 1988) for the system of controlling the object with single output and single control input. REFERENCES
Proposition 4 Controller (8) of the limit adaptive system (1)t (8) of given structure at conditions A)-D), R=O shall be invariant on
ex * .
Proposition 5 A-priori knowledge of matrix Bo is a sufficient condition of the invariance of the controller ( 8) of the limit adaptive system (1), ( 8 ) of given structrure at RtO on ex -¥ • Proposition 5 clarifies the commonness of occurance of a-priori knowledge of Bo assumption in the works on adaptive control with identification. Statistic modelling results cited in (Vorchik, 1988) for the object with single input-single output show that at R=O losses in control on 0{ l< may considerably exceed losses at
e.
l'-
Then invariant stability on ~ of the limit adaptive control system with identification is considered. A description of the controller on at lI- in the form of
Astrom, K.(1970). Introduction to stohastic control theory. Academic Press, New York. Keviczky, L., Kumar K.S.P.(1981). Multivariable self tuning re gu lator with generalised cost-function. Intern. J.Control, 33, 913-921. Vorchik, B.G. (1985). Identifiability of linear parametric stohastic systems.I. Identifiability equations. Avtomatika i Telemekhanika. No.5,
64-78.
Vorchik, B.G.(1988). Limit properties of adaptive control system with identification (application of identifiability equations.I.Object with single output, single control input. Avtomatika i Telemekhanika. No.6, 96-112.
LIMIT BEHAVIOUR OF A MULTIVARIABLE ADAPTIVE ST ABILIZATION SYSTEM WITH IDENTIFICATION WITH AN UNIDENTIFIABLE MODEL B. G. Vorchik and O. A. Gaysin Central R esearch IlI stitllte of COlllprehensh'f Automatioll, Alascoli', USSR
Abstract. Influence of unidentifiability of a model upon limit (with the increase of time) properties of the multiveriable adaptive stabilization system with identification is analyzed. A procedure of carrying out an a-priori analysis of limit properties of the system is described. For procedure to be executed, one only requires the knowledge of the object structure and control algorithm structure. The identification algorithm is not concretized when an a-priori analysis is carried out. But it is assumed that the estimates of unknown parameters of the identification model converge in a certain probability sense with the increase of time: possibilities of carrying out an a-priori analysis of limit properties of the system are illustrated with an example of an adaptive stabilization system with identification and a locally optimal control strategy. As a working apparatus used are the equations of identifiability (Vorchik, 198 5), which make it possible to reduce solving the problem to solving simple algebraic operations over the sy s tem operators. Key words. Adaptive stabilization system with identification, limit properties; multi variable system; identifiability equations. I NTRODUCTION
with a concrete adaptation algorithm. It seems that solving this pro blem with the help of an a-posteriori analysis may be found difficult for the following reasons.
A procedure of examining limit properties of adaptive stabilization system with identification is offered hereby (Vorchik, 1988). The attention shall be drawn to a practically unstudied aspect of the problem, i.e. limit properties of the adaptive system are being analysed, with the object being unidentifiable.
Firstly, the analysis of the system behaviour with a concrete algorithm does not give answer to this question. The received result is determined by general properties of the adaptive system or particular properties of the concrete algorithm of adaptation are inherited by it; or, moreover, the superposed conditions have determined by the ap proved proving circuit;
Persistent excitation of a desired reference trajectory in the adaptive control s ystem is one of the basic sufficient con ditions of identifiability of an object. 'l 'herefore, a class of adaptive control systems with identification, for which the analysis of limit properties, with the model being unidentifiable, is of interest, i.e. the systems with a constant desired trajectore, and preCisely this class of adaptive sys t ems is dealt with in this paper.
secondly, and this especially re nders the a-posteriori analysis of common properties of the adaptive control sy s tem difficult, and even in case of a linear stationary object, such an anal ysis is not a trivial problem.
When limit properties of adaptive control systems with identification are analy zed, as a rule one concretizes the algorithm of calculating the estimates of unknown parameters of the object (further referred to as adaptation algorithm). Let's call such a traditional analysis of limit properties as an a-posteriori analysis.
To take account of the influe n ce of the changes posted to the structure of the object, controller, to the control criteria etc. upon the limit properties of the adaptive system during a-posteriori analysis, serious analy tical efforts are required. Therefore, an a-posteriori approach is difficult to be used as a working apparatus of synthesis, of deSigning adaptive stabilization systems with identification; unidentifiability of the object renders the a-posteriori analysis more difficult and as a result of it,the latter difficulty in works on adaptive control with identification is simply excluded from the conSideration; when boundary properties of the adaptive cont-
An a-posteriori analysis makes it possible
to analyze the efficiency of concrete adaptation algorithms for an adaptive control system of a desired structure, but nonetheless it is of importance as well to establish general properties of the adaptive control system with identification, that are not interconnected
177
178
B. G. Vorchik and O. A. Gaysin
rol system with identification are analyzed, the assumed unidentifiability is in advance eliminated by this or that method used, most often by introducing a desired trajectory of the output variable or by feeding excitation to the controller, often without due substantiation. Evidently due to these circumstances no reply to the main problem - in which conditions does the unidentifiability of the object influence upon the optimality of the adaptive system controller; no link between the limit properties of the system and the structure of the object and controller as well as a priori knowledge of a number of parameters of the object. The paper is devoted to a description of the procedure of analyzing the limit properties of adaptive stabilization s ystems with identification, with the object being unidentifiable. To implement t he offered analysis procedure, one uses only the knowledge of the structure of the object and control algorithm structure. The adaptation algorithm is not concretized, but at the same time it is assumed that the estimates of unknown parameters of the object do converge in a certain probability sense with the increase of time (not obligatorily to true values). The accepted assumption, when analyzing identifiability of the limit adaptive control systems makes it possible to consider the m as stationary parametric stochastic systems. To describe parametriC set of limit points of successive estimates of unknown parameters of the object, to be called furtheron, when the object is not identifiable, as set of unidentifiability. In this case rather developed methods of the theory of unidentifiability of stochastic stationary parametriC systems may be used. I dentifiability (Vorchik, 1985) equation apparatus, that makes it possible to reduce the solution of the problem to simple algebraic operations over the system operators is used in this paper. Let's call t he offered procedure of analyzing the limit properties of the adaptive stabilization systems with identification an a-priori examination. The a-priori examination procedure does not depend upon a concrete type of the object equation,upon the accepted control strate gy, but the concretizing substantially simplifies the statement and at the same time the concretizing of the object and control strategy incidentally allow to get interesting results. Possibilities of an a-priori analysis of limit properties of adaptive stabilization systems with identification are illustrated in this paper with an example of adaptive systems with the simplest version of widely-spread locally-optimal strategies of control, without any constraints to the control signal and to the output variable. This paper is an extension of the results (Vorchik, 1988) on multivariable adaptive stabilization systems with identification. STATEMENT OF THE PROBLEM Let the behaviour of the multivariable control object be described by the following equations:
A(q)y t = qkB(q)ut+G(q)
Et
(1)
In this case Y € Rn, u E. Rn - the observed vector output vari~ble and control Signal t: t E..Rn - unobserved stationary (in a wide sense) white noise; E E t =0, E Et = diag ( 6} , ... , 0;-),0< ()/ < 00
E;
i=1,n; E - symbol of mathematical expec- f tation; t - discrete time; q - backward shift operator ( qY t=Y t - 1 );
A(q)=IIAij(q)lI~ , B(q)= /lBij(q)lI~ ,
UG ij (q)11 ~ tors;
- matrice polynomial opera-
Aij(q)=~ij+ ~ij
aij(l)ql,Bij(q)=
1=T:" . lJ
B .. ~J
b . . (1) ql
lJ
1=0
G"
+g~L(e) lj, tOff,; }
t
Gij(q)=
~ij+
.
)
dij - Kroneker's symbol; aij(l), 1= r-ij , Aij; bij(l), 1=0, Bij ; gij(l), l=~i~ Gij , 1 ~ i, j ~ n - unknown parameters of the object; Aij , Bij , k, 1:ij~1'fij ~ 1, 1 ~ i, j ~ n - known orders and delays of the object. The paper as well uses the description of the object operators in the form of : A(q)=I n +A1 q+ ••• + An q B(q)=Bo+B UG
f
nA
;
A nB
q+ ... +B nB q
; G(q)=In+G f q+ •••
-
+ Gn q ; Ai' i = 1, n A; Bi, i=O, nB; G __ Gi , i=1, nG - numerical real matrices of size n; In - single matrix of size n n A, nB' nG - known orders of the object (nA= max (A ij ) nB=max(B ij ), nG=max(G ij ).
Let's designate the vector of unknown parameters of the object as Le. the vector of unknown coefficient of polynomials of the object matrix operators A(q), B(q), G(q), the element of a limit parametriC set
e ,
o-f..".
Let the control strategy in the adaptive system be described by equation as follows: (2) n
H
where Ht (q)=I n +H 1t q+ ••• +H q nHt nL Lt(q)=Lot+L1t q + ••• +Ln t q ; Hit' L
i= I,nH; Lit' i=O, n L - numerical real matrices of size n, the elements of which are interconnected with functional dependences with et, calculated at the moment of time t by a certain algorithm of adaptation and estimation vector of unknown parameters of the object (I) :
Multivariable Adaptive Stabilization Sys tem
~,nL
- known orders of the controller.
One may always admit that the controller parametrization in the adaptation control system wi til identification coincides with the object parametrization. Then et - vector of the controller (2) parameters at the moment of time t. Similarly (Vorchik, 1988), limit properties of the adaptive control system with identification shall be considered without the adaptation algorithm being concretized, i.e. without the calculation algorithm being concretized at each tact of vector et. It is assumed that met is only the condition: A) Controller parameter vector of the limit (with the increase of time) adaptive control system with identification on the set of full probabilistic measure is an element of some parametric set V'(,"'" (set 0(,. -I< is entered and discussed in (Vorchik,1988). In (Vorchik, 1985) suggests an a-priori before the adaptation algorithm was concretized - analysis of limit properties of the adaptive control system with identification. A control object with single input - single output was considered. This paper is meant to sum up the results of (Vorchik, 1988) in case there is a multivariable control object. A number of conditions that will be formulated in the next section, when controller operators are concretized are supposed to be met alingside with condition A). OBJE CT IDENTIFIABILITY EQUATIONS IN THE LIMIT ADAPTIVE CONTROL SYSTEM WITH IDENTIFICATION The procedure of describing set with the help of identifiability equations does not depend upon a concrete control algorithm (2), but algorithm concretization simplifies the procedure of describing. Therefore furtheron, as in (Vorchik, 1988), a widely spread class of locally optimal control strategies without any limitations of control was considered as well. Let for the control problem be chosen the following criterium: J= E (Yt+k Q Yt+k+Ut Ru t l1;)
(3)
where Q= diag (q1 , ••• qn»O' R= diag (r 1 ,... r ) ~ 0 - diagonal numerin ccal real matrices of size n; :F t - 6'"-algebra generated by co' ••• , Et
allows to put down: A(q) (Yt+k -F (q) ct+k=B(q)ut+C(q)Ct (4 )
Reversibility of the matrix A(q) q=o =I n involves reversibility of the operator A(q), therefore (4) may be expressed as Yt+k=F(q) et+k+A-" (q)(B(q)ut+C(q) Et)
(5) Let alongside with the condition A) the operators of object (1) meet the conditions: B) operators - Bo ' Bo+(BoQ)-1 R, C(q) are not degenerate; C) matrix operators G(q), B(q), A(q) are pairwise simple; D) matrix Gk _ -1 Ak _i-A k _ II F 1 - A-1 Fk_:t. has no null elements. Conditions C) , D) exclude the set of the null parametric measure from the consideration; condition D) involves equality de g Fij (q)=k - 1, where F (q)=/lFij(q)\I
1
Furtheron the limit adaptive system is considered on {)(,If- • Matrices and parameters of the controller on ~ -I' are marked with upper index
*.
Minimization of (3) with respect to Ut' wit h (5) taken into account, makes it possible to get on : 1{ k ~ Yt - F ( q) t - q S Ut =0 (B ~ (q)+A
F(q)=In+F -1 q+ ••• +F k _1 qk-1, C(q)=C o + + C-1 q+ ••• +C nc qnC, satisfying the polynomial equation C(q)=A(q) F(q)+ qkC (q). Substitution of this equation into (1)
e * (q)S * )ut+C ¥ (q)
Ct=O,
S -ll=(B* Q)-1 R o
(6)
Here reversibility of matrix Bo follows from the condition B). Thus, we 'll get equation of the controller on 0( ~ • Following (Keviczky, Kumar~ 1981)~let's input polynomial matrices u (q), F (q), 'ir (q), satisfying the equation "',..,
k
G(q) =F(q) A(q)+ q C(q) with F (q) C(q)= =C (q) F(q) det F(q) = det F(q) (7) Having excluded £ from (6), with (7) taken into consid~ration we'll get the equation of the controller on ¥
_{
~¥
If
-¥
*
(F (q) B (q)+G (q) S )u t + ~* +S '*" ) - 4 C ( q) Yt =0 (8)
(Bo+S) + (B 0
The design of the control minimizing criterion (3) for object (1), is sufficiently formalized (Astrom, 1970). Following (Astrom, 1970), let's introduce matrix polynomial operators
179
Reversibility of the matrix B; + S~ in (8) results from condition B). Equation (1) and (8) make it possible, with condition A) in force, to analyze the parametric identifiability of object (1) in the limit adaptive system of control with identification. Identifiability equations (Vorchik,1985) are the formal apparatus of describing the set CJ-{" Qf unidentifiability • To simplify the
ISO
B. G. Vorchik and O. A. Gaysin
the equations of identifiability, let's exclude the vector variable Yt from (1) and (8). With (6) taken into consideration, the equations (1) and (8) may be put down as k
w
are independent as well.
~
q (B(q)+A(q)S )ut+(G(q)-A(q)F (q»Ct=O
(9)
q k¥(B (q)+A ¥ (q)S -tf )ut+(G-I'- (q)-A * (q) F¥(q») (10) ~C t =0
x
Let's scrutinize the parametric identifiability of the object (9) in the limit system (9), (10). Let ~t be vector of coefficient of polynomials - elements of matrix operators B (q)+ A(q) S·, G(q) - A(q) F*(q) of object (9). I t follows (Vorchik, 1985) that parametrization of object (9) by vector 9 1 is not the uniqueness iff the variables of the object, with the parameters - coordinates of vector 9., being unknown, are linearly dependant. The said linear dependence of object (9) variables may be caused by the object structure independently of the controller structure (10). The linear dependence of the object variables may be as well caused by the controller (10).
The above stated is informal explanation of the equation of object (9) identifiability in the limit system (9), (10), which is as a matter of fact another form of putting down the equations (11), (12) as follows: r;k lq (B¥(q)+A,,(q)S ~\:G,,(q)-A,,(q) F • (q) ] = =[N (q);M 1 (q)]x x q k (B(q)+A(q)S ¥ IG (q)-A(q) F (q)
Lk(B~(q)+A"*(q)S\
*]'
G(q)_A-l«q) F""(q) (1.3)
where by the lower index ~ marked are the operators of object (9) at the vector of parameters Oi. ~ (set 0(..* is put in and discussed in (Vorchik,1988». Heplacing vector f)~ with vector e I( € 0< * makes it possible to receive the description of the set ()( *" • Let's put down (1.3) with e~=9"'as follows: (I -M 1 ( q» (B~\q)+Av-( q)S"")=N( qXB( q) + n -I' +A(q)S) (14) (In-M~(q»(G
~
~
¥
(q)-A (q) F (q»=N(q)(G(q)-
- A(q) F*(q»
The parametrization of the object of known structure, being not the only one with any structure of the controller, is called incorrect. Let the parametrization of object (9) be incorrect. Then its behaviour will be identically described by the equation: k ~ w N(q)(q (B(q)+A(q)S )ut+(G(q)F (q»€t)=O ( 11)
where N(q) is the matrix of size n, the elements of which are the polynomials of q with coefficients-coordinates of arbitrary vector n. Matrix N(q) meets the following conditions: N(q)$O; N(q)=1, Hf the object is correctly parametrized; multiplying by N(q) from the left won't change the structure of object (9) operators. Let's further assume, that the said linear dependence of object variables is caused by the controller (10). Then the behaviour of object (9) will be identically described by the equation k ~ f( q (B(q)+A(q)S )ut+(G(q)-A(q)F (q)St+ + M (q)(q k (B If (q)+A~ (q)S * JUt +(G ~ (q)1 - A* (q) F*(q) Ct)=O (12) where M1 (q) is a polynomial matrix of size n, the elements of which are polynomials of q with coefficients-coordinates of the arbitrary vector m1 • Matrix M,,(q) meets the following conditions; multiplying by M~ (q) from the left results in no change of the structure of object (9) operators. Note. Vectors i.e. there is between their properties of
(15) Ko
Definition
n and m. are independent, no functional dependence coordinates, and hence the operators N( q) and M 1 (q)
f
Describing 0{ as (14), (15) may be simplified. Let N(q)=N +N 1 q + ••• + nN 0 nM +N q M-f(q)=MfO+M H q+M 1n q nN M Ni' i=O,nN, Mi' i=O, n M are numer~cal real matrices. Let's prove that F (q)= .B' (q). From (14) with M1 (q)=0 we get: B'I + S~=N (B +S*). AS per condition B) o 0 0 ~ metrices Bt + S*, Bo + S are reversible, which involves reversibility of matrix No with Mt(q)=O. Due to independence of vectors m1 and n on matrix No is reversible on (J( ~ with any operator M1 (q). As ~x~Q{~ matrix No is reversible on ot X • From (15) we get equality (I -M 1 (q» qkC(q)_N(q) qkC(q)= n ~ = N(q)A(q)(F(q)- F (q}), hence No(F:- Ff )=0, which due to non-degenerait tion of No leads to F.= Ff • Thus, we have No(F: - F~)=O, which re~ults in F; = F~ etc. ThUS, we have F (q)=F(q) with any k (with k=I we'll have F~(q)= =F(q)=I), Le. set Q{ ~ is described with equations (H), (15) with F*(q)= = F(q). With F~q)=F(q) taken into consideration we'll have (14), (15) as: 11... ,.,. ¥ B (q)+ A (q) S = M(q)(B(q)+A(q) S) (16)
G~(q)- A~(q) F(q)=M(q) (G(q)-A(q)F(q» (17) -1
where M(q)=(In-Mt(q» N(q). The reversibility of the operator (I-Mi(q) )
jVlulli\'ariable Adapli\'e Slabilizalion Syslem
follows from the reversibility of N(q), conditioned by reversibility of No' as well as reversibility due to condition B), operators Bo + S ,B + S~ • o ¥ ~ From (16), (17) we get M(q)=(B (q)+A (q)~ 'S") (B(q)+A(q) S,,)-1 =(G¥(q)-A·(q) F(q» 1\ -'f ~(G(q)-A(q) F(q». Owing to condition C) and dependence of nulls of the operator ~ B(q) + A(q) S upon vectors m( , n , operaif tors B(q)+A(q) S • G(q)-A(q) F(q) are re~. ~ ciprocally simple on subset 0{ C al. Since matrix operator M(q) poles coincide with common nulls of operators (B(q) + ~ + A(q) S ), (G(q)-A(q) x F(q». hence it appears that operator M(q) is a matrix polynominal operator on the set ()-(. ~ of a full parametric measure. Thus, with the accuracy of the null parametric measure the set 0<.. is described by equations (16), (17), where M(q) is a polynomial matrix with the coefficients of polynomials-coordinates of the arbitrary vector m1 •
.,
'"
LIMIT PROPERTIES OF THE ADAPTIVE SYSTEM ON 0<.. if'The description of the set ot. -11- in the form of (14), (15) or in an equivalent form in the shape of (16), (17), cited in above allows getting object (1) identifia bility conditions in the limit system (1), (6) as well as studying the controller invariance and invariant stability of the limit system on set O£~. The notion of the controller invariance on (X ¥. set. corresponding to the notion of strong adaptivity. and the notion of an invariant stability of the limit system on O(~ were put in and analyzed in (Vorchi k , 1988) for the object with single output and single control input. Thus, we'll get sufficient conditions of the parametric identifiability of object (1) in the limit adaptive system (1),(8), i.e. the sufficient cond itions of soleliness of solving the identifiability equations (16), (17). Proposition In the limit adaptive control system wi t h identification (1),(8) meeting the requirements of the conditions A)-D) and if the inequality k >I + max deg G.. (q)=n G + I (1 8 ) ~J
is true, then [A'" ( q ) : B" ( q) : Gj< ( q )] =1,1( q ) [A ( q ) ,: B( q ) ~ G( q) ]
•
(19) where M(q) is a matrix polynomial operator. meeting the requirements of (16).(17). Equation (19) is an equation of identifia bility of a multivariable open-looped object (1) and in general has not a unique solution.
18 1
Let's consider the conditions of the uniqueness solution of equation (19), i.e. the conditions of the equation M( q) I . Let's take advantage of the notion of n object (1) element. put in (Vorchik.1985). Definition A part of the system of equations (1),the behaviour of which is described by its i-m equation (1 in), is called i-m element of the system. Proposition 2 The first element of the open-looped object (1) is parametrically identifiable iff for all S=2, n executed are the inequations min (A ij (n-S)-A s /n-S),j=1:'S; B1j(n-S ) Bi . (n-S ). j = 1. n; G-1 • (n-S ) -G . (n-S) , J_ ~ J sJ j=1,n) - , (n-S) ('0 t"(n-S)=max deg(A .(n-S q). j=;:S ; -2J _ BSj(n-S . q), j=1,n ; GSj (n-S,q),j=1,n) Aij(n-S )=de g Aij(n-s .q), ••• ,Gij(n-S)= =deg Gij(n-S. q ) Aij(l,q) = =A ij (1-1,q) An-l+1.n-l+1 (1-1,q- Ai ,n_l+1(1-1,q) x An _ l + 1 ,j (1-1.q). i=1, (n-l+1), j=1. In-l+1) Bij (1,q)=B ij (1-1.q) An _ l + 1 ,n_l+1(1-1. q ) - Ai , n-l+1 (1-1,q) x x Bn _ l + 1 ,j(1-1. q ), j= 1;n Gij (1.q)=G ij (1-1.1) An _ l + 1;n_l+1(1-1. q ) - A.~.n- 1+1 (1-1,q) xG n- 1+1 ,J. (1-1,q),j=1,n. Aij(o.q)=Aij(q). Bij(o.q)=Bij(q), Gij(o.q) = Gij(q), 1~i, j~n. No argument to prove proposition 2 is g iven. Since rearrangement of columns and rows of operators A(q), B(q) a nd G(q) may put any element of the object to be the first, the inequation (20) solves the problem of uniqueness of equation (19) solution. We'll furnish sufficient conditions of uniqueness of equation (19) solution: a) deg Aij(q)=nA for any couple i.j b) deg Gij(q)=n G for any couple i,j ; c) deg Bij(q)=nB for any couple i.j ; d) object (1) is unconditionally parametrically indentifiable. The condition d) is met, for example, when at least one operator equality is justified: A(q)=diag (A 11 (q), ••• ,A nn (q). B(q)=diag ( B 1~(q) •••• , Bnn(q), G(q)=diag (G 11 (q) •••• , Gnn(q) ). Proposition 3 Object (1) in the limit adaptive control
IS2
B. G. Vorchik and O. A. Gaysin
system with identification (1), (8), meeting the conditions A)-D), is parametrically identifiable, if alongside with inequation (18), at least one of the conditions a), b), c) or d) is met. Then, invariance of the controller of the limit adaptive control system with identification on set ~~ is considered. Let's put down the controller equation (8) as follows: ~ -(F(q) BIf (q)+(F(q) A-)( (q)+(q k C (q) S '* )u t + """"K ~ ... oK. +C (q)Yt=F(q)(B (q)+A (q)S »u t + ~* k If + C (q) (q S Ut +y t ) =0 ( 21 ) With (16) taken into account, we'll get F(q) (B"(q)+A*(q) Sit)=F(q) M(q) (B (q) + +A(q) S*). The equallty F(q)C"(q) = ,...,v
,.....~
=C (q) F(q), C (q)=M(q) C(q) and owing to condition B) of the operator C(q) reversibility allows us to put it down as follows: ,.., F(q) M(q) =C * (q) F(q)C - 4 (q). -(
Th~
receixed equation, with F(q) C (q)= = C- f (q) F(q) taken into account, makes it possible to put (21) down as follows: -4 _( ,.., it(C (q)C (q)F(q)( B(q) +A(q) S »u t + -* (q) (q k" C SUt +y t) =0 (22) N
k
-
Owing to equality G(q)=F(q)A(q)+q C(q) we'll get from (22) controller equation (8) on OC", as follows: "" '" (B +S 11 )- 1 (F(q)B(q)+G(q) SIf )u t + + &- (q) Yt)= 0 (23) Hence it becomes clear from (23) that at R=O or when there is a-priori knowledge of B , the behaviour of controller (8) on 0(- 0 shall be determined by the vector of true parameters of the object.
equation (23) makes it possible to put down a characteristic polynomial of the system (1), (8) as follows "" ",,-It P(q)=det A(q) det (F(q)B(q)+G(q)S 1 _ qk U(q)A- (q)B(q» x (det«B +S-k»-t ~ 0 As at R=O we have S· =0, hence from (24) results a proposition. Proposition 6 The limit adaptive system (1), (8) of a given structure at conditions A)-D) and R=O is invariantly stable on O<~ Proposition 7 The limit adaptive system (1), (8) of a given structure at conditions A)-D) ~nd RtO, with the control object being unidentifiable, is unstable on the set of nonnull parametric measure _)I'" C O-ll( • The availability of unstability set JY is not interconnected with the ideal system stability, i.e. at the conditions of proposition 7 the limit adaptive system won't possess the~roperty of invariant stabili ty on 0< • CONCLUSION The a-priori-prior-to-concritizing-adaptation-algorithm approach to analysis of limit properties of the adaptive control system with identification, offered in (Vorchik,1988) has been summed up in this paper for a multivariable case. Multivariability object has not influenced upon the basic properties of a limit adaptive control s ys tem with identification given in (Vorchik, 1988) for the system of controlling the object with single output and single control input. REFERENCES
Proposition 4 Controller (8) of the limit adaptive system (1)t (8) of given structure at conditions A)-D), R=O shall be invariant on
ex * .
Proposition 5 A-priori knowledge of matrix Bo is a sufficient condition of the invariance of the controller ( 8) of the limit adaptive system (1), ( 8 ) of given structrure at RtO on ex -¥ • Proposition 5 clarifies the commonness of occurance of a-priori knowledge of Bo assumption in the works on adaptive control with identification. Statistic modelling results cited in (Vorchik, 1988) for the object with single input-single output show that at R=O losses in control on 0{ l< may considerably exceed losses at
e.
l'-
Then invariant stability on ~ of the limit adaptive control system with identification is considered. A description of the controller on at lI- in the form of
Astrom, K.(1970). Introduction to stohastic control theory. Academic Press, New York. Keviczky, L., Kumar K.S.P.(1981). Multivariable self tuning re gu lator with generalised cost-function. Intern. J.Control, 33, 913-921. Vorchik, B.G. (1985). Identifiability of linear parametric stohastic systems.I. Identifiability equations. Avtomatika i Telemekhanika. No.5,
64-78.
Vorchik, B.G.(1988). Limit properties of adaptive control system with identification (application of identifiability equations.I.Object with single output, single control input. Avtomatika i Telemekhanika. No.6, 96-112.