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A Parameter Optimization Approach to the Design of Structurally Constrained Regulators for Discrete-Time Systems
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A PARAMETER OPTIMIZATION APPROACH TO THE DESIGN OF STRUCTURALLY CONSTRAINED REGULATORS FOR DISCRETE-TIME SYSTEMS R. Scattolini and N. Schiavoni lh/Jllllllllllllu
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Abstract. The problem dealt with in this paper i s t he design of a regula t or for a discre t e- time system. The re~ulator should supply a satisfacto r y dynamic behavior to the ove r all cont rol sys t em and guarantee robus t zero - e r ror regulation for constant exogenous si -
gnals. The here presen t ed technique is based on parameter op t im i za t i on and, being very flexible, is particularlY suited to cope with var i ous constra i n t s which may be imposed on the structure of the r egulator (e.g. , d ecentralizat i on of the overall regulation system , order of t he lucal devices) . The design problem is first s t a t ed as an optima l control prob l em . The n, its reformulation as a mathemat i cal prog r amming pr oblem is given and some computational rema r ks a r e presented . The effect i veness of the p r oposed technique i s ve r ified on an i l lustra t ive example . Keywords. Dece nt ralized control , parameter optimiza ti on , multivari able regulator theo r y , computer - aided design.
satisfying the info r matlon and comp l ex it y constraints . The second is t un i ng the r egu l a t o r pa r a meters , by op t imiz in g a n i ndex we i ght i n g t he dynamic behavio r of the cont r o l system , wh il e t a kin g
INTRODUCTlO~:
Discre t e -ti me systems , as opposed to continuous - ti me systems , are ~oing to pl ay an ever i ncreasing
role i n control theory and practice. This is due both to the nowadays available results of modern sys t em identifica t ion and to the widesp r ead diffu sion of digital con t rollers . The aim of this paper is t o con t r i bute to th e development of dis cre te - time system control t heory , by presenting an effective and flex i b l e des i gn technique.
into accoun t the ze r o- erro r r egu l a ti o n cons tr a int .
Specia l at t ent ion wi l l be pa i d , in t he f o ll owin g sections , t o the second s t e p . In f a c t, de spit e the problem of choosi ng a n app r opr i a t e st r uc t u r e ha s not vet been satisfactorily fo r ma li zed ( s ee, a nyhow , Locate ll i and Co l leagues , 1977) , th e expe rience of a sk i l l ed control e ng in ee r i s i n gene r a l sufficient to sug ge st the use of a l i mit e d numbe r o f tentative s t ructu r es. On th e con tr a r y , th e f ina l selection wit hin th i s se t would ent a il a n unb ea rable tr i al and error procedure , wh i ch ca n b e avo ided bv comparing the performances obta in e d when th e optimal tuni n gs of t he var i ous s tru c t u r es a r e us ed . Hence , an efficien t t uning algo r i t hm i s the bas is of an inte r active package which al l ows one t o d e termine the most app r op r ia t e so l u ti on among t he feasible ones.
The particu l ar problem dealt with consists uf synthesizing a li near regulator for a linear plant . The regula t or sho ul d supply a sat i sfacto r y dynamic behavior to th e overall control system, whil ee nsur ing r obust zero - e rror r egu l ation for constant e xo -
genous signals. The design pro ce du r e put forward here is based on parameter optimizatio n and presents a number of similarities with a technique which has already p r oved to be effective i n the desi gn of co ntinuous - time control svstems(Guardabassi and Colleagues, 1979 1982a , 1983a , 1983b; Davison and ferQ,uson , 1981; Loca t elli a nd Colleagues , 1981; Davison a nd Chang , 1982) . The main feature distinguishing it from the majority of other standard design methods is its cap ability to handle in a natural wav some frequentlY encounte r ed constra i n t s on the structure of the regulat o r . Among t hem are : (i) the requirement t hat the regulator be decentra li zed, that is , composed by various elements (local regulators) interconnect ed in such a way tha t , in general , any singl e i nput of the plant is made to depend on part only of all the informat i on available (outputs of the plant, disturbances, and set - poin t s); ( i i) an a priori partial specifica t ion of the functional relation ships the local regulators can establish bet~ ee n the i r inputs and outputs , e.g. , t he urder of the local regulators .
The layout of the paper is as fo l lows. Nex t sec tion is devo t ed to fo r mal l y state the pr oblem. Then , a prel i min a r Y a nalys i s on the r egu lat o r s tru c ture is carr i ed out i n the t hird sectio n . It s a im is t o ascertain whe t he r t he r e a r e r egu la t o r s , e ndowed with the structure a t hand , wh i c h r ende r t he con t rol sys t em asymptotica ll y s t ab l e a nd ma ke the steady - state error t o robus t ly vanish. In pa rticular , a class of structu r es , to be ca ll e d p r oper, is introduced , the use of which gr ea t ly simplifies the parameter opt i mi za t io n phase . Th e f orth s ec ti on presents the problem r eformu l at i on i n t e rms of a mathematical prog r amming problem as we ll a s s ome rema r ks on computat i onal as pects . fin a ll y , th e desi gn of a regulator fo r a hea d-box of a pap e r ma ch ine is deal t wi t h in t he f i fth sec ti on.
The overall design procedure consists of two main steps. The first is choosing a regu l ator stru c ture
1159
R. Scattolini and N. Schiavoni
1160
PROBLEM STATEMENT This section is devoted to introduce the problem dealt with in the rest of the paper. Assume that the plant under control is described by the linear, time-invariant system ( La)
CpRxp(k)+ NPRd(k)
(1.b)
CpFxp(k)+ NpFd(k),
(1. c)
Os c 0 be the set of all the q' s such that all the eingevalues of system (1),(3) lie within the unit circle, the tunings 0.(q) of interest are only those such that
(4)
q€ Q • S
Moreover, the tuning must guarantee that the mean value of the output YR asymptotically track the asymptotic mean value of the reference signal yO, in spite of the presence of the disturbance d, i.e.,
lim E{e(k) } where xp€ Rn, u€ Rm, and d€ R~ are the state,control, and disturbance vectors, respectively. Two different output vectors have been shown: yp€ RP is the vector of variables which should track a reference signal yO, whereas YF€ Rr is the vector of other measured variables to be possibly exploited for regulation purposes. Both the reference signal yO and the disturbance dare modeled as coloured noises with an asymptotically rational spectrum. HencP"they can be described by (2.a)
CISx I (k) + NISv(k)
(2.b)
= CIDx I (k) + NIDv(k)
(2.c)
=
d(k)
where xIS Ra is the state vector, AI is Hurwitz (i.e., its eigenvalues all lie within the unit circle) and v(') is a white noise with known mean value ;:;€ R ~ and intensity v, that is, v(·) "- WN(;:;, V).
(2.d)
As for the regulator, it is represented, as a whole, by the linear, time-invariant system X
u(k)
where E{'} is the expected value operator. In addition, it is required that the regulation constraint (5) be fulfilled in a robust way, that is: i) for all perturbations of the plant matrices Ap,Bp,Mp,CpR,NpR,C pF , and NpF ' which preserve, as for Ap,Bp,CPR,C Pf the asymptotic stability of system (1),(3); ii) for all perturbations of the filter matrices AI,MI,CIS,NIS,CID, and NID, which preserve, as for AI, the asymptotic stability of system (2); iii) for all perturbations of the mean value v of v(·). Note that constraint (5) is satisfied in a robust way if and only if the control system (1) ,(3), when subject to arbitrary constant (step) exogenous signals yO and d, exhibits an asymptotically vanishing error, in spite of sufficiently small perturbations of the plant matrices. As for the control system dynamic behavior, it will be evaluated through the value taken on by a linear combination of the covariances of the steady-state processes asymptotically approached by the error e, the control u and the state of both the plant and the regulator. Specifically,
+[2«k)-;o'
(3.a)
CRxR(k)+DREe(k)+DRFYF(k)+DRsyO(k)+
u
~
lim k-
x
~
Ixp '
x € RV, v being a given constant, and R
e(k) ~ yO(k) - YR(k)
+
T[!:(k)- ~ } ,
(6.a)
where (3.b)
where
(5)
J ~ lim E{e'(k)Re(kl+G(k)-G]' sG(k)-li] k-+ 00
R(k+ 1) = ARx R(k) + BREe (k) + BRFYF (k) + BRSYO (k) + + MRd(k)
0,
k-+ 00
(3.e)
is the system error. It is apparent that the regulator can incorporate, in general, both feedback and feedforward actions. In order to synthetically take into account the possibility that the regulator be made out of subsystems (e.g., standard PID regulators) interconnected among them, all the matrices appearing in eqs.(3) are assumed to be known co ntinuously differentiable functions of a design parameter vector q ranging into an admissible parameter set QCRP. For any q € 0, the ordered set of matrices 0.(q) ~ i AR(q) ,BRE(q) ,BRF(q) ,BRS(q) ,~(q) ,CR(q), DRE(q) ,DRF (q) ,DRS(q) ,NR(q) ) represents a particular tuning of the regulator. Of course, the set of all possible tunings is completely described by the couple (Q, 0 (')) ,which will henceforth be called structure of the regulator. The first property that the control systemis desired to have is asymptotic stability. Then, letting
x (:, lim k-
E{u(k)} , xR' I ' E{x(k) } ,
(6.b) (6.c)
(6.d)
whereas R,S, and T are symmetric positive semidefinite matrices. Note that the value of J mainly depends on the system behavior in the frequency band where the spectra of the processes asymptotically approached by yO and d are located. Hence, shaping these spectra (through an appropriate choice of the matrices appearing in eqs. (2» allows the designer to define the frequency band where the system behavior has to be particularly taken care of, and constitutes an indirect way of qualifying the system transients.
In conclusion, the design problem dealt with here can be synthetically stated as follows: Main Problem Find a tuning 0.( q) of a regulator with given structure (0, 0. ('» such that, for system (1)-(3): (i) the stability constraint (4) is met;
11 61
Design of Structurally Constrained Regulators
constr a int (5) is me t in a robust way.
( ii) the re gulation constraint (5) is met in a robust way; (iii) the performance index (6) is minimized.
REGULATION
fu~D
STABI LITY CONSTRAINTS
Thi s sect i on is d evo t ed t o dis c us s th e feasibi l ity of the Main Problem. Conditions will be given under which ther e exis t regulators e ndowed with a stipulated structure such that the stability co nstraint (4) i s me t an d the re gu l a ti on constraint (5) i s met in a robust wa y . As fo r the r egu l atio n constraint, th e Internal Model Principle (see Francis and Wonham, 19 76 , for th e continuous -ti me case) supplies a comp l e t e characterization of al l re gula tors which mak e it satisfied in a robust way. For th e case at hand, th e tunin g must be such that th e co rrespondin g system (3) in co rp orates a number of "appropriat e integral actions" not l ess than the numb e r ot th e plant r egu lat ed var i ab l es . Then, if system (I), (3) is asymptotically s table, th e r egulati on const r a int (5) is verified in a robust way . This va gue s tat ement of th e Internal Model Principl e is specified in Theorem 1 below (see Guardabassi and Col l e a gues, 19 83a , fo r the conti nuous-time ca se). Hereafter, for any positive i nt ege r a , l a deno t es th e a Xa id e ntit y ma tri x. Th eo r em 1 Th e tuning D(q) , q€ OS, guarant ees that th e re gula ti on constra int (5) is met in a robust way i f and on l y if ther e ex i s ts a nonsingular matrix Le VxV € R such tha t
o I
P
o
Thi s th eo rem, th ough fa irl y obvious, est ablis hes important fact: wh e ne ver the regulator s tru cture is proper, th e robu s t re gulation co nstraint (5) does not need to be exp li c it e l y tak en into account.
0
Con si de r now t he c l ass of structures depicted in Fig.l, where Vl", simply consi s t s of p para l le l dis c r ete inte g r ato r s (i . e., th e z-transfer - function ma trix of ~ is diagonal and its nonzero e ntri es are all equal t o (z -I ) -I ), whereas ~1 (q) and ~2(q) a r e lin ea r, time-invariant sys t ems of given o rder. All the s tructyr es of this class a r e pr oper . In fact, l et tin g xk , i = 0 ,1,2, be the state vector of subsystem :Ri (g) , i = 0, 1, 2, re s pect iv ely , and xR -_ !x l' x 0 ' R R 1 holds with L= I .
x 2 "' , then R
Th eo r em
u
It is wo rth noti c in g that the c l assof proper structur es of Fig.l is very lar ge . In fact , the subsy stems OC 1 (q) and ~ 2(q) can be dynamical or non dynamical , centralized or decen tr alized , and pa r a me tri zed at will . Then, any possible a priori in forma ti o n as we ll as complexity con s traints ca n easily be handl ed , th e only s i gni f i cant constraint impli cit in th e choice of this c la ss of s tru c tur es being that th e o rd er of the subsystems " preceding" and "f o ll ow in g " th e blocK of int egrato rs is ind e pendent of th e specif i c val ue of q. As for the possibility of choosing a nond ynamical ~1 (q) , see Gua rd abass i and Colleagues (1983a , 1984). Suppose, in parti c ul a r, that t he s tructure o f th e r egula t o r is assigned by means of a block di ag r am whi ch des cr ib es how various single -in put single output loca l regulators are int erconn ec ted among them. In this case , a simp l e a l gori thm a ll ows one to asce rtain wh e th e r it b elo ngs to the c lass of Fig.1, so that it is prope r. In fact, it is suf ficien t to verify the two followin g points: i) there ex i st (at l eas t) p blocks having, for a ll q€ Q, transf e r-fun c tion matrix (Z -I ) -I Ip between th e ir inputs and outputs; ii\ the tr an sfer - function matr ix be tween !YF' yo' d' t ' a nd th e outpu ts of the above blocks is zero for all q € O.
0
In view of th e above discussion , it is reasonable to r es tri c t attention t o prop e r regulato r structur es . This is what will be done from now on. The prob l em of establishing whether there exist regulators endowed with a given structure which render system (1), (3 ) asymtotically stab l e i s,
o o
o
in the gene ral case , unso l ved . The absence of unstable fixed modes (see , e.g. , Wa ng and Davi son,
Ch eck in g wh ether , for a consid e r ed q € OS' the con dit i on of Theorem 1 is satisfied i s an all but trivi al task f r om a numerica l point of v i ew . Fort una t ely e nough , this di ff i cul t y ca n be overcome by approp riat ely se l ect ing the regulator structure along th e li nes described be l ow (Guardabassi and Colleagues , 1979, 198 3a ). Let (C ,A,B) denote th e lin ear , time -invariant system th e ou tput, dynamic a nd input matrices of which are C, A, and B, r espectivel y . Then, the structure (Q , D( ' )) is said t o be pr ope r if ma tri x L c all ed for in Theorem 1 exis t s for any q € 0*, Q* being th e se t of a ll qe € Q such th a t the uncon tr ol l ab l e and/ or unobservab l e part of system (CR(q), AR(q), : BRE(q) BRF(q) : ) is asymp toti ca ll y s t able.
1973 ; Corfmat and Morse , 1976; Locatelli and Col l eagues 1977; Anderson and Clements , 1981; Anderson , 1982) i s on ly a necessary cond ition for Os to be nonempty , because a structure (Q, D(')) , besides specifying the info rmation flow within the regu lator a l so bounds the regulator comp l ex ity.For discrete time s y stems , suff i c i e nt condit i ons can be given when the p lant i s a symp totically stab l e (for con tinuous - time sys tem s , see Davison and Ferguson, 1981 ;Guardabassi and Colleagues,1982b;Locatelli and Schiavoni ,1 982, 1983 ,whe re al so some cases of unstable plants a r e cons idered). Let the number o f regulated va riab l es egual the number o f control variables , L e . , m=p.Mo r eove r assume that the r egulator is purely
The definitions themselves o f prope r structure and Q* i mp l y that Q*cQS ' Then , the following re sult can immediately be proved .
i . e ., v=p . Then it ca n be described by x (k) + e(k ) x (k+l) R R u( k ) CR (g ) xR( k)
Th eo r em 2
Th e e xtreme case of completely decen tralize d in formation f l ow will be considered . Let (8) C (g) = diag (g l' g2 " " , gp ) R
If th e s tru c tur e (Q, 0 ( ' )) is prope r, th en any tuning ~ (q), q € QS ' guarantees that the re gulation
integral,which implies that its order is minimum,
(7.a) (7.b)
R. Scattolini and N. Schiavoni
1162
and assume that Q contains an open neighborhood of the ori g in of RP , where P = p. Furthermore, let (P t, ( 1,2, ... ,p ) , {) be any permutation (i , i , ... ,i'p) of the set (j' and, for any 1 2 j € 6',
B. (-;J)
J C. (
J
tr U r . +tr Z A. +n' n. 1 1 1 where Z and n are the unique solutions of the equations Z- F'(q)Z F(q)= H'(q)U H(q)
!bi
~
~
1
1 c.
b. '2
b.
c.
c.
'2
'1
1 .
I,
(10.a)
and
J
J}n+ v+o- F'(q2!n =H'(q)U H(q)~- F'(q)ZG(q)v,(10.b)
I 1.
whereas
J
where b. and c! are the i-th column of B and the i-th ro~ of Cp~, respectively. The folloeing result then holds.
~C~H(q)+H(q)[!- ~C~ aH'(q) aq.
+
1
Theorem 3 If
i) matrix Ap is Hurwitz; ii) there exists a permutation
J of
~
m]" (~) t, det{C . (J) (I-A )-l B. (J»" 0, = J P J
such that j €
then there exists £: > 0 such that, for any e(o,£ ) ,s ystem (1),(7),(8) with j €
where
6'; £
aG' (q)
aq:-
+
1
€
lP,
asymptotically stable. and matrices F(q), G(q), H(q), K(q), U, P, and ~ are the same as in Theorem 5. c
PROBLEM REFORMULATION AND SOLUTION In this section, restricting attention to proper regulation structures, it is shown how the Main Problem can be reformulated as a mathematical programmin g problem, the peculiar features of which make it fairly easy to solve. The reformulation of the Main Problem is presented in Theorem 4 below. Theore 4 If the structure of the regulator is proper, the Main Problem can equivalentl y be restated as follows: min
tr U{ H(q)[!- C~~H'(q)+K(q)VK'(q) } ,
q € Q S
where P and
~
are the unique solutions of the
equations
and
Un +v +a -
The evaluation at any q € Q of both the perfo~ mance index and its gradieng essentially calls for solving the four equations (9) and (10). The asymptotic stability of s ystem (1)-(3) guarantees that they always admit one and only one solution. An efficient solution algorithm for the Liapunov equations (9.a),(10.a) can be obtained by suitable modifying the algorithm proposed by Bartels and Stewart (1972), also taking into account the suggestions of Kleinman and Krishna Rao (1978). The main step consists of carryin g matrix F(q) into its real Schur canonical form FS(q) by means of an orthogonal transformation. The computations are simplified by takin g into account the fact that F(q) is block-trian gular. Once FS(q) is known, also solving eqs. (9.b),(10.b) is strai ghtforward. Moreover, checking whether q € Os is immediate, since FS exhibits the eingevalues of the control system on the (scalar or bidimensiona l) diagonal blocks. Among the various iterative te c hniqu e s which can be adopted to solve the mathematical programmin g problem stated in Theorem 4, particularl y efficient is the Fletcher and Powell conju gate gradient algorithm. Since it directl y applies to unconstrained problems only, it can be used when Q=RP and must be slightl y modified in order to be sure that, during the optimization, q remains in
F(ql]C
(9.b)
and matrices F(q) ,G(q) ,H(q) ,K(q), and U are explicit functions of the data, the expressions of which are reported in the Appendix . C Moreover, the gradient of the performance index J with respect to the design parameter q can also be evaluated in an easy way at any q € OS' Theorem 5 At any q€ QS' the partial derivative with respect to the i-th parameter qi of the performance index J is given by
OS' Apart from very critical cases, the meanin g itself of the performance index actuall y guarantees that q moves away from the boundary of QS' However, if the step size of the algorithm is exceedingly large, during the opti mization it might happen that stabilit y is lost and yet the performance index, as evaluated in the mathematical programming problem of Theor em 4, is improved. This improvement is clearly absolutel y meaningless. Stab1lity, and henc e significance of th e evaluation of the performance index throu gh the equations of the mathematical programmin g problem of Theorem 4, is recovered in this case by simpl y reducing the step size of the al gorithm. Of course, the algorithm must be initialized with a qe € QS' For that, Theorem 3 can help.
Design of Structurally Constrained Regulators
A~
ILLUSTRATIVE EXA"PLE
The
desi~n
procedure described in the preceding
sections is now applied to svnthesize a regulator
for the head-box of a paper machine considered bv Borison (1979). . The plant is described bv a second order sYstem with two inouts (airflow and stockflow), two rep,ulated outputs (total pressure and stock level~ no free outouts and two additive disturbances. By resorting to normalized variables, the corresoonding model (I) is therefore characterized bv - 0.99
0. 0088
-0.81
- 0.77
1
j '
Bp =
I
1-0.0046
0.90
0.88
19.39
I I
~;PR = O.
The disturbance d is a zero - mean white noise, whe-
reas the r e ference signal v O is purely deterministic,specificallv 0=0
iJ :::
4,
o
o
o , i'J
o v =
[I
V = dia~
I:
o
o
0
0
0 0
i
ID
o
(j
o
o o
o
=
i'
I ' 10 .
02
I, 10.35
0 . 35 1 7.6
I
The considered regulator structure is consistent with the block diagram shown in Fig.2, where the dynamics of the local regulators are expressed in termS of z-trasfe r functions . Notice that the regulator structure is proper, so that any stabilizing tuning also guarantees asymptotic zero-error regulation for constant exogenous signals.
Since the plant is asymptotically stable and, for (1,2), ml(J)= - 0.96 and m2(J)=-96, the optimization algorithm can be initialized by exploiting the result slated in Theorem 3 . Specifically, it is possible to a ssume ql=02=Q 5=q6=q7=q8=0, whereas q , = -1. 04 and q,= 0.01.
~=
The performance of the control system has been evaluated through the index (6) with R=S=I , 2 T = d ia g (0,0, I , I , I , I ) • The tuning phase has supplied the following values for the parameters qis: 0 ) =-0.056, q 2=-0.004, q3= - 3.3, q_=0.035, Q5=0.057, q . =O. 11, q7=0.16, q , =0.89. The step responses of the optimal control system are shown in Figs.3 and 4. CONCLL"DgG RE'!ARKS A design procedure for discrete -ti me control svstems has been presented in this paper. The aim was getting a satisfactory dynamic behavior and a robust asymptotic zero - error regulation for con -
stant exogenous signals. The sug?ested technique is based on parameter optimization and consists of
t,,'o steps. The first is choosing the rep.ulator structure so as t o meet with information and complexity constraints. The second is tuning the regulator parameters by means o f an optimization algorithm .
Since this second step can be performed in an efficient way, it is very easy to compare the perfor-
mances which can be obtained by various tentative structures. The presented design procedure is susceptible of many extensions. Just to quote someof them, it is possible to handle explicit con trol ef-
J J63
fort constraints and to design control systems robust with respect to large plant parameters variations or reliable with respect to actuator and sensor failures. REFERENCES Anderson, B.D.O. (1982) Transfer function matrix description of decentralized fixed modes. IEEE Trans. Automat. Contr., AC-27, 11761182. Anderson, B.D.O., and D.J. Clements (1981). Algebraic characterization of fixed modes in decentralized control. Automatica, 17, 703712. Bartels, R.M., and G.W. Stewart (1972).Solutionof thematrix equation AX+XB=C. Commun. ACM, 15, 820-826. Boriso~ U.(1979). Self-tuning regulators for a class of multivariable systems. Automatica, 15, 209-215. Corfma~ J.P. , and A.S. Morse (1976). Decentralized control of linear multivariable systems. Automatica, 12, 479-495. Davison, E.J.(1976~ The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Automat. Contr., AC-21, 25-34. Daviso~. ,~T. Chang (1982). The design of decentralized controllers for the robust servomechanism problem using parameter optimization methods. Proc. American Control Conference, Ill, 905-909. Daviso~, and I. Ferguson (1981). The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods. IEEE Trans. Automat. Contr., AC-26, 93-100. Franci~A., and W.M. Wonham (1976). The internal model principle of contro l theory. Automatica, 12, 457-465. Guardabassi, G.,~. Locatelli, C. Maffezzoni,andN. Schiavoni (1979). Parameter optimization in decentralized process co ntrol: a unified setting for multivariable industrial regulator design. M.A. cuenod (Ed.), Proc. IFAC Symposium on Computer-Aided Design of Control Systems, Pergamon Press, Oxford,87-92. Guardabassi, G., A. Locatelli, C. Maffezzoni, and N. Schiavoni, (1982a). A parameter optimization approach to the computer-aided design of structurally constrained multivariable regulators. Control and Computers , 10, 3949. Guardabassi, G., A. Locatelli, C. Maffezzoni, and N. Schiavoni (1983a). Computer-aided design of structurally constrained multivariable re gu lators. Part I: problem statement, analysis, and solution. Proc. IEE- Part D, 130, 155-164. Guardabassi, G., A. Locatelli, C. Maffezzoni, and N. Schiavoni (1983b). Computer-aided desi gn of structurally constrained multivariable regulato rs. Part 11: applications. Proc. lEE - Part D, 130, 165-172. Guardabassi, G.,~ Locatelli, and N. Schiavoni (1982b). On the initialization problem in the parameter optimization of structurally constrained industrial regulators. Large Scale Systems, 3, 267-277. --Guardabassi, G., A. Locatelli, and N. Schiavoni (1984). Further results on the structu re of the robust decentralized regulator. These Proceedings. Kleinman, D.L., and P. Krishna Rao (1978). Extensions to the Bartels-Stewart algorithm for linear matrix equations. IEEE Trans. Automat. Contr., AC-23, 85-87. Locatelli, A., Y.I. Peng, F. Romeo, R. Scattolini, ~ . Schiavoni, and J. Xiao (1981). A computer -aided design technique for decentralized
J J 64
R. Scattolini and N. Schiavoni
process controllers. Proc. 8th IFAC Triennial World Congress, Pergamon Press, Oxford, Xl, 7-12. Locatelli, A. , and N. Schiavoni (1982). A computer -aided solution to the initialization problem in the parameter optimization of structurally constrained regulators. G. Leininger (Ed.) Proc. IFAC Symposium on Computer-Aided Design of Multivariable Technological Systems, Per gamon Press, Oxford, 377-387. Locatelli, A., and N. Schiavoni (1983). On the initialization problem in the design of structurally constrained regulators. Int. J. Control 38, 717-733. LocateDi, A., N. Schiavoni, and A. Tarantini (1977). Pole placement: role and choice of the underlying information pattern. Ric. Autom., 8, 107-126. Wang, S.H., and E.J. Davison (1973). On the stabilization of decentralized control systems. IEEE Trans. Autom. Contr., AC-18, 473-478.
I,
Fig. 2 - Block diagram of the regulator for the head-box of a paper machine.
Y, 1
APPENDIX: DEFINITION OF F, G, H, K, and U
,.'
In this appendix the explicit expression of matrices F, G, H, K, and U are reported. The dependence on q is omitted.
F ~
F12
F
F21
F22
F
0
0
F33
G 3
0
0
H 13
Kl
Y2 0
H22
H 23
H32
0
Hll H ~
QS
G l
F 11
H21 H31
13 G
23
..................
.,'
G 2
(,
K ~
10
20
30
10
20
30 ",
K2
."
40
....
"
,
0
40
"
-2 F
(, B
13= P
•
'{DRE(CIS-NpRCID)+DRSCIS+(DRFNpF+NR)CID)+MpCID;
F21~BRFCPF-BRECPR;
F22~AR;
F23~BRE(CIS- NpRC ID )+
+BRSCIS+(BRFNpF+~)CID; F33~AI; Gl~Bp(DRE(NIS
+
-4
"
o Fig. 3 - Responses to a unitary step of Yl'
-NpRNID)+DRSNIS+(DRFNpF+NR)NID)+MpNID; G2~ BRE ' .(NIS-NpRNID)+BRSNIS+(BRFNpF+MR)NID; G3~ MI ; Hll~- CpR; H13~CIS- NpRC ID ; H21~DRFCPF- DREC pR ; H22~CR;
"
H23~DRE(CIS- NpRCID)+DRFNpFCID+DRSCIS+
+NRC ID ; H31~ IIn
01'; H32~ I O
Iv l'; Kl~NIS-NpRNID;
Q02
K2~DRE(NIS- NpRNID)+DRFNpFNID+DRSNIS+ NRN ID ;
"
0
U~ diag(R, S, T) .
10
20
'"
....... 30
40
ACKNOWLEDGEMENTS This paper has been partially supported by MPI and Centro di Teoria dei Sistemi, CNR. The authors are grateful to Prof. A. Locatelli for many useful comments and to G. Lombardi, O. Marzorati and E. Rossi for developing the computer programs. d
Yl 1
......... ........
........................
as
y'
0r---~1~0----~2~0~--~3LO-----4~0-------
Fig. 1 - A proper
class of regulators.
o
Fig. 4 - Responses to a unitary step of Y2'
\l
{"I'\lll.!,lrl
'[
Huol,qJt",1.
11 11111..;,11 \ .
'Ilh
1'1."
I 111"1I111,d \\'>lld ( "II~lt'''' 1
A PARAMETER OPTIMIZATION APPROACH TO THE DESIGN OF STRUCTURALLY CONSTRAINED REGULATORS FOR DISCRETE-TIME SYSTEMS R. Scattolini and N. Schiavoni lh/Jllllllllllllu
tI,
F I' '''''"I(II. 1',,111,,1111"
rll
.\I dllll". 1' ,11':..:11 1.11<11I/lflll do \ '1111 1 ")l. ]U/)") .\ ld(ll/fI,
I llfh
Abstract. The problem dealt with in this paper i s t he design of a regula t or for a discre t e- time system. The re~ulator should supply a satisfacto r y dynamic behavior to the ove r all cont rol sys t em and guarantee robus t zero - e r ror regulation for constant exogenous si -
gnals. The here presen t ed technique is based on parameter op t im i za t i on and, being very flexible, is particularlY suited to cope with var i ous constra i n t s which may be imposed on the structure of the r egulator (e.g. , d ecentralizat i on of the overall regulation system , order of t he lucal devices) . The design problem is first s t a t ed as an optima l control prob l em . The n, its reformulation as a mathemat i cal prog r amming pr oblem is given and some computational rema r ks a r e presented . The effect i veness of the p r oposed technique i s ve r ified on an i l lustra t ive example . Keywords. Dece nt ralized control , parameter optimiza ti on , multivari able regulator theo r y , computer - aided design.
satisfying the info r matlon and comp l ex it y constraints . The second is t un i ng the r egu l a t o r pa r a meters , by op t imiz in g a n i ndex we i ght i n g t he dynamic behavio r of the cont r o l system , wh il e t a kin g
INTRODUCTlO~:
Discre t e -ti me systems , as opposed to continuous - ti me systems , are ~oing to pl ay an ever i ncreasing
role i n control theory and practice. This is due both to the nowadays available results of modern sys t em identifica t ion and to the widesp r ead diffu sion of digital con t rollers . The aim of this paper is t o con t r i bute to th e development of dis cre te - time system control t heory , by presenting an effective and flex i b l e des i gn technique.
into accoun t the ze r o- erro r r egu l a ti o n cons tr a int .
Specia l at t ent ion wi l l be pa i d , in t he f o ll owin g sections , t o the second s t e p . In f a c t, de spit e the problem of choosi ng a n app r opr i a t e st r uc t u r e ha s not vet been satisfactorily fo r ma li zed ( s ee, a nyhow , Locate ll i and Co l leagues , 1977) , th e expe rience of a sk i l l ed control e ng in ee r i s i n gene r a l sufficient to sug ge st the use of a l i mit e d numbe r o f tentative s t ructu r es. On th e con tr a r y , th e f ina l selection wit hin th i s se t would ent a il a n unb ea rable tr i al and error procedure , wh i ch ca n b e avo ided bv comparing the performances obta in e d when th e optimal tuni n gs of t he var i ous s tru c t u r es a r e us ed . Hence , an efficien t t uning algo r i t hm i s the bas is of an inte r active package which al l ows one t o d e termine the most app r op r ia t e so l u ti on among t he feasible ones.
The particu l ar problem dealt with consists uf synthesizing a li near regulator for a linear plant . The regula t or sho ul d supply a sat i sfacto r y dynamic behavior to th e overall control system, whil ee nsur ing r obust zero - e rror r egu l ation for constant e xo -
genous signals. The design pro ce du r e put forward here is based on parameter optimizatio n and presents a number of similarities with a technique which has already p r oved to be effective i n the desi gn of co ntinuous - time control svstems(Guardabassi and Colleagues, 1979 1982a , 1983a , 1983b; Davison and ferQ,uson , 1981; Loca t elli a nd Colleagues , 1981; Davison a nd Chang , 1982) . The main feature distinguishing it from the majority of other standard design methods is its cap ability to handle in a natural wav some frequentlY encounte r ed constra i n t s on the structure of the regulat o r . Among t hem are : (i) the requirement t hat the regulator be decentra li zed, that is , composed by various elements (local regulators) interconnect ed in such a way tha t , in general , any singl e i nput of the plant is made to depend on part only of all the informat i on available (outputs of the plant, disturbances, and set - poin t s); ( i i) an a priori partial specifica t ion of the functional relation ships the local regulators can establish bet~ ee n the i r inputs and outputs , e.g. , t he urder of the local regulators .
The layout of the paper is as fo l lows. Nex t sec tion is devo t ed to fo r mal l y state the pr oblem. Then , a prel i min a r Y a nalys i s on the r egu lat o r s tru c ture is carr i ed out i n the t hird sectio n . It s a im is t o ascertain whe t he r t he r e a r e r egu la t o r s , e ndowed with the structure a t hand , wh i c h r ende r t he con t rol sys t em asymptotica ll y s t ab l e a nd ma ke the steady - state error t o robus t ly vanish. In pa rticular , a class of structu r es , to be ca ll e d p r oper, is introduced , the use of which gr ea t ly simplifies the parameter opt i mi za t io n phase . Th e f orth s ec ti on presents the problem r eformu l at i on i n t e rms of a mathematical prog r amming problem as we ll a s s ome rema r ks on computat i onal as pects . fin a ll y , th e desi gn of a regulator fo r a hea d-box of a pap e r ma ch ine is deal t wi t h in t he f i fth sec ti on.
The overall design procedure consists of two main steps. The first is choosing a regu l ator stru c ture
1159
R. Scattolini and N. Schiavoni
1160
PROBLEM STATEMENT This section is devoted to introduce the problem dealt with in the rest of the paper. Assume that the plant under control is described by the linear, time-invariant system ( La)
CpRxp(k)+ NPRd(k)
(1.b)
CpFxp(k)+ NpFd(k),
(1. c)
Os c 0 be the set of all the q' s such that all the eingevalues of system (1),(3) lie within the unit circle, the tunings 0.(q) of interest are only those such that
(4)
q€ Q • S
Moreover, the tuning must guarantee that the mean value of the output YR asymptotically track the asymptotic mean value of the reference signal yO, in spite of the presence of the disturbance d, i.e.,
lim E{e(k) } where xp€ Rn, u€ Rm, and d€ R~ are the state,control, and disturbance vectors, respectively. Two different output vectors have been shown: yp€ RP is the vector of variables which should track a reference signal yO, whereas YF€ Rr is the vector of other measured variables to be possibly exploited for regulation purposes. Both the reference signal yO and the disturbance dare modeled as coloured noises with an asymptotically rational spectrum. HencP"they can be described by (2.a)
CISx I (k) + NISv(k)
(2.b)
= CIDx I (k) + NIDv(k)
(2.c)
=
d(k)
where xIS Ra is the state vector, AI is Hurwitz (i.e., its eigenvalues all lie within the unit circle) and v(') is a white noise with known mean value ;:;€ R ~ and intensity v, that is, v(·) "- WN(;:;, V).
(2.d)
As for the regulator, it is represented, as a whole, by the linear, time-invariant system X
u(k)
where E{'} is the expected value operator. In addition, it is required that the regulation constraint (5) be fulfilled in a robust way, that is: i) for all perturbations of the plant matrices Ap,Bp,Mp,CpR,NpR,C pF , and NpF ' which preserve, as for Ap,Bp,CPR,C Pf the asymptotic stability of system (1),(3); ii) for all perturbations of the filter matrices AI,MI,CIS,NIS,CID, and NID, which preserve, as for AI, the asymptotic stability of system (2); iii) for all perturbations of the mean value v of v(·). Note that constraint (5) is satisfied in a robust way if and only if the control system (1) ,(3), when subject to arbitrary constant (step) exogenous signals yO and d, exhibits an asymptotically vanishing error, in spite of sufficiently small perturbations of the plant matrices. As for the control system dynamic behavior, it will be evaluated through the value taken on by a linear combination of the covariances of the steady-state processes asymptotically approached by the error e, the control u and the state of both the plant and the regulator. Specifically,
+[2«k)-;o'
(3.a)
CRxR(k)+DREe(k)+DRFYF(k)+DRsyO(k)+
u
~
lim k-
x
~
Ixp '
x € RV, v being a given constant, and R
e(k) ~ yO(k) - YR(k)
+
T[!:(k)- ~ } ,
(6.a)
where (3.b)
where
(5)
J ~ lim E{e'(k)Re(kl+G(k)-G]' sG(k)-li] k-+ 00
R(k+ 1) = ARx R(k) + BREe (k) + BRFYF (k) + BRSYO (k) + + MRd(k)
0,
k-+ 00
(3.e)
is the system error. It is apparent that the regulator can incorporate, in general, both feedback and feedforward actions. In order to synthetically take into account the possibility that the regulator be made out of subsystems (e.g., standard PID regulators) interconnected among them, all the matrices appearing in eqs.(3) are assumed to be known co ntinuously differentiable functions of a design parameter vector q ranging into an admissible parameter set QCRP. For any q € 0, the ordered set of matrices 0.(q) ~ i AR(q) ,BRE(q) ,BRF(q) ,BRS(q) ,~(q) ,CR(q), DRE(q) ,DRF (q) ,DRS(q) ,NR(q) ) represents a particular tuning of the regulator. Of course, the set of all possible tunings is completely described by the couple (Q, 0 (')) ,which will henceforth be called structure of the regulator. The first property that the control systemis desired to have is asymptotic stability. Then, letting
x (:, lim k-
E{u(k)} , xR' I ' E{x(k) } ,
(6.b) (6.c)
(6.d)
whereas R,S, and T are symmetric positive semidefinite matrices. Note that the value of J mainly depends on the system behavior in the frequency band where the spectra of the processes asymptotically approached by yO and d are located. Hence, shaping these spectra (through an appropriate choice of the matrices appearing in eqs. (2» allows the designer to define the frequency band where the system behavior has to be particularly taken care of, and constitutes an indirect way of qualifying the system transients.
In conclusion, the design problem dealt with here can be synthetically stated as follows: Main Problem Find a tuning 0.( q) of a regulator with given structure (0, 0. ('» such that, for system (1)-(3): (i) the stability constraint (4) is met;
11 61
Design of Structurally Constrained Regulators
constr a int (5) is me t in a robust way.
( ii) the re gulation constraint (5) is met in a robust way; (iii) the performance index (6) is minimized.
REGULATION
fu~D
STABI LITY CONSTRAINTS
Thi s sect i on is d evo t ed t o dis c us s th e feasibi l ity of the Main Problem. Conditions will be given under which ther e exis t regulators e ndowed with a stipulated structure such that the stability co nstraint (4) i s me t an d the re gu l a ti on constraint (5) i s met in a robust wa y . As fo r the r egu l atio n constraint, th e Internal Model Principle (see Francis and Wonham, 19 76 , for th e continuous -ti me case) supplies a comp l e t e characterization of al l re gula tors which mak e it satisfied in a robust way. For th e case at hand, th e tunin g must be such that th e co rrespondin g system (3) in co rp orates a number of "appropriat e integral actions" not l ess than the numb e r ot th e plant r egu lat ed var i ab l es . Then, if system (I), (3) is asymptotically s table, th e r egulati on const r a int (5) is verified in a robust way . This va gue s tat ement of th e Internal Model Principl e is specified in Theorem 1 below (see Guardabassi and Col l e a gues, 19 83a , fo r the conti nuous-time ca se). Hereafter, for any positive i nt ege r a , l a deno t es th e a Xa id e ntit y ma tri x. Th eo r em 1 Th e tuning D(q) , q€ OS, guarant ees that th e re gula ti on constra int (5) is met in a robust way i f and on l y if ther e ex i s ts a nonsingular matrix Le VxV € R such tha t
o I
P
o
Thi s th eo rem, th ough fa irl y obvious, est ablis hes important fact: wh e ne ver the regulator s tru cture is proper, th e robu s t re gulation co nstraint (5) does not need to be exp li c it e l y tak en into account.
0
Con si de r now t he c l ass of structures depicted in Fig.l, where Vl", simply consi s t s of p para l le l dis c r ete inte g r ato r s (i . e., th e z-transfer - function ma trix of ~ is diagonal and its nonzero e ntri es are all equal t o (z -I ) -I ), whereas ~1 (q) and ~2(q) a r e lin ea r, time-invariant sys t ems of given o rder. All the s tructyr es of this class a r e pr oper . In fact, l et tin g xk , i = 0 ,1,2, be the state vector of subsystem :Ri (g) , i = 0, 1, 2, re s pect iv ely , and xR -_ !x l' x 0 ' R R 1 holds with L= I .
x 2 "' , then R
Th eo r em
u
It is wo rth noti c in g that the c l assof proper structur es of Fig.l is very lar ge . In fact , the subsy stems OC 1 (q) and ~ 2(q) can be dynamical or non dynamical , centralized or decen tr alized , and pa r a me tri zed at will . Then, any possible a priori in forma ti o n as we ll as complexity con s traints ca n easily be handl ed , th e only s i gni f i cant constraint impli cit in th e choice of this c la ss of s tru c tur es being that th e o rd er of the subsystems " preceding" and "f o ll ow in g " th e blocK of int egrato rs is ind e pendent of th e specif i c val ue of q. As for the possibility of choosing a nond ynamical ~1 (q) , see Gua rd abass i and Colleagues (1983a , 1984). Suppose, in parti c ul a r, that t he s tructure o f th e r egula t o r is assigned by means of a block di ag r am whi ch des cr ib es how various single -in put single output loca l regulators are int erconn ec ted among them. In this case , a simp l e a l gori thm a ll ows one to asce rtain wh e th e r it b elo ngs to the c lass of Fig.1, so that it is prope r. In fact, it is suf ficien t to verify the two followin g points: i) there ex i st (at l eas t) p blocks having, for a ll q€ Q, transf e r-fun c tion matrix (Z -I ) -I Ip between th e ir inputs and outputs; ii\ the tr an sfer - function matr ix be tween !YF' yo' d' t ' a nd th e outpu ts of the above blocks is zero for all q € O.
0
In view of th e above discussion , it is reasonable to r es tri c t attention t o prop e r regulato r structur es . This is what will be done from now on. The prob l em of establishing whether there exist regulators endowed with a given structure which render system (1), (3 ) asymtotically stab l e i s,
o o
o
in the gene ral case , unso l ved . The absence of unstable fixed modes (see , e.g. , Wa ng and Davi son,
Ch eck in g wh ether , for a consid e r ed q € OS' the con dit i on of Theorem 1 is satisfied i s an all but trivi al task f r om a numerica l point of v i ew . Fort una t ely e nough , this di ff i cul t y ca n be overcome by approp riat ely se l ect ing the regulator structure along th e li nes described be l ow (Guardabassi and Colleagues , 1979, 198 3a ). Let (C ,A,B) denote th e lin ear , time -invariant system th e ou tput, dynamic a nd input matrices of which are C, A, and B, r espectivel y . Then, the structure (Q , D( ' )) is said t o be pr ope r if ma tri x L c all ed for in Theorem 1 exis t s for any q € 0*, Q* being th e se t of a ll qe € Q such th a t the uncon tr ol l ab l e and/ or unobservab l e part of system (CR(q), AR(q), : BRE(q) BRF(q) : ) is asymp toti ca ll y s t able.
1973 ; Corfmat and Morse , 1976; Locatelli and Col l eagues 1977; Anderson and Clements , 1981; Anderson , 1982) i s on ly a necessary cond ition for Os to be nonempty , because a structure (Q, D(')) , besides specifying the info rmation flow within the regu lator a l so bounds the regulator comp l ex ity.For discrete time s y stems , suff i c i e nt condit i ons can be given when the p lant i s a symp totically stab l e (for con tinuous - time sys tem s , see Davison and Ferguson, 1981 ;Guardabassi and Colleagues,1982b;Locatelli and Schiavoni ,1 982, 1983 ,whe re al so some cases of unstable plants a r e cons idered). Let the number o f regulated va riab l es egual the number o f control variables , L e . , m=p.Mo r eove r assume that the r egulator is purely
The definitions themselves o f prope r structure and Q* i mp l y that Q*cQS ' Then , the following re sult can immediately be proved .
i . e ., v=p . Then it ca n be described by x (k) + e(k ) x (k+l) R R u( k ) CR (g ) xR( k)
Th eo r em 2
Th e e xtreme case of completely decen tralize d in formation f l ow will be considered . Let (8) C (g) = diag (g l' g2 " " , gp ) R
If th e s tru c tur e (Q, 0 ( ' )) is prope r, th en any tuning ~ (q), q € QS ' guarantees that the re gulation
integral,which implies that its order is minimum,
(7.a) (7.b)
R. Scattolini and N. Schiavoni
1162
and assume that Q contains an open neighborhood of the ori g in of RP , where P = p. Furthermore, let (P t, ( 1,2, ... ,p ) , {) be any permutation (i , i , ... ,i'p) of the set (j' and, for any 1 2 j € 6',
B. (-;J)
J C. (
J
tr U r . +tr Z A. +n' n. 1 1 1 where Z and n are the unique solutions of the equations Z- F'(q)Z F(q)= H'(q)U H(q)
!bi
~
~
1
1 c.
b. '2
b.
c.
c.
'2
'1
1 .
I,
(10.a)
and
J
J}n+ v+o- F'(q2!n =H'(q)U H(q)~- F'(q)ZG(q)v,(10.b)
I 1.
whereas
J
where b. and c! are the i-th column of B and the i-th ro~ of Cp~, respectively. The folloeing result then holds.
~C~H(q)+H(q)[!- ~C~ aH'(q) aq.
+
1
Theorem 3 If
i) matrix Ap is Hurwitz; ii) there exists a permutation
J of
~
m]" (~) t, det{C . (J) (I-A )-l B. (J»" 0, = J P J
such that j €
then there exists £: > 0 such that, for any e(o,£ ) ,s ystem (1),(7),(8) with j €
where
6'; £
aG' (q)
aq:-
+
1
€
lP,
asymptotically stable. and matrices F(q), G(q), H(q), K(q), U, P, and ~ are the same as in Theorem 5. c
PROBLEM REFORMULATION AND SOLUTION In this section, restricting attention to proper regulation structures, it is shown how the Main Problem can be reformulated as a mathematical programmin g problem, the peculiar features of which make it fairly easy to solve. The reformulation of the Main Problem is presented in Theorem 4 below. Theore 4 If the structure of the regulator is proper, the Main Problem can equivalentl y be restated as follows: min
tr U{ H(q)[!- C~~H'(q)+K(q)VK'(q) } ,
q € Q S
where P and
~
are the unique solutions of the
equations
and
Un +v +a -
The evaluation at any q € Q of both the perfo~ mance index and its gradieng essentially calls for solving the four equations (9) and (10). The asymptotic stability of s ystem (1)-(3) guarantees that they always admit one and only one solution. An efficient solution algorithm for the Liapunov equations (9.a),(10.a) can be obtained by suitable modifying the algorithm proposed by Bartels and Stewart (1972), also taking into account the suggestions of Kleinman and Krishna Rao (1978). The main step consists of carryin g matrix F(q) into its real Schur canonical form FS(q) by means of an orthogonal transformation. The computations are simplified by takin g into account the fact that F(q) is block-trian gular. Once FS(q) is known, also solving eqs. (9.b),(10.b) is strai ghtforward. Moreover, checking whether q € Os is immediate, since FS exhibits the eingevalues of the control system on the (scalar or bidimensiona l) diagonal blocks. Among the various iterative te c hniqu e s which can be adopted to solve the mathematical programmin g problem stated in Theorem 4, particularl y efficient is the Fletcher and Powell conju gate gradient algorithm. Since it directl y applies to unconstrained problems only, it can be used when Q=RP and must be slightl y modified in order to be sure that, during the optimization, q remains in
F(ql]C
(9.b)
and matrices F(q) ,G(q) ,H(q) ,K(q), and U are explicit functions of the data, the expressions of which are reported in the Appendix . C Moreover, the gradient of the performance index J with respect to the design parameter q can also be evaluated in an easy way at any q € OS' Theorem 5 At any q€ QS' the partial derivative with respect to the i-th parameter qi of the performance index J is given by
OS' Apart from very critical cases, the meanin g itself of the performance index actuall y guarantees that q moves away from the boundary of QS' However, if the step size of the algorithm is exceedingly large, during the opti mization it might happen that stabilit y is lost and yet the performance index, as evaluated in the mathematical programming problem of Theor em 4, is improved. This improvement is clearly absolutel y meaningless. Stab1lity, and henc e significance of th e evaluation of the performance index throu gh the equations of the mathematical programmin g problem of Theorem 4, is recovered in this case by simpl y reducing the step size of the al gorithm. Of course, the algorithm must be initialized with a qe € QS' For that, Theorem 3 can help.
Design of Structurally Constrained Regulators
A~
ILLUSTRATIVE EXA"PLE
The
desi~n
procedure described in the preceding
sections is now applied to svnthesize a regulator
for the head-box of a paper machine considered bv Borison (1979). . The plant is described bv a second order sYstem with two inouts (airflow and stockflow), two rep,ulated outputs (total pressure and stock level~ no free outouts and two additive disturbances. By resorting to normalized variables, the corresoonding model (I) is therefore characterized bv - 0.99
0. 0088
-0.81
- 0.77
1
j '
Bp =
I
1-0.0046
0.90
0.88
19.39
I I
~;PR = O.
The disturbance d is a zero - mean white noise, whe-
reas the r e ference signal v O is purely deterministic,specificallv 0=0
iJ :::
4,
o
o
o , i'J
o v =
[I
V = dia~
I:
o
o
0
0
0 0
i
ID
o
(j
o
o o
o
=
i'
I ' 10 .
02
I, 10.35
0 . 35 1 7.6
I
The considered regulator structure is consistent with the block diagram shown in Fig.2, where the dynamics of the local regulators are expressed in termS of z-trasfe r functions . Notice that the regulator structure is proper, so that any stabilizing tuning also guarantees asymptotic zero-error regulation for constant exogenous signals.
Since the plant is asymptotically stable and, for (1,2), ml(J)= - 0.96 and m2(J)=-96, the optimization algorithm can be initialized by exploiting the result slated in Theorem 3 . Specifically, it is possible to a ssume ql=02=Q 5=q6=q7=q8=0, whereas q , = -1. 04 and q,= 0.01.
~=
The performance of the control system has been evaluated through the index (6) with R=S=I , 2 T = d ia g (0,0, I , I , I , I ) • The tuning phase has supplied the following values for the parameters qis: 0 ) =-0.056, q 2=-0.004, q3= - 3.3, q_=0.035, Q5=0.057, q . =O. 11, q7=0.16, q , =0.89. The step responses of the optimal control system are shown in Figs.3 and 4. CONCLL"DgG RE'!ARKS A design procedure for discrete -ti me control svstems has been presented in this paper. The aim was getting a satisfactory dynamic behavior and a robust asymptotic zero - error regulation for con -
stant exogenous signals. The sug?ested technique is based on parameter optimization and consists of
t,,'o steps. The first is choosing the rep.ulator structure so as t o meet with information and complexity constraints. The second is tuning the regulator parameters by means o f an optimization algorithm .
Since this second step can be performed in an efficient way, it is very easy to compare the perfor-
mances which can be obtained by various tentative structures. The presented design procedure is susceptible of many extensions. Just to quote someof them, it is possible to handle explicit con trol ef-
J J63
fort constraints and to design control systems robust with respect to large plant parameters variations or reliable with respect to actuator and sensor failures. REFERENCES Anderson, B.D.O. (1982) Transfer function matrix description of decentralized fixed modes. IEEE Trans. Automat. Contr., AC-27, 11761182. Anderson, B.D.O., and D.J. Clements (1981). Algebraic characterization of fixed modes in decentralized control. Automatica, 17, 703712. Bartels, R.M., and G.W. Stewart (1972).Solutionof thematrix equation AX+XB=C. Commun. ACM, 15, 820-826. Boriso~ U.(1979). Self-tuning regulators for a class of multivariable systems. Automatica, 15, 209-215. Corfma~ J.P. , and A.S. Morse (1976). Decentralized control of linear multivariable systems. Automatica, 12, 479-495. Davison, E.J.(1976~ The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Automat. Contr., AC-21, 25-34. Daviso~. ,~T. Chang (1982). The design of decentralized controllers for the robust servomechanism problem using parameter optimization methods. Proc. American Control Conference, Ill, 905-909. Daviso~, and I. Ferguson (1981). The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods. IEEE Trans. Automat. Contr., AC-26, 93-100. Franci~A., and W.M. Wonham (1976). The internal model principle of contro l theory. Automatica, 12, 457-465. Guardabassi, G.,~. Locatelli, C. Maffezzoni,andN. Schiavoni (1979). Parameter optimization in decentralized process co ntrol: a unified setting for multivariable industrial regulator design. M.A. cuenod (Ed.), Proc. IFAC Symposium on Computer-Aided Design of Control Systems, Pergamon Press, Oxford,87-92. Guardabassi, G., A. Locatelli, C. Maffezzoni, and N. Schiavoni, (1982a). A parameter optimization approach to the computer-aided design of structurally constrained multivariable regulators. Control and Computers , 10, 3949. Guardabassi, G., A. Locatelli, C. Maffezzoni, and N. Schiavoni (1983a). Computer-aided design of structurally constrained multivariable re gu lators. Part I: problem statement, analysis, and solution. Proc. IEE- Part D, 130, 155-164. Guardabassi, G., A. Locatelli, C. Maffezzoni, and N. Schiavoni (1983b). Computer-aided desi gn of structurally constrained multivariable regulato rs. Part 11: applications. Proc. lEE - Part D, 130, 165-172. Guardabassi, G.,~ Locatelli, and N. Schiavoni (1982b). On the initialization problem in the parameter optimization of structurally constrained industrial regulators. Large Scale Systems, 3, 267-277. --Guardabassi, G., A. Locatelli, and N. Schiavoni (1984). Further results on the structu re of the robust decentralized regulator. These Proceedings. Kleinman, D.L., and P. Krishna Rao (1978). Extensions to the Bartels-Stewart algorithm for linear matrix equations. IEEE Trans. Automat. Contr., AC-23, 85-87. Locatelli, A., Y.I. Peng, F. Romeo, R. Scattolini, ~ . Schiavoni, and J. Xiao (1981). A computer -aided design technique for decentralized
J J 64
R. Scattolini and N. Schiavoni
process controllers. Proc. 8th IFAC Triennial World Congress, Pergamon Press, Oxford, Xl, 7-12. Locatelli, A. , and N. Schiavoni (1982). A computer -aided solution to the initialization problem in the parameter optimization of structurally constrained regulators. G. Leininger (Ed.) Proc. IFAC Symposium on Computer-Aided Design of Multivariable Technological Systems, Per gamon Press, Oxford, 377-387. Locatelli, A., and N. Schiavoni (1983). On the initialization problem in the design of structurally constrained regulators. Int. J. Control 38, 717-733. LocateDi, A., N. Schiavoni, and A. Tarantini (1977). Pole placement: role and choice of the underlying information pattern. Ric. Autom., 8, 107-126. Wang, S.H., and E.J. Davison (1973). On the stabilization of decentralized control systems. IEEE Trans. Autom. Contr., AC-18, 473-478.
I,
Fig. 2 - Block diagram of the regulator for the head-box of a paper machine.
Y, 1
APPENDIX: DEFINITION OF F, G, H, K, and U
,.'
In this appendix the explicit expression of matrices F, G, H, K, and U are reported. The dependence on q is omitted.
F ~
F12
F
F21
F22
F
0
0
F33
G 3
0
0
H 13
Kl
Y2 0
H22
H 23
H32
0
Hll H ~
QS
G l
F 11
H21 H31
13 G
23
..................
.,'
G 2
(,
K ~
10
20
30
10
20
30 ",
K2
."
40
....
"
,
0
40
"
-2 F
(, B
13= P
•
'{DRE(CIS-NpRCID)+DRSCIS+(DRFNpF+NR)CID)+MpCID;
F21~BRFCPF-BRECPR;
F22~AR;
F23~BRE(CIS- NpRC ID )+
+BRSCIS+(BRFNpF+~)CID; F33~AI; Gl~Bp(DRE(NIS
+
-4
"
o Fig. 3 - Responses to a unitary step of Yl'
-NpRNID)+DRSNIS+(DRFNpF+NR)NID)+MpNID; G2~ BRE ' .(NIS-NpRNID)+BRSNIS+(BRFNpF+MR)NID; G3~ MI ; Hll~- CpR; H13~CIS- NpRC ID ; H21~DRFCPF- DREC pR ; H22~CR;
"
H23~DRE(CIS- NpRCID)+DRFNpFCID+DRSCIS+
+NRC ID ; H31~ IIn
01'; H32~ I O
Iv l'; Kl~NIS-NpRNID;
Q02
K2~DRE(NIS- NpRNID)+DRFNpFNID+DRSNIS+ NRN ID ;
"
0
U~ diag(R, S, T) .
10
20
'"
....... 30
40
ACKNOWLEDGEMENTS This paper has been partially supported by MPI and Centro di Teoria dei Sistemi, CNR. The authors are grateful to Prof. A. Locatelli for many useful comments and to G. Lombardi, O. Marzorati and E. Rossi for developing the computer programs. d
Yl 1
......... ........
........................
as
y'
0r---~1~0----~2~0~--~3LO-----4~0-------
Fig. 1 - A proper
class of regulators.
o
Fig. 4 - Responses to a unitary step of Y2'