An Algorithm for Design of Low Order Dynamic Compensators for Large-Scale Systems

Copyright © IFAC Software for Computer Control Madrid , Spain 1982

COMPUTER AIDED DESIGN OF DIGITAL COMPUTER CONTROL SYSTEMS II

AN ALGORITHM FOR DESIGN OF LOW ORDER DYNAMIC COMPENSATORS FOR LARGE-SCALE SYSTEMS M. Rakic* and Dj. Petkovski** *Faculty of Electrical Engineering, Belgrade, Yugoslavia * *Faculty of Technical Scienc es, Novi Sad, Yugoslam'a

Abstract. A design-oriented methodology is presented for the construction of decentralized low order dynamic compensators for large-scale systems. Computationally, the method is very feasible. The proposed algorithm for design of low order dynamic compensators involves only the calculation of pseudo-inverse matrices. In other words, the algorithm proposed is very effective in solving high dimensional control problems and leads to easy implementation in practice. It is shown that the design procedure is oriented towards microcomputer implementation, Control system does not require complex signal processing and therefore, it can be easily implemented on a microcomputer. The practicality and good performance of control system developed, using this approach, is demonstrated on the mathematical model of a plate absorption column. Keywords . Decentralized control, dynamic compensators, disturbances compensation, computer aided design, absorption column. I NTRODUCTI ON

policies. A realistic control scheme in such cases involves the use of decentralized low order dynamic compensators where the integral action is explicitly incorporated.

Optimal control has not provided a means of designing simple dynamic compensators for high dimensional linear plants. To date, most attempts to determine optimum, reduced order compensators gains (Levin, Johnson, Athans, 1971; Newmann, 1969; Fortmann, Williamson, 1972; Roman, Bullock, 1975) have led to highly nonlinear algebraic relations, which are difficult to solve. An exception is the minimal-order observer-based compensator (Luenberger, 1971; Rom, Sarachik, 1973; Miller, 1973), but its dimension (number of plant states less number of outputs) is too high to be useful in many applications. Furthermore, the application of these compensators is handled in a centralized way. Such designs have typically been implemented on large computers, so these techniques do not seem to be practical when applied to large-scale systems. A major difficulty in the implementation of centralized feedback control for large-scale systems arises from the fact that the subsystems are often rather widely distributed in space. In large-scale systems consisting of several coupled subsystems, such as power systems, technological processes, etc., there may be a desire to control the system with local feedbacks on different subsystems, eventually with the addition of a small number of interconnections. Therefore, the increased current trends are towards control applications including small-scale computers. These facts justify the demands for simpler control

This paper presents a new algorithm for design of low order dynamic compensators for large-scale linear systems. The result of this method is a solution which enables to control the system by a set of dynamic compensators each having different information and control variables . The optimal state dynamic compensator solution is taken as the starting point. This solution can not be implemented since only the measured outputs are available and construction of high order dynamic compensator is to be avoided, if possible . The step taken is to fit this control into another "similar" control with a predefined decentralized structure. The idea behind this fit is to minimize appropriate selected matrix norm. Computationally, the method is very feasible. The proposed algorithm for design of decentralized low order dynamic compensators involves only the calculation of pseudoinverse matrices. In other words, the algorithm proposed is very effective in solving high-dimensional control problems and leads to easy implementation in practice. The results are employed for the computer-aided design of practical decentralized dynamic compensators for control of large-scale systems. The practicality and good performance of control systems developed using this approach, is demonstrated on the mathematical 203

M. Rakic and Dj. Petkovski

204

model of a plate absorption colum~. The.example indicates that propose9 al~or1thm.g1~es a control configuration wh1ch 1S rea11st1c to apply and simple to implement.

'V

PC 0T A = [A

o

'V

S

PROBLEM FORMULATION It is assumed that the process to be controlled can be adequately represented in the region of normal operation, by the controllable linear state model ( 1) x(t)=Ax(t)+Bu(t)+Sw(t), x(to)=xo y(t) Cx(t) (2) y( t) = Dx( t)

and

x(t) is the nxl state vector, u(t) is the 1xl control vector, w is the sxl vector of unmeasurable constant or slowly varying input disturbances, y (t) is the pxl vector of measurable outputs, y (t) is the qxl vector of the measurable outputs, formed as linear combinations of states, required to have zero steady-state value. A,B,C,D and S are constant matrices of appropriate dimensions, and q::p:: l ::n; s::n.

T

It is desire to design a simple decentralized dynamic compensator acting directly on ~he outputs, which explicitly incorporates 1ntegral actions, to compensate the effect of constant or slowly varying disturbances w(t), i.e. to eliminate the steady-state offset in selected output variables. It is assumed that each element ui(t) of u(t), i=l,2, . . , 1 and that each element wi (t) of w(t), i=l,2, . . ,r is restricted to be a function of certain specified outputs: u( t )=G~11 C. x( t )+G ~ ~ vi (t) +G ~ 31,zi ( t ), i =1,2 , . . ,1 i 1 1 11 1

P

0

~]

[~ ~J ; 0 I 0

'V

B

[~ 1 o

[i'

(10)

,0 ]

~2' .~

[1:]

C

o "P r

[i'

o .. 0 ~2' . ~

J

o .. Tr

; 0

~

[::]

(11 )

(12 )

The block diagram of the open-loop augmented system is shown in Fig . 1. It is easily verified that the local as well as hierarchical types of control systems are included in this formulation. Due to increasing complexity of modern technological process the local as well as hierarchical control'schemes, have become an essential requirement in the design of large-scale systems . In many practical situations, this requirement is imposed by bandwidth limitations on communication channals, or by limitations on the effective bandwidth of computer interface. In the next section the design of the centralized feedback law will be accomplished with an eye towards decentralization. COMPENSATOR WITH COMPLETE STATE FEEDBACK

(4)

w.1(t )=G~11 C.1x( t )+G~~V i (t )+G?~z 11 11 i (t), i =1,2, .. ,1 (5 )

where vi(t), i=1,2, .. ,1, is the subvector of the compensator state v(t), v s~ Vi(t)=PiCi X(t)+TiVi(t)+wi(t); (6) vi (to)=v oi ' i=1,2, . . ,r and where zi(t) is the subvector of the additional state vector z(t), zsRq:

zi (t)=

J Dix(t)dt+Z~. zi (to)=z~, to

i=l ,2, .. ,q (7)

The (n+r+q) order state model of the system augmented by the r compensator states and the q integral states is: ~

'V

'V

'V

'V 'V 'V ( x(t)=Ax(t)+Bu(t)+Sw(t); x to ) =x'V o

where X= [xvz]

T

, u

(8)

(9 )

In this section we consider the multivariable system (1) with the control vectors u(t) and w(t) which are constrained to have the following form: u(t)=Gllx(t)+G12v(t)+G13z(t)

(13 )

w(t)=G21x(t)+G22v(t)+G23z(t)

(14 )

where v, v s~, is the compensator state, v(t) = PCx(t) + Tv(t) + w(t) z, zsR q , is the additional state vector

(15 )

t

z(t) = J Dx(t)dt + Zo

(16)

to and where the matrices P,C,T and 0 are defined by (11) and (12). r, wilT are It the variables 'VX= [x v z] T and '"u= lU J defined such that the origin represents the normal steady-state operating point of the unforsed system, then the optimal control

An Algorit hm for Design of Low Order Dynamic Compen sators

problem for the augmented system is that of determining into the class of continuous functions , a control ~(t) which minimizes the functio nal: J =

f (~TQX + ~TR~)dt

( 17)

205

associa ted with the reduced, decent ralized control law. Rewrite eqs . (4) and (5) as: ~1 ~

(t), u*1·(t) = G.cJ* 1 1

i=1,2, .. , t

(24)

wilt) = GiCi~*(t),

i=1,2, .. ,r

(25)

~2 ~

to (26 )

where 0:

(27)

(18 )

and : is symmetric nonnegative defini te matrix , and R R1 E R( t xt ) (19 ) R2 E R(qxq) is symmetric positiv e defini te matrix. It is well known (Anderson , 1100re, 127.1,), under the assumption that the pair (A,B) is completely contro llable (Porte r, 1971), there exists a unique optimal control law ~

~

u(t) = Gx(t);

-1 ~T

(20)

G = -R B K

and K is symmetric, positiv e defini te matrix which is the solutio n of Riccati equation 1~ ~ ~ ~ (21) KA + ATK - KBTR- BTK + Q = 0

~1· · ~ .. ~

.

.

.

1

~ .. ~ i .. ~

C.= 1

.

.

.

o . .0

(28)

i=1,2, . . , t for eq.(4) , i=1,2, .. ,r for eq.(5) (29)

2 .. I i

matrices The matricesI~1 and I~1 are diagonal . where the i-th diagonal element i ~i=1, j=1,2 if the corresponding ~tate is included in the control vector and i ~11. =O if the corresponding state is not included in the control vector . Now the augmented, decent ralized closed-loop system can be described by:

such that ~ (t) minimizes (17). The matri x G can be now partiti oned as (22)

where the matri x B is partiti oned as ~

~

~

~

~

B=[B1···B t Bt +1···B t +r] so that the submatrices Gij , i=1,2, ; j=1,2,3 , have approp riate dimensions according to (13) and (14). The block diagram of this centra lized, optimal, complet state feedback system, with dynamic compensator in which the integral action is explic itly incorp orated , is shown in Fig . 2. This system is guverned by the equation: cC

~

~

x = (A + BG) ~ (t) + Sw(t),

~

~

x(t o ) = Xo

(23)

The optimal dynamic compensator will be used as the startin g point in the deriva tion of the output constrained dynamic compensator with decent ralized structu re. The methodology described in the following section provides an organized way for selecti ng the parameters of the decent ralized dynamic compensator. DECENTRALIZED DYNAMIC COMPENSATOR In order to simpli fy the notatio n the star (*) will be used to indica te proper ties

("31 )

and BH J. =

[r ..] , .JJ

j=1,2, .. ,r

(32)

o When there are some states which are not feedback into the control system, it is impossib le to obtain the optimal feedback control which agrees with the optimal one for state feedback system. Then the traject ory of the decent ralized control system becomes differe nt from that of the state feedback one. Thus, the objecti ve is to determine the control laws (4) and (5) so that the transie nt responses of the systems (23) and (30) are "near" each other. One obvious possibil ity is to define the traject ory error vector as: (33 ) e(t) = ~(t) - ~*(t), e(t o ) = 0 that is:

M. Rakic and Dj. Petkovski

206

'"

e(t)=(A+

r '"

£ '"

I B.G~+ I B .G~)e(t)+d(t) i=1 1 1 i=1 £+1 1

(34)

where d(t)=

r '"

'" '"

I B. (G~-G~C. )~*(t)+ . I-1 B1 +1. (G~-G?C. )~*(t) . -1 1 1 1 1 1 1 1 11(35) £ '"

'" '"

and where the matrix G G11 G12 · .G 12 G13 · .G 13 1q 11 11 1r 11 :

G=

G11 G12 G12 £1 £1" £r 21 12 G21 11 G11 ·· G1r

13 13 G£1" G£q 13 13 G11 · .G 1q

(36 )

:

:

G21 r1

21 .. G21 G23 .. G23 Gr2 rq rr r1

is a partition of the state feedback matrix (22), compatible with the partition of the control vectors u*(t) (24) and w*(t) (25). It is obvious that if d(t)=O then e(t)=O for all t >O, since e(t 0 )=0. However, this case is excluded and therefore approximations must be made. There is a variety of quantities one can choose to minimize the influence of d(t). For instance, an obvious possibility is to choose u*(t) and w*(t) i.e. '"GiC 1'" and "'2'" GiC i , so as to minimize i £ '"

1 "'1'"

inf~1 11

L B.(G .-G.C.)x* 11 i=1 1 1 1 1

inf~2 11

I B . (G.-G.C · )~*I I i=1 1+1 1 1 1

i

i

r '"

2 "'2'"

(37)

(38)

where 11 ' 11 denotes the ordi nary encl i di an quadratic norm. It can easily be shown that the problem reduces to determine . '" 1 "'1'" . '" 2 "'2'" m1nG1 .1 1I G.-G.C 1 1 1·II , m1nG21 .1 IG.1-G,C1· 1 11

(39)

For this condition the decentralized feedback matrix is '"G.1=G.C. 1"'T '" "'T -1 , i=1,2, .. , £ (40) 1 1 1(C 1.C.) 1 "'2 2"'T '" "'T -1 (41) Gi=GiC i (CiC i ) , i=1,2, .. ,r Notice that the suggested approach is quite general and is not crucially dependent on any particular method for a state feedback compensator design. Besides the linear quadratic method, considered in this paper, other methods (for example pole assignment method) can be rationally and systematically involved in order to achieve good transient response characteristics of the starting feedback system. COMPUTATIONAL CONSIDERATION The proposed method provides an organized

way for selecting the parameters of the decentralized dynamic compensators. Computationally, the method is very feasible, it involves only calculation of pseudo-inverse matrices. Numerical algorithms exist to find such inverses. In other words, the algorithm proposed is very effective in solving high dimensional control problems. Further, the control system does not require complex signal processing and therefore, it can be easily implemented on microcomputers. It is obvious that the proposed method is far from being over specified. The solution depends on the values of the matrices Q and R in performance index and especially on the dimensions, structures and values of the matrices P and T. The dimensions, structures and values of the matrices P and T can be considered as design parameters, which can be employed for the computer-aided design of practical decentralized dynamic compensatores for control of large-scale systems. The dimension r of the dynamic compensator is an additional free parameter, to be specified during the design procedure . The case r=O corresponds to the proportional plus integral solution, while the case r=n-p corresponds to a compensator of the same dimension as a minimal order observer, with addition of integral action. Of course, lower baunds on the allowed values of r are implicit in design specifications of desired properties, as stability for example. An outline of the proposed design procedure is as fo 11 ows: (a) determine the number of the integrators q according to the dimension of the measurable outputs, formed as linear combinations of states, required to have zero steady-state val ues; (b) initialize r>1; (c) choose the matrices P and T, (d) compute the decentralized, dynamic compensator. If the resultant decentralized closed-loop system has unsatisfactory properties (e.g. the system is unstable or it has unacceptable transient response characteristics), let r=r+1 and repeat step (c), otherwise let the compensator dimension be r. NUMERICAL EXAMPLE To illustrate the performance of the derived decentralized dynamic compensator, model of plate absorption column described by six linearized differential equations with two inputs, is taken as an example. A detailed description of this process, as well as all parameters and their values can be found in Lapidus, E.Shapiro, S.Shapiro and Stillman (1961). The physical system considered consists of a six plate absorption column with changes permitted in the gas and liquid feed compositions. Here, the problem of the column control is otaining a constant concentration of the outlet of the column, subject to constant or slowly varying disturbances input.

An Algorit hm for Design of Low Order Dynamic Compen sators

From the technic al and economical point of view, optimization of column using complet feedback, i.e. measuring of all plate concentration s, is not a practic al solutio n. A realistic control scheme can be obtained by the applic ation of decent ralized dynamic compensator. Three types of dynamic compensators are considered (a) state feedback compensator; (b ) output feedback compensator with C1

C

oJ

[1 00 00 00 00 1 0

J,

P1

P

[~~

D1

D

[ 0 0 0 0 0 1] ;

T1

~ ~

= T = [- - ]

(c) decent ralized compensator C1 = [1 0 0 0 0

oJ,

P1=1, T1=-1, D1 =0

C2 = D2 = @0 0 0 0], P2=1, T2=-1. Numerical values for matrice S and input disturbance ware: w = 0,1 S = [ 0 0 0 0 0 1J T, For matrices in performance index Q=I and R = 1/ 91 are chosen. Fig . 3 gives the time response for the output variab le of the plant Y2(t)= x6 (t), of the open-loop and the closed -loop system with dynamic compensators. It is seen that the time response of the closed loop systems is a significa nt improvement over the open loop system. On the other hand the time responses of the output and the decent ralized control systems with dynamic compensators are satisfa ctory indica ting that decent ralized dynamic compensator will be suffic ient in this case. The shematic diagram of considered absorption column contro lled via decent ralized dynamic compenzator is shown in Fig. 4. The exelle nt performance of the derived decent ralized compensator is of specia l interr est, as due to increas ing comple xity of modern technological proces ses, decent ralized control schemes have become an essent ial requirement in the design of large-s cale system. CONCLUSION A design -orient ed methodlogy has been developed for the constru ction of decent ralized low order dynamic compensators where the integral action is explic itly incorp orated. Unlike most other methods for design of dynamic compensators, the proposed procedure has given a control config uration realis tic to apply and simple to impelemnt. Engineering constrain ts can easily be imposed for decent ralized control scheme considered. Control system does not require complex signal proces sing and therefo re, it can be easily

207

implemented on microcomputers. Computationally, the method is very feasib le, it involves a very simple algorithm . The method can be employed for the computer-aided design of practic al decent ralized dynamic compensators of large-s cale systems. An example illustr ates the procedure. REFERENCES Anderson,B.D.O., and J.B.Moore (1971). Linear optimal tontro l. Prentic e-Hall , New Jersey . Davison, E.J., and H.W.Smith (1974). A note on the design of indust rial regula tors: integra l feedback and feedforward controllers. Automatica, Vol. 10, pp. 329-332. Fortmann, T.E., and D.Williamson (1972) . Design of low order observers for linear feedback control laws . IEEE Trans.Autom. Control, Vol. AC-17, pp. 301-308 . Johnson, C.D. (1968) . Optimal control of linear regula tor with consta nt disturb ances. IEEE Trans. Autom. Control, Vol. AC-13, pp. 416-421. Lapidus, L., E.Shapiro, S.Shap iro, and R.E. Stillman (1961). Optimization of process performance. AIChe.J., Vol. 7, pp.2 88-294. Levine, W.S., T.L. Johnson, and M.Athans (1971). Optimal limited state variab le feedback contro llers for linear systems . IEEE Trans.Autom.Control, Vol. AC-17, No. 6, pp. 785-793 . Luenberger, D.G. (1971). An introdu ction to observ ers. IEEE Trans.Autom.Control, Vol. AC-17, pp. 596-602. Miller , R.A. (1973). Specif ic optimal control of the linear regula tor using a minimalorder observer. Int.J .Control, pp.139-159. Newmann, ~1.M. (1969). Optimal and suboptimal control using an observer when same of the state variab les are not measurable. Int.J.C ontrol , Vol. 9, pp. 281-290. Porter , B.(1971). Optimal control of multivariab le linear systems incorp orating integra l feedback . Electron . Lett., Vol. 7, pp. 170-172. Rom, D.B., and P.E.Sarachik (1973). The design of optimal compensators for linear consta nt systems with inacce ssible states. IEEE Trans.Autom.Control, Vol. AC-18, pp. 509-512. Roman, J . R., and T.E. Bullock (1975). Design of minimal order state observers for linear functions of the state via realization theory. IEEE Trans.Autom .Contr ol, Vol.AC-20, pp . 613-622 .

M. Rakic and Dj. Petkovski

208

Fig.1.

~ig.2.

The augmented open-loop system

The optimal control system with state feedback dynamic compensator

0--

1-

I

I

10

l'

12

t(mi~)

--4-__xL-l-

I

I

I

--4--!~-+-

;

---!--!~-+-

- 0, 02

i

i

- 0,04

-0,06

;

i i

i

;

-0,08

liquid

gas

0,02

;

i ; -0,10

a. open-loop system b. closed-loop system with state feedback dynamic compensator c. closed-loop system with output feedback dynamic compensator d. closed-loop system with decentralized dynamic compensator

- - 4--!~-+---!--~~--I---

t-----t-t--.-l-__xL -l-

w disturbance

gas

- 0,12

Fig.4. -0,14

Fig.3.

Time response Y2(t)=x6(t) of the open-loop and closed-loop system

Schematic diagram of considered absorption column controlled via decentralized dynamic compensator