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Discrete time control of linear distributed parameter systems
Automatica, Vol. 6, pp. 409-417. Pergamon Press, 1970. Printed in Great Britain.
Discrete Time Control of Linear Distributed Parameter Systems" Commande fi temps diseret de syst~mes lindaires h param&res repartis Zeitdiskrete Steuerung von linearen Systemen mit verteilten Parametern YlIpaBJIeHHe C ~J,IcKpeTHblM BpeMeHeM JIHHe~I, IMH eHCTeMaMH paclIpe~te0IeHHr~rMH napaMeTpaM~
C
M. A. H A S S A N and K. O. S O L B E R G I "+
Computers control distributed systems at discrete intervals. A sequential procedure to compute an approximation to the optimal control policy is given. The method presented is useful for complex objects whose dynamics are obtained experimentally. Smmnmry--The problem of optimum discrete time control for linear distributed parameter systems with quadratic cost function is considered. The system behaviour at discrete instants is experssed in terms of recursive functional expressions involving Green's function matrices. For the case of point controls the technique of dynamic programruing is used to derive an expression for feedback control in terms of an auxiliary spatially dependent variable. This variable is shown to satisfy a Ricatti type functional equation with known final value. Using orthogonal series expansion, this equation can be transformed to a recursive algebraic equation hi the coefficients of the expansion. A computational comparison of the straight diseretization in time and space and the use of Green's function orthogonal expansion for a simple case is included.
1. INTRODUCTION IN THE last few years various aspects o f optimum control for distributed parameter systems were considered. However, most of the available literature is concerned with the continuous time problem. General necessary conditions for the continuous problem were derived by WANe [1], using the technique of dynamic programming. He also applied his results to the special case of linear plant and quadratic cost function. For this special case WIBERC [2] and for the case of boundary control KIM and ERZBERGER [3] gave simpler direct derivations and considered methods of solving the resulting differential equations. The ease of discrete time control of infinite dimensional systems was not treated. This paper considers discrete time control of linear plants which is particularly important in computer control. The approach used is different from the known one, that is approximating the infinite dimensional system by a finite number of nodes and then using the usual optimum control relationships. Instead an exact, recursive, optimality equation is derived, then an approximation is applied to reduce it to a computationally useful expression. The system describing equations are rewritten in a form more suited for complex systems, leading to recursive relations involving Green's functions. The optimum feedback controller is derived using the principle of optimality as a formal generalization of the lumped parameter case, treated by GUNKEL and FRANKLIN [4]. The feedback policy is expressed in terms of an auxiliary spatially dependent variable satisfying a functional Ricatti type
NOTATIONS Capital letters
Small letters Small Bold letters Subscripts Superscripts I J N T
(Except N , / , J, T) denote matrices with elements as constants or scalar functions of x, x'. Denote scalars. Denote vectors. Indicate value of a variable at a certain point of time or space. Indicate element location in an array of similar elements. Value of cost function. Optimum value of cost function. Total number of control intervals. Used as superscript to indicate transpose of a matrix or a vector.
* Received 9 May 1969; revised 8 September 1969. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by associate editor L. MEmR. t Imtitutt for Atomenergi, Kjeller, Norway on leave from the Atomic Energy Establishment, Cairo, U.A.R. +*Institutt for Atomenergi, Kjeller, Norway. 409
410
M. A. HASSANand K. 0. SOLBERC
equation. The method of series expansion in orthonormal functions, COLLATZ [5], is used to reduce the functional equation to a recursive algebraic equation in the expansion coefficients. This type of recursive relation could be easily mechanized for computer solution. An example of optimal feedback regulation of the flux density in a slab type nuclear reactor with two control rods is used to illustrate the procedure.
2. FORMULATION OF THE PROBLEM To keep the idea and the derivations clear, our formulation will not aim at the greatest possible generality, but will be directed to a special case which is very important in nuclear reactor control. However, for all linear cases with quadratic cost, the given technique could be applied. This will be implied in the proper place. Consider the plant given by the matrix partial differential equation. $=Lu+f
(1)
where 6 is the spatial Dirac Delta function. The same procedure that will be developed could be applied to continuous spatial control. 3. PLANT DISCRETE TIME MODEL In the derivations to follow integral representation of the plant equations is adopted. For a plant described by equation (I) it is possible to write the following integral relation. u(t, x)=
G(t, t,, x, x’)u(I,, x’)dR’ s f-l t +
1st” o
G(t, t’, x, x’)f(t’, x’)da’dt
where G(t, t’, x, x’) is the Green’s function matrix satisfying the imposed boundary conditions. If we discretize time and choose two instants t, = kz and t, = (k + 1)t in which f(t, x) is held at the constant value of f(kz, x), then (4) can be written as
where u[(k+ l)r, x] =
L is an (n x n) spatial differential operator.
u&(t, x) is an (n) vector representing the state. fAf(t, x) is an (n) vector representing the control.
sR
1)r
where x&2,
cl
G[(k f l)z, t’,
kr
x, x’]dt’ f(kz, x’)dn’ .
u(t, x,)=0
x) ;
(k+ J(s
G[(k+ l)r, kz, x, x’]u(kT, x’)
da’+
The associated initial and boundary condition are Wdu(t,,
(4)
and R, is the boundary of 0 the domain
For the special case of point control by q elements, equation (3) can be written in matrix form
X.
The control effort will be constant over each discrete interval 17,
22,
---kT---,
NT.
It is required to find the feedback control policy f,4W,
n)
which minimizes the following performance terion
ZN=
cri-
ki1 n R CukT(x)Q(x, x’)u&‘) JJ + fk’_,(x)R(x, x’)f,_ ,(x’)]dRdR’ .
f= D*m where
D is an (n xq) matrix with elements &S(xj-x’) and m is a (q) vector. Using subscripts to indicate values over different sampling intervals and substituting from equation (6) into (2), the following relationships are obtained for point control.
uk+r(x) = (2)
JnGk(X, x’)“dx’)dfi’ + Hk(X)
f’(t, x’) =
jjyld”6(Xj -x’)
(3)
* mk
where
J
(k+l)r
Q(x, x’) and R(x, x’) are matrices whose elements are, in general, scalar functions of x, x’. Specific computational relations will be derived for the special case of point control
(6)
Hk(X)
=
G,[(k+
l)t, t’, x, xj]dt’
kr
and G,[(k+ l)~, t’, x, xj] =
G[(k + l)z, t’, x, x’] . DdR’ .
sR
(7)
Discrete time control of linear distributed parameter systems Similar relation holds for continuous spatial control namely
ua+ l(x) = j" [G(x, x'). ua(x') + Ha(x, x')" fk(x')]dfl'
411
where P(x, x') is a symmetric matrix and define SN-a as SN_k(X, X')= Q(x, x')+PN_a(x, x')
(13)
(8) substituting equations (9), (12) and (13) into (11) we have
where (k + 1)1:
Ha(x, x')= dkr
G[(k+1)r, t', x, x']dt'. ..f ur(x)G,_k(x, x')u~,(x')d~d~'
As will be seen later, Green's functions are not used by themselves in the final computations for the optimal policy but rather the coefficients of their expansion in terms of orthonormal functions. This offers a way for optimization of complex distributed systems whose dynamics are to be obtained from experimental measurements without reference to the exact describing equations. If Green's function is interpreted as the change at the observer x' due to point source at x then an array of numerical values could be obtained experimentally to represent the Green's function of the plant and a multidimensional orthogonal series could be fitted and the coefficients of the expansion evaluated computationally. 4. OPTIMUM FEEDBACK CONTROLLER SYNTHESIS Consider a plant with point control Uk+ 1(X) = f
=m2n{faft~[ft G,(x,x')uk(x')dtT +Hk(x)" ma]rsN-rr~(x, x')[;t G(x', x)uadf~
Equation (14) can be written as follows by grouping the integral signs.
fa faUrk(X)PN_k(X,x')uk(x')df~d[~' = min mk
{iofoioio Uk(X)Gk (X, x,
SN_f-+--I(x, x')Gk(x', x')uk(x")dfld~'d[~"
Gk(X,X')Uk(X')d~' + nk(x)mk(9)
dO+foIofo
and the corresponding quadratic cost functional
SN- r~"f(x, x')nk(x')mk dgld~'d~"
i = l f o f o u (x)Q(x, x')ua(x')dfad.' + mr- 1Rma- 1~ J
+
(lO)
where Q, R are symmetric positive definite matrices of dimension (n x n) and (q x q) respectively. We will use the principle of optimality to derive an expression for ma which minimizes IN. Let JN denote the optimum value of IN; then for the last N - k stages
JN_a[uk(x)]=mintf ~ ur+~(x)Q(x, x')dfldt"l'
ioioio mk nk (x)SN-r~--f(x, x,o,x x, uk(x")d~df~'d"+ftafamrH~(x) SN- w~--l(x, x')Hk(x')mkdf~df~' + mkrRmk} •
(15) Differentiating the right side of equation (15) with respect to and setting the result equal to zero, one obtains the optimum policy:
mk
mk I~J fl J ~'1
+ mrRmk + dN_g-~-i[Uk+ t(X)]}
(11)
assume JN-k can be written in the form
mk=-IfnfnHr(x)SN-r~f(
+ R]-lI.II..I
JN_a[Ua(X)]=f afau~(X)PN-k(X,x')ua(x')dOx:l"'
(12)
x, x')Hk(x')dIld[~'
n~(x)SN_k + ~(X, X')
Gk(x', x")d~d f~']nk(x")d['l" •
(16)
412
M . A . HASSAN and K. O. SOLBERG
Substituting for mk from (16) in the performance identity (14) the following equation for SN-k(X, Xt) results.
SN_k(X,X')--~-Q(X,X')+;t~fII ~T(,.s
k , X ", X)
SN-~r(x", x")Gk(X', x')df~"dfF'
fofoG:,x x,s,
x)
where Z, IV, A are composite matrices. The matrix Z is of dimension (nr x n) and the matrices IV, A ale of dimension (nr x nr). Their block elements Z ~j, W ij, A ~j are of dimension (n), (n x n) (n x n) respectively. Z is a quasidiagonal matrix with the column vector Z " on its diagonal blocks and zeros elsewhere. W, A have blocks representing the expansion coefficients of S(x, x') and G(x, x') respectively.
1
Zll
Ldfldn
Z 22
S~- Z~"r(x, x')H~(x')df~dfV +R
l-'Iofo
r . )SN-r~-f(x . . . Hk(X , x")
Gk(x", x ' ) d ~ " d f l "
Z 33
Z=
(17) 0
where
z~n
So Q. This equation is a recursive functional relationship which represents the counterpart of the Ricatti recursive algebraic relationship in lumped parameter discrete time problems.
_
m
All A2t A=
A12 A22
[
I I I
5. APPROXIMATION IN TERMS OF ORTHOGONAL EXPANSIONS
I I I
I
The recursive solution of equation (17) would be computationally very difficult. For this reason we will reduce it to an algebraic form mor~ suitable for computer solution. If the elements of Gk(x, x') and Sk(x, x') have certain regularity properties, for instance they can be assumed square integrable, then a set of orthonormal functions may be used to express them in series form to any required accuracy. The choice of an orthonormal set reduces the complexity of the resulting relations and thus the amount of required computations. Assume a complete set of orthonormal functions zl(x ), z2(x), z~(x) of which r members is enough for the required accuracy. The elements of matrix S k could be written as follows:
-t
A,,
Wit
W12
W 21
W 22
I
W=
I I I I
I I I
W nn
Furthermore
f zq(x)zr(x)d~= 1 /=1 q = l
for q = l and 0 for q # L Making use of the above property it is not difficult to verify the following equalities.
or in matrix form
s'~J(x, x')= ziT(x)WiJzJ(x ')
s~(x, x') = ZT(x) WkZ(x')
(18)
similarly
(i)
-,k~A ,
'')
Gk(x", x')d~"d~" Gk(x, x') = Z r(x)A,Z(x')
(19)
= X ~(x) W,,_r,-r Xk(x')
(20)
Discrete time control of linear distributed parameter systems where
Xk(X') = fn Z(x)Zr(x)AkZ(x')d~ = FkZ(X'). (ii)
fnfaH[(x)SN_k+ t(x, x')Hk(X')dt'~LQ' -- B~WN-I-+'IBk
(21)
where , ~'(k+ 1)~
z (x,la (t,) Z(x~)Ddt~'dt' .
x")n,(x")m',m"
= Xr(x)WN_F,-TBk . (iv) S~-k(x, (v) Q(x,
(22)
x')=Zr(x)WN-kZ(X ')
(23)
x')=Zr(x)YZ(x').
(24)
Substituting from equations (20)--(24) in (17) and dropping zr(x), Z(x') from both sides we get an equation of the same form as the familiar recursive Ricatti relation.
wN- = Y + eIWM_r+-r[Vl-8@IWN_
-zB
(25)
+ R)- t BrWN_t- F k].
Equation (25) can be solved in backward time by repetitive application of simple matrix algebra.
413
Presumably a relationship similar to equation (25) may be obtained if we first approximate the spatial differential operator in terms of its eigenfunctions. In this way a partial differential equation is replaced by an infinite set of ordinary differential equations usually called modal equations. Now the problem is ready for discretizing the time variable and using optimal control theory. The above approach should be compared to approximating equation (17) in terms of the eigenfunctions. However, obtaining the eigenfunctions is a difficult job which is in general possible only numerically. The method presented in this paper overcomes this difficulty by approximating the exact optimality relationship (17) in terms of general orthonormal sets which leads to reduced effort in the design process.
Example To illustrate the application of the proposed method we will consider the design of an automatic regulator for the flux pattern in a slab nuclear reactor [6]. The example will also serve as a numerical experiment which is useful in the evaluation and comparison of the different possible approximation schemes. Consider a slab type homogeneous reactor with two control rods Fig. 1. Assume that the reactor is operating at a given neutron flux level with a steady state spatial pattern. Let a spatial disturbance u(t, x) in the flux shape be introduced at t = 0. The two control rods have to be moved to reduce the deviation to zero. This is done automatically by taking spatial measurements of u(t, x) at different time intervals and computing an optimal movement for the rods as a function of the measured value.
"~
¢,i
20
i
2
~
Z
5
B
7
Mesh divisions
Fro. 1. Slab reactor with initial flux disturbance and its steady state shape (no control exerted).
B
~I
I
414
M.A. HASSANand K. O. SOLBERG
In the analysis to follow we will make the simplifying assumption that the movement of the regulating rods does not disturb the operating pattern, but will only affect the deviation• The following diffusion equation relates u(t, x) with the reactor parameters and the regulating rods movements•
OU__ 2(~2U 2 -~ -- O "~XX2"~ CU + E dJ~(xJ-- x ) m j . j = l
The boundary conditions for the above equation
are
u(t, 0)=u(t, h)=0 for all t.
In this example G and H will be independent of" k and no composite matrices are involved.
Uk+1(X)= g(X, x')uk(X')dx' + H(x).mk. Now we choose our orthonormal set as ~/-2Sin-nx x/~Sinh2x, h h . . . .
.
~/~Sin~q/, .
And express S(x, x') as a series in the four first above functions. The matrix B will be a (4 × 2) matrix with elements
The criterion of optimality is the minimization of _ bij,
In= ~.,
u~dx+mr-lRmk-t .
1 ' ~ - t z + c)zl~/h 7t2a2i2 crlL -- e x p/, - ~2a2•
--
k = 1 (do
h2
The investigation of the problem is divided, for clarity, to three parts. In the first part the method of this paper, from now on called pol;cy 1, is used to compute W. Different degrees of approximation r = 1,4, 5 are used to compute W. In the second part the system dynamics are approximated by a lumped system of four ordinary differential equations representing the behaviour at four nodes• The usual lumped system approach is then used to compute W. This we will call policy 2. In the third part the performance index is evaluated for the different approximations used above• To do this the system is simulated by an eight node approximation, an initial value for u(t, x) assumed and the different control policies tried. The performance index is calculated numerically. (i) To compute W for policy one, we use the Green's function for the object equation
G(t, t', x,
~f { x')= 2 V exp -
hq~lL
\
•
/t
Sm-~ixj "djj . The matrix F will be a (4 x 4) diagonal matrix with elements
f"= exp(----~q-~2022 i C)'~ using the following numerical values aZ=1600, c=0.252, x=0.1 Sec.
dJ=-lOSinnxj,
h=250 cm,
x~=0.25,
x2=0.6.
The operating pattern is normalized to 100 units at the center• Then B, F will have the following values m
22
~-a'q2 h2 B=
m
-0.045
-0.081
-0.061
+ 0.049
-0.042
+ 0-048
9
0
-0.068
h~=-~f:i+')~q~=l[exp(- g e q 2 + c ) [ k + l ] z m
-")]Sinhqx
1
Sinhqx,dt' F=
replacing t ' = t* + kT we can write
' h~-
~ q2~°° - 1 19 2 [01 - 2e x pq ( 2--20t2"-~" / q C h2
+ c)z]Sinhqx Sinhqxj •
0
0.927 0.819 0
using
Q=[1] ,
0"687
Discrete time control of linear distributed parameter systems The steady value of W was found to be
Then 1.28
0
0
y= 0
0
Computation gives the steady state value of W as follows 8.88 x 101
0
l ' 7 0 x 1 0 -2
0
0
0
0
0
1.70 x 10-2
0
0
0
W=
1"29 x 10° 0
- 6 . 0 1 x 10 -4
- 0.021
- 2 . 8 4 x 10 -4
- 4 . 2 x 10 -4
-0.904
- 7.05 x 10- 5
- 0.028
0"904
0"058
0"002 3"9 x 10 -4
0"058
0"906
0"058 1"85 x 10 -3
0"002
0"058
0"906 0"058
3'9 x 10 -4 1"85 × 10 -3 0"058 0.904 and m
0 1
Q=
R = [ 56 1
0 m
1
23.7
250.7
16.94
-9.88
1.71
16.94
-1"06
-9.88
82.786 17"64
17.64 257.63
(iii) To compute the performance index in various eases we use the evaluated W. The control policy for the method of orthogonal approximation (policy 1) involves integration over space.
where
0
- 0"666
1
- 1"06
do
FUk+ Bmk
_
1"71
0
k= 1 -5uk ~uk "~ m k - l K m k - 1
B=
23.7
m -- - [8 T w B + R] -1BT W f ' X ( y ) u ( y ) d y
Notice that we are now controlling directly only the 1st and 3rd harmonics and the Ricatti equation actually uses (2 x 2) matrices. (ii) To compute W for policy 2, the slab is divided into five sections and four ordinary differential equations used to represent the flux at the four non-boundary nodes. Usual lumped parameter methods are used to obtain the transition matrix and the distribution matrix which will be denoted by F and B. The system equations and the performance criterion are Uk + I =
96.3 W=
0.424
F=
415
50].
X(y) is a (4 x 1) matrix with elements
_F x,,-texp
/
-
n202.2
, +~) 1~~
S
inhix
and u(y) is a scalar function of the spatial location, the system state, while the control policy for discretization in space, policy 2, is as follows m=
-[BrWB+ R]-IBTWFu.
u is a vector representing the state at the nodes. Policies (I) and (2) were tried on an eight node approximation of the slab reactor with an assumed initiaI disturbance. Figure 1 shows the shape of the disturbance at t = 0 and its steady state shape if the regulator is not working. Figure 2 shows the shape of the disturbance at t = 1 see for both policy 1 (r = 4) and policy 2. Table 1 gives the values of the performance index for policy 1 with r = 1, 4, 5 and for policy 2. From the above table it is seen that the values of J for policy 1 are in general smaller than those for policy 2. The value of J for policy 1 decreases with the increase in r. However, for the ease considered the reduction is small and the use of low values of r, with the resulting reduction in computational requirements, is advantageous. 6. CONCLUSIONS A new approach to the problem of optimum, discrete time, control of distributed parameter systems is given. A simple computational expression for obtaining an approximation to the optimal policy, in the case of point control, is derived. Similar results could be easily derived for other linear cases. The approach makes use of Green's function and its expansion in orthogonal series. This offers an opportunity for treating complex distributed systems whose describing relations are obtained experimentally. The results obtained ale useful for the design of computer control for distributed systems.
416
M . A . HASSAN and K. O. SOLBERG
cE
i
{z
II
2 o -
/~
~ot,cy 1 •
=
X 12-
.c x
o
Mesh divisions Slob
width
-~
h = 250 cm
FIG. 2. Effect of control calculated by different policies on initial disturbance after 1 sec. TABLE 1 Performance
Policy 1 r=l
J
fhu~(x)Quk(x)dx
r=4
Policy 2 r=5
10.6x104
lO.5xlO 4
10.45x104
11.2xi04
4"3x104
4"2x104
4"15x104
8"4x104
k=ldo
The procedure was tested on a simple example and found to give good results with reduced computational requirements. Further work need to be done on the application to complex systems. REFERENCES [1] P. K. C. WANO: Control of distributed parameter f systems. Chapter in, Advances in Control Systems, Vol. 1, |(Ed. C. T. I.Y.ONDES). Academic Press, New York (1964). [2] D. M. WmEgG: Feedback control of linear distributed parameter systems. Trans. ASME, J. Bas. Engng, 319384 (June 1967). [3l M. I(AMand H. EP,Z~gGER: On the design of optimum distributed parameter systems with boundary control functions. IEEE Trans. Aut. Control AC-12, 22-28 (1967). [4] T. L. GUNCKELand G. F. FgANKL~: A general solution for linear sampled-data control. Trans. ASME, J. Bas. Engng, 197-201 (1963). [5] L. COLLATZ: The Numerical Treatment of Differential Equatians, Springor-Vexlag, Berlin, (1960). [6] D. M. WmwG: Optimal control of nuclear reactor systems. Chapter in Advances in Control Systems, Vol. 5, (Ed. C. T. Lig>NDES). Academic Press, New York (1967).
~L'article consid6re le probl~ne de commande optimale ~ temps discret pour des syst~nes lin~aires param6tres repartis avec des fonctions de cofit quadrafiques. L¢ comportement du syst&n¢ ~ des instants discrets est exprim6 par des expressions fonctionnelles rocurrentes comprenant des matrices de fonctions de Green. Pour le cas de commande par points, la technique de programmation dynamique est suivie aria de deduire une expression pour la commande/~ r6action en function d'une variable auxiliaire d6pendant de l'espace. I! est montr6 clue cette variable satisfait/t une &luation fonctionneHe du type de Ricatti avec valeur finale connue. En utilisant un d6veloppement en scrie orthogonal, cette 6quation peut etre transform6e en une 6quation alg6brique recurrent¢ scion les coefficients du d6velop~ment. L'article comprend une comparaison num6rique de l'6chantillonna~ direct dam le temps et dam l'espace avec l'emploi du d6veloppement orthogonal de la fonction de Green dans un cos simple. Z t m m m e m ~ - - B e t r a c h t e t wird das Problem der optimalen zeitdisckreten Steuerung von lineaxen Systemen mit verteiRen Paxametern bei einem quadratischen Gfitekriterium. Das Systemverhalten in diskreten Zeitpunkten wird ausgedrflckt in Termen yon rekursiven Funktionalausdriicken, die Matrizen Greenscher Funktionen enthalten. Im Fall von Punktsteuerungen wird die Technik der d ~ Programmicruag benutzt, um ¢inen Ausdruck
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417
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Discrete Time Control of Linear Distributed Parameter Systems" Commande fi temps diseret de syst~mes lindaires h param&res repartis Zeitdiskrete Steuerung von linearen Systemen mit verteilten Parametern YlIpaBJIeHHe C ~J,IcKpeTHblM BpeMeHeM JIHHe~I, IMH eHCTeMaMH paclIpe~te0IeHHr~rMH napaMeTpaM~
C
M. A. H A S S A N and K. O. S O L B E R G I "+
Computers control distributed systems at discrete intervals. A sequential procedure to compute an approximation to the optimal control policy is given. The method presented is useful for complex objects whose dynamics are obtained experimentally. Smmnmry--The problem of optimum discrete time control for linear distributed parameter systems with quadratic cost function is considered. The system behaviour at discrete instants is experssed in terms of recursive functional expressions involving Green's function matrices. For the case of point controls the technique of dynamic programruing is used to derive an expression for feedback control in terms of an auxiliary spatially dependent variable. This variable is shown to satisfy a Ricatti type functional equation with known final value. Using orthogonal series expansion, this equation can be transformed to a recursive algebraic equation hi the coefficients of the expansion. A computational comparison of the straight diseretization in time and space and the use of Green's function orthogonal expansion for a simple case is included.
1. INTRODUCTION IN THE last few years various aspects o f optimum control for distributed parameter systems were considered. However, most of the available literature is concerned with the continuous time problem. General necessary conditions for the continuous problem were derived by WANe [1], using the technique of dynamic programming. He also applied his results to the special case of linear plant and quadratic cost function. For this special case WIBERC [2] and for the case of boundary control KIM and ERZBERGER [3] gave simpler direct derivations and considered methods of solving the resulting differential equations. The ease of discrete time control of infinite dimensional systems was not treated. This paper considers discrete time control of linear plants which is particularly important in computer control. The approach used is different from the known one, that is approximating the infinite dimensional system by a finite number of nodes and then using the usual optimum control relationships. Instead an exact, recursive, optimality equation is derived, then an approximation is applied to reduce it to a computationally useful expression. The system describing equations are rewritten in a form more suited for complex systems, leading to recursive relations involving Green's functions. The optimum feedback controller is derived using the principle of optimality as a formal generalization of the lumped parameter case, treated by GUNKEL and FRANKLIN [4]. The feedback policy is expressed in terms of an auxiliary spatially dependent variable satisfying a functional Ricatti type
NOTATIONS Capital letters
Small letters Small Bold letters Subscripts Superscripts I J N T
(Except N , / , J, T) denote matrices with elements as constants or scalar functions of x, x'. Denote scalars. Denote vectors. Indicate value of a variable at a certain point of time or space. Indicate element location in an array of similar elements. Value of cost function. Optimum value of cost function. Total number of control intervals. Used as superscript to indicate transpose of a matrix or a vector.
* Received 9 May 1969; revised 8 September 1969. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by associate editor L. MEmR. t Imtitutt for Atomenergi, Kjeller, Norway on leave from the Atomic Energy Establishment, Cairo, U.A.R. +*Institutt for Atomenergi, Kjeller, Norway. 409
410
M. A. HASSANand K. 0. SOLBERC
equation. The method of series expansion in orthonormal functions, COLLATZ [5], is used to reduce the functional equation to a recursive algebraic equation in the expansion coefficients. This type of recursive relation could be easily mechanized for computer solution. An example of optimal feedback regulation of the flux density in a slab type nuclear reactor with two control rods is used to illustrate the procedure.
2. FORMULATION OF THE PROBLEM To keep the idea and the derivations clear, our formulation will not aim at the greatest possible generality, but will be directed to a special case which is very important in nuclear reactor control. However, for all linear cases with quadratic cost, the given technique could be applied. This will be implied in the proper place. Consider the plant given by the matrix partial differential equation. $=Lu+f
(1)
where 6 is the spatial Dirac Delta function. The same procedure that will be developed could be applied to continuous spatial control. 3. PLANT DISCRETE TIME MODEL In the derivations to follow integral representation of the plant equations is adopted. For a plant described by equation (I) it is possible to write the following integral relation. u(t, x)=
G(t, t,, x, x’)u(I,, x’)dR’ s f-l t +
1st” o
G(t, t’, x, x’)f(t’, x’)da’dt
where G(t, t’, x, x’) is the Green’s function matrix satisfying the imposed boundary conditions. If we discretize time and choose two instants t, = kz and t, = (k + 1)t in which f(t, x) is held at the constant value of f(kz, x), then (4) can be written as
where u[(k+ l)r, x] =
L is an (n x n) spatial differential operator.
u&(t, x) is an (n) vector representing the state. fAf(t, x) is an (n) vector representing the control.
sR
1)r
where x&2,
cl
G[(k f l)z, t’,
kr
x, x’]dt’ f(kz, x’)dn’ .
u(t, x,)=0
x) ;
(k+ J(s
G[(k+ l)r, kz, x, x’]u(kT, x’)
da’+
The associated initial and boundary condition are Wdu(t,,
(4)
and R, is the boundary of 0 the domain
For the special case of point control by q elements, equation (3) can be written in matrix form
X.
The control effort will be constant over each discrete interval 17,
22,
---kT---,
NT.
It is required to find the feedback control policy f,4W,
n)
which minimizes the following performance terion
ZN=
cri-
ki1 n R CukT(x)Q(x, x’)u&‘) JJ + fk’_,(x)R(x, x’)f,_ ,(x’)]dRdR’ .
f= D*m where
D is an (n xq) matrix with elements &S(xj-x’) and m is a (q) vector. Using subscripts to indicate values over different sampling intervals and substituting from equation (6) into (2), the following relationships are obtained for point control.
uk+r(x) = (2)
JnGk(X, x’)“dx’)dfi’ + Hk(X)
f’(t, x’) =
jjyld”6(Xj -x’)
(3)
* mk
where
J
(k+l)r
Q(x, x’) and R(x, x’) are matrices whose elements are, in general, scalar functions of x, x’. Specific computational relations will be derived for the special case of point control
(6)
Hk(X)
=
G,[(k+
l)t, t’, x, xj]dt’
kr
and G,[(k+ l)~, t’, x, xj] =
G[(k + l)z, t’, x, x’] . DdR’ .
sR
(7)
Discrete time control of linear distributed parameter systems Similar relation holds for continuous spatial control namely
ua+ l(x) = j" [G(x, x'). ua(x') + Ha(x, x')" fk(x')]dfl'
411
where P(x, x') is a symmetric matrix and define SN-a as SN_k(X, X')= Q(x, x')+PN_a(x, x')
(13)
(8) substituting equations (9), (12) and (13) into (11) we have
where (k + 1)1:
Ha(x, x')= dkr
G[(k+1)r, t', x, x']dt'. ..f ur(x)G,_k(x, x')u~,(x')d~d~'
As will be seen later, Green's functions are not used by themselves in the final computations for the optimal policy but rather the coefficients of their expansion in terms of orthonormal functions. This offers a way for optimization of complex distributed systems whose dynamics are to be obtained from experimental measurements without reference to the exact describing equations. If Green's function is interpreted as the change at the observer x' due to point source at x then an array of numerical values could be obtained experimentally to represent the Green's function of the plant and a multidimensional orthogonal series could be fitted and the coefficients of the expansion evaluated computationally. 4. OPTIMUM FEEDBACK CONTROLLER SYNTHESIS Consider a plant with point control Uk+ 1(X) = f
=m2n{faft~[ft G,(x,x')uk(x')dtT +Hk(x)" ma]rsN-rr~(x, x')[;t G(x', x)uadf~
Equation (14) can be written as follows by grouping the integral signs.
fa faUrk(X)PN_k(X,x')uk(x')df~d[~' = min mk
{iofoioio Uk(X)Gk (X, x,
SN_f-+--I(x, x')Gk(x', x')uk(x")dfld~'d[~"
Gk(X,X')Uk(X')d~' + nk(x)mk(9)
dO+foIofo
and the corresponding quadratic cost functional
SN- r~"f(x, x')nk(x')mk dgld~'d~"
i = l f o f o u (x)Q(x, x')ua(x')dfad.' + mr- 1Rma- 1~ J
+
(lO)
where Q, R are symmetric positive definite matrices of dimension (n x n) and (q x q) respectively. We will use the principle of optimality to derive an expression for ma which minimizes IN. Let JN denote the optimum value of IN; then for the last N - k stages
JN_a[uk(x)]=mintf ~ ur+~(x)Q(x, x')dfldt"l'
ioioio mk nk (x)SN-r~--f(x, x,o,x x, uk(x")d~df~'d"+ftafamrH~(x) SN- w~--l(x, x')Hk(x')mkdf~df~' + mkrRmk} •
(15) Differentiating the right side of equation (15) with respect to and setting the result equal to zero, one obtains the optimum policy:
mk
mk I~J fl J ~'1
+ mrRmk + dN_g-~-i[Uk+ t(X)]}
(11)
assume JN-k can be written in the form
mk=-IfnfnHr(x)SN-r~f(
+ R]-lI.II..I
JN_a[Ua(X)]=f afau~(X)PN-k(X,x')ua(x')dOx:l"'
(12)
x, x')Hk(x')dIld[~'
n~(x)SN_k + ~(X, X')
Gk(x', x")d~d f~']nk(x")d['l" •
(16)
412
M . A . HASSAN and K. O. SOLBERG
Substituting for mk from (16) in the performance identity (14) the following equation for SN-k(X, Xt) results.
SN_k(X,X')--~-Q(X,X')+;t~fII ~T(,.s
k , X ", X)
SN-~r(x", x")Gk(X', x')df~"dfF'
fofoG:,x x,s,
x)
where Z, IV, A are composite matrices. The matrix Z is of dimension (nr x n) and the matrices IV, A ale of dimension (nr x nr). Their block elements Z ~j, W ij, A ~j are of dimension (n), (n x n) (n x n) respectively. Z is a quasidiagonal matrix with the column vector Z " on its diagonal blocks and zeros elsewhere. W, A have blocks representing the expansion coefficients of S(x, x') and G(x, x') respectively.
1
Zll
Ldfldn
Z 22
S~- Z~"r(x, x')H~(x')df~dfV +R
l-'Iofo
r . )SN-r~-f(x . . . Hk(X , x")
Gk(x", x ' ) d ~ " d f l "
Z 33
Z=
(17) 0
where
z~n
So Q. This equation is a recursive functional relationship which represents the counterpart of the Ricatti recursive algebraic relationship in lumped parameter discrete time problems.
_
m
All A2t A=
A12 A22
[
I I I
5. APPROXIMATION IN TERMS OF ORTHOGONAL EXPANSIONS
I I I
I
The recursive solution of equation (17) would be computationally very difficult. For this reason we will reduce it to an algebraic form mor~ suitable for computer solution. If the elements of Gk(x, x') and Sk(x, x') have certain regularity properties, for instance they can be assumed square integrable, then a set of orthonormal functions may be used to express them in series form to any required accuracy. The choice of an orthonormal set reduces the complexity of the resulting relations and thus the amount of required computations. Assume a complete set of orthonormal functions zl(x ), z2(x), z~(x) of which r members is enough for the required accuracy. The elements of matrix S k could be written as follows:
-t
A,,
Wit
W12
W 21
W 22
I
W=
I I I I
I I I
W nn
Furthermore
f zq(x)zr(x)d~= 1 /=1 q = l
for q = l and 0 for q # L Making use of the above property it is not difficult to verify the following equalities.
or in matrix form
s'~J(x, x')= ziT(x)WiJzJ(x ')
s~(x, x') = ZT(x) WkZ(x')
(18)
similarly
(i)
-,k~A ,
'')
Gk(x", x')d~"d~" Gk(x, x') = Z r(x)A,Z(x')
(19)
= X ~(x) W,,_r,-r Xk(x')
(20)
Discrete time control of linear distributed parameter systems where
Xk(X') = fn Z(x)Zr(x)AkZ(x')d~ = FkZ(X'). (ii)
fnfaH[(x)SN_k+ t(x, x')Hk(X')dt'~LQ' -- B~WN-I-+'IBk
(21)
where , ~'(k+ 1)~
z (x,la (t,) Z(x~)Ddt~'dt' .
x")n,(x")m',m"
= Xr(x)WN_F,-TBk . (iv) S~-k(x, (v) Q(x,
(22)
x')=Zr(x)WN-kZ(X ')
(23)
x')=Zr(x)YZ(x').
(24)
Substituting from equations (20)--(24) in (17) and dropping zr(x), Z(x') from both sides we get an equation of the same form as the familiar recursive Ricatti relation.
wN- = Y + eIWM_r+-r[Vl-8@IWN_
-zB
(25)
+ R)- t BrWN_t- F k].
Equation (25) can be solved in backward time by repetitive application of simple matrix algebra.
413
Presumably a relationship similar to equation (25) may be obtained if we first approximate the spatial differential operator in terms of its eigenfunctions. In this way a partial differential equation is replaced by an infinite set of ordinary differential equations usually called modal equations. Now the problem is ready for discretizing the time variable and using optimal control theory. The above approach should be compared to approximating equation (17) in terms of the eigenfunctions. However, obtaining the eigenfunctions is a difficult job which is in general possible only numerically. The method presented in this paper overcomes this difficulty by approximating the exact optimality relationship (17) in terms of general orthonormal sets which leads to reduced effort in the design process.
Example To illustrate the application of the proposed method we will consider the design of an automatic regulator for the flux pattern in a slab nuclear reactor [6]. The example will also serve as a numerical experiment which is useful in the evaluation and comparison of the different possible approximation schemes. Consider a slab type homogeneous reactor with two control rods Fig. 1. Assume that the reactor is operating at a given neutron flux level with a steady state spatial pattern. Let a spatial disturbance u(t, x) in the flux shape be introduced at t = 0. The two control rods have to be moved to reduce the deviation to zero. This is done automatically by taking spatial measurements of u(t, x) at different time intervals and computing an optimal movement for the rods as a function of the measured value.
"~
¢,i
20
i
2
~
Z
5
B
7
Mesh divisions
Fro. 1. Slab reactor with initial flux disturbance and its steady state shape (no control exerted).
B
~I
I
414
M.A. HASSANand K. O. SOLBERG
In the analysis to follow we will make the simplifying assumption that the movement of the regulating rods does not disturb the operating pattern, but will only affect the deviation• The following diffusion equation relates u(t, x) with the reactor parameters and the regulating rods movements•
OU__ 2(~2U 2 -~ -- O "~XX2"~ CU + E dJ~(xJ-- x ) m j . j = l
The boundary conditions for the above equation
are
u(t, 0)=u(t, h)=0 for all t.
In this example G and H will be independent of" k and no composite matrices are involved.
Uk+1(X)= g(X, x')uk(X')dx' + H(x).mk. Now we choose our orthonormal set as ~/-2Sin-nx x/~Sinh2x, h h . . . .
.
~/~Sin~q/, .
And express S(x, x') as a series in the four first above functions. The matrix B will be a (4 × 2) matrix with elements
The criterion of optimality is the minimization of _ bij,
In= ~.,
u~dx+mr-lRmk-t .
1 ' ~ - t z + c)zl~/h 7t2a2i2 crlL -- e x p/, - ~2a2•
--
k = 1 (do
h2
The investigation of the problem is divided, for clarity, to three parts. In the first part the method of this paper, from now on called pol;cy 1, is used to compute W. Different degrees of approximation r = 1,4, 5 are used to compute W. In the second part the system dynamics are approximated by a lumped system of four ordinary differential equations representing the behaviour at four nodes• The usual lumped system approach is then used to compute W. This we will call policy 2. In the third part the performance index is evaluated for the different approximations used above• To do this the system is simulated by an eight node approximation, an initial value for u(t, x) assumed and the different control policies tried. The performance index is calculated numerically. (i) To compute W for policy one, we use the Green's function for the object equation
G(t, t', x,
~f { x')= 2 V exp -
hq~lL
\
•
/t
Sm-~ixj "djj . The matrix F will be a (4 x 4) diagonal matrix with elements
f"= exp(----~q-~2022 i C)'~ using the following numerical values aZ=1600, c=0.252, x=0.1 Sec.
dJ=-lOSinnxj,
h=250 cm,
x~=0.25,
x2=0.6.
The operating pattern is normalized to 100 units at the center• Then B, F will have the following values m
22
~-a'q2 h2 B=
m
-0.045
-0.081
-0.061
+ 0.049
-0.042
+ 0-048
9
0
-0.068
h~=-~f:i+')~q~=l[exp(- g e q 2 + c ) [ k + l ] z m
-")]Sinhqx
1
Sinhqx,dt' F=
replacing t ' = t* + kT we can write
' h~-
~ q2~°° - 1 19 2 [01 - 2e x pq ( 2--20t2"-~" / q C h2
+ c)z]Sinhqx Sinhqxj •
0
0.927 0.819 0
using
Q=[1] ,
0"687
Discrete time control of linear distributed parameter systems The steady value of W was found to be
Then 1.28
0
0
y= 0
0
Computation gives the steady state value of W as follows 8.88 x 101
0
l ' 7 0 x 1 0 -2
0
0
0
0
0
1.70 x 10-2
0
0
0
W=
1"29 x 10° 0
- 6 . 0 1 x 10 -4
- 0.021
- 2 . 8 4 x 10 -4
- 4 . 2 x 10 -4
-0.904
- 7.05 x 10- 5
- 0.028
0"904
0"058
0"002 3"9 x 10 -4
0"058
0"906
0"058 1"85 x 10 -3
0"002
0"058
0"906 0"058
3'9 x 10 -4 1"85 × 10 -3 0"058 0.904 and m
0 1
Q=
R = [ 56 1
0 m
1
23.7
250.7
16.94
-9.88
1.71
16.94
-1"06
-9.88
82.786 17"64
17.64 257.63
(iii) To compute the performance index in various eases we use the evaluated W. The control policy for the method of orthogonal approximation (policy 1) involves integration over space.
where
0
- 0"666
1
- 1"06
do
FUk+ Bmk
_
1"71
0
k= 1 -5uk ~uk "~ m k - l K m k - 1
B=
23.7
m -- - [8 T w B + R] -1BT W f ' X ( y ) u ( y ) d y
Notice that we are now controlling directly only the 1st and 3rd harmonics and the Ricatti equation actually uses (2 x 2) matrices. (ii) To compute W for policy 2, the slab is divided into five sections and four ordinary differential equations used to represent the flux at the four non-boundary nodes. Usual lumped parameter methods are used to obtain the transition matrix and the distribution matrix which will be denoted by F and B. The system equations and the performance criterion are Uk + I =
96.3 W=
0.424
F=
415
50].
X(y) is a (4 x 1) matrix with elements
_F x,,-texp
/
-
n202.2
, +~) 1~~
S
inhix
and u(y) is a scalar function of the spatial location, the system state, while the control policy for discretization in space, policy 2, is as follows m=
-[BrWB+ R]-IBTWFu.
u is a vector representing the state at the nodes. Policies (I) and (2) were tried on an eight node approximation of the slab reactor with an assumed initiaI disturbance. Figure 1 shows the shape of the disturbance at t = 0 and its steady state shape if the regulator is not working. Figure 2 shows the shape of the disturbance at t = 1 see for both policy 1 (r = 4) and policy 2. Table 1 gives the values of the performance index for policy 1 with r = 1, 4, 5 and for policy 2. From the above table it is seen that the values of J for policy 1 are in general smaller than those for policy 2. The value of J for policy 1 decreases with the increase in r. However, for the ease considered the reduction is small and the use of low values of r, with the resulting reduction in computational requirements, is advantageous. 6. CONCLUSIONS A new approach to the problem of optimum, discrete time, control of distributed parameter systems is given. A simple computational expression for obtaining an approximation to the optimal policy, in the case of point control, is derived. Similar results could be easily derived for other linear cases. The approach makes use of Green's function and its expansion in orthogonal series. This offers an opportunity for treating complex distributed systems whose describing relations are obtained experimentally. The results obtained ale useful for the design of computer control for distributed systems.
416
M . A . HASSAN and K. O. SOLBERG
cE
i
{z
II
2 o -
/~
~ot,cy 1 •
=
X 12-
.c x
o
Mesh divisions Slob
width
-~
h = 250 cm
FIG. 2. Effect of control calculated by different policies on initial disturbance after 1 sec. TABLE 1 Performance
Policy 1 r=l
J
fhu~(x)Quk(x)dx
r=4
Policy 2 r=5
10.6x104
lO.5xlO 4
10.45x104
11.2xi04
4"3x104
4"2x104
4"15x104
8"4x104
k=ldo
The procedure was tested on a simple example and found to give good results with reduced computational requirements. Further work need to be done on the application to complex systems. REFERENCES [1] P. K. C. WANO: Control of distributed parameter f systems. Chapter in, Advances in Control Systems, Vol. 1, |(Ed. C. T. I.Y.ONDES). Academic Press, New York (1964). [2] D. M. WmEgG: Feedback control of linear distributed parameter systems. Trans. ASME, J. Bas. Engng, 319384 (June 1967). [3l M. I(AMand H. EP,Z~gGER: On the design of optimum distributed parameter systems with boundary control functions. IEEE Trans. Aut. Control AC-12, 22-28 (1967). [4] T. L. GUNCKELand G. F. FgANKL~: A general solution for linear sampled-data control. Trans. ASME, J. Bas. Engng, 197-201 (1963). [5] L. COLLATZ: The Numerical Treatment of Differential Equatians, Springor-Vexlag, Berlin, (1960). [6] D. M. WmwG: Optimal control of nuclear reactor systems. Chapter in Advances in Control Systems, Vol. 5, (Ed. C. T. Lig>NDES). Academic Press, New York (1967).
~L'article consid6re le probl~ne de commande optimale ~ temps discret pour des syst~nes lin~aires param6tres repartis avec des fonctions de cofit quadrafiques. L¢ comportement du syst&n¢ ~ des instants discrets est exprim6 par des expressions fonctionnelles rocurrentes comprenant des matrices de fonctions de Green. Pour le cas de commande par points, la technique de programmation dynamique est suivie aria de deduire une expression pour la commande/~ r6action en function d'une variable auxiliaire d6pendant de l'espace. I! est montr6 clue cette variable satisfait/t une &luation fonctionneHe du type de Ricatti avec valeur finale connue. En utilisant un d6veloppement en scrie orthogonal, cette 6quation peut etre transform6e en une 6quation alg6brique recurrent¢ scion les coefficients du d6velop~ment. L'article comprend une comparaison num6rique de l'6chantillonna~ direct dam le temps et dam l'espace avec l'emploi du d6veloppement orthogonal de la fonction de Green dans un cos simple. Z t m m m e m ~ - - B e t r a c h t e t wird das Problem der optimalen zeitdisckreten Steuerung von lineaxen Systemen mit verteiRen Paxametern bei einem quadratischen Gfitekriterium. Das Systemverhalten in diskreten Zeitpunkten wird ausgedrflckt in Termen yon rekursiven Funktionalausdriicken, die Matrizen Greenscher Funktionen enthalten. Im Fall von Punktsteuerungen wird die Technik der d ~ Programmicruag benutzt, um ¢inen Ausdruck
Discrete time control of linear distributed parameter systems ftir die Regelung in Termen einer ortsabh~ngigen Hilfsvariablen abzuleiten. Es wird gezeigt, dab diese Variable einer Funktionalgleichung yore Riccati-Typ mit bekanntem Endwert geniigt. Wenn man die Entwicklung in Orthogonalreihen benutzt, kann diese Gleichung in eine rekursive algebraische Gleichung in den Koeff~enten der Entwicklung traasformiert werden. Ftir einen einfachen Fall ist ein reclmerischer Vergleich der direkten Diskretisierung in Zeit und Raum und der Benutzung einer Orthogonalentwicklung der Greenschen Funktion eingeschlossen. Pe3xoMe---CTaT~S paccMaTpHBaeT npo6J~¢My OHTHMaHbHOrO ynpaB~eH~ c ~J~c~erH~M BpeMeHeM ~ mmettm~ cHcreM c pacnpe~cneHHbiMH napaMerpaMH H KBa~paTHqH]~IMH ~
~
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