Accuracy of time-optimal control of the diffusion equation using a spatial discretisation technique
3
Accuracy of time-optimal control of the diffusion equation using a spatial discretisation technique A.S.I. Zinober. B.Sc.(Eng.), M.Sc.(Eng.), Ph.D.. C.Eng., M.I.E.E., A.F.I.M.A. Indexing terms: Control theory, Optimal control, Optimisation, Computer simulation Abstract: A well known numerical technique for calculating the optimal control of distributed parameter systems is the spatial discretisation (SD) procedure. In the paper an improved formulation of the SD scheme is studied in the case of the boundary control of the diffusion equation. Significantly more accurate results are obtained with no additional computational complexity. 1 Introduction The approximate optimal control of systems described by partial differential equations may be determined by the well known technique of spatial discretisation (SD) [1-3]. This yields differential difference equations which simplify the technique for finding the optimal solution by allowing the standard optimal methods of ordinary differential equations to be applied. The accuracy of the SD approx- imation is clearly of great importance in obtaining a solu- tion which is close to the optimal of the original distributed-parameter system. Clearly greater accuracy can be obtained by increasing the number of grid points in the spatial discretisation. However, the actual form of the dif- ference approximation has tended to be ignored in the control literature. A careful choice of this approximation can lead to greatly improved accuracy with a small number of grid points. In this paper, we shall concentrate on the optimal boundary control of the one-dimensional diffusion (or heat) equation: dy dt o-£ (0 < x < 1, 0 ^ t < oo) with derivative boundary conditions dy dy /(0, t)= -u(t) / ( I , 0 = 0 ox ox (1) (2) The arguments may be extended to other linear and non- linear distributed-parameter systems with suitable modifi- cations to the analysis. The time-optimal problem (PI) is to determine the control u(t) such that, in minimum time T, the final distribution y(x, T) = 0 is attained from an initial distribution y(x, 0) = y o (x). This problem has been discussed by numerous authors [1, 3, 4, 5] using a variety of approximation techniques. The SD method [1-3] is conceptually simple and yields suitably accurate results with correct application. The optimal control of eqns. 1 and 2 minimising a quadratic performance criterion (P2) can also be calculated using discretisation techniques [2, 6]. 2 Spatial discretisation An nth-order differential difference equation may be obtained from eqns. 1 and 2 using the central difference Paper 4072D (Cl, C3, C7), first received 16th July 1984 and in revised form 15th April 1985 The author is with the Department of Applied & Computational Mathematics, Uni- versity of Sheffield, Sheffield S10 2TN, United Kingdom formula, giving the approximation to the original system y(t) = Ay(t) + bu(t) (3) where the components y t of the vector y(t) are the values y(ih, t) along the x-axis with the spacing h between adjac- ent grid points. We obtain — a 1 0 b -2 1 0 1 -2 0 1 0 1 0 -2 1 0 1 -2 b 0 1 — a. (4) and -(c 0 0 0) T (5) The values a, b and c are specified by the particular approximation used to model the derivative boundary conditions of eqns. 2. Sage [2] and Mahapatra [3], among others, approximate (dy/dx)(0, t) at the boundary x = 0 by A x y(O,t)/h = {y(h,t)-y(O,t)}/h which yields y(h, t) - y(0, t) = hu(t) (6) (7) yielding a truncation error of order h 2 when determining y(0, t) [7], where a = b = c = 1, h= l/{n + 1) and y = (y^i '"' y n ) T - This is the formula generally used in the control literature [2, 3]. A far better approximation to the boundary condition (eqns. 2) at x = 0 is attained by introducing the fictitious exterior mesh point y( — h, t) = y- l (t) and using the divided mean central difference approximation = {y(h,t)-y(-h,t)}/2h which gives y(h,t)-y(-h,t) = 2hu(t) (8) (9) with a truncation error of order h 3 [7]. The fictitious value y_ x can be eliminated from eqn. 9 using an appropriate numerical analysis technique on the mesh points y t {t) (see Reference 7). Similar results are obtained by symmetry at the boundary point x = 1. We obtain the parameters a = b = c = 2,h=\/(n-l)andy = (j^i • • • y n -i) T . The accuracy of these approximations can better be assessed in the control framework by comparing the dominant eigenvalues of the original and the approx- IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985 239
Transcript of Accuracy of time-optimal control of the diffusion equation using a spatial discretisation technique
Accuracy of time-optimal control of thediffusion equation using a spatial
discretisation techniqueA.S.I. Zinober. B.Sc.(Eng.), M.Sc.(Eng.), Ph.D.. C.Eng., M.I.E.E., A.F.I.M.A.
Indexing terms: Control theory, Optimal control, Optimisation, Computer simulation
Abstract: A well known numerical technique for calculating the optimal control of distributed parametersystems is the spatial discretisation (SD) procedure. In the paper an improved formulation of the SD scheme isstudied in the case of the boundary control of the diffusion equation. Significantly more accurate results areobtained with no additional computational complexity.
1 Introduction
The approximate optimal control of systems described bypartial differential equations may be determined by thewell known technique of spatial discretisation (SD) [1-3].This yields differential difference equations which simplifythe technique for finding the optimal solution by allowingthe standard optimal methods of ordinary differentialequations to be applied. The accuracy of the SD approx-imation is clearly of great importance in obtaining a solu-tion which is close to the optimal of the originaldistributed-parameter system. Clearly greater accuracy canbe obtained by increasing the number of grid points in thespatial discretisation. However, the actual form of the dif-ference approximation has tended to be ignored in thecontrol literature. A careful choice of this approximationcan lead to greatly improved accuracy with a smallnumber of grid points.
In this paper, we shall concentrate on the optimalboundary control of the one-dimensional diffusion (orheat) equation:
dydt o-£ (0 < x < 1, 0 ̂ t < oo)
with derivative boundary conditions
dy dy/ ( 0 , t)= -u(t) / ( I , 0 = 0ox ox
(1)
(2)
The arguments may be extended to other linear and non-linear distributed-parameter systems with suitable modifi-cations to the analysis. The time-optimal problem (PI) isto determine the control u(t) such that, in minimum timeT, the final distribution y(x, T) = 0 is attained from aninitial distribution y(x, 0) = yo(x). This problem has beendiscussed by numerous authors [1, 3, 4, 5] using a varietyof approximation techniques. The SD method [1-3] isconceptually simple and yields suitably accurate resultswith correct application. The optimal control of eqns. 1and 2 minimising a quadratic performance criterion (P2)can also be calculated using discretisation techniques[2, 6].
2 Spatial discretisation
An nth-order differential difference equation may beobtained from eqns. 1 and 2 using the central difference
Paper 4072D (Cl, C3, C7), first received 16th July 1984 and in revised form 15thApril 1985
The author is with the Department of Applied & Computational Mathematics, Uni-versity of Sheffield, Sheffield S10 2TN, United Kingdom
formula, giving the approximation to the original system
y(t) = Ay(t) + bu(t) (3)
where the components yt of the vector y(t) are the valuesy(ih, t) along the x-axis with the spacing h between adjac-ent grid points. We obtain
— a10
b-2
1
01
- 201 0
10
- 210
1- 2
b
01
— a.
(4)
and
- (c 0 0 0)T (5)
The values a, b and c are specified by the particularapproximation used to model the derivative boundaryconditions of eqns. 2. Sage [2] and Mahapatra [3], amongothers, approximate (dy/dx)(0, t) at the boundary x = 0 by
Axy(O,t)/h = {y(h,t)-y(O,t)}/h
which yields
y(h, t) - y(0, t) = hu(t)
(6)
(7)
yielding a truncation error of order h2 when determiningy(0, t) [7], where a = b = c = 1, h= l/{n + 1) and y =(y^i '"' yn)
T- This is the formula generally used in thecontrol literature [2, 3].
A far better approximation to the boundary condition(eqns. 2) at x = 0 is attained by introducing the fictitiousexterior mesh point y( — h, t) = y-l(t) and using thedivided mean central difference approximation
= {y(h,t)-y(-h,t)}/2h
which gives
y(h,t)-y(-h,t) = 2hu(t)
(8)
(9)
with a truncation error of order h3 [7]. The fictitious valuey_x can be eliminated from eqn. 9 using an appropriatenumerical analysis technique on the mesh points yt{t) (seeReference 7). Similar results are obtained by symmetry atthe boundary point x = 1. We obtain the parametersa = b = c = 2,h=\/(n-l)andy = (j^i • • • yn-i)
T.The accuracy of these approximations can better be
assessed in the control framework by comparing thedominant eigenvalues of the original and the approx-
IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985 239
imating systems. The exact eigenvalues of the system ofeqns. 1 and 2 are A,- = —i2n2(i = 0, 1, 2, ...). The approx-imation I (eqn. 6) yields eigenvalues A{ = — 4(n + I)2 sin2
{in/(2ri)} (see the Appendix), while the improved formula-tion II (eqn. 8) has eigenvalues A" = — 4(n — I)2 cos2
{{n - i - l)7t/2(n - 1)} for i = 0, 1, 2 , . . . , n - 1 [9].For i -4 n we obtain the dominant eigenvalues. The
errors in the values of Aj and A" can be shown for i ^ 0 tobe
and
e\ = A,- - A|- ~ ;27r22/n
and
ef = Xt - AJ1 ~ -i4n4/i2(n - I)2
(10)
(11)
The percentage errors lOOe'i/A^/o and lOOe'i/^1% a r e
plotted in Fig. 1. The approximation II is seen to be far
30r
20
10
SDH 500
-101-
Fig. 1 Percentage errors in eigenvalues of approximations SD I andSD IIlOOeV/,% SD IlOOe'.V/.o/o SD II
more accurate than I. For a required accuracy of less than1% in the value of X1 we need n = 11 for approximation IIand n = 201 for formulation I. Similar results hold fori = 2 , 3 , . . . .
For the time-optimal problem, Mahapatra [3] uses theSD approximation I with n = 19. The resulting system(eqn. 3) is reduced to a low-order system of n*th order byretaining the dominant eigenvalues, using the modelreduction technique of Davison [8]. The model reductionsimplifies the determination of the switching times. Thereduced-order systems for n* = 4 with state vector yR =(^4 J>8 yn yief are, respectively,
-9 .2332.32
-3.59-5.38
+
-
-1 .79-66.10
37.8016.40
0.438 •
0.410
1.6403.276.
u(t)
16.4037.38
-66.10-1.79
-5 .38"-3.5932.32
- 9 . 2 3 .
-12.6427.02
-3.38-2.79
+
-
_
i: 72-55.23
31.597.72
1.866"0.7900.2580.134.
u(t)
7.7231.59
-55.237.72
-2.79-3.3827.02
-12.64
In Table 1 for n* = 4, the approximate SD I [3] and SD IIvalues of the switching times 7] are listed for yo(x) = 1. Atthe final time (T = T4) the integral squared error is /* =jo[y(*> T)~\2 dx. Ideally y(x, T) = 0. The accurate resultsfrom Galerkin's method [5] are also listed in the Table.Method II yields suitably accurate results. The final dis-tributions y(x, T) shown in Fig. 2 are calculated from theanalytical solution of eqns. 1 and 2.
0.02
y(x, T) distributions— x — x — accurate [5]— # — • — SD II (this paper)
SD I Mahapatra [3]
3 Conclusions
The accuracy in the determination of the (approximate)optimal control clearly depends on the accuracy of theapproximating system (eqn. 3). In the time-optimalproblem, in particular, the sensitivity of the nominalsystem of eqns. 1 and 2 to 'parameter variations', in thiscontext the inaccuracies in the SD approximation of eqn.3, is very large [10-12]. As the formulation II leads to nogreater computational effort than I, it should be used whenapplying spatial discretisation procedures in the calcu-lation of optimal control.
The reader should, however, take care when applyingnumerical approximation techniques in optimal controlcomputations. Huntley [6] has shown that the less accu-
Table 1 : Eigenvalues, time-optimal switching times and final error integralA o A , A2 A 3 7", 7*2 7 ^ T^ I*
Accurate [5] 0SD II (this paper) 0Mahapatra SD I [3] 0
-9.87 -39.48 -88.83 1.074 1.167 1.192-9.85 -39.08 -86.82 1.074 1.168 1.194
-10.91 -43.35 -96.42 1.068 1.153 1.176
1.199 1.42 x i o - 5
1.201 1.44 x i o - 5
1.182 1.82 X 1 0 - 5
= 19
r, T)]2dx
240 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985
rate trapezoidal (integration) rule should be employed inthe quadratic cost function problem (P2) rather than theinherently more accurate Simpson's rule, as the latter leadsto spurious oscillations when solving the matrix Riccatiequation.
4 References
1 BUTKOVSKI, A.G.: 'Some approximate methods for solving prob-lems of optimal control of distributed parameter systems', Autom.Remote Control, 1961, 22, pp. 1429-1438
2 SAGE, A.P.: 'Optimum systems control' (Prentice-Hall, 1968)3 MAHAPATRA, G.B.: 'Time optimal control of linear diffusion
systems by a spatial discretization procedure', IEEE Trans., 1977,AC-22, pp. 481-482
4 McCAUSLAND, I.: 'Time optimal control of a linear diffusionprocess', Proc. IEE, 1965,112, pp. 543-548
5 PRABHU, S.S., and McCAUSLAND, I: 'Time optimal control oflinear diffusion processes using Galerkin's method', Proc. IEE, 1970,117, pp.1398-1404
6 HUNTLEY, E.: 'A note on the application of the matrix Riccatiequation to the optimal control of distributed parameter systems',IEEE Trans., 1979, AC-24, pp. 487-489
7 HILDEBRAND, F.B.: 'Finite-difference equations and simulations'(Prentice-Hall, 1968)
8 DAVISON, E.J.: 'A method for simplifying linear dynamic systems',IEEE Trans., 1966, AC-11, pp. 93-101
9 MITCHELL, A.R., and GRIFFITHS, D.F.: 'The finite differencemethod in partial differential equations' (Wiley, 1980)
10 ZINOBER, A.S.I., and FULLER, A.T.: 'The sensitivity of nominallytime-optimal control systems to parameter variation', Int. J. Control,1973,17, pp. 673-703
11 RYAN, E.P.: 'On the sensitivity of a time-optimal switching function',IEEE Trans., 1980, AC-25, pp. 275-277
12 BECKER, N.: 'A note on performance index sensitivity of time-optimal control systems', IEEE Trans., 1980, AC-25, pp. 819-821
5 Appendix
The n eigenvalues X of the matrix A/(n + I)2 of the SD Iapproximation satisfy
D =
0 + 1 11 0 10 1 0
= 0 (12)1 0 1 0
1 0 1
1 0 + 1
where 0 = — (A + 2). Expanding successively about thefirst and last rows leads to
D = (0 + l)2Tn_2(A) - 2(0 + l)Tn_3(A) + Tn_4(A) = 0 (13)
where Tm(X) is the mth-order determinant:
0 11 0 1
0 1 0 1
1 0 1 0
1 0 1
1 0
Expanding eqn. 14, we obtain the relationship
and substitution in eqn. 13 yields after simplification
(14)
Using an identity relationship for TH_i(X) [9] gives
D = -X(- (4 cos2 nj/2n + X)
(15)
(16)
(17)
So the eigenvalues of the matrix A are
X\ = -4{n + I)2 sin2 {in/In) i = 0, 1, 2, . . . , n - 1 (18)
IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985 241
