Ind. Eng. Chem. Res. 1995,34, 2077-2083
2077
A Simultaneous Iterative Solution Technique for Time-Optimal Control Using Dynamic Programming Solomon k Dadebo and K. B. McAuley* Chemical Engineering Department, Queen’s University, Kingston, Ontario, Canada K7L 3N6
A simultaneous iterative solution technique based on dynamic programming is presented as a reliable procedure for time-optimal control. By a simple transformation of the independent variable, t , the problem is converted to one of combined optimal parameter selection and optimal control. This approach allows us to find the optimal control policy that drives the system to the desired final state while searching for the optimal final time, tf, simultaneously. Admissible values of the final time are randomly generated within a search region, and a direct search for the optimal final time, tfo,is done iteratively only at the final time stage, with systematic region contraction, until convergence is attained. Significant savings in computation time can be realized using this approach instead of the traditional sequential approach.
Introduction In time-optimal control, the objective is to find a control policy that will drive a given system from its initial state t o a desired state in minimum time. This type of problem occurs very often in chemical engineering practice, especially in process start-up and batch operations. Some of the well-known methods available for time-optimal control are the calculus of variations, Pontryagin’s maximum principle, dynamic programming, linear programming, and some suboptimal methods based on Lyapunov’s direct method. Edgar and Lapidus’ (1972a,b) discrete dynamic programming algorithm utilizes a limiting process which obtains the singularhang-bang solution as the limit of a series of nonsingularhon bang-bang solutions. Their method involves solving the optimal control problem for each admissible candidate of the final time until a bang-bang control policy is obtained. Obviously, this approach could be very time-consuming if the initial guess is far from the optimum. Until quite recently, methods based on dynamic programming were limited to systems with low dimensionality. However, with the introduction of iterative dynamic programming (IDP) (Luus, 1989) which employs systematic contraction of the search region and thus dispenses with the need for a fine grid, the problems posed by dimensionality and the menace of grid expansion have been largely overcome. The successful application of IDP to time-optimal control problems has recently been demonstrated by Bojkov and Luus (1994a,b). However, our approach is different in the sense that the performance index is a direct function of the final time and is therefore expected to be more sensitive to the optimization scheme. Secondly, the sequential approach of Bojkov and Luus (1994a) is computationally less attractive since a “one-at-a-time” optimization scheme is employed. Essentially, a different optimal control problem is solved completely for each final time candidate before using a Regula Falsi optimization technique to improve upon the estimate of the final time. Their (Bojkov and Luus, 1994b) second approach is a simultaneous solution technique but differs from our proposed approach in three important ways: ~
_
_ _
_
~
~
~
* To whom
correspondence should be addressed. E-mail:
[email protected]. 0888-5885/95/2634-2077$09.00/0
1. Their performance index is not a direct function of the final time, and final state constraints are implicitly enforced via continuously differentiable penalty functions. Since the original performance index is a direct function of the final time, we prefer t o use it and preserve the sensitivity of the solution technique. We have chosen to use exact (absolute error) penalty functions because they have been shown to be superior to continuously differentiable penalty functions because they have been shown to be superior to continuously differentiable penalty functions (Dadebo and McAuley, 1995; Ledyaev, 1994). 2. To keep the problem computationally simple, they examine only the extreme values to effect bang-bang control based on the assumption that the optimal control is bang-bang in the entire time interval. This assumption is not generally valid for nonlinear systems, as we will demonstrate with the second example. 3. Their optimization technique involves finding optimal stage lengths (as decision variables) for bangbang control. Our decision variables are the final time (considered only at the last time stage) and the values of the manipulated variables. The aim, in this article, is to solve efficiently for the optimal control policy and optimal final time simultaneously for time-optimal control problems that exhibit bang-bang or more complicated control policies. The class of problems that can be handled with Bojkov and Luus’ (1994b)second method is restricted to those which do not exhibit either boundary control (where state constraints are enforced) or singular control. Since the economy of dynamic programming lies in its backward direct search optimization procedure, it is required that the final time, t f , be known during each iteration. However, for time-optimal control problems, tf is unknown. By a simple variable transformation, the problem is converted to one of a combined optimal parameter selection and optimal control. Here, the independent time variable is normalized and the final time becomes a decision variable only at the final stage of the IDP procedure. This approach allows us to solve the optimal control problem while simultaneously searching for the optimal value of tf. The same variable transformation approach has also been used by Bell and Katusiime (1980) (an integral form), Chew and Goh (1989),Chen and Hwang (1990),Rosen and Luus (19911, Teo et al. (19911, and more recently by Bojkov and Luus (1994b).
0 1995 American Chemical Society
2078 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995
Using IDP, the search is done by randomly generating admissible values of tf, within a search region. The procedure is performed iteratively, with systematic region contraction, until convergence is attained. The search for tf is done simultaneously with the optimal control problem which is also solved iteratively using dynamic programming. The optimal control policy and the minimum time may be continuously refined by using the control policy obtained after a chosen number of iterations as the initial policy for the next pass. The uncertainty as t o whether the principle of optimality is really satisfied vanishes as the search region becomes smaller. In fact, even after starting with a relatively coarse grid, the search region begins to collapse into a small one after only a few iterations and thus the fine grid requirement is satisfied.
performance index. Applying Bellman’s principle of optimality,P single stages are optimized in turn starting at the last stage instead of optimizing all P stages simultaneously. Thus, whenever the final K stages of the optimal control have been established, the control policy for the precedmg stage may be obtained by simply considering one new stage and then continuing with the already established control policy for the remaining k stages. A clipping technique is used to ensure that only allowable values of the control vector u are used for integration: UJ = UJ
if if
uJ aa. 5 uJ 5 b’,
j = 1 , 2 , ..., m
(10)
To satisfy the state constraints, we construct the following augmented performance index,
System Model and Problem Formulation Consider the system
dx(t)/dt = f(x(t),u(t)); x(0) = xo (1) where x(t) is an ( n x 1) state vector and u(t)is an ( m x 1)control vector which is bounded:
aj Iuj(t) IPj; j = 1, 2, ..., m
(2) The associated performance index to be minimized is I[X(O),tfI = tf subject t o some final state constraints
(3)
Xi(tf)= Xfi i = 1, 2, ..., 11; 11 In (4) and some continuous algebraic inequality constraints on the states h,[x(t),tl I0 i = 1, 2, ..., /I (5) Additional control move suppression constraints could also be imposed as in Dadebo and McAuley (1995). The problem is to obtain the control policy which will drive the system to the required final state in minimum final time, tf, so that the constraints given by eqs 2,4, and 5 are satisfied. Application of IDP to Time-Optimal Control Problems To use IDP it is required that the final time, tf, be known so that the time interval can be divided into stages. Since tf is unknown in the minimum time problem, we shall overcome this problem by normalizing the independent time variable: t=-
t
- to
tf - t o Without any loss of generality, we shall assume that t o = 0. Thus 0 It 5 1 and eq 1 becomes dx(t)/dt = tff(x(t),u(t)) where tf is expected to lie in some region:
(7)
(8) tf min tf 5 tf max Let us divide the given time interval ( 0 , l )into P equal subintervals (0, td,(TI,tal, ..., ( ~ ~ - 2tP-d, , (+I, 1) each of length L,such that
L = 1/P (9) The problem is to find a piecewise constant control , u(P-l), which minimizes the policy, d o ) , u(l),~ ( 2 1...,
where Pik is used to enforce state constraints: if hi[x(tk),tk1> 0 (12) if hi[x(tk),tJ I0 T,I is the number of terminally constrained state variables, xfi is the desired state of the ith state variable, and A is the number of state constraint functions. 5, Qi’s and ai’s are penalty function factors or weighting factors. Here, the original objective function, tf, was scaled using tfmax.Note that the augmented penalty function is a direct function of penalized (unknown)final time. The application of penalty functions to final state constrained optimal control problems was introduced by Luus and Rosen (1991) for use in the IDP algorithm. Luus (1991) also used penalty functions to solve problems with continuous state constrained problems. The work by Luus and Rosen (1991) and Luus (1991) employed continuously differentiable penalty functions. Since the objective function does not have to be continuously differentiable (smooth) when using IDP, it allows us t o employ absolute error (exact) penalty functions. The choice of a nonsmooth objective function to enforce constraints increases the probability of obtaining the exact solution (Dadebo and McAuley, 1995; Ledyaev, 1994). The analysis of exact penalty functions can be found in Fletcher (1987),Clarke (1979,1989),and Polak et al. (1983). A detailed IDP algorithm, including a graphical description, can be found in Dadebo and McAuley (1995) and will not be reproduced for the sake of brevity. However, the required modification for timeoptimal control is presented here. At the last time stage, during the backward pass, we allow the control u(P-1) and decision variable tf t o take on M allowable values inside the respective regions ru and rtf. The values of U P - 1 ) are randomly paired with M allowable values of tf for integration of the state equation (eq 7). For each grid point integration, we pick and store u(P1) and tf which minimize J. However, since the exact value oft? is still unknown, we take as the best estimate of tfo the one which gave the least value of J for the N grid points at the last time stage P. This choice will not necessarily make the set (u(P-l), tf),which is stored for the ith grid point optimal. To ensure uniformity, the optimization is continued by using the best estimate of t f for all the remaining time stages and grid points
hi[x(tk),tk1
P. I k = (0
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2079 1.2
0.8
0.4 3
2 +
0.0
c
0
0 -0.4
-0.8
0.0
0.2
0.4
0.6
0.8
Dimensionless Time Figure 1. Control policy for example 1.
-2
-4
1 .o
0 XI
Figure 2. Time-optimal phase plane for example 1
as we proceed in a backward direction. The estimate of tfo becomes the nominal value (to be perturbed) at the next iteration.
4.5 1
I
--
Illustrative Examples and Discussions Example 1: A Linear Time-Invariant TimeOptimal Problem. Let us consider the linear system given by Sage and White (1977). The problem is to find the minimum time control for the system:
dx,(t) - -&&I - x,(t) dt such that the cost function
--
+ u(t)
I = t,
~ ~ (=01) (13)
(14)
0
is minimized, with
I
4.0 1 5
10
15
20
Iteration Number
Xl(tf) = 0
lu(t)l 5 1
Figure 3. Final time versus iteration number for example 1
(15)
Preliminary runs with tf = 4 and tf = 10 had shown that the optimal final time was between these two times. We therefore selected these values as the limits for the final time: 4 I t f I 10 (16) (For practical problems, one may have good estimates of tfmin and tfmax based on experience or physical knowledge of the system.) Using IDP with P = 20, N = 27, A4 = 3, y = 0.70, E = 0.10, and the initial guess for the optimal control policy as u = -0.25 with a corresponding region of 0.50, we obtained a minimum time of 4.41. The initial guess for the minimum time was 4.2 with a corresponding search region of 1.0. The control policy is shown in Figure 1. The analytical solution to this problem is bang-bang. The numerical solution is almost bang-bang except for an intermediate value of u between 0.85 Iz I0.90 due to discretization of the time scale. This sort of behavior
will always occur unless the switching time for the analytical solution corresponds exactly to the beginning (or end) of a discretization time stage. It is our experience, however, that such intermediate values for control do not significantly affect the optimal solution if sufficiently fine discretization of the time scale is used (see for example, Dadebo and McAuley, 1994a, 1995). Alternatively, if a true bang-bang solution is required, the procedure proposed by Bojkov and Luus (1994b) which searches for the optimal stage lengths can be used. The state trajectory corresponding t o Figure 1 is shown in Figure 2. Consecutive open circles correspond t o the beginning and end of the time stages. The final time is plotted against the iteration number and shown in Figure 3. The control policy was refined after 15 iterations by doubling the number of stages, P, to 40. The final state constraints were satisfied to within 0.000 000 1for both states after 25 iterations. Although we used a nonsmooth objective function in obtaining the optimal control and the minimum time, it was noted that since the final time is unknown, it is advisable to make the penalty function factor, E, relatively small to increase the sensitivity for final constraint violations.
2080 Ind. Eng. Chem. Res., Vol. 34,No. 6 , 1995
In this example we used w 1 = w2 = 4, in agreement with our previous guidelines for choosing absolute error penalty function factors for cases where tf is not free (Dadebo and McAuley, 1993, 1994a,b, 1995). Example 2: A Nonlinear Time-Optimal Drug Displacement Problem. This is an interesting drug displacement problem considered by Aarons and Roland (19811, Bell and Katusiime (19801, Maurer and Wiegand (19921, and more recently by Bojkov and Luus (1994a,b). The problem is based on a kinetic model which simulates the interaction of two drugs, warfarin and phenylbutazone, in a patient’s bloodstream:
&,-- D2[C,(0.02 - x,) dt
+ 46.4X1(u - 2x2)l
(17)
c 3
Table 1. Comparison of Results with Those Reported by Maurer and Wiegand (1992) tfmax tfmln tflnltlai 212 0.0287c 240 180 212 240 180 0.0285 230 280 180 0.027 250 300 200 0.026 300 350 250 0.025 310 350 250 0.0247 320 380 300 0.0245 460 500 400 0.023 660 700 620 0.022 c
+ O.&, + O.&, C, = 0’+ 232 + 4 6 . 4 ~ ~ C, = 0’+ 232 + 4 6 . 4 ~ ~
15 20 20 15 20 20 25 20 20
optimal t f MWQ IDPb 221.466 222.587 240.061 262.637 299.279 314.295 325.347 461.809 674.252
219.772 220.066 240.983 263.619 298.016 311.12 325.553 463.47 672.429
no. of
iterations 17 15 14 16 16 16 18 7
1
a MW, Maurer and Wiegand (1992). Final state constraints Maximum unconstrained warfarin satisfied at least within concentration (0.028 704 64).
I
4-46.k2(o.02 - XI)] &, - D2[cl(u - a,) -(18) dt c 3
D
rtf
I
=1
6
a
C3 = C,C, - ( 4 6 . 4 ) 2 ~ 1 ~ ,
(19) The initial condition and the final time constraints on the states and control are = X1(tf)= 0.02
(20)
x,(O) = 0; X2(tf)= 2
(21)
X,(O)
0 Iu(t) I8 (22) In addition, we have a continuous state constraint on
2
0
x1(t):
x,(t) I E
v 0 I t It f ( €’0.02)
.o
Figure 4. Comparison of control policy for example 2 with
6
(23)
and x2 are the concentrations of unbound warfarin and phenylbutazone, respectively. We note, here, that Bojkov and Luus (1994a,b) considered the case without a state constraint (eq 23). When the anticoagulant drug warfarin is injected into a patient’s bloodstream, only a portion circulates throughout the body while the remainder resides in plasma and tissue protein binding sites. However, if a pain-killing drug, such as phenylbutazone, is then injected into the same patient, it displaces the bound warfarin which consequently results in an increase of the unbound warfarin concentration. It is therefore necessary to ensure that the warfarin concentration is lower than a certain toxic level, E . It is also known that the faster the rate of injection of phenylbutazone, u, the higher the resulting peak concentration of warfarin. The problem, therefore, is to control the rate of infusion of phenylbutazone so that the drugs reach steady state in the blood stream in minimum time while ensuring that the warfarin concentration does not rise above a certain specified level (eq 23). Initially, the unbound warfarin concentration is assumed to be at steady state. The optimal solution exhibits boundary subarcs by virtue of the state constraint. In applying the maximum principle, the formulation of the Hamiltonian requires the inclusion of a Lagrange multiplier and a slack variable in addition to the adjoint variables in order to
XI
1
0.0
0.2
0.4
0.6
0.8
Dimensionless Time
=
0.026.
handle the continuous state constraint on XI. As noted by Bell and Katusiime (1980) and Maurer and Wiegand (1992),the solution of this problem is nontrivial and “the numerical solution exhibits some unusual features”. Maurer and Wiegand, who solved this problem for different values of 6, concluded that the failure to find any existing sufficient conditions suitable for this problem should constitute a challenge to devise new types of sufficient conditions for control problems with control appearing linearly. Using IDP, after scaling the time, with P = 20, N = 21, M = 7, y = 0.70, we obtained the results shown in Table 1. It is noted that our present results are very similar to those obtained by Maurer and Wiegand and very few iterations are required for convergence. The optimal final time is very sensitive to the final state specification as was noted by Rosen et al. (1987). Our goal is not to determine optimal final times which may be better than those reported by Maurer and Wiegand. Rather, we wish t o demonstrate that the IDP algorithm is very fast, reliable and robust. The control policy and the state trajectories for warfarin and phenylbutazone are shown in Figures 4, 5, and 6 respectively for E = 0.026 as reported in Table 1. The final states for X I and xz were 0.020 003 7 and
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2081 270 0.026
265
0.024 X
-
J 260
E
.-
k
-
0.022
0
.-
255
LL
250
0.020
0.0
0.2
0.4
0.6
0.8
1 1 .o
Time Figure 5. State trajectory for warfarin with 6 = 0.026 for example Dimensionless
2.
4
2
1
0
5
10
15
20
Iteration Number Figure 7. Final time versus iteration number for example 2 with E = 0.026.
used by Bojkov and Luus (1994a). As shown in Table 2, we were able to obtain the solution considerably faster. Example 3: A Nonlinear Two-Stage CSTR in Series. Consider the nonlinear CSTRs in series described by Siebenthal and A r i s (1964)with heat transfer controllers and used by Edgar and Lapidus (1972),Luus (19741, Rosen and Luus (19911, and recently by Bojkov and Luus (1994a) for optimal control studies:
3
x"
245
dx,/dt = -3X1
i/
dx,/dt =
+R,
+
- 1 1 1 . 1 5 5 8 ~ ~ R, - 8.1558(x2
(24)
+ 0 . 1 5 9 2 ) ~(25) ~
0 0.0
0.2
0.4
0.6
0.a
1 .o
Dim en s i 0 n less Time
Figure 6. State trajectory for phenylbutazone with 6 = 0.026 for example 2.
2.000 000, respectively. The convergence rate for c = 0.026 is shown in Figure 7.' In all the cases considered the convergence was very rapid and no computational difficulties were encountered. Also, for all the different values of c shown in Table 1, the constraints were satisfied at least within The average CPU time for all the cases considered for the drug displacement problem was no more than 36 s per iteration on a 4861 33 Packard Bell computer. The piecewise constant control policy is very similar t o the one reported by Maurer and Wiegand. It is noted that the non bang-bang (boundary) optimal control policy for this state constrained case is about 60% of the final time. Therefore, the advantage of this proposed procedure over the method proposed by Bojkov and Luus (199413)which does not consider both singular and boundary control is demonstrated. We have also demonstrated that the simultaneous solution technique is considerably faster than the sequential technique
where A, = 1.5 x lo7; A, = 1.5 x 10,'
B , = x,
+ 0.6932;
B , = x4 + 0.6560
(28) (29)
R, =A,(0.5251 - x,) exp(-lOIB,) A2(0.4748 + x,) exp(-15/B1) - 1.4280 R, = A,(0.4236 - x,) exp(-lO/B,) A,(0.5764
-
+ x 3 ) exp(-15/B2) - 0.5086
(30)
The problem is to find the constrained control, -1 4 ui 5 1, i = 1, 2, which drives the system from the initial state ~ ( 0=) [0.1962, -0.0372, 0.0946, 0.0000IT to the final state (with precision tolerance), Ixi(tf) 5 (i = 1,2, 3, 4), in minimum time. x1 and x3 are normalized concentration variables, x2 and x4 are normalized temperature yariables, and u1 and u2 are normalized controls. Edgar and Lapidus (1972), Luus (1974),and Rosen and Luus (1991)obtained a minimum
2082 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 Table 2. Comparison of Results with Those Obtained by Bojkov and Luus (1994a) parameters present work present work" Drug Displacement Problem (Example 2) 21 3 N 7 15 R 20 10 P [180, 2401 [220, 227.51 [tf min, tf maxl 10.2 min 2.0 min CPU time 219.772 220 optimal final time, tfo Two-Stage CSTR Problem (Example 3) 17 5 N 15 15 R 20 10 P [0.2, 1.01 r0.32, 0.331 [tf minr tf max1 Microsoft DOS 5.0 operating system LAHEY FORTRAN 5.01 FORTRAN compiler Runge -Kutta 4(5) integration subroutine RND (built-in) random number generator 486DX, 33 MHz machine 63 min 8.64 minb CPU time 0.321427 0.327131b optimal final time, tfo
Bojkov and Luus 3 15 10 [220,227.51 19 min 221.956 5 15 10 10.32. 0.331 IBM osi22.0 Microsoft FORTRAN 5.0 DVERK URAND (built-in) 486DX2,66 MHz 180 minb 0.37215b
Same initial conditions as used by Bojkov and Luus (1994a). Final state precision tolerance = 1.0 x lo-*.
a
optimization method is at least 21 times faster. We expect that this difference will become more pronounced for higher dimensional nonlinear systems.
0.35
Conclusions .+.
-
0.30
.-? k-
0
.-LLC
0.25
0.20
0
4
0
12
16
20
Iteration Number
Firmre 8. Final time versus iteration number for examde 3.
time of 0.342, 0.325, and 0.320 (using 20 time stages with a nonlinear programming formulation), respectively. Using P = 20, N = 17, 6 = 0.01, wi = 1.0 (i = 1, 2, 3, 41, y = 0.70, and u1 = u2 = -0.25 as the initial guess of the optimal control policy, we obtained a final time of 0.321, as shown in Figure 8, which is in agreement with the previously reported results. The initial control region was 0.50 and R, the number of randomly generated values for control and final time, was chosen to be 15. The initial guess for the range of the final time was between 0.20 and 1.0. The CPU time for 15 iterations was 63 min. The optimal control policy is very similar to that reported by Rosen and Luus (1991) using nonlinear programming. In order to show the computational efficiency of the proposed simultaneous procedure, a comparison with the results obtained by Bojkov and Luus (1994a) (with precision tolerance, Iri(tf)5 i = 1,2,3,4), using their method of searching for optimal control policy and final time sequentially, is shown in Table 2. Even with a very wide initial region of uncertainty in the unknown final time, the proposed procedure is very reliable and computationally more attractive. Note that even on a slower computer the proposed simultaneous
The variable transformation approach used in this article dispenses with the need t o know the final time before IDP could be used. Therefore, by transforming the problem to one of a combined optimal parameter selection and optimal control, significant savings in computation time can be realized compared to the case where a different optimal control problem is solved for each admissible candidate of the final time. The proposed algorithm converges very quickly even with a very large initial region of uncertainty of the final time. For all the cases considered, no difficulty was encountered with convergence to the optimum. There was no need to go through the mathematical rigor of establishing sufficient conditions for the boundary control problems encountered in example 2. It is envisaged that the proposed IDP procedure could well be used for solving minimax optimal control problems where some function of the final states is to be optimized in minimum time. For two of the examples studied, the proposed simultaneous optimization technique was at least 9 and 21 times faster than the sequential approach used by Bojkov and Luus (1994a). The proposed approach is computationally attractive as is the bang-bang method of Bojkov and Luus (1994b). However, the proposed approach is applicable to a larger class of problems and is more sensitive to finding the optimal final time, even for a reasonably large size of the initial range of uncertainty in the value of the final time. Therefore, for state-constrained problems for which the solutions are not known, the proposed procedure is recommended.
Nomenclature f = vector of n functions of x ( t ) and u(t) I = performance index to be minimized J = augmented performance index k = index used to denote time stage L = length of time stage m = dimension of the control vector M = odd number of allowable values of each control variable at each time stage n = dimension of the state vector
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2083
N = odd number of grid points for x P = number of time stages chosen R = number of randomly generated allowable values for control or final time ru = region allowed for the control rtf = region allowed for the final time t = time u = ( m x 1)control vector x = ( n x 1)state vector q = initial condition of the state vector Greek Letters a = lower bound on control ,8 = upper bound on control y = amount by which the control region is contracted after each iteration E = upper bound on warfarin drug concentration, X I = number of final state constrained variables A = number of constrained states e = penalty function factor for enforcing state constraints 6 = penalty function factor on final time w = absolute error penalty function factor Subscripts
f = final i = index for entry in vector k = index for time at stage k 0 = initial Superscripts
’ = optimal T = transpose Literature Cited Aarons, L. J.; Roland, M. Kinetics of Drug Displacement Interactions. J . Pharmacokinet. Biopharm. 1981,8, 181-190. Bell, D. J.; Katusiime, F. A Time-Optimal Drug Displacement Problem. Optimal Control. Appl. Methods 1980, 1, 217-225. Bojkov, B.; Luus, R. Application of Iterative Dynamic Programming to Time Optimal Control. Chem. Eng. Res. Des. 1994a, 72, 72-80. Bojkov, B.; Luus, R. Time-Optimal Control by Iterative Dynamic Programming. Znd. Eng. Chem. Res. 1994b, 33, 1486-1492. Chen, C.-T.; Hwang, C. Optimal Control Computation for Differential Algebraic Process Systems with General Constraints. Chem. Eng. Commun. 1990,97,9-26. Chew, E. P.; Goh, C. J. On Minimum Time Optimal Control of Batch Crystallization of Sugar. Chem. Eng. Commun. 1989, 80, 225-231. Clarke, F. H. Optimal Control and the True Hamiltonian. SZAM Rev. 1979,21 (21, 157-166. Clarke, F. H. Optimization and Nonsmooth Analysis; Les Publ. CRM: Montreal, PQ, 1989. Dadebo, S. A,; McAuley, K. B. Minimum Energy Control of Timedelay Systems Via Iterative Dynamic Programming. Proc. Am. Control Conf. 1993,2, 1261-1265.
Dadebo, S. A.; McAuley, K. B. Iterative Dynamic Programming for Minimum Energy Control Problems with Time-Delay. Optimal Control Appl. Methods 1994a, in press. Dadebo, S. A.; McAuley, K. B. Dynamic Optimization of Constrained Chemical Engineering Problems Using Dynamic Programming. Book of Abstracts, 44th Canadian Chemical Engineering Conference, Calgary, AB, 1994b. Dadebo, S. A.; McAuley, K. B. Dynamic Optimization of Constrained Chemical Engineering Problems Using Dynamic Programming. Comput. Chem. Eng. 1995,19, 513-525. Edgar, T. F.; Lapidus, L. The Computation of Optimal Singular Bang-Bang Control I: Linear Systems. M C h E J . 1972a, 18, 774-779. Edgar, T. F.; Lapidus, L. The Computation of Optimal Singular Bang-Bang Control 11: Nonlinear Systems. AZChE J . 197213, 18, 780-785. Fletcher, R. Practical Methods of Optimization, 2nd ed.; John Wiley & Sons: New York, 1987. Ledyaev, Y. Private communication, Queen’s University, Kingston, Ontario, 1994. Luus. R. ODtimal Control bv Direct Search on Feedback Gain Matrix. khem. Eng. Sci. i974,29, 1013-1017. Luus, R. Optimal Control By Dynamic Programming Using Accessible Grid Points and Region Contraction. Hung. J . Znd. Chem. 1989,17, 523-543. Luus, R. Application of Iterative Dynamic Programming to State Constrained Problems. Hung. J . Znd. Chem. 1991, 19, 245254. Luus, R.; Rosen, 0.Application of Iterative Dynamic Programming to Final State Constrained Problems. Znt. Eng. Chem. Res. 1991,30, 1525-1530. Maurer, H.; Wiegand, M. Numerical Solution of a Drug Displacement Problem with Bounded State Variables. Optimal Control Appl. Methods 1992, 13, 43-55. Polak, E.; Mayne, D. Q.; Ward, Y. On the Extension of Constrained Optimization Algorithms from Differentiable to Nondifferentiable Problems. SZAM J . Control Optim. 1983, 21 (21, 179203. Rosen, 0.;Luus, R. Evaluation of Gradients for Piecewise Constant Optimal Control. Comput. Chem. Eng. 1991, 15, 273-281. Rosen, 0.;Imanudin; Luus, R. Final State Sensitivity for TimeOptimal Control Problems. Znt. J . Control 1987, 45, 13711381. Sage, A. P.; White, C. C., 111. Optimum Systems Control; PrenticeHall: Englewood Cliffs, NJ, 1977. Siebenthal, C. D.; Aris, R. The Application of Pontryagin’s Methods to the Control of a Stirred Tank. Chem. Eng. Sei. 1964, 23, 729-746. Teo, K. L.; Goh, C. J.; Wong, K. H. A Unified Computational Approach to Optimal Control Problems; Longman: Harlow, England, 1991.
Received for review August 11, 1994 Revised manuscript received March 1, 1995 Accepted March 22, 1995@ IE940485Y Abstract published in Advance A C S Abstracts, May 1, 1995. @
