Nonetheless, computational times have been about a n order of magnitude better than previously published values in some cases.
Y
Acknowledgment
E,
During a portion of this work Carl D. Eben was a National Science Foundation Cooperative Fellow. Much of the computational work was carried out at the University of Delaware Computing Center. The authors express thanks t o both organizations.
P
0
x P
U
INDICES
i, j , k , m, n, N , r
constants or dependent variables, dimensionless chemical species arithmetic mean and its conjugate activation energy, cal. function, dimensionless geometric mean and its conjugate chemical rate constant, appropriate dimensions invariant, dimensionless product function; chemical species chemical species gas constant, 1.987 cal./g. mole K. sum function temperature, K. real variable, dimensionless chemical species GREEK LETTERS a
P
= weighting factor = ky,Equation 21 = reactor volume, liters = invariant = value of species A relative to B = algebraic operators = ratio of chemical rate constants = +1 or - 1 ; dimensionless rate constant
17
- constant; adjoint variable - adjoint variable -
= integers within specified limits
literature Cited
Ark, Rutherford, “Optimal Design of Chemical Reactors,” p. 85, Academic Press, New York, 1961. Denbigh, K. G., Chem. Eng. Sci. 8, 125 (1958). Denn, M. M., Ark, Rutherford, IND.ENG.CHEM.FUNDAMENTALS 4, 7, 213, 248 (1965). Duffin, R. J., Peterson, E. L., Zener, C. M., “Geometric Programming,” Wiley, New York, 1967. Eben, C. D., Ferron, J. R., A.1.Ch.E.J. 14, 32 (1968). Eben, C. D., Ferron, J. R., J. Opt. Theory A p p l . , in press, 1969. Fan, L. T., Wang, C. S., “The Discrete Maximum Principle,” Wiley, New York, 1964. Ilildebrand, F. B:, “Introduction to Numerical Analysis,” p. 443,McGraw-Hill, New York, 1956. Passy, Ury, Wilde, D. J., S.I.A.M.J. Appl. Math. 16, 1344 (1967). Scarborough, J. B., “Numerical Mathematical Analysis,” pp. 199-31 1, 557-8, Johns Hopkins, Baltimore, 1958. Stanton, R. G., “Numerical Methods for Science and Engineering, p. 81, Prentice-Hall, Englewood Cliffs, N. J., 1961. Storey; C., Chem. Eng. Xci. 17,-45 (1962). RECEIVED for review June 28, 1968 ACCEPTEDFebruary 10, 1969 Division of Industrial and Engineering Chemistry, 155th meeting, ACS, San Francisco, Calif., April 1968.
T I M E - O P T I M A L CONTROL OF A C L A S S OF L I N E A R D I S T R I B U T E D - P A R A M E T E R PROCESSES H E N R Y C. LIM School of Chemical Enganeering, Purdue University, West Lafayette, Ind. 47907 Time-optimal controls of a class of linear distributed-parameter processes having a transfer function of the form n
[I -
e--(a+a)bI/
H (s + p
~ ,
pt
2
o
i=l
are derived. Once the output has been brought to a desired level in minimum time by a “bang-bang’’ type control, a periodic type of control with period b and decaying amplitude (therefore, an infinite number of switches between nonextremal values) is required to maintain the output at the desired level. Illustrative examples are given, and, in some simple cases, exact conditions are given on the magnitude of new operating levels which correspond to minimum time being greater than b as well as less than b.
TIME-OPTIMAL control or regulation of lumped-parameter processes has been treated very extensively by a number of workers, and one may refer to Pontryagin et al. (1962), for example, or t o the survey paper by Athans (1966). Timeoptimal “bang-bang” control of time-invariant linear lumpedparameter processes has been treated by various approaches (Athanassiades and Smith, 1960; Bellman et al., 1956; Desoer, 1959; Eaton, 1962; Knudsen, 1964; LaSalle, 1959;
Neustadt, 1960; Schmidt, 1959). However, very little is known on time-optimal controls of linear distributed-parameter processes. Goldwyn et al. (1967) treated time-optimal control of one-dimensional diffusion processes with a n amplitude constraint on the control. They assumed that the timeoptimal control is bang-bang, and also used the concept of entire function tro calculate finite numbers of switching times among the infinite number required. Koppel (1966b) VOL.
8
NO.
4
NOVEMBER
1969
757
showed that the time-optimal control of a tubular process with a transfer function, (1 - e'+)/s, required an infinite number of switchings and stated that this is because the transfer function results from a distributed-parameter system and may be crudely regarded as a lumped system of infinite order. Koppel (1967) also considered time-optimal control of distributed-parameter processes with transfer function C (s j -Jf (s)
- k3[l
-
(s
kl)
+
Equation 1 may be written in the time domain dnc ( t ) dt"
+
-
an-1
d"'c ( t ) -. . . dtn-'
dc (t 1 + a1 + aoc ( t ) = dt
k[m ( t ) - e-abm( t - b)u (t - b ) ]
1 t=O (s
+ k2)
for limited cases where the minimum time was less than unity. The time-optimal control was constructed from that of a related lumped-parameter process, and it was stated that there are some difficulties associated with this approach and that with this method it was not clear how to proceed if the minimum time is greater than unity. In the present paper a simpler approach is presented which will lead to complete solutions for cases where the minimum time is greater or less than unity. Using the very definition of time-optimal control-Le., to drive the output to a new desired operating level in minimum time and to maintain the output there thereafter-time-optimal controls for a class of linear distributed-parameter processes are derived in a simple and straightforward manner. This approach can handle a more general class of processes with transfer function
where all a, are real nonnegative constants and u (t - b ) is a delayed unit step function. The coefficients ai are related by p , , in that - p i are the roots of the characteristic equation sn
+ an-lsn--l +. . . + + a, = 0
(4)
a1s
A state representation of Eqbation 3 may be written as
x2(t),
1 0
0
1
...
0
fo IO
+ 1.
where p , and z3 are real and nonnegative. However, to make the analysis as simple as possible, we restrict it to the processes with the following transfer function
where p, are real and nonnegative and a and b are positive.
The processes which can be represented by the transfer function of Equation 1 have been vel1 documented by Koppel (1967), and we refer to his work for details. Harriot (1964) showed that Equation 1 with n = 2 , pl = a closely represents the transfer function relating the exit temperature of a double-pipe heat exchanger t o the steam temperature. Koppel (1966a) has shown that a class of chemical reactors and heat exchangers, when subjected to changes in flow rate, shows a transfer function represented by Equation 1 with n = 1, pl = 0,a = 0. This same transfer function was also shown by ICoppel (1967) to represent the response of the exit temperature of a tubular heat exchanger due to wall flux manipulation. He also show-ed that the response due to wall temperature manipulation can be represented by Equation 1 with n = 1, p1 = 1, a = 1. Thus, a broad class of distributed parameter processes can be represented by Equation 1. Time-Optimal Control Problem
The problem is to drive the output c ( t ) of a tubular process whose transfer function is given by Equation 1 from its initial value of zero to a new desired operating level, r, in minimum time, t*, and maintain it there thereafter by manipulating a control variable m ( t ), which is constrained by -h 0 the timeoptimal m* (1) is given by
m* ( t ) = H [ u ( t ) - u ( t - tl)] - h[u (t - t l ) - u (t - t 2 ) ]
H [ ~ ( t - t t , ) - ~ ( ( t -f 3 ) ] - h [ u ( t - - 3 ) - ~ ( t - - t l ) J + H [ u ( t - t,+1) - u (t -
t*)]
+
+
...+
n
aj-lzj*u(t
- t*),
0 5 t 5 t*
(8)
j=1
Once x* is known, the switching times and minimum time can be determined, for instance, by the method proposed by Schmidt (1959). The determination of x* = x(t*), which corresponds to c ( t * ) = r and dic(t*)/dti = 0, i = 1, 2, . . . , n - 1 can be made by solving Equation 6 and its ( n - 1) derivatives a t t = t*. At t = t*, Equation 6 becomes
- e-abzl(t* - b ) u ( t * - b ) ] = r
c ( t * ) = k[zl(t*)
(9)
If t* is less than b, the second term in the right-hand side of Equation 9 drops out, and XI* can be readily obtained, while if t* 2 b, the determination of z1* is not a simple matter. Hence, we now treat these two cases separately. Case I. t* < b. For this case the solution of Equation 9 is z l ( t * ) = zl*= r/k (10) and z1( t * ) , i = 2, 3, . . . , n, are obtained by differentiating Equation 6 (i - 1) times and evaluating a t t = t*. They are zi(t*) = 0, i = 2 , 3 , . . ., n (11) Combination of Equations 10 and 11 yields x ( t * ) = x* = ( r / k , 0, . . ., O)T
(12)
Hence, when t* < b, the determination of m* ( t ) , 0 5 1 5 t*, is an ordinary time-optimal set point change of a lumped process, and is completely determined by Equation 5 with its initial conditions, the final conditions given by Equation 12, and the time-optimal control form given by Equation 7 or 8, whichever is appropriate. Case 11. t* 2. 6. For this case Equation 9 may be written as c ( t * ) = k[zl (t*) - c G b z(t* 1 - b)] = r (13) and therefore, t.he corresponding derivatives of c ( t ) are dic (t*)/dti =
(t*)
- e-%i+l (t* - b ) ] = 0 , i = 1 , 2 , ..., ( n - 1)
or xi+l(t*) = e-'%i+l(t*
- b),
i = 1, 2, . . ., (n - 1)
(14)
Thus, the desired state x ( t * ) = x* must be determined from Equations 13 and 14. However, to do this, one must know ~ ( t over ) the interval 0 _< t _< t*, so that x(t* - b ) can be expressed in terms of x ( t " ) , or both x ( t * ) and x (t* - b ) can be evaluated so that Equat'ions 13 and 14 can be solved for t*, which, in turn, allows one to evaluate x(t*). However, the time function x ( t ) , 0 5 t 5 t*, depends upon the actual
m"(t) used. Hence, we propose the following approach, Let us assume, for the time being, that x(t*) = x* has been determined. Then, the time-optimal control, m* ( t ) , which will drive x ( t ) from its initial zero value to the desired point, x*, is of bang-bang type because in terms of the state representation given by Equation 5 this is a time-optimal control of time-invariant lumped-parameter process. Hence, the time-optimal control is given by Equation 7 or 8 depending upon whether n is even or odd. Using this as an input, Equation 5 can be integrated to obtain x(t) in terms of t,, i = 1, 2, . . . , n, t", and x*. However, as stated earlier, t, and t* are functions of x*-Le., t z ( x * ) and t * ( x * ) . Therefore, x ( t ) can be expressed as x ( t ; x*). Substitution of this into Equations 13 and 14 yields x*. Once x* is determined in this manner, all t,, i = 1, . . . , (n - l ) ,and t* also become known. Hence, m * ( t ) , 0 _< t 5 t*, t* > b, is determined completely. Of course, if the initial state is on switching boundary, requiring less than ( n - 1 ) switchings, some of ti are zero and the above process becomes more complex, since without knowing the terminal state, one does not know if the initial state is on switching boundary. However, in reality, for low order process, one can possibly check all possibilities and pick the one which satisfies all equations. For instance, if n = 2 and r > 0, one can try m ( 1 ) = H[u ( t ) - u (t - t l ) ] - h[u ( t - t l ) - u ( t - t * ) ] or m ( t ) = H [ u ( t ) - ~ ( -t t * ) ] . There is a difference between the cases where t* < b and t* 2 6. When t* < b, the time-optimal control t o bring the output to a desired level is bang-bang and switching times and the minimum time are determined by the usual methods for lumped-parameter system. Although a similar type of control is required for the case t* 2 b, somewhat involved analysis is required, since the terminal state is not known explicitly. Control to Maintain Output at Desired level
We now proceed to derive a suitable control, m*(t), to maintain the output a t the new desired level, r. For convenience, the cases where t* < b and t* 2 b are dealt with separately. Case I. t* < b. Since dic(t)/dti = 0, i = 1, 2, . , n - 1 and c ( t ) = r for all t > t*, we obtain from Equation 3
aor = k [ m ( t ) - e-abm(t - b ) u ( t - b ) ]
t
> t*
which is equivalent to
m(t)=
i
(aurlk 1
(a,r/k)
t* t* In accordance with Equation 17, the time-optimal control for the interval t* < t < t* + b can be obtained by multiplying m ( t ) = (a,r/k)
by a factor of e-ab the time-optimal control for the time interval t* - b < t t* and then adding a,r/k to the resultant. The time-optimal controls for t* nb < t 5 t* (n f l ) b , n = I , 2 , . . . , 03 are obtained from the time-optimal control of the previous time interval, t* ( n - 1 ) b 5 t < t* nb, i = 1,2, . . . , 0 3 , by the same procedure. Equation 17 can be realized under the conditions given by Equation 16. There is a difference between this and that for t* < b as given by Equation 15. T h e n t* < b, m*(t) is a periodic-type function over the entire interval 12 0, while a periodic type control for the interval t > t* - b is required when t* 2 b.
t*, required t o maintain the output a t T . Thus, Case I. t* < b
<
t*, into Equation 21,
t 2 t*
plr = k[m* ( t ) - m* (t - b ) u ( t - b ) ]
t26
where ti, i = 1, 2 , . . . , ( n - l ) , and t* are determined completely by the usual procedure. Case 11. t* 2 b
m*(t) =
which implies
which is equivalent to
m*(t) =
rlr’k plr/k
t*
(19)
where t i , i = 1, 2 , . . . , (n - l ) , and t* must be determined by the somewhat involved analysis given in the previous section.
t*< t < b
+ ePabm*(t - b )
Some examples of time-optimal control of tubular processes are presented below. In all examples, the objective is to determine the time-optimal control of the process with the given transfer function, which will transfer the output from its initial zero steady state to a new desired operating point, r > 0, in minimum time. In some cases exact conditions on r for which t* 2 b are given, as well as some cases in which t* cannot be greater than b.
2b
In view of Equation 25, this control is subject to the following constraint, r kH (1 - e-ab)/pl (291
b
I
m * ( t ) = .ar/k
4b
3b
(a) r < b k H , t*< b; r -
b
0
' H [ z L ( ~.- ) ~ ( -t t * ) ]
I
0 _< t
< t*
t*_< t
(0
>
By combining Equations 68 and 71, the time-optimal control obtained and given by
IS
t
H [ u ( t ) - u (t -- t l ) ] - h[u (t - t l ) - u ( t - t * ) ] 0 5 t < t*
=
ITl*(t)
t*
