T I M E OPTIMUM CONTROL OF SECOND-ORDER OVERDAMPED SYSTEMS WITH TRANSPORTATION LAG L. B. KOPPEL AND P.R. LATOUR Purdue Uniuersity, Lafayette, Ind. Time optimum control of a second-order overdamped system with transportation lag was studied for given maximum and minimum constraints on the manipulated variable. Pontryagin's maximum principle and a phase plane analysis were used to derive the two-position control law which would drive the system output to its dlesired value in the shortest time. A two-position programmed controller, based on switching time, was synthesized on an analog computer. This controller verified the optimum control law and provided significant response time improvement over a well tuned proportional-integral-derivative industrial controller. The degree of improvement is best for large set point changes on processes with large t h e constants, large transportation lags, or narrow limits of asymmetric saturation. The results of this study are useful for startups and for transitions between steady-state operating conditions, for the class of processes whose dynamics are adequately represented as second-order with delay.
process dynamics, as distinguished from those of mechanical or electrical systems, are characterized by large time constants, distributed parameters, and sluggish response. Although thi: criterion for optimum control in the chemical industry is generally maximum profit, a criterion of minimum time would seem to be almost equivalent and is clearly more convenient for analysis. The time optimum programmed control function specifies the manipulated variable, M ( t ) , that drives the process output, C ( t ) , to the desired value as rapidly as poscible. The optimum feedback control law is the manipulated variable specified as a function of the output, rather than of time. I n this work, control of single manipulated input-single controlled output processes, where dynamics may be represented by a transfer function of the form HEMICAL
is considered. We assume a unity process gain with no loss of generality. C(s) and M ( s ) are the normalized, transformed process output and input variables, T is the predominant process time constant, b T is the smaller process time constant (0 < b < I ) , and U T if3 the transportation lag or dead time. Since the effects of distributed parameters and higher order dynamic characteristics may often be lumped with the delay constant, U T ,this transfer function has been found to represent adequately the behavior of many chemical process operations, including a liquid-liquid extraction column (2), a cracking furnace (729, heat exchangers, pneumatic transmission lines, agitated mixing vessels (70), and large distillation columns (75). T h e three dynamic parameters, T , 6 , and a, are assumed to be known. Methods for fitting these parameters by practical dynamic testing, to processes whose true (unknown) dynamics are more complex than the transfer function of Equation 1 , have received extensive recent attention as reported in the monograph of Hougen (70). In contrast with recent studies on synthesis of time optimum controllers (4, 9), a major objective of the present study is a practically implementable conbroad class of processes, without troller, applicable to s~
necessity for knowledge of the theoretical process dynamics, which are usually impossible to obtain. Since Equation 1 has been found to be broadly applicable and experimental procedures for fitting the parameters have been established, the results of the present study should be applicable on a practical basis to a reasonably broad class of process control problems (5, 7). However, it is clear that processes which cannot be approximated by Equation 1: such as a highly exothermic chemical reactor, will require specific formulation and solution for the time optimum control law; this topic receives no attention in the present work, but has been investigated in a t least two recent papers (4, 9 ) . T h e manipulated variable in a real process is always constrained between some maximum limits; hence we assume k 5 M 5 K , where k and K are known. We neglect constraints on dM/dt. If saturation occurs elsewhere in the loop, k and K are determined by the most narrow limits. The block diagram is shown in Figure 1 . R is the set point or desired value of the output C. The error is defined to be e R - C. The problem of controller design is to specify M such that e is reduced to zero in minimum time. Although the process itself is linear, the constraints on M introduce nonlinearity to the control loop. The investigation of this problem without transportation lag was pioneered by Bushaw in 1 3 5 2 (3) and, independently, by Feldbaum (8). In 1 9 5 6 , a principle leading to the solution of the general problem of finding a n optimum control function for this type of constraint was formulated by Pontryagin (74). Desoer (6) has shown from Pontryagin's maximum principle that, for
K
k
Figure 1 .
IL
-1
Block diagram of the system
VOL. 4
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463
linear processes, the minimum time solution always results in a control law \\-ith two-position action-that is, on a time optimum trajectory, M is always a t the boundary of its limits (either R or k ) with a possible finite number of switches between them. except when the system returns to the origin. For nth-order linear systems with real roots, Feldbaum (8) has sho\vn that the optimum number of switches between K and k (discounting the final switch to maintain equilibrium a t the origin) is a t most n - 1, and the optimum switching points depend upon the initial error and its first (n - 1) initial time derivatives. The solution to the time optimum control problem for second-order overdamped systems is well known, but little work has been done for the case with dead time. Kurzweil ( 7 7) has investigated the control of multivariable systems with time delays for fairly general process transfer functions. His algorithm for minimum settling time control is complex and requires a digital computer for implementation and storage of past process output history. Further, it assumes accurate knolyledge of process transfer functions and all state variables. I n the present study, these requirements are relaxed to provide a simple implementation of minimal time control of processes which are adequately represented by the transfer function given in Equation 1. This study provides the analytical solution for this case, a comparison between the optimum control and well tuned P I D control, and a simple suggestion for practical realization of the optimum control.
'
M=R (a)
M=k (b)
M =K (C)
I
(d) Figure 2. Phase plane analysis of second-order overdamped systems 1. 2. 3.
Trajectory from Figure 2b, M = k Trajectory from Figure 2c, M = K Origin of Figure 20, M = R
Application of Maximum Principle
The equation relating e and M from Equation 1 is
bT2 a ( t )
+ (6 + 1) T i ( t ) + e ( t ) =
-
-M(t
at)
+R
(2)
if R is piecelvise constant. The initial conditions are e(Oi) = e l , i ( O + ) = 0, and an initial forcing function M ( t - U T )= R,, 0 _< t < U T . Now define X I = e, x2 = d, and form the state equations dl = x2 =
F1
(x,f,t )
(3)
T h e forcing function, f, corresponds to --M(t cector x to the pair (XI,xg). We form a conjugate system defined by
-
+ R,
UT)
Pontryagin's maximum principle states (74) that a necessary condition for the time optimum M is that H ( M , x, X) be a maximum, with respect to M , for all t along the trajectory. I t can be seen that H i s maximum if
M(t
- U T )=
K , Xz < 0 k , Xz > 0
(9)
It is clear that X 2 ( t ) cannot change sign more than once. Since H is maximum when M is maximum in magnitude, the maximum principle establishes the form of the control function as two-position with no more than one switch. To determine the time for the single switch from full on, M = K , to full off, M = k, or vice versa, we next study the phase plane trajectories. Phase Plane Analysis
Equation 2 will be studied for three forcing functions (4)
--M 7 R =
For the present system
-K
Cib(t)Fi(x,f,t )
+ R -~ (6 + 1)
- M ( t - UT)
bT
464
I&EC FUNDAMENTALS
x2
1'
- b T2
minimum maximum
(10)
For the present, we neglect transportation lag and consider M = M ( t ) , to = 0. The phase plane trajectories are shown in
(7)
Figure 2 for the three forcing functions. Time is a parameter along the trajectories. In each case, a stable node occurs on the e axis at the value of the forcing function. At time t = 0, a step change is made in the set point from R, to R. For time t < 0, e = e = 0 = R, - M . The conditions immediately after the set point change are e (0) = el = R R,, and i ( 0 ) = 0. The desired final conditions are e ( t J = & ( t 5 )= 0, where t j is unspecified,but is to be minimized. The heavy trajectories of Figure 2, b and c, which pass exactly through the origin, constitute the desired final paths, and hence are switching lines. When e l is finite and negative, M should be k (because the system output must be reduced)
to get
H=
set point equilibrium
+R +R
(5)
Next formulate the Hamiltonian of the conjugate system
HE
["
-k
(8)
-
and the trajectory of Figure 26 beginning a t (el, 0) will be followed. This trajectory has been redrawn on Figure 2d. If, a t the instant this tr2ijectox-y crosses the heavy switching line of Figure 2c, M is switched to K , the switching line will be followed to the origin, and the trajectory of Figure 2d results, A final change to M == R is made a t the origin to keep the system a t the origin; this is not counted as a required twoposition switch, since the system is a t equilibrium. For positive el, the values of k and K are interchanged, so that M = K for any state above the complete switching line, shown in Figure 4,and M = k below this line. When the transportxtion lag is included, the effect of a switch in M ( t ) is not observed until a n elapse of time a T . As shown in Appendix I, the optimum programmed control function for a system wi t h dead time is the same as for the case with no dead time. Lsing this M ( t ) , the first trajectory will not move from the point ( e l , 0) in Figure 3 until a time aT after application of M ( t ) = k. Further, M ( t ) must be switched to K while on the first trajectory, a t a time aT before reaching the final trajectory. This switching time is shown as t 2 , and t 3 = tz aT. The locus of the points tz for various e l constitute the phase plane switching line in the presence of a delay. If the final switch of M ( t ) to M = R is made a t t l , a time a T before reaching the origin, the system will stop and remain a t the origin. This leads to synthesis of the control law for initial conditions of the form given with Equation 2. As shown by Balakirev (7), the control law for arbitrary initial conditions is more difficult to realize.
Figure 3. Phase plane trajectory for secondorder overdamped system with transportation lag M(f)
Time Interval to-11 f1-t2
+
h-f4
K K
f4-k tr-
R R
f2-f3
-
M(f
k k
- aT) Ro k k
K K R
-
el) exp (aT t d / T and (k R f el) exp (aT substituted into Equation 13 to give
- t3)/bT and
Derivation of Switching Equations
For negative el, the parametric equations for the first part of the trajectory from ( e l , 0) to the point a t t 3 , determined from solution of Equation 2 with M = k , are: e =
O sk1 - b
[ e x p ( F >
-
-
bexp(s)]
and for the second part of the trajectory from are :
t3
k
+R
(11)
to the origin
e =
K A R1 - b e'=
[exPC+)
T(l
-
bexpcG)]
-K
+R
(12)
- b)
We have assumed a set point change from Ro to R, so that e l = R - R,. Simultaneous solution at t 3 , the phase plane intersection, gives two equations in t 3 , t j , and e l . These in turn can be solved simultaneous1.y to eliminate t 5 and give a n implicit equation for t 3 and e l :
which is the equation for the switching line in the left-hand - i plane. Subscript s is used to designate that e and are on the switching line. K and k should be interchanged to give the right-hand plane equation for positive e l . [Balakirev (7) has derived a less general form of this equation.] If the process of Equation 1 has a steady-state gain K,, K and k in Equation 14 are replaced by K K , and kK,, respectively. If continuous measurement of e and i is available, the first switch should occur when e and e' solve Equation 14. This equation should be used when lei1 is somewhat greater than the intersection of the switching line and the e axis (given by Equation 14 when i s = 0), to ensure that tz > tl. This may not be satisfied when errors are small and dead time is large. For the processes we have studied, however, switching time was greater than dead time. An example switching line is shown in Figure 4. An expression for t ~ the , time of the first switch, can be obtained from Equation 13 by direct substitution of t z = t S - aT to give e
[E:+(
k -KR- + Rel)exp(s)]=
-
K - R
____ K - k K-I?+
K - R+
e x p t T )
(13)
Equations 1 1 of the first part of the curve, evaluated a t t = tz = t 3 - aT, give equations for the switching error and error derivative. These can 'be solved for the quantities ( k - R
+
+ ( k -K - R
+
"'> (9) exp
(15)
If e' is not readily available, the controller design can be based on time. Equation 15 gives tz implicitly for initial conditions $1 = 0, e l < 0. When e l > 0, K and k should be interchanged. The switching time, t 2 , is independent of the dead time. This important result allows synthesis of a programmed timing controller for process startups and transitions, without the need VOL. 4 NO. 4
NOVEMBER 1 9 6 5
465
of specifying a basic process parameter, the transportation lag. T h e error a t the final switch can be determined from Equation 12 when t d > ta and is given by
K - R [exp (u) 1 - b
e(tJ =
for e
~
- b exp(a/b)]
-K
+R
< 0.
Performance of Two-Position Controller The performance of the time optimum two-position controller was compared with that of a well tuned proportionalintegral-derivative controller on a n analog computer. Equation 1 5 was the basis for logic circuits to provide the switching time controller, A second-order Pad6 approximation was used to simulate the process transportation lag. Latour (13) gives the details of the circuits. To assure fair comparison for each set of process constants, u and 6, “optimum” P I D controller settings were determined.
I
-60
T = 1.0 K = 40
I
60
-30
-60-
k E -100 R - -20 b = 0.5
Figure 4.
Example phase plane switching line
A repetitive operation computer was used to find settings which minimized the integral of absolute value of error for step load changes, with an oscillation constraint of no more than thre: visible changes in the sign of the error derivative. This was no saturationdone in the linear range of operation-Le., to ensure some theoretical significance of the results. Minimum response time or settling time for load changes was not a useful criterion for optimum P I D controller settings, because the “optimum” transients showed undesirable characteristics. As an example, if one chooses to minimize the 5% settling time, t j % , settings can be found for some processes (small u ) for which the error never exceeds 0.05L, so that ts% = 0. However, such a transient had excessive oscillation and was not acceptable for practical applications. Latour (73) gives more details of this topic. T h e optimum P I D settings based on absolute area under the error curve gave transients, such as that shown in Figure 5, which were more likely to be considered as acceptable for industrial practice. For convenience, the responses generated by the two-position control are labeled BB on the diagrams (for bang-bang). A series of comparisons between the well tuned P I D controller and the two-position controller was made. The output of the PID controller was now limited a t K and k for fair comparisons. Figure 6 is an example phase plane resulting from one of these tests. Each trajectory begins a t e l = -30, e‘ = 0, and remains a t this position for an elapse of time U T after application of M = k . T h e first switch on the twoposition trajectory, a t t = tz, is indicated. At t = t 4 , the final switch from two-position back to PID control occurred. This was done to approximate the action that would occur in practice if a control room operator forced this two-position control by manual set point adjustment. (Also, the theoretical final switch to M = R is difficult to achieve in practice, because of inaccuracies in knowledge of transduction constants, such as change in flow per unit change i n pneumatic pressure.) This imperfect final switch caused slight undershoot past the origin, since a small nonzero error is applied to the P I D controller a t t 4 . The reported theoretical minimum response time, t s , was not affected by this final switch approximation. I t is important to emphasize that the switch a t t z is from full forward to full reverse (k to K ) to give maximum deceleration and bring the system rapidly to the origin. For simple on-off control, the switch occurs at e = 0 rather than a t the indicated switching line, and a limit cycle results. I n comparing the trajectories of Figures 5 and 6, it may be seen that the P I D trajectory initially follows the two-position
2or
I
3 -10
/
-v
b=
I
4
t
Q5
R = -30 K = 100
-20
Figure 5. 466
l&EC FUNDAdENTALS
Error transient comparison between two-position and PID control for test 20
60
f
\
K =Kx) =-I00
k ..
$:
Figure 6.
Phase plane comparison between two-position and PID control for test 20
trajectory, since each is saturated a t k . Maximum deceleration is not available i n the P I D case, and hence considerable overshoot results. T h e point at which the 5% settling time occurs is indicated on each trajectory. T h e error transients, e ( t ) us. t , corresponding to Figure 6 , are shown in Figure 5. T h e time optimum two-position transient is clearly superior. T h e manipulated variable transients in Figure 7 indicate the initial saturation of both controllers. These tests were repeated over a range of possible conditions. Table I shows the results of comparisons for various levels of set point changes, process parameter values, saturation range, and controller output asymmetry levels. Some of these results are presented graphically in Figures 8 to 12. The two-position co:ntroller gives significant improvement even for small set point changes, as indicated in Figure 8 . Greater improvement is realized for large set point changes. (The P I D t57o is a funciion of step size because of manipulated variable saturation, as shown in Figure 7 . ) On this graph t j % for P I D is compared with what is effectively to% for twoposition. As shown in Figure ‘9, response is slower and the improvement of two-position over P I D increases when the minor time constant, 6, is large. T h e minimum response time appears to 0.87b) in this be approximately linear in b (ts/T = 0.54 case. Transportation lag appears as a n additive factor i n t 5 , as The shown in Figure 10. For this case t j / T = 0.91 Q. switching time t 2 is independent of the dead time (see Figure 3). For the conditions listed in Figure 10, the final switch point, t 4 , is 0.91 T for all Q , and the delay is additive to give t 5 . T h e degree of improvement appears to be better when a is large. T h e total available range of manipulated variable, K - k , affected each controller performance as shown in Figure 11. As expected, response was faster when the spread between the maximum and minimum M was wide. However, the twoposition improvement over P I D is best when the saturation limits are narrow.
Figure 7. Manipulated variable transient comparison between two-position and PID control for test 20
at = PI0 tSol0- BE ttj, a= b= K=
0.1
DEGREE OF
IMPRWEMENT
05 40
k =-100
+
+
1
I
-
I
-20 SET POINT CHANGE,
A
I
I
40
I
-60
-&
Figure 8. Effect of initial error size on response time for two-position and PID controllers VOL. 4
NO. 4
NOVEMBER 1 9 6 5
467
Table 1. Variable Set point
Minor time constant
Dead time
Saturation level
Asymmetry
Results of Comparison between Two-Position and PID Control
Test
R
b
1 2 3 4 5 6 7 3 8 9 10 11 3 12 13 14 3 15 16
-90 -60 -30 -10 - 3 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30 -30
0.5 0.5 0.5 0.5 0.5 0.12 0.3 0.5 0.7 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
17 3 18 19 20
K
a 0.1 0.1 0.1 0.1 0.1
+ 40 + 40 + 40 + 40 + 40 + 40 + 40 + 40 + 40 + 40 + 40 + 40 ++ 40 40 + 40 + 10 + 40 + 70
0.1 0.1 0.1 0.1 0.1 0 0.05 0.1 0.2 0.5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
$100
++ 4010 + 70 +lo0 +lo0
k -100
-100 -100 -100 -100
-100 -100 -100 -100
-100 -100 -100
-100 -100 -100 - 70 -100 -130 -160 -130 -100 - 70 - 40 -100
K
-k
BBtj 3.10 1.60 1.00 0.60 0.40 0.62 0.83
140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 80 140 200 260 140 140 140 140 200
1.00 1.15 1.30 0.91
0.97 1.00 1.13 1.44 1.30 1.00 0.92 0.77 0.96 1.00 1.21 2.08 0.98
BBtj, 4.56 2.35 1.72 1.30 1.14 1.18 1.57 1.72 1.81 1.90
PIDtj, 15.8 4.8 2.85 1.89 1.50 1.70 2.35 2.85 3.10 3.32
0.91 0.97 1.72 2.41 4.25 2.02 1.72 1.57 1.45 1.64 1.72 1.90 3.17 1.65
2.62 2.85 3.31 2.87 3.40 2.85 2.49 2.29 1.98 2.85 3.58 7.03 2.83
Atss, 11.2 2.5 1.10 0.59 0.36 0.52 0.78 1.10 1.38 1.42 Limit cvcle 1. ~ 6 5 - 1.10 0.90 -1.38 1.38 1.10 0.92 0.84 0.34 1.10 1.68 3.86 1.18
PIDta,
- ts
12.7 3.2 1.8 1.3 1.1 1.08 1.52 1.80 1.95 2.02
~~
1 .65 1.80 2.19 1.43 2.10 1.80 1.57 1.52 1.02 1.80 2.37 4.95 1.85
T = l
T h e effect of asymmetry on response time is shown in Figure
12. Asymmetry occurs when R is not midway between k and
K . Asymmetry in the K direction (refer to Figure 7 once again) means shifting K and k upward while maintaining K - k The reconstant a t 140, with fixed R, = 0 and R = -30. sponse time becomes very large near k - R = 0, since this set point requires the saturation minimum, k , a t steady state. T h e two-position improvement is better for greater asymmetry. I n summary, the time optimum controller is significantly superior to a well tuned PID controller for nearly all condi-
l
I
,
I
,
I
0.5 MINOR TIME CONSTANT,
I
,
I
I.o
b
Figure 9. Effect of minor time constant on response time for two-position and PID controllers 468
I&EC FUNDAMENTALS
~
tions studied. For test 3, Table I, if T = 1 hour, position response would be 1.8 hours faster than tb%. The degree of improvement is best for large changes on processes with large a and b or narrow asymmetric saturation.
the twothe PID set point limits of
Applications
An immediate application of this work is the use of the time switching equation for manual set point control on a typical industrial controller. An experienced operator often applies full on or off saturating action for some period of time before
0
0.2
0.I TRANSPORTATION LAG,
a
Figure 10. Effect of transportation lag on response time for two-position and PID controllers
3.5r
7
At
a
3.0
’ PID ~ ~ O / ~ -t5B B a
b=
-
R
O\
6
:-30
\
I
I-
z
+
w‘ 2.0\
O
g
c
3
I-
‘“Y
v) W
z
0.1
4
c
\
2 2
a
b = 0.5 R =-30
5
c
z
a
K-k m I40
PID t5el0 2.5 -
i
At = PID ~ ~ O / ~ -t5B B
0.1 0.5
w
O
1.5-
i2
U
a
v)
0
L? t5
1.0-
I
‘0
I
80
l
l
1
140
I
I
I
I
200
SATURATION RANGE,
K
-k
I
0 I
--
I
0
260
100
40
Figure 1 1 . Effect of saturation range on response time for two-position and PID controllers
-70
k- R K-R
70
-
40
-10
100
130
ASYMMETRY IN K DIRECTION
putting the control set point to a new desired value. If parameters T , b, K , and k can be adequately approximated, Equation 15 can be used to make a table of values for t z for various el = R - R,. Thus, to make a set point change, the pointer can be moved to cause full force in one direction u p to t 2 , then full force in the opposite direction until t 4 . At t 4 , the pointer is returned to the desired set point value, R. This procedure was found to function well if the integral action was turned off during the transient, and subsequently reapplied when the process reached the vicinity of its new control point. This avoids the undesired integration of the errors during the two-position transient. If use of this simple two-position action with full reversal is made for process startups and transitions, significant time savings will be made in bringing the process to a new steady state. Lapse (72) found the transfer function of a cracking furnace to be exp (-1.06s) (6.9s f 1) (1.03s f 1) where the constants are in minutes. For this case, T = 6.9 minutes, a = 0.154, and b = 0.15. Figure 13 indicates the switching time, t 2 , for various set point changes from R, = 0 to R. Saturation was assumed to occur a t K = 1 inch and k = - 1 inch from the center of the chart, R,. If the actual process b deviated from the assumed b = 0.15 by loyo,t 4 was i n error by approximately 0.1 minute. If the process is subject to loads during programmed twoposition control, a different trajectory will be followed. Since the operator will switch to a linear feedback controller near the origin, deviations due to loads will not greatly affect this procedure. Also, calculations indicate the position of the switching line to be somewhat insensitive to loads when K k is large. If the switching time is consistently in error, a trial and error adjustment of parameter b or T may give improvement. Equation 1 5 might even be used to determine or confirm these model parameters. We show in Appendix I1 that the optimum control function
-
Figure 12. Effect of asymmetry on response time for two-position and PID controllers
r
15 -
-
Ro= 0 @I=R K = I,O in. k =-IO in. b = 0.150
.
a = 0.154 T = 6.9min.
v)
W e z 3
5
10-
a r
e5 -
v)
c
5
v)
5-
a
: Y c -ID
-a5
0
SET POINT CHANGE, R INCHES OF CHART
Figure 13. Time for first switch for set point changes on Lapse’s cracking furnace
is also valid for a system with transportation lag in the feedback measuring element. T h e results of this work provide an upper bound on the possible response time improvement which can be made over “good” P I D control of processes with transportation lag. I n addition, they indicate the class of control situations for which possible response time improvement is greatest. This information forms the basis for further research on the development of simple, suboptimal control methods, such as the suggested VOL. 4
NO. 4
NOVEMBER 1 9 6 5
469
manual set point manipulation, for process control. In particular, Equations 14 and 16 provide the basis for a dual mode feedback controller, providing two-position action when the error exceeds some predetermined value, and standard control otherwise. T h e results are restricted to the class of processes whose transfer functions may be approximated by Equation 1. Conclusions
y ( t ) = x(t
(7). Additional research on time optimum control of chemical processes may be profitably directed toward establishing procedures for fitting Equation 1 to experimental process dynamics in such a manner that the control function of Equation 15 gives rapid control. Appendix I
Proposition. If M l * ( t ) , t > 0 is the time optimum control function for a stable, completely controllable system, G(s), with no dead time, and if Mz*(t),t > 0 is the time optimum control function for the same system with dead time, G(s) exp(-aTs), then Mz*(t) = M1*(t), t > 0 .
Y(t)
G(s 1
exp(- aTs)
c
Suppose a dynamical system is governed by the differential equation
;(t) = Ax@) k
+ bM(t)
< M ( t ) 6 K , all t x
(AI)
(-UT)= x,
-
If the vectors (b, Ab, A A b , A3b, . . . . ., An-'b) are linearly independent, the system is said to be completely controllable. This ensures that it is possible to bring any initial state xo to the origin in finite time, t 4 ; and the system equation can be written in the form of a single nth-order differential equation:
(-43)
t4*.
We must specify M 2 * ( t ) ,t > 0 such that y ( t J = R, the desired equilibrium position, in minimum time t5*-that is, t j * < t j , for k M ( t ) K . y ( t ) ,0 < t a T i s predetermined by M , and is not influenced by Mz*(t). Now, if M ( t ) = M 1 * ( t ) ,t > 0, y(t4* U T )= R from Equation A3 because x(t4*) = R. Furthermore, y ( t ) cannot reach R in a time t 5 < t 4 * aT, since this would imply that x(t) reached R in a time less than t l * . Hence, Mz*(t) = Ml*(t), t > 0 and t j * = t 4 * a T . The time optimum control function is independent of dead time, since M1*(t)is determined solely by G(s). The equivalence of M I * and M2* does not carry over to the feedback case where M * is to be written as a function of the , than of time. This is the point state variables, M [ y ( t ) ]rather a t which Balakirev ( 7 ) left the problem, in an unsuccessful attempt to construct a feedback control law for arbitrary x,.
