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An adaptive pole placement controller for chemical processes with variable dead time
Compur. them. Engng, Vol. 12, No. I, pp. 15-26, 1988
Printedin Great
Britain. All rights reserved
Copyright 0
0098-1354/88 93.00 + 0.00
1988 Pergamon Journals Ltd
AN ADAPTIVE POLE PLACEMENT CONTROLLER FOR CHEMICAL PROCESSES WITH VARIABLE DEAD TIME E. F. Control
Systems
Engineering,
Tennessee
VOGEL
Eastman
Company,
Kingsport,
Texas, U.S.A.
and T. F. EDGAR Department
of Chemical
Engineering,
The University
of Texas at Austin,
(Received 10 December 1986; final revision received 16 April 1987; received
Austin,
TX 78712-1062,
U.S.A.
forpublication 26 June 1987)
AI&rat--A pole placement adaptive controller/dead-time compensator for multiple-input, multipleoutput systems with unknown or variable dead time is presented. The algorithm does not require an explicit estimate of the dead time and is suitable for nonminimum-phase processes. Parameter estimation for the control algorithm is performed with a recursive least squares algorithm. Sensitivity to model errors is maintained by covariance resetting rather than a forgetting factor. Experimental application of the single-loop adaptive controller on a pilot scale heat exchanger temperature control loop has demonstrated improved performance over a PI controller. Simulation results for a two-input-two-output distillation column are also described. Scope-The performance d f conventional digital feedback controllers on chemical processes is limited by the presence of dead times and control-loop interactions. Performance may be improved by applying a controller which provides dead-time compensation. However, for processes which exhibit variable dynamics and/or dead times, feedback controllers must be tuned for the worst case in order to provide satisfactory performance over all operating conditions. Consequently, performance is often sluggish. Higher levels of performance may be achieved and maintained by employing an adaptive controller/dead time compensator. Typically, with an adaptive algorithm, the parameters of a process model arc evaluated during the sampling period by an online parameter estimation algorithm. From the computed estimates of the process model parameters, new controller parameters are calculated and the controller is updated Thus, the controller is retuned during each sampling period according to variations in the process dynamics and dead times. In this paper, we present an adaptive control algorithm which is applicable to both single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) processes. The design of the multivariable controller is based on pole placement and avoids many of the disadvantages of existing adaptive control algorithms. Conelusions and Sllni6ea-The adaptive pole placement controller/dead-time compensator presented here has several desirable features. It can adapt to unknown or variable process dead times without explicit estimation of the time delays or time-consuming controller design calculations. It is suitable for nonminimum-phase processes. Further, the cpntroller does not require additional model parameters for application to processes with different time delays between the input-output combinations. The estimator uses covariance resetting for maintaining sensitivity to modeling errors and it performs well for tracking sudden parameter variations. When applied to a temperature control loop with variable gain and dead time, the adaptive algorithm demonstrated improved performance compared to a conventional PI controller. The capability of the adaptive multivariable controller/dead-time compensator to adjust to process parameter variations has also been demonstrated by application to a two-input-twooutput distillation column simulation. In both applications, the adaptive algorithm maintained a constant level of performance in spite of considerable variations in the process gains and dead times.
INTRODUCT’ION Adaptive control @gorithms are useful’on control loops where the process or operating conditions change such that conventional, constant-parameter
feedback controllers (e.g. PID) cannot maintain a consistent level of performance. Many adaptive control techniques are now available and each has its own advantages and disadvantages. Papers by Seborg et al. [l] and Astrijm [2] have provided extensive surveys of the field. With a few exceptions, most algorithms assume that the process dead time is constant and known.
This paper presents an adaptive pole placement control algorithm for processes with unknown or variable dead time. Its primary advantage is that it does not require an explicit estimate of the process dead time. Additionally, the controller is simple to implement and is suitable for nonminium-phase processes. Nonminimum-phase processes are an important consideration because discrete time modeling of continuous processes can often lead to nonminimum-phase models, even if the continuous process is minimum phase [3-S]. This often is due to the occurrence of fractional time delays. 15
16
E. F. VOGEL and T. F. EDGAR
The SISO compensator
pole
placement
controller/&ad-time
The SISO pole placement controller is based on an input-output process model of the form given below. J.%-%(r)
= g(q-‘)u(t)
(1)
where u(t) and y(t) are the process input and output, respectively, and
B(q-‘)=(b,q_‘+b,q_Z+
... +bA-‘)z-k,
where: n r k 4 -’
= = = =
order of the A (q -I) polynomial, order of the polynomial in B(q-I), minimum expected process dead time, backwards shift operator.
In the B polynominal, r is selected large enough so that dead time in addition to k is represented by leading coefficients in B which are zero. Thus, online estimation of the ais and b,‘s allows this model to adapt to a process with an unknown or variable dead time without requiring an explicit estimate of the dead time. Note that k is specified initially and is not estimated online. As is the normal practice in design of digital controllers, the desired closed-loop transfer function is specified for derivation of the SISO controller (the q-l operator is dropped for convenience below): G,,=$
general pole placement design procedure [6]. The controller, which is a special case of internal model control [7], is similar to Dahlin’s control algorithm [8] (identical for first-order systems). However, for second and higher order processes, Dahlin’s controller attempts to cancel process zeros, and consequently, is . . subject to nngmg. Gva does not ring because it does not attempt to cancel process zeros. Previous performance comparisons between GvB and Dahlin’s controller [9] have demonstrated the superiority of this new algorithm. However, for small values of z and system order greater than two, G,, can generate excessive control action [lo]. As given, G, is not suitable for integrating or unstable processes. However, a simple modification to permit implementation of GvE on such processes is to include a proportionalonly feedback loop around the process and design G,, for the process with the feedback loop. The multivariable pole placement controller /dead-time compensator The multivariable controller/dead-time compensator can be developed by using a procedure analogous to the SISO controller derivation above. The multivariable process model for a p-input-p-output system is given by: &I-‘)y(r)
= k%-‘M(r)
where u(t) is a p x 1 vector of inputs and y(t) is a p x 1 vector of outputs. A(q-‘) and B(q-‘) arep x p matrices of polynomials with elements defined as follows.
Aii(q~‘)=Si,+a’iq-‘+a~q-2+
where G, F and Q are:
Bii(q-‘)=(b’iq-‘+bijq-*+
G = B/A,
F _ (1 - e-T/‘)q-’ 1 _ e-“‘q-l
big-‘.
... +a~oq-“U . . . +bt,q-‘il)q-kri
where:
’
nV= order of the A,(q-‘) polynomial, r,j = order of the polynomial in Bii(q -‘), k,= minimum expected dead time for the ijth element. 6, = Kronecker delta defined by: ’
and ZB is defined by: ZB = i
(9)
(6)
6, = l(i =j),
i-l
T is the sampling interval and 7 is the controller tuning parameter analogous to a time constant for the closed-loop response. In terms of the process model, G and the desired controller, denoted by GvE, the closed-loop transfer function is: GCL
=
GVEG 1+
G,G’
Substituting equation (2) for GcL in equation (7) and solving for Gva yields the following controller. F C,E=~--FG. The control law of equation (8) can also be derived from a special case of Astrom and Wittenmark’s
6, = O(i #j). The q-l operator is again dropped for convenience. Similar to the SISO case, rii is selected large enough so that dead time in the ijth element in addition to k, is modeled by leading coefficients in Bij which are 1 zero. Thus, through estimation of the a and b coefficients of the A and B polynomial matrices, the multivariable model can be adapted to unknown or variable dead times without explicit estimation of the delays. Prager and Wellstead [l 1] have developed pole placement algorithms that employed a model similar to that given by equation (9), with nU= R, rii = r and k, = k for all i and j. To extend their model to allow for different delays and variable delays, Prager and
17
Adaptive placement controller Wellstead suggested increasing the order of the polynomials represented by B. However, because of the form of their model, the orders of all of the polynomials in B must be equal. Therefore, the orders of all of the polynomials in %tmust be increased equally, even if the time delays in some of the elements are constant. This may lead to the estimation of many unnecessary model parameters, since the order of the PI polynomials must be selected large enough to include the ranges of all the process delays. In contrast, the more general model given by equation (9) automatically allows for different time delays and the number of parameters in each element of II depends only on the expected range of the dead time of that element. Thus, the model of equation (9) has the advantage that it is likely to require estimation of fewer parameters compared to the model used by Prager and Wellstead. McDermott et al. [12] have also used this approach in designing an adaptive controller for a tubular reactor. Chien et al. [13] have modified the self-tuning controller design procedure based on minimum variance [14] to handle variable dead times using the basic framework presented here. For derivation of the multivariable pole placement controller, a multivariable closed-loop transfer function analogous to that given by equation (2) is employed: a&_ = GQ-‘IF, where 6 and Q are definded
(10)
as:
16 = A-%,
(11)
Q = A-‘ZB
(12)
A is given in equation defined as:
(9) and the elements of ZB are
EB,=
2
( m=*
q-’
b!
>
(13)
F is selected as a diagnoal matrix to reduce interactions between the input-output pairs. Similar to F of the SISO case, the diagonal elements of IF are chosen to be:
Comparison of equations (8) and (17) verifies that the single loop and multivariable controllers are analogous. Both the SISO and MIMO controllers have several desirable features for application as an adaptive algorithm: (1) the Vogel-Edgar controller can be adapted to unknown or varible dead times without explicit estimation of the process delays; (2) the controller parameters are easily updated from new parameter estimates with no time-consuming controller design calculations, because the process model parameters appear in the controller; (3) through specification of minimum expected time delays, large process time delays do not increase the complexity of the algorithm; (4) nonminimum-phase processes and processes with poorly damped zeros are readily handled since. the algorithm does not attempt to cancel process zeros; (5) the controller includes integral action and an online tuning parameter for each input-output pair; (6) the multivariable algorithm decouples the input-output pairs at steady state; and (7) it allows for different time delays between all of the process input-output combinations without requiring additional process model parameters. Multivariable parameter estimation Implementation of the multivariable controller/ dead-time compensator as an adaptive algorithm requires the online estimation of the process model parameters given by equation (9). For this purpose, the multivariable model is first written in recursive form in the time domain, with a bias term included for each output to represent unmeasured disturbances entering the process. In recursive form, the multivariable model consists of p difference equations. For the ith output, the difference equation is:
y,(t) = i
2 UQ-y,(t
- m)]
j-lm=l
+
‘f
2
biu,(t
-m
-k,)+di.
(18)
j-lm-I
Equation
(18) may be written in vector form as: J+(t) = #T(t - 1%
where ri is the tuning parameter for the ith input-output pair. Writing the closed-loop transfer function in terms of the process model, G, and the desired controller, denoted by GvE, yields: 6,, Equating
= [I + GGvs]-‘GG”s.
equations GQ-‘F
(15)
= [I + GGvs]-‘GG”s.
(16)
Solving for 6,s yields the multivariable pole-zero placement controller/dead-time compensator below. GYE =
[as-
FG]-‘6.
where:
qiT(t - 1) = [-yl(t
(17)
- 1), . . . ,
-yp(t -n,),u,(t Up0 - ‘,p -k,), tY:=[a:‘,._.,
(10) and (15) then gives:
(19)
a’&,b:‘,...,
- 1 -_kil)..
..,
lb bk, dil.
The parameter vectors, Bi, of the multivarlable model are estimated one at a time with the recursive least squares algorithm shown below [15].
Pit/t .
- 1), = IPit. - 1), + IL.,
(201 . ,
E. F. VOGEL and T. F. EDGAR
18 Pi(f)
= P&/r
x [4:(t
- 1) - P&/t
- l)P,(t/t - 1)&(t - 1)
+ I]-‘4T(t
- 1)P#/t
4(t) = 40 - 1) + P,0)4& -8$‘(t
- I)&(? - 1)
- l)&
- 1)
(21)
- l)[n(r)
- l)].
(22)
&t) are the parameter estimates for the ith output time t. P,(f) is proportional to the covariance matrix of the parameter estimate vector. ED,is a positive definite matrix. The estimation algorithm is implemented p times each sampling interval, once for each of the parameter vectors. The addition of lIDimaintains the sensitivity of the estimation algorithm to parameter changes. However, to avoid excessive sensitivity, IDIis added in equation (20) only if: (1) the absolute value of the estimation error [y,(t) - C&T@- l)d,(r - 1)] exceeds a user specified limit; and (2) the trace of p,(t) is below a user-specified limit. This approach, called covariance resetting, has been found in this research to be preferred over the use of a forgetting factor [16, 17J Modeling errors are detected by monitoring the estimation error, y,(t) - a5T(t - l)&(t - 1). If the absolute value of the estimation error exceeds the user-specified limit, D is added, otherwise it is not. When D is added, it should not be added again until the elements of P have again become small. When the elements of P are not small, addition of D is not necessary but causes unnecessary fluctuation of the parameter estimates. Therefore, D is added only if the trace of P is below a user-specified limit. With this technique, the elements of P are allowed to become small, as long as no model mismatch is indicated. Thus, there is no danger of estimator windup. Additionally, when the elements of P are small, the parameter estimates are insensitive to noise, making the continuous addition of a perturbation signal
This method for maintaining sensitivity performs well for tracking large, sudden parameter changes. Selection of appropriate values for the user-specified trigger limits on the estimation error and the trace of P is important. The performance of the algorithm does not appear to be particularly sensitive to the values selected for the elements of ID. Normally, D is chosen as a diagonal matrix with the magnitude of the elements depending upon the expected rate of variation of the respective parameters. Addition of perturbation signals to the inputs of the multivariable process is necessary to obtain satisfactory parameter estimates when the elements of P,(t) are not small. In the closed loop, the perturbation signals may be added either to the controller outputs or to the set points. With the adaptive controller, addition of perturbation signals to the set points is preferred since, as the process gains change, the perturbations affect the process outputs to the same degree. For multivariable parameter estimation, it is important that the perturbation signals added to each input be unwrrelated in order to ensure reliable parameter estimates. For two-input-two-output systems, Gauthier and Landau [18] suggested using a pseudorandom binary sequence (PRBS) as one perturbation and the same PRBS delayed by half its length for the other perturbation. Using the PRBS given below for one perturbation and the same PRBS delayed 9 intervals for the other perturbation gives satisfactory results, based on experience to date: [+l, +1, -1,
fl,
-1, +l,
-1,
-1, -1,
1.
-1,
-1,
+1, -1, +l,
fl,
-11.
(23)
Since each output is generally affected by all of the process inputs, the PRBS signals are added to all of the process inputs whenever the elements of any of the P,(t)‘s are not small, i.e. when the sum of the traces. of the P,(t)‘s exceeds a user-specified limit.
REFLUX
Fig.
+1, +1,+1,
Pilot scale distillation column (total reflux).
Adaptive placement controller The parameter estimates are tested and filtered at each time step before updating the controller to avoid erroneous estimates in the controller. The tests include calculation of the process model gain for each input-output combination and location of the roots of the A, polynomials. If the model gains or roots of the A, polynomials are not in acceptable regions, the parameters of the corresponding controller parameter vectors are not updated that sampling interval. A first-order filter is used to smooth changes in the parameter estimates as shown below:
mented in discrete follows:
19 time
in the velocity
form
as
u(t) = n(t - 1) + K,{[e(r) - e(t - l)] + (T/T,)e(t)}, (25)
where K, and T, are the controller parameters; e(t) and u(t) are the controller input and output, respectively. Figure 1 shows the equipment schematic for this application. The column was run at total rellux with pure water in the reboiler to simplify reproduction of geltt) =P@L-~(~ - ]I + tl.O - P)4(r), (24) the operating conditions. The ease of reproducing the where 6,(t) is the vector of current parameter esti- I operating conditions permitted an accurate comparison of the two control algorithms. The boil-up mates and d=,(t) is the vector of parameter estimates rate, determined by the steam rate to the reboiler, used by the adaptive controller, p is the filter factor with a value between 0.0 and 1.0. served as a load disturbance to the cooling water outlet temperature. The parameters of the A, polynomials do not need to be estimated in the closed loop if small variations The following test case was used as a basis for in those parameters are expected. This still allows for comparing the performance of the PI controller and the adaptive controller. With each controller, the adaptation to gain and dead-time changes, while steam valve was initially positioned to provide a ensuring that the controller performance will not reflux rate of approximately 0.090 gal/min (low boilbecome unsatismctory due to unnecessary fluctuup rate). The condenser cooiing water outlet temations in the A, model parameters. A reduction in perature was maintained at 40°C by each controller. computation time is also realized by not estimating With the boil-up rate held constant, the cooling water the A, parameters in the closed loop. outlet temperature set point was moved to 42°C and Procedures for implementing the adaptive controlback to 40°C after 2min. After the temperature ler and for specifying the parameters of the algorithm returned to 4O”C, the boil-up rate was increased by have heen discussed in detail by Vogel [9]. Although opening the steam valve to provide a reflux rate of the performance of the adaptive algorithm is not 0.155 gal/min (high boil-up rate). This change reextremely sensitive to the values selected for many of sulted in a sharp increase in the cooling water outlet the parameters, some of the parameters do require temperature and each controller increased the cooling proper selection to obtain the best performance. The water flow rate to bring the temperature back to four parameters which have a significant effect on the 40°C. At the high boil-up rate, the gain on the cooling performance are: (1) the estimation error limit above which EDis added; (2) the trace of IFDbelow which D water outlet temperature loop was a factor of 3 or 4 less than that at the low boil-up rate. The dead time can be added; (3) the PRBS amplitude; and (4) the was also reduced, due to the increase in the cooling trace of P above which the PRBS is added. While water flow rate. To test the performance of each values for these parameters are not necessarily controller at the new operating point, the cooling difficult to determine, they normally require some water outlet temperature set point was again moved initial adjustments based on the performance of the to 42°C and then back to 40°C. estimation algorithm. The PI controller had to be tuned for the worst case in order to ensure satisfactory performance over Experimental application of the SISO controller all operating conditions. In this example, the worst The superior performance of the Vogel-Edgar case was at the low boil-up rate for which the process gain was higher and the dead time was longer. With adaptive algorithm has been demonstrated in both a sampling interval of 2s, tuning online by trial and computer simulations and experimental applications [9, 191. In the experimental application presented error yielded the following PI controller parameters: here, the adaptive algorithm was used to manipulate KC= 0.1 ,and TI = 20.0 s. Figure 2 shows the dynamic response of the coolcooling water flow rate to control the cooling water outlet temperature on an overhead vapor condenser. ing water outlet temperature and the corresponding The gain and dead time for this loop both vary with controller output for the PI controller applied with the cooling water flow rate, permitting a demonthe test case. The first two set point changes occurred at the low boil-up rate, the operating conditions for stration of the capability of the adaptive algorithm. which the PI controller was tuned. For these set point As a basis for evaluating the adaptive algorithm, changes, the PI controller performed well. The reits performance on the cooling water temperature loop was compared to that of a conventional sponse was a little sluggish because the dead time proportional-integral (PI) controller with constant prohibited tight tuning. After the increase in the tuning parameters. The PI controller was impleboil-up rate and corresponding gain and dead-time
20
E.
F. Vcos~ and T.
Time
F. EDGAR
(min)
Fig. 2. Cooling water outlet temperature
response and controller output for a PI controller applied to the coding water control loop.
decreases, the performance of the PI controller was quite slow for both the load disturbance and the subsequent set point changes. While tighter tuning would improve performance at the high boil-up rate, it would lead to oscillatory behavior at the low boil-up rate. Implementation of the adaptive algorithm on the cooling water outlet temperature control loop first required specification of the algorithm parameters/ decisions listed in Table 1. The parameters of the process model, T, II, r and k were selected from observation of the open-loop response of the cooling water outlet temperature to changes in the cooling water flow rate. The response exhibited first order dynamics with a time constant of only a few seconds.
From this information, n was selected to be 1 and the sampling interval T was set at 2 s. The dead time also was only a few seconds. Therefore k, the minimum expected dead time, was chosen to be 1 and r was selected to be 5. These values allowed for a dead time range of 1 to 5 sampling intervals (2-10s). After specification of the model parameters, the estimation algorithm was employed in the open-loop to verify the model parameters and to aid in the selection of values for the other parameters of the estimation algorithm. P was initialized as a diagonal matrix with all diagonal elements equal to 10,000. A PRBS of constant amplitude (4% of full scale) was added to the cooling water valve position as necessary to excite the process dynamics. The parameter
Table 1. User specified paramexen/decisions for application of the adaptive algorithm on the cooling water control
loop
Processmodel and
related parameters Sampling interval, T Number of denominator parameten, n Number of numerator parameters, I Minimum expected process dead time, k Initial values for the parameter vectors, 8(O) and 8,(O) Closed loop estimation of the denominator parameters Controller and related parameters Controller tuning, parameter, 7 Filter factor for updating controller parameter vector, p High process model gain limit Low process mode1 gain limit Estimation algorithm and related parameters IpfO1 F&‘&inn factor. A
iI-
-
Estimation error limit above which 0 is added Trace of P below which Ct can be added PRBS kvxtion PRBS amplitude Trace of b above which the PRBS is added
2s I 5
5.0 0.7 -0.5 -9.0 1o.ooo I 1:o 1001 1.0
5.0 Set Doint 1.k 2.5
Adaptive placement controller
Incrwso
in boll-up
I 6
21
rate I 10
I a
Time
I 12
I 14
(min)
Time (min
1
Fig. 3. Cooling water outlet temperature response and controller output for the adaptive controller applied to the cooling water control loop. vector, 6, was initialized with zeros. Appropriate values for the remaining estimation algorithm parameters were determined by observing the behavior of the estimation error, the trace of P, the process gain calculated from the parameter estimates, and the effect of the PRBS. For application of the adaptive algorithm with the test case, the initial parameter estimates, for the controller were obtained by implementing the par-
Or
,ameter estimation algorithm in the open-loop at the low boil-up rate. The 4% PRBS signal was added to the process input for the open-loop estimation. The model parameter a, was estimated during the openloop estimation. Once satisfactory estimates were obtained, estimation of a, was discontinued, addition of the PRBS to the valve position was discontinued and the loop was closed with the adaptive controller. Figure 3 shows the cooling water outlet tempera-
ii! Process model gain chxhted from ,tle pamma+sr esiimares used by the adaptive ConttDkW.
Process model grrin calculated from the current pammetw estimates.
II
II
I
I
I
I
6
6
10
12
I 14
Time (mid
Time (min)
Fig. 4. F%cess model gain and absolute value of the estimation error for the adaptive controller applied to the cooling water control loop.
22
E. F.
5rbz
5
.
and
T. F. EDGAR
5rb5
I
OH-
-510
VOGEL
I
10
I 15
Time ( min)
I
OkI--
-50V
15
Time (min 1
Fig. 5. Parameter estimates for the adaptive controller applied to the cooling water control loop.
ture response and the controller output for the test case with the adaptive controller. The performance of the adaptive algorithm did not deteriorate with the decrease in gain and dead time as did the PI controller. The adaptive controller performed well for the set point changes at both the low and high boil-up rates and gave a fast, smooth response to the step change in the boil-up rate. Additionally, the dead-time compensation in the adaptive controller allowed it to provide a faster response than the PI controller for the set point changes at the low boil-up rate. Figure 4 shows both the process model gain calculated from the current parameter estimates, f&t) and the process model gain calculated from the parameter estimates used by the controller, @C(t). When the increase in boil-up rate occurred, there was a sharp decrease in the magnitude of the process model gain. The large, rapid fluctuations in the gain calculated from the current parameter estimates illustrate both the importance of testing new parameter estimates before updating the controller and the benefit of filtering acceptable sets of parameter estimates. As shown in Fig. 4, the tllter factor of 0.7 allowed the controller to be updated reasonably quickly but smoothed out the sudden changes seen in the current parameter estimates. Also shown in Fig. 4 is the absolute value of the estimation error. Figure 5 shows the estimates for the bias term, d and the five process model numerator parameters, b, through b5. Initially, b, , b, and b3 were close to zero and b, was the first significant term, indicating that the process had approximately four sampling intervals (8 s) of dead time. After the decrease in gain occurring with the increase in the boil-up rate, it was
much more difficult to determine which numerator parameters are non-zero. However, close examination of the estimates showed that, after the change in operating conditions, b, was the most significant term, implying that the process then had about 3 sampling intervals (6 s) of dead time. Thus, the parameter estimates verified that the gain and dead time of the cooling water loop decreased with the increase in the cooling water flow rate. While an estimate of the process dead time has been inferred from examination of the numerator parameter estimates, the adaptive algorithm did not require an explicit estimate of the dead time and did not attempt to estimate it directly. Simulation test of the adaptive MIMO controller In order to conveniently implement the multivariable control algorithm given in equation (17), it was first rearranged as follows: [CD- FGlll(t) = k(t).
(26)
Next, equations (11) (12) and (14) were substituted, respectively, for G, Q and IF in equation (26). The terms which represent the current control actions appear as the unknowns ofp simultaneous equations. The solution of these equations required inversion of the ZB matrix, defined in equation (13). However, since ZB is a matrix of constants, its inversion was simple for the two-input-two-output case. Thus, for the usual case in which p is not large, the solution of the simultaneous equations can be determined analytically, yielding explicit expressions for the current control actions. The multivariable control algorithm was then implemented by converting these
Adaptive placement controller expressions to difference equations, which were employed at each sampling time to calculate the current control actions from past control actions as well as current and past process output errors. While the multivariable control algorithms was not difficult to implement in its general form, implementation was simplified considerably by selecting the n;s of the process model such that A was a diagonal matrix (tz, = 0 for all i #j). In this case the effect of outputs on each other was neglected. G and 69 are given by
(27)
Gw
After substituting equations (27), (28) and (14) for 6, Q and F in equation (26), solving for the terms representing the current control actions yields the following set of simultaneous equations: + =12%(r)
= m,(t).
(2%
%(f) + =22%(r)
= m,(t),
(30)
=1,%(r) %,
Berry [20] for a pilot scale distillation column which separates a mixture of methanol and water. This model consists of first-order plus time-delay transfer functions and has been employed in multivariable control studies by Wood and Berry [20], Shah and Fisher [21] and Ogunnaike and Ray [22]. The distillation column model relates the process inputs and outputs through continuous time transfer functions as given below: Y(s) = G(s)U(.T) + G,(s)L(s)
The transfer given by
G(s) =
x[A,,e,(t)
+ &u,(t)
+ &d)]r
(31)
+ 4k~lit)
+
4Arll.
(32)
Note that m,(t) and m*(t) are functions of the process output errors and the past control actions. Solution of equations (29) and (30) yields the explicit expressions for the current control actions:
u2O)
=
r 12.8e-‘,” 16.7s + 1 6 k-7.0
G(s) and G,(s),
- 18.9e-‘.O” 1 21.0s + 1 19&y-3.”
-
’
14.4s + 1 1
I 1 3 ge-8h
+ (1 - e-nTi)q-’
+CBzzuz(r)] + (1 - ecTlr2)q-’
u,(t) =
matrices,
14.9s + 1 4 9e-3.4”
are
(3’5)
(37)
.
13.2s + 1
m*(t) = e-~‘2q-‘[~BZ,u1(t)
x [&e,(t)
function
10.9s + 1 l----
ml(t)=e-“~lq-‘[ZB,,ul(t) +ZB,,u,(t)]
(35)
where the inputs and outputs are deviation variables defined as: y, = overhead product composition deviation, y2 = bottoms product composition deviation, u, = reflux flow rate deviation, u2 = steam flow rate deviation, L = feed flow rate deviation.
GAS) =
where m,(r) and m,(f) are given by
23
~&m,(f) - W2m2(G ZB,,XB,
- Z&EB2,
--Z&mIO)+~~~ImA~) EB,,ZB,, - I;B,,ZB,,
(33)
’
.
(34)
Remember that all of the ZBii factors in the equations above are constants, as apparent from their definition in equation (13). Conversion of equations (31)-(34) to the time domain yields the difference equations necessary for implementation of the simpler form of the two-input-two-output multivariable controller/ dead-time compensator. The simulated process chosen to test the MIMO adaptive controller was the two-input-two-output distillation column model developed by Wood and
The time units in the model are minutes. To investigate the ability of the adaptive multivariable algorithm to adjust the variable process parameters, we assumed that the gains and dead times of the distillation column model given above varied inversely with the feed flow rate, as defined below. These are reasonable assumptions since the dead time in a measurement taken on a process stream generally varies inversely with the flow rate of that stream, while a nonlinearity in the process gain may be inherent in the process. The following test case was employed for examining the performance of the adaptive control strategy. The set point of either the overhead or bottoms composition was first increased 0.5% and then decreased 0.5% to the original value. Following the set point changes, the feed flow rate was increased by 0.34 Ib/min. Simultaneous with the feed flow rate increase, the process gains and time delays decreased such that G(s), the process model initially defined by equation (36), became: 6.1 G(s) =
16.7s + 1 3.5e-4.0J
-SAe-*~~ 21.0s + 1 -g.ge-l.~
1 .
(38)
E. F. VOGEL and T. F. EDGAR
24
Table 2. User specifwl parameters for the adaptive multivariabk controller/dead-time compensator Estimation algorithm and controller parameters Sampling interval, T Initial P,, P,(O) Initial valuea for the elements of the process model parameter vec1ors. e,(O) and 6,, (0) Q Absolute value of the estimation error above which D, is added Trace of P, below which D, can be added PRBS location PRBS amplitude (wt%) Sum of the traces of the P,‘s which yield addition of the PRBS signals Closed loop estimation of the parameters of
1.Omin l,OOO,OO4l I
0.0 1,OOO,Ooa I 0.10 10,000 Set point 0.10 500
the
no
A, polynomials
r, = 3.0 z*= 5.0 Controller tuning parameters, ‘TV Parameters specified for each polynomial element of the process model: Element (i,j) y f!$ f&y +!I Number of au parameters, nti 4 4 8 3 Number of bU parameters, rr/ 0 0 0 0 Minimum expected dead time, k, 15.0 -2.0 -2.0 25.0 High process model gain limit -40.0 1.5 -40.0 2.0 Low process model gain limit
Implementation of the multivariable adaptive algorithm initially required selection of values for the user specified parameters of the adaptive algorithm. The values of the user specified parameters chosen for this example are given in Table 2; n,2 and nzl were chosen to be zero, yielding the simpler form of the multivariable controller given by equations (31)-(34). In this case, with all of the k,‘s equal to zero, the dead time between the ith input and jth output of the process could vary from zero to r,,- 1 sampling intervals. For start-up of the adaptive multivariable algorithm, initial estimates for the elements of 8, and &, were obtained by implementing the parameter esti97.0
5
1
mation in the open loop at the initial operating conditions of the test case. For the open loop estimation, PRBS signals were added to the process inputs, both with a constant amplitude of 0.4 lb/min. After 50 sampling intervals, estimation of the parameters in the A, polynomials of the process model employed by the controller was discontinued and the loops were closed with the multivariable adaptive algorithm. Also at this point, the location of the PRBS signals was switched to set points with the amplitudes defined as in Table 2. Figures 6 and 7 show, respectively, the responses of the contrblled variables for set point changes in the overhead and bottoms compositions with the adap-
r \
0 9!5.so
I 100
I
200
‘nyaseasc~
feed I
300
I
400
Time (min)
Time (min)
Fig. 6. F’rocess responses to set point changes in y, with an adaptive multivariable controller applied to the distillation column simulation (variable gains and dead times).
Adaptive placement controller
25
Composition
i
flow ratI I
'0
I
I
200
100
I 400
300
Timetmin)
I-
, -
Increase In lead I
flow
100
mm
I
I 400
300
200
Time (min)
Fig. 7. Process responses for set point changer in y, with the adaptive multivariable controller applied to the distillation column simulation (variable gains and dead times). tive multivariable controller for the test case discussed above. These figures also include the responses for the initial open loop parameter estimation. In contrast to the responses for PI controllers [9], the multivariable controller yields slightly faster responses at the initial operating conditions. However, after the decreases in the process gains and time delays occurring with the increase in the feed flow rate, the performance of the PI controllers .became sluggish. In contrast, Figs 6 and 7 show that the multivariable algorithm maintained its previous level of performance after the decreases in the process gains and dead times as it adapted to the process parameter changes. The responses of the MIMO controller for the unmeasured load was sluggish (although comparable to that for a PI controller). Modifications to the algorithm are currently under investigation to see if disturbance estimation can be used to improve the load response. As seen in Figs 6 and 7, interactions occur with the multivariable controller because the inputoutput pairs are not dynamically decoupled. The interactions arise primarily from different time delays between the input-output combinations. While interactions occur in the process outputs, the benefit of dead-time compensation and steady-state decoupling provided by the multivariable controller can be observed in the controller outputs [9]. For set point changes with the multivariable controller, the controller outputs move directly to their new steady-state values, after their initial actions, without noticeable oscillation. A significant advantage of the multivariable controller over conventional single loop control is that the multivariable controller requires a minimal amount of tuning compared to PI controllers. Tuning
PI controllers in the presence of the interactions of a multivariable process can be quite tedious. After implementation of the parameter estimation algorithm, the multivariable controller requires only adjustment of the t;s for each loop to achieve the desired performance.
REFRRENCB
1. D. E. Seborg, S. L. Shah and T. F. Edgar, Adaptive control strategies for process control: a survey. AICXE
Jl32, 881 (1986). 2. K. J. Astriim, Theory and application of adaptive control-a survey. Automatica 19, 471 (1983). 3. P. E. Wellstead, D. Pragcr and P. Zanker, Pole assignment self-tuning regulator: Froc. IEE 126, 781 (1979). 4. K. J. Astriim, P. Hagander and J. Stemby, Zeros of sampled systems. Automatica 20, 31 (1984). 5. D. W. Clarke, Self-tuning control of nomninimumphase systems. Automatica 20, 501 (1984). 6. K. J. AstrGm and B. Wittemnark, Self-tuning controllers based on pole-zero placement. Proc. IEE 127, Part D, 120 (1980). 7. C. E. Garcia and M. Morari, Internal model control unifying review and some new results. Ind. figag Chem., Process Des. Deu. 21, 308 (1982). 8. E. B. Dahlin, Designing and tuning digital controllers. Imtrumen~s and Control Systems 41, 77 (1968). 9. E. F. Vogel, Adaptive control of chemical processes with variable dead time. Ph.D. Thesis, University of Texas at Austin (1982). 10. E. Zafiriou and M. Morari, Digital controllers for SING systems: a review and a new algorithm. Int. .I. Control 42, 855 (l(985). 11. D. L. Prager and P. E. Wellstead, multivariable poleassignment self-tuning regulators. Proc. IEE 128, Part D, 9 (1980). 12. P. E. McDermott, D. A. Mellichamp and R. G. Rinker, Multivariable self-tuning control of a tubular authothermal reactor. Proc. ACC, 1614 (1984).
26
E. F. VOOEL and T. F. EDGAR
13. I. L. Chien, D. A. Mellichamp and D. E. Scborg, Self-tuning controllers for variable time-delay processes. Int. J. Control 949 (1985). 14. H. N. Koivo, A multivariable self-tuning controller. Aufomatica 16, 351 (1980). 15. P. C. Young, Applying parameter estimation to dynamic systems-part I. Control Engng 16,119 (1969). 16. B. Wittenmark and K. J. Astrom, Practical issues in the implementation of self-tuning control. Automatica 20, 595 (1984). 17. T. R. Fortescue, L. S. Kershenbaum and B. E. Ydstie, Implementation of self-tuning regulators with variable forgetting factors. Automatica 17, 831 (1981). 18. A. Gauthier and I. D. Landau, On the recursive
19.
20.
21.
22.
identification of multi-input, multi-output systems. Automatica 14, 609 (1978). J. P. Gerry, E. F. Vogel and T. F. Edgar, Adaptive control of a pilot-scale distillation column. AZChE Spring Natnl hftg, Houston, Texas (1983). R. K. Wood and M. W. Berry, Terminal composition control of a binary distillation column. Gem. Engng Isci. 28, 1707 (1973). S. L. Shah and D. G. Fisher, A multivariable frequency domain design method for feedforward disturbance minimization. Proc. JACC 4, 1 (1978). B. A. Ogunnaike and W. H. Ray, Multivariable controller design for linear systems having multiple time delays. AIChE JI 25, 1043 (1979).
Printedin Great
Britain. All rights reserved
Copyright 0
0098-1354/88 93.00 + 0.00
1988 Pergamon Journals Ltd
AN ADAPTIVE POLE PLACEMENT CONTROLLER FOR CHEMICAL PROCESSES WITH VARIABLE DEAD TIME E. F. Control
Systems
Engineering,
Tennessee
VOGEL
Eastman
Company,
Kingsport,
Texas, U.S.A.
and T. F. EDGAR Department
of Chemical
Engineering,
The University
of Texas at Austin,
(Received 10 December 1986; final revision received 16 April 1987; received
Austin,
TX 78712-1062,
U.S.A.
forpublication 26 June 1987)
AI&rat--A pole placement adaptive controller/dead-time compensator for multiple-input, multipleoutput systems with unknown or variable dead time is presented. The algorithm does not require an explicit estimate of the dead time and is suitable for nonminimum-phase processes. Parameter estimation for the control algorithm is performed with a recursive least squares algorithm. Sensitivity to model errors is maintained by covariance resetting rather than a forgetting factor. Experimental application of the single-loop adaptive controller on a pilot scale heat exchanger temperature control loop has demonstrated improved performance over a PI controller. Simulation results for a two-input-two-output distillation column are also described. Scope-The performance d f conventional digital feedback controllers on chemical processes is limited by the presence of dead times and control-loop interactions. Performance may be improved by applying a controller which provides dead-time compensation. However, for processes which exhibit variable dynamics and/or dead times, feedback controllers must be tuned for the worst case in order to provide satisfactory performance over all operating conditions. Consequently, performance is often sluggish. Higher levels of performance may be achieved and maintained by employing an adaptive controller/dead time compensator. Typically, with an adaptive algorithm, the parameters of a process model arc evaluated during the sampling period by an online parameter estimation algorithm. From the computed estimates of the process model parameters, new controller parameters are calculated and the controller is updated Thus, the controller is retuned during each sampling period according to variations in the process dynamics and dead times. In this paper, we present an adaptive control algorithm which is applicable to both single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) processes. The design of the multivariable controller is based on pole placement and avoids many of the disadvantages of existing adaptive control algorithms. Conelusions and Sllni6ea-The adaptive pole placement controller/dead-time compensator presented here has several desirable features. It can adapt to unknown or variable process dead times without explicit estimation of the time delays or time-consuming controller design calculations. It is suitable for nonminimum-phase processes. Further, the cpntroller does not require additional model parameters for application to processes with different time delays between the input-output combinations. The estimator uses covariance resetting for maintaining sensitivity to modeling errors and it performs well for tracking sudden parameter variations. When applied to a temperature control loop with variable gain and dead time, the adaptive algorithm demonstrated improved performance compared to a conventional PI controller. The capability of the adaptive multivariable controller/dead-time compensator to adjust to process parameter variations has also been demonstrated by application to a two-input-twooutput distillation column simulation. In both applications, the adaptive algorithm maintained a constant level of performance in spite of considerable variations in the process gains and dead times.
INTRODUCT’ION Adaptive control @gorithms are useful’on control loops where the process or operating conditions change such that conventional, constant-parameter
feedback controllers (e.g. PID) cannot maintain a consistent level of performance. Many adaptive control techniques are now available and each has its own advantages and disadvantages. Papers by Seborg et al. [l] and Astrijm [2] have provided extensive surveys of the field. With a few exceptions, most algorithms assume that the process dead time is constant and known.
This paper presents an adaptive pole placement control algorithm for processes with unknown or variable dead time. Its primary advantage is that it does not require an explicit estimate of the process dead time. Additionally, the controller is simple to implement and is suitable for nonminium-phase processes. Nonminimum-phase processes are an important consideration because discrete time modeling of continuous processes can often lead to nonminimum-phase models, even if the continuous process is minimum phase [3-S]. This often is due to the occurrence of fractional time delays. 15
16
E. F. VOGEL and T. F. EDGAR
The SISO compensator
pole
placement
controller/&ad-time
The SISO pole placement controller is based on an input-output process model of the form given below. J.%-%(r)
= g(q-‘)u(t)
(1)
where u(t) and y(t) are the process input and output, respectively, and
B(q-‘)=(b,q_‘+b,q_Z+
... +bA-‘)z-k,
where: n r k 4 -’
= = = =
order of the A (q -I) polynomial, order of the polynomial in B(q-I), minimum expected process dead time, backwards shift operator.
In the B polynominal, r is selected large enough so that dead time in addition to k is represented by leading coefficients in B which are zero. Thus, online estimation of the ais and b,‘s allows this model to adapt to a process with an unknown or variable dead time without requiring an explicit estimate of the dead time. Note that k is specified initially and is not estimated online. As is the normal practice in design of digital controllers, the desired closed-loop transfer function is specified for derivation of the SISO controller (the q-l operator is dropped for convenience below): G,,=$
general pole placement design procedure [6]. The controller, which is a special case of internal model control [7], is similar to Dahlin’s control algorithm [8] (identical for first-order systems). However, for second and higher order processes, Dahlin’s controller attempts to cancel process zeros, and consequently, is . . subject to nngmg. Gva does not ring because it does not attempt to cancel process zeros. Previous performance comparisons between GvB and Dahlin’s controller [9] have demonstrated the superiority of this new algorithm. However, for small values of z and system order greater than two, G,, can generate excessive control action [lo]. As given, G, is not suitable for integrating or unstable processes. However, a simple modification to permit implementation of GvE on such processes is to include a proportionalonly feedback loop around the process and design G,, for the process with the feedback loop. The multivariable pole placement controller /dead-time compensator The multivariable controller/dead-time compensator can be developed by using a procedure analogous to the SISO controller derivation above. The multivariable process model for a p-input-p-output system is given by: &I-‘)y(r)
= k%-‘M(r)
where u(t) is a p x 1 vector of inputs and y(t) is a p x 1 vector of outputs. A(q-‘) and B(q-‘) arep x p matrices of polynomials with elements defined as follows.
Aii(q~‘)=Si,+a’iq-‘+a~q-2+
where G, F and Q are:
Bii(q-‘)=(b’iq-‘+bijq-*+
G = B/A,
F _ (1 - e-T/‘)q-’ 1 _ e-“‘q-l
big-‘.
... +a~oq-“U . . . +bt,q-‘il)q-kri
where:
’
nV= order of the A,(q-‘) polynomial, r,j = order of the polynomial in Bii(q -‘), k,= minimum expected dead time for the ijth element. 6, = Kronecker delta defined by: ’
and ZB is defined by: ZB = i
(9)
(6)
6, = l(i =j),
i-l
T is the sampling interval and 7 is the controller tuning parameter analogous to a time constant for the closed-loop response. In terms of the process model, G and the desired controller, denoted by GvE, the closed-loop transfer function is: GCL
=
GVEG 1+
G,G’
Substituting equation (2) for GcL in equation (7) and solving for Gva yields the following controller. F C,E=~--FG. The control law of equation (8) can also be derived from a special case of Astrom and Wittenmark’s
6, = O(i #j). The q-l operator is again dropped for convenience. Similar to the SISO case, rii is selected large enough so that dead time in the ijth element in addition to k, is modeled by leading coefficients in Bij which are 1 zero. Thus, through estimation of the a and b coefficients of the A and B polynomial matrices, the multivariable model can be adapted to unknown or variable dead times without explicit estimation of the delays. Prager and Wellstead [l 1] have developed pole placement algorithms that employed a model similar to that given by equation (9), with nU= R, rii = r and k, = k for all i and j. To extend their model to allow for different delays and variable delays, Prager and
17
Adaptive placement controller Wellstead suggested increasing the order of the polynomials represented by B. However, because of the form of their model, the orders of all of the polynomials in B must be equal. Therefore, the orders of all of the polynomials in %tmust be increased equally, even if the time delays in some of the elements are constant. This may lead to the estimation of many unnecessary model parameters, since the order of the PI polynomials must be selected large enough to include the ranges of all the process delays. In contrast, the more general model given by equation (9) automatically allows for different time delays and the number of parameters in each element of II depends only on the expected range of the dead time of that element. Thus, the model of equation (9) has the advantage that it is likely to require estimation of fewer parameters compared to the model used by Prager and Wellstead. McDermott et al. [12] have also used this approach in designing an adaptive controller for a tubular reactor. Chien et al. [13] have modified the self-tuning controller design procedure based on minimum variance [14] to handle variable dead times using the basic framework presented here. For derivation of the multivariable pole placement controller, a multivariable closed-loop transfer function analogous to that given by equation (2) is employed: a&_ = GQ-‘IF, where 6 and Q are definded
(10)
as:
16 = A-%,
(11)
Q = A-‘ZB
(12)
A is given in equation defined as:
(9) and the elements of ZB are
EB,=
2
( m=*
q-’
b!
>
(13)
F is selected as a diagnoal matrix to reduce interactions between the input-output pairs. Similar to F of the SISO case, the diagonal elements of IF are chosen to be:
Comparison of equations (8) and (17) verifies that the single loop and multivariable controllers are analogous. Both the SISO and MIMO controllers have several desirable features for application as an adaptive algorithm: (1) the Vogel-Edgar controller can be adapted to unknown or varible dead times without explicit estimation of the process delays; (2) the controller parameters are easily updated from new parameter estimates with no time-consuming controller design calculations, because the process model parameters appear in the controller; (3) through specification of minimum expected time delays, large process time delays do not increase the complexity of the algorithm; (4) nonminimum-phase processes and processes with poorly damped zeros are readily handled since. the algorithm does not attempt to cancel process zeros; (5) the controller includes integral action and an online tuning parameter for each input-output pair; (6) the multivariable algorithm decouples the input-output pairs at steady state; and (7) it allows for different time delays between all of the process input-output combinations without requiring additional process model parameters. Multivariable parameter estimation Implementation of the multivariable controller/ dead-time compensator as an adaptive algorithm requires the online estimation of the process model parameters given by equation (9). For this purpose, the multivariable model is first written in recursive form in the time domain, with a bias term included for each output to represent unmeasured disturbances entering the process. In recursive form, the multivariable model consists of p difference equations. For the ith output, the difference equation is:
y,(t) = i
2 UQ-y,(t
- m)]
j-lm=l
+
‘f
2
biu,(t
-m
-k,)+di.
(18)
j-lm-I
Equation
(18) may be written in vector form as: J+(t) = #T(t - 1%
where ri is the tuning parameter for the ith input-output pair. Writing the closed-loop transfer function in terms of the process model, G, and the desired controller, denoted by GvE, yields: 6,, Equating
= [I + GGvs]-‘GG”s.
equations GQ-‘F
(15)
= [I + GGvs]-‘GG”s.
(16)
Solving for 6,s yields the multivariable pole-zero placement controller/dead-time compensator below. GYE =
[as-
FG]-‘6.
where:
qiT(t - 1) = [-yl(t
(17)
- 1), . . . ,
-yp(t -n,),u,(t Up0 - ‘,p -k,), tY:=[a:‘,._.,
(10) and (15) then gives:
(19)
a’&,b:‘,...,
- 1 -_kil)..
..,
lb bk, dil.
The parameter vectors, Bi, of the multivarlable model are estimated one at a time with the recursive least squares algorithm shown below [15].
Pit/t .
- 1), = IPit. - 1), + IL.,
(201 . ,
E. F. VOGEL and T. F. EDGAR
18 Pi(f)
= P&/r
x [4:(t
- 1) - P&/t
- l)P,(t/t - 1)&(t - 1)
+ I]-‘4T(t
- 1)P#/t
4(t) = 40 - 1) + P,0)4& -8$‘(t
- I)&(? - 1)
- l)&
- 1)
(21)
- l)[n(r)
- l)].
(22)
&t) are the parameter estimates for the ith output time t. P,(f) is proportional to the covariance matrix of the parameter estimate vector. ED,is a positive definite matrix. The estimation algorithm is implemented p times each sampling interval, once for each of the parameter vectors. The addition of lIDimaintains the sensitivity of the estimation algorithm to parameter changes. However, to avoid excessive sensitivity, IDIis added in equation (20) only if: (1) the absolute value of the estimation error [y,(t) - C&T@- l)d,(r - 1)] exceeds a user specified limit; and (2) the trace of p,(t) is below a user-specified limit. This approach, called covariance resetting, has been found in this research to be preferred over the use of a forgetting factor [16, 17J Modeling errors are detected by monitoring the estimation error, y,(t) - a5T(t - l)&(t - 1). If the absolute value of the estimation error exceeds the user-specified limit, D is added, otherwise it is not. When D is added, it should not be added again until the elements of P have again become small. When the elements of P are not small, addition of D is not necessary but causes unnecessary fluctuation of the parameter estimates. Therefore, D is added only if the trace of P is below a user-specified limit. With this technique, the elements of P are allowed to become small, as long as no model mismatch is indicated. Thus, there is no danger of estimator windup. Additionally, when the elements of P are small, the parameter estimates are insensitive to noise, making the continuous addition of a perturbation signal
This method for maintaining sensitivity performs well for tracking large, sudden parameter changes. Selection of appropriate values for the user-specified trigger limits on the estimation error and the trace of P is important. The performance of the algorithm does not appear to be particularly sensitive to the values selected for the elements of ID. Normally, D is chosen as a diagonal matrix with the magnitude of the elements depending upon the expected rate of variation of the respective parameters. Addition of perturbation signals to the inputs of the multivariable process is necessary to obtain satisfactory parameter estimates when the elements of P,(t) are not small. In the closed loop, the perturbation signals may be added either to the controller outputs or to the set points. With the adaptive controller, addition of perturbation signals to the set points is preferred since, as the process gains change, the perturbations affect the process outputs to the same degree. For multivariable parameter estimation, it is important that the perturbation signals added to each input be unwrrelated in order to ensure reliable parameter estimates. For two-input-two-output systems, Gauthier and Landau [18] suggested using a pseudorandom binary sequence (PRBS) as one perturbation and the same PRBS delayed by half its length for the other perturbation. Using the PRBS given below for one perturbation and the same PRBS delayed 9 intervals for the other perturbation gives satisfactory results, based on experience to date: [+l, +1, -1,
fl,
-1, +l,
-1,
-1, -1,
1.
-1,
-1,
+1, -1, +l,
fl,
-11.
(23)
Since each output is generally affected by all of the process inputs, the PRBS signals are added to all of the process inputs whenever the elements of any of the P,(t)‘s are not small, i.e. when the sum of the traces. of the P,(t)‘s exceeds a user-specified limit.
REFLUX
Fig.
+1, +1,+1,
Pilot scale distillation column (total reflux).
Adaptive placement controller The parameter estimates are tested and filtered at each time step before updating the controller to avoid erroneous estimates in the controller. The tests include calculation of the process model gain for each input-output combination and location of the roots of the A, polynomials. If the model gains or roots of the A, polynomials are not in acceptable regions, the parameters of the corresponding controller parameter vectors are not updated that sampling interval. A first-order filter is used to smooth changes in the parameter estimates as shown below:
mented in discrete follows:
19 time
in the velocity
form
as
u(t) = n(t - 1) + K,{[e(r) - e(t - l)] + (T/T,)e(t)}, (25)
where K, and T, are the controller parameters; e(t) and u(t) are the controller input and output, respectively. Figure 1 shows the equipment schematic for this application. The column was run at total rellux with pure water in the reboiler to simplify reproduction of geltt) =P@L-~(~ - ]I + tl.O - P)4(r), (24) the operating conditions. The ease of reproducing the where 6,(t) is the vector of current parameter esti- I operating conditions permitted an accurate comparison of the two control algorithms. The boil-up mates and d=,(t) is the vector of parameter estimates rate, determined by the steam rate to the reboiler, used by the adaptive controller, p is the filter factor with a value between 0.0 and 1.0. served as a load disturbance to the cooling water outlet temperature. The parameters of the A, polynomials do not need to be estimated in the closed loop if small variations The following test case was used as a basis for in those parameters are expected. This still allows for comparing the performance of the PI controller and the adaptive controller. With each controller, the adaptation to gain and dead-time changes, while steam valve was initially positioned to provide a ensuring that the controller performance will not reflux rate of approximately 0.090 gal/min (low boilbecome unsatismctory due to unnecessary fluctuup rate). The condenser cooiing water outlet temations in the A, model parameters. A reduction in perature was maintained at 40°C by each controller. computation time is also realized by not estimating With the boil-up rate held constant, the cooling water the A, parameters in the closed loop. outlet temperature set point was moved to 42°C and Procedures for implementing the adaptive controlback to 40°C after 2min. After the temperature ler and for specifying the parameters of the algorithm returned to 4O”C, the boil-up rate was increased by have heen discussed in detail by Vogel [9]. Although opening the steam valve to provide a reflux rate of the performance of the adaptive algorithm is not 0.155 gal/min (high boil-up rate). This change reextremely sensitive to the values selected for many of sulted in a sharp increase in the cooling water outlet the parameters, some of the parameters do require temperature and each controller increased the cooling proper selection to obtain the best performance. The water flow rate to bring the temperature back to four parameters which have a significant effect on the 40°C. At the high boil-up rate, the gain on the cooling performance are: (1) the estimation error limit above which EDis added; (2) the trace of IFDbelow which D water outlet temperature loop was a factor of 3 or 4 less than that at the low boil-up rate. The dead time can be added; (3) the PRBS amplitude; and (4) the was also reduced, due to the increase in the cooling trace of P above which the PRBS is added. While water flow rate. To test the performance of each values for these parameters are not necessarily controller at the new operating point, the cooling difficult to determine, they normally require some water outlet temperature set point was again moved initial adjustments based on the performance of the to 42°C and then back to 40°C. estimation algorithm. The PI controller had to be tuned for the worst case in order to ensure satisfactory performance over Experimental application of the SISO controller all operating conditions. In this example, the worst The superior performance of the Vogel-Edgar case was at the low boil-up rate for which the process gain was higher and the dead time was longer. With adaptive algorithm has been demonstrated in both a sampling interval of 2s, tuning online by trial and computer simulations and experimental applications [9, 191. In the experimental application presented error yielded the following PI controller parameters: here, the adaptive algorithm was used to manipulate KC= 0.1 ,and TI = 20.0 s. Figure 2 shows the dynamic response of the coolcooling water flow rate to control the cooling water outlet temperature on an overhead vapor condenser. ing water outlet temperature and the corresponding The gain and dead time for this loop both vary with controller output for the PI controller applied with the cooling water flow rate, permitting a demonthe test case. The first two set point changes occurred at the low boil-up rate, the operating conditions for stration of the capability of the adaptive algorithm. which the PI controller was tuned. For these set point As a basis for evaluating the adaptive algorithm, changes, the PI controller performed well. The reits performance on the cooling water temperature loop was compared to that of a conventional sponse was a little sluggish because the dead time proportional-integral (PI) controller with constant prohibited tight tuning. After the increase in the tuning parameters. The PI controller was impleboil-up rate and corresponding gain and dead-time
20
E.
F. Vcos~ and T.
Time
F. EDGAR
(min)
Fig. 2. Cooling water outlet temperature
response and controller output for a PI controller applied to the coding water control loop.
decreases, the performance of the PI controller was quite slow for both the load disturbance and the subsequent set point changes. While tighter tuning would improve performance at the high boil-up rate, it would lead to oscillatory behavior at the low boil-up rate. Implementation of the adaptive algorithm on the cooling water outlet temperature control loop first required specification of the algorithm parameters/ decisions listed in Table 1. The parameters of the process model, T, II, r and k were selected from observation of the open-loop response of the cooling water outlet temperature to changes in the cooling water flow rate. The response exhibited first order dynamics with a time constant of only a few seconds.
From this information, n was selected to be 1 and the sampling interval T was set at 2 s. The dead time also was only a few seconds. Therefore k, the minimum expected dead time, was chosen to be 1 and r was selected to be 5. These values allowed for a dead time range of 1 to 5 sampling intervals (2-10s). After specification of the model parameters, the estimation algorithm was employed in the open-loop to verify the model parameters and to aid in the selection of values for the other parameters of the estimation algorithm. P was initialized as a diagonal matrix with all diagonal elements equal to 10,000. A PRBS of constant amplitude (4% of full scale) was added to the cooling water valve position as necessary to excite the process dynamics. The parameter
Table 1. User specified paramexen/decisions for application of the adaptive algorithm on the cooling water control
loop
Processmodel and
related parameters Sampling interval, T Number of denominator parameten, n Number of numerator parameters, I Minimum expected process dead time, k Initial values for the parameter vectors, 8(O) and 8,(O) Closed loop estimation of the denominator parameters Controller and related parameters Controller tuning, parameter, 7 Filter factor for updating controller parameter vector, p High process model gain limit Low process mode1 gain limit Estimation algorithm and related parameters IpfO1 F&‘&inn factor. A
iI-
-
Estimation error limit above which 0 is added Trace of P below which Ct can be added PRBS kvxtion PRBS amplitude Trace of b above which the PRBS is added
2s I 5
5.0 0.7 -0.5 -9.0 1o.ooo I 1:o 1001 1.0
5.0 Set Doint 1.k 2.5
Adaptive placement controller
Incrwso
in boll-up
I 6
21
rate I 10
I a
Time
I 12
I 14
(min)
Time (min
1
Fig. 3. Cooling water outlet temperature response and controller output for the adaptive controller applied to the cooling water control loop. vector, 6, was initialized with zeros. Appropriate values for the remaining estimation algorithm parameters were determined by observing the behavior of the estimation error, the trace of P, the process gain calculated from the parameter estimates, and the effect of the PRBS. For application of the adaptive algorithm with the test case, the initial parameter estimates, for the controller were obtained by implementing the par-
Or
,ameter estimation algorithm in the open-loop at the low boil-up rate. The 4% PRBS signal was added to the process input for the open-loop estimation. The model parameter a, was estimated during the openloop estimation. Once satisfactory estimates were obtained, estimation of a, was discontinued, addition of the PRBS to the valve position was discontinued and the loop was closed with the adaptive controller. Figure 3 shows the cooling water outlet tempera-
ii! Process model gain chxhted from ,tle pamma+sr esiimares used by the adaptive ConttDkW.
Process model grrin calculated from the current pammetw estimates.
II
II
I
I
I
I
6
6
10
12
I 14
Time (mid
Time (min)
Fig. 4. F%cess model gain and absolute value of the estimation error for the adaptive controller applied to the cooling water control loop.
22
E. F.
5rbz
5
.
and
T. F. EDGAR
5rb5
I
OH-
-510
VOGEL
I
10
I 15
Time ( min)
I
OkI--
-50V
15
Time (min 1
Fig. 5. Parameter estimates for the adaptive controller applied to the cooling water control loop.
ture response and the controller output for the test case with the adaptive controller. The performance of the adaptive algorithm did not deteriorate with the decrease in gain and dead time as did the PI controller. The adaptive controller performed well for the set point changes at both the low and high boil-up rates and gave a fast, smooth response to the step change in the boil-up rate. Additionally, the dead-time compensation in the adaptive controller allowed it to provide a faster response than the PI controller for the set point changes at the low boil-up rate. Figure 4 shows both the process model gain calculated from the current parameter estimates, f&t) and the process model gain calculated from the parameter estimates used by the controller, @C(t). When the increase in boil-up rate occurred, there was a sharp decrease in the magnitude of the process model gain. The large, rapid fluctuations in the gain calculated from the current parameter estimates illustrate both the importance of testing new parameter estimates before updating the controller and the benefit of filtering acceptable sets of parameter estimates. As shown in Fig. 4, the tllter factor of 0.7 allowed the controller to be updated reasonably quickly but smoothed out the sudden changes seen in the current parameter estimates. Also shown in Fig. 4 is the absolute value of the estimation error. Figure 5 shows the estimates for the bias term, d and the five process model numerator parameters, b, through b5. Initially, b, , b, and b3 were close to zero and b, was the first significant term, indicating that the process had approximately four sampling intervals (8 s) of dead time. After the decrease in gain occurring with the increase in the boil-up rate, it was
much more difficult to determine which numerator parameters are non-zero. However, close examination of the estimates showed that, after the change in operating conditions, b, was the most significant term, implying that the process then had about 3 sampling intervals (6 s) of dead time. Thus, the parameter estimates verified that the gain and dead time of the cooling water loop decreased with the increase in the cooling water flow rate. While an estimate of the process dead time has been inferred from examination of the numerator parameter estimates, the adaptive algorithm did not require an explicit estimate of the dead time and did not attempt to estimate it directly. Simulation test of the adaptive MIMO controller In order to conveniently implement the multivariable control algorithm given in equation (17), it was first rearranged as follows: [CD- FGlll(t) = k(t).
(26)
Next, equations (11) (12) and (14) were substituted, respectively, for G, Q and IF in equation (26). The terms which represent the current control actions appear as the unknowns ofp simultaneous equations. The solution of these equations required inversion of the ZB matrix, defined in equation (13). However, since ZB is a matrix of constants, its inversion was simple for the two-input-two-output case. Thus, for the usual case in which p is not large, the solution of the simultaneous equations can be determined analytically, yielding explicit expressions for the current control actions. The multivariable control algorithm was then implemented by converting these
Adaptive placement controller expressions to difference equations, which were employed at each sampling time to calculate the current control actions from past control actions as well as current and past process output errors. While the multivariable control algorithms was not difficult to implement in its general form, implementation was simplified considerably by selecting the n;s of the process model such that A was a diagonal matrix (tz, = 0 for all i #j). In this case the effect of outputs on each other was neglected. G and 69 are given by
(27)
Gw
After substituting equations (27), (28) and (14) for 6, Q and F in equation (26), solving for the terms representing the current control actions yields the following set of simultaneous equations: + =12%(r)
= m,(t).
(2%
%(f) + =22%(r)
= m,(t),
(30)
=1,%(r) %,
Berry [20] for a pilot scale distillation column which separates a mixture of methanol and water. This model consists of first-order plus time-delay transfer functions and has been employed in multivariable control studies by Wood and Berry [20], Shah and Fisher [21] and Ogunnaike and Ray [22]. The distillation column model relates the process inputs and outputs through continuous time transfer functions as given below: Y(s) = G(s)U(.T) + G,(s)L(s)
The transfer given by
G(s) =
x[A,,e,(t)
+ &u,(t)
+ &d)]r
(31)
+ 4k~lit)
+
4Arll.
(32)
Note that m,(t) and m*(t) are functions of the process output errors and the past control actions. Solution of equations (29) and (30) yields the explicit expressions for the current control actions:
u2O)
=
r 12.8e-‘,” 16.7s + 1 6 k-7.0
G(s) and G,(s),
- 18.9e-‘.O” 1 21.0s + 1 19&y-3.”
-
’
14.4s + 1 1
I 1 3 ge-8h
+ (1 - e-nTi)q-’
+CBzzuz(r)] + (1 - ecTlr2)q-’
u,(t) =
matrices,
14.9s + 1 4 9e-3.4”
are
(3’5)
(37)
.
13.2s + 1
m*(t) = e-~‘2q-‘[~BZ,u1(t)
x [&e,(t)
function
10.9s + 1 l----
ml(t)=e-“~lq-‘[ZB,,ul(t) +ZB,,u,(t)]
(35)
where the inputs and outputs are deviation variables defined as: y, = overhead product composition deviation, y2 = bottoms product composition deviation, u, = reflux flow rate deviation, u2 = steam flow rate deviation, L = feed flow rate deviation.
GAS) =
where m,(r) and m,(f) are given by
23
~&m,(f) - W2m2(G ZB,,XB,
- Z&EB2,
--Z&mIO)+~~~ImA~) EB,,ZB,, - I;B,,ZB,,
(33)
’
.
(34)
Remember that all of the ZBii factors in the equations above are constants, as apparent from their definition in equation (13). Conversion of equations (31)-(34) to the time domain yields the difference equations necessary for implementation of the simpler form of the two-input-two-output multivariable controller/ dead-time compensator. The simulated process chosen to test the MIMO adaptive controller was the two-input-two-output distillation column model developed by Wood and
The time units in the model are minutes. To investigate the ability of the adaptive multivariable algorithm to adjust the variable process parameters, we assumed that the gains and dead times of the distillation column model given above varied inversely with the feed flow rate, as defined below. These are reasonable assumptions since the dead time in a measurement taken on a process stream generally varies inversely with the flow rate of that stream, while a nonlinearity in the process gain may be inherent in the process. The following test case was employed for examining the performance of the adaptive control strategy. The set point of either the overhead or bottoms composition was first increased 0.5% and then decreased 0.5% to the original value. Following the set point changes, the feed flow rate was increased by 0.34 Ib/min. Simultaneous with the feed flow rate increase, the process gains and time delays decreased such that G(s), the process model initially defined by equation (36), became: 6.1 G(s) =
16.7s + 1 3.5e-4.0J
-SAe-*~~ 21.0s + 1 -g.ge-l.~
1 .
(38)
E. F. VOGEL and T. F. EDGAR
24
Table 2. User specifwl parameters for the adaptive multivariabk controller/dead-time compensator Estimation algorithm and controller parameters Sampling interval, T Initial P,, P,(O) Initial valuea for the elements of the process model parameter vec1ors. e,(O) and 6,, (0) Q Absolute value of the estimation error above which D, is added Trace of P, below which D, can be added PRBS location PRBS amplitude (wt%) Sum of the traces of the P,‘s which yield addition of the PRBS signals Closed loop estimation of the parameters of
1.Omin l,OOO,OO4l I
0.0 1,OOO,Ooa I 0.10 10,000 Set point 0.10 500
the
no
A, polynomials
r, = 3.0 z*= 5.0 Controller tuning parameters, ‘TV Parameters specified for each polynomial element of the process model: Element (i,j) y f!$ f&y +!I Number of au parameters, nti 4 4 8 3 Number of bU parameters, rr/ 0 0 0 0 Minimum expected dead time, k, 15.0 -2.0 -2.0 25.0 High process model gain limit -40.0 1.5 -40.0 2.0 Low process model gain limit
Implementation of the multivariable adaptive algorithm initially required selection of values for the user specified parameters of the adaptive algorithm. The values of the user specified parameters chosen for this example are given in Table 2; n,2 and nzl were chosen to be zero, yielding the simpler form of the multivariable controller given by equations (31)-(34). In this case, with all of the k,‘s equal to zero, the dead time between the ith input and jth output of the process could vary from zero to r,,- 1 sampling intervals. For start-up of the adaptive multivariable algorithm, initial estimates for the elements of 8, and &, were obtained by implementing the parameter esti97.0
5
1
mation in the open loop at the initial operating conditions of the test case. For the open loop estimation, PRBS signals were added to the process inputs, both with a constant amplitude of 0.4 lb/min. After 50 sampling intervals, estimation of the parameters in the A, polynomials of the process model employed by the controller was discontinued and the loops were closed with the multivariable adaptive algorithm. Also at this point, the location of the PRBS signals was switched to set points with the amplitudes defined as in Table 2. Figures 6 and 7 show, respectively, the responses of the contrblled variables for set point changes in the overhead and bottoms compositions with the adap-
r \
0 9!5.so
I 100
I
200
‘nyaseasc~
feed I
300
I
400
Time (min)
Time (min)
Fig. 6. F’rocess responses to set point changes in y, with an adaptive multivariable controller applied to the distillation column simulation (variable gains and dead times).
Adaptive placement controller
25
Composition
i
flow ratI I
'0
I
I
200
100
I 400
300
Timetmin)
I-
, -
Increase In lead I
flow
100
mm
I
I 400
300
200
Time (min)
Fig. 7. Process responses for set point changer in y, with the adaptive multivariable controller applied to the distillation column simulation (variable gains and dead times). tive multivariable controller for the test case discussed above. These figures also include the responses for the initial open loop parameter estimation. In contrast to the responses for PI controllers [9], the multivariable controller yields slightly faster responses at the initial operating conditions. However, after the decreases in the process gains and time delays occurring with the increase in the feed flow rate, the performance of the PI controllers .became sluggish. In contrast, Figs 6 and 7 show that the multivariable algorithm maintained its previous level of performance after the decreases in the process gains and dead times as it adapted to the process parameter changes. The responses of the MIMO controller for the unmeasured load was sluggish (although comparable to that for a PI controller). Modifications to the algorithm are currently under investigation to see if disturbance estimation can be used to improve the load response. As seen in Figs 6 and 7, interactions occur with the multivariable controller because the inputoutput pairs are not dynamically decoupled. The interactions arise primarily from different time delays between the input-output combinations. While interactions occur in the process outputs, the benefit of dead-time compensation and steady-state decoupling provided by the multivariable controller can be observed in the controller outputs [9]. For set point changes with the multivariable controller, the controller outputs move directly to their new steady-state values, after their initial actions, without noticeable oscillation. A significant advantage of the multivariable controller over conventional single loop control is that the multivariable controller requires a minimal amount of tuning compared to PI controllers. Tuning
PI controllers in the presence of the interactions of a multivariable process can be quite tedious. After implementation of the parameter estimation algorithm, the multivariable controller requires only adjustment of the t;s for each loop to achieve the desired performance.
REFRRENCB
1. D. E. Seborg, S. L. Shah and T. F. Edgar, Adaptive control strategies for process control: a survey. AICXE
Jl32, 881 (1986). 2. K. J. Astriim, Theory and application of adaptive control-a survey. Automatica 19, 471 (1983). 3. P. E. Wellstead, D. Pragcr and P. Zanker, Pole assignment self-tuning regulator: Froc. IEE 126, 781 (1979). 4. K. J. Astriim, P. Hagander and J. Stemby, Zeros of sampled systems. Automatica 20, 31 (1984). 5. D. W. Clarke, Self-tuning control of nomninimumphase systems. Automatica 20, 501 (1984). 6. K. J. AstrGm and B. Wittemnark, Self-tuning controllers based on pole-zero placement. Proc. IEE 127, Part D, 120 (1980). 7. C. E. Garcia and M. Morari, Internal model control unifying review and some new results. Ind. figag Chem., Process Des. Deu. 21, 308 (1982). 8. E. B. Dahlin, Designing and tuning digital controllers. Imtrumen~s and Control Systems 41, 77 (1968). 9. E. F. Vogel, Adaptive control of chemical processes with variable dead time. Ph.D. Thesis, University of Texas at Austin (1982). 10. E. Zafiriou and M. Morari, Digital controllers for SING systems: a review and a new algorithm. Int. .I. Control 42, 855 (l(985). 11. D. L. Prager and P. E. Wellstead, multivariable poleassignment self-tuning regulators. Proc. IEE 128, Part D, 9 (1980). 12. P. E. McDermott, D. A. Mellichamp and R. G. Rinker, Multivariable self-tuning control of a tubular authothermal reactor. Proc. ACC, 1614 (1984).
26
E. F. VOOEL and T. F. EDGAR
13. I. L. Chien, D. A. Mellichamp and D. E. Scborg, Self-tuning controllers for variable time-delay processes. Int. J. Control 949 (1985). 14. H. N. Koivo, A multivariable self-tuning controller. Aufomatica 16, 351 (1980). 15. P. C. Young, Applying parameter estimation to dynamic systems-part I. Control Engng 16,119 (1969). 16. B. Wittenmark and K. J. Astrom, Practical issues in the implementation of self-tuning control. Automatica 20, 595 (1984). 17. T. R. Fortescue, L. S. Kershenbaum and B. E. Ydstie, Implementation of self-tuning regulators with variable forgetting factors. Automatica 17, 831 (1981). 18. A. Gauthier and I. D. Landau, On the recursive
19.
20.
21.
22.
identification of multi-input, multi-output systems. Automatica 14, 609 (1978). J. P. Gerry, E. F. Vogel and T. F. Edgar, Adaptive control of a pilot-scale distillation column. AZChE Spring Natnl hftg, Houston, Texas (1983). R. K. Wood and M. W. Berry, Terminal composition control of a binary distillation column. Gem. Engng Isci. 28, 1707 (1973). S. L. Shah and D. G. Fisher, A multivariable frequency domain design method for feedforward disturbance minimization. Proc. JACC 4, 1 (1978). B. A. Ogunnaike and W. H. Ray, Multivariable controller design for linear systems having multiple time delays. AIChE JI 25, 1043 (1979).