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Ind. Eng. Chem. Res. 1987,26, 774-781
Derbyshire, F. J.; Odoefer, G. A.; Varghese P.; Whitehurst, D. D. EPRI Report AP-2117, Vol. 1, 1981. Derbyshire, F. J.; Odoefer, G. A,; Varghese P.; Whitehurst, D. D. EPRI Report AP-2912, 1983. McMillen, D. F.; Malhotra, R.; Chang, S. J.; Ogier, W. C.; St. John, G. A.; Butrill, S. E.; Fleming, R. H. Proceedings of the 8th Contractor’s Conference on Coal Liquefaction, 1985; EPRI Report AP-2917. McNeil, R. I.; Young, D. C.; Cronauer, D. C. Fuel 1983, 62, 806. Silver, H. F.; Corry, R. G.; Miller, R. L. EPRI Report AP-2779,1982.
Spratt, M. P.; Dorn, H. C. Anal. Chem. 1984, 56, 2038. Taylor, L. T.; Hellgeth, J. W.; EPRI Report AP-4026, 1985. Taylor, L. T.; Hellgeth, J. W.; Squires, A. M. EPRI Report AP-2980, 1983. von Hausigh, D.; Koelling, G.; Ziegler, F. Brennst.-Chen. 1969,50, 8.
Received for review October 30, 1986 Accepted December 12, 1986
Adaptive Control Scheme for SISO Processes with Delays 0. E. Agamennoni, A. C. Desages, and J. A. Romagnoli* Planta Piloto de Ingenieria Quimica, UNS-CONICET, 8000 Bahia Blanca, Argentina
An adaptive control scheme for SISO processes with delays is presented. It uses an improved Smith Predictor with a dynamic filter to improve the disturbance attenuation properties of the system. Two different approaches to the filter formulation are presented. The identification algorithm evaluates an on-line Hessian matrix of the model-plant square error. Alternative schemes for the Hessian matrix evaluation are proposed based on parallel models and pseudosensitivity models. The strategy for controller and filter design allows us to handle the robustness of the adaptive control scheme against model uncertainties. The stability and dynamic performance of the control system are studied in relation to the model-plant parameters mismatch. Several simulation results illustrate the behavior of the proposed scheme, and a practical implementation in real time is shown by controlling an electrical furnace.
I. Introduction In process control, time delays are always a series obstacle which complicate the analytical aspects of control design and make good control more difficult to achieve. Smith (1957) proposed a compensation technique to eliminate the delay term in the closed-loop characteristic equation which is known world-wide as the Smith Predictor (SP). Among the special control techniques proposed for processes with delays, the SP has probably received the most attention. Further studies include experimental as well as theoretical investigations (Nielsen, 1969; Meyer et al., 1976; Marshall, 1980; Watanabe and Ito, 1981; Horowitz, 1983). Meyer et al. (1976) compared the SP control scheme with a conventional control. They omit, however, proper discussion of the effect of modelplant mismatch. Donoghue (1977) compared with SP with an optimal design approach. Marshall (1980) developed an adaptive scheme by using a SP, where the parameter identification subsystem uses an off-line Hessian matrix approximation of the model-plant square error. Palmor and Shinnar (1981) showed a systematic approach to advanced controller design suitable for processes with complex transfer functions by using the SP control scheme. Watanabe and It0 (1981) investigated the disadvantage of the SP in the presence of load changes and presented a basic solution for the disturbance rejection problem. Recently Horowitz (1983) concentrated on the sensitivity of the SP to plant uncertainty: uncertain plant parameters, and uncertain plant disturbances. Basically, he distinguished between the fundamental feedback and filter properties of the SP. This work presents an adaptive control scheme based on the SP with a filter to improve the disturbances attenuation properties of the system. The objective was to develop an adaptive algorithm in which the robustness property could be easily managed by the operator. Furthermore, attention was paid to the effect of imperfect modeling. The parameter identification subsystem eval0888-5885/87/2626-0774$01.50/0
uates a real-time approximation of the Hessian matrix of the model-plantsquare error. This allows fast convergence, even with initial errors larger than 100% in the model parameters. Two alternative ways to evaluate the Hessian matrix are considered, one based on parallel modeling and the other on the use of pseudosensitivity models. The approach may be used for the corresponding parametric adaptive problem and the mixed case, where temporal and parametric adaptive control are used simultaneously. An alternative filter formulation to that developed by Watanabe and Ito (1981), simpler to implement, is also proposed. It is based on a combination of the inverse of the free delay plant and a rational function. In addition, a strategy for the evaluation of the controller and filter parameters is also presented. It allows manipulation of the reference input-output and disturbance input-output transfer function poles. The stability and dynamical performance of the control system is studied in relation with the model-plant parameter mismatch to find “good” initial points for the model parameters. Finally, simulation results illustrate the behavior of the proposed scheme, and practical implementation is shown by the control of an electrical furnace.
11. Modified Smith Predictor The SP control system scheme for a SISO (single input-single output) system is shown in Figure 1; G,(s)T,(s) is the transfer function of the process which is assumed to consist of a rational stable transfer function, G,(s), and a dead time, T,(s) = exp(-sT,). Similarly, G,(s)T,(s) = G,(s) exp(-s.r,) is the model transfer function. The free delay part of the model (Gm(s)) predicts the effect of the controller, Gc(s), in the process. The closed-loop transfer function, between the reference input (r(s))and the process output (y(s)) is given by GcGpT, (1) Gr(s) = 1 + GcGm + Gc(GpTp- G,T,) 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 775
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Figure 1. Smith Predictor control system.
nd q u r t michiniim
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Figure 3. Adaptive control scheme with disturbance filter.
I
Figure 2. Smith Predictor control system with disturbance filter.
If the model and the process match perfectly (G,T, = GpTp),the delay term is eliminated from the characteristic polynomial of the closed-loop system. In this case, the design problem for the process with delay can be converted into one without delay. This is the most important advantage of the SP. On the other hand, the transfer function between the disturbance input (d(s)) and the process output Cy(s)) is
If the process and model match perfectly, eq 2 can be rearranged to give (3) where Gd' (s) is given by Gd' (s) = G,G,( 1 - T,) (4) The poles of Gd(S) are the zeros of 1+ GcGmand the poles of GJ (s). The controller, Gc(s),can be designed to place the zeros of 1 + G,G, in any desired position. However, if G,(s) has a large time constant (a pole near the origin), there is no way to decrease it in Gd' (s). In the limit, if G,(s) has a pole in the origin, a disturbance input will produce a steady-state error. Watanabe and Ito (1981) proposed the addition of a feedback filter ( F ( s ) )to the SP control system (Figure 2) to compensate the disturbance effects. Considering now the new scheme of Figure 2 and assuming no modeling errors, the effect of the disturbance input on the output is given by
transient response to unmeasurable step disturbances for all times greater than the time delay ( t > T ~ ) . Horowitz (1983) showed that this is not actually true since getting any desired transient response for t > T~ involves ideal prediction. However, it is still possible to use a scheme such as that in Figure 1if the claim is suitably redefined, since an approximation suffices. In this work, a robust adaptive control system which makes use of the control scheme shown in Figure 3 is suggested. The identification algorithm evaluates the Hessian matrix of a performance index in real time. An alternative filter to Watanabe's filter is also proposed. It makes use of an approximation of the inverse of the free delay plant. A strategy of design for the controller and filter parameter is presented. The basic solution of the design problem is given in terms of one adjustable parameter that allows us to define the location of the reference input-output and the disturbance input-output transfer function poles. That is, the resulting scheme will have one tuning parameter which can be easily adjusted on-line by the operator. The value of this parameter is limited by the degree of uncertainty. The final goal is to obtain an approximation of the acceptable region for the filter-controller parameters to establish a robust operating zone. 111. Identification Algorithm For the accurate control of time delay systems, both the delay elements and the free delay dynamic elements should be well modeled. A time delay control system works essentially on the basis of prediction, thus requiring that accurate models be available. A methodology is discussed in what follows which is based on using sensitivity point techniques to generate sensitivity coefficients. This adaptive strategy will be added to the SP to change the model parameters, so that they continually match the changing plant parameters. The strategy makes use of the convexity of the integral J ( a ) = ~ 0 t 0 + A i [ e ( t , dt a)]2
(7)
where e(t,a)is the mismatch between the model output and the plant output and (Y is the n-dimensional vector of model parameters to be estimated. The Taylor expansion of J ( a ) is and the transfer function from the reference input to the output is equal to that one of the system in Figure 1. Watanabe and Ito claimed that this modified control system can yield zero steady-state error and desired
J(ao+ A a ) = J(ao)+
Aa'VJ(cy0)
1 + -Aa'H(ao)Aa + ... 2 (8)
where V J = [gi] and H =
[hij](i, j =
1, 2,
..., n) are the
776 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
gradient vector and the Hessian matrix, respectively. From the last expression, the Newton's minimization method establishes that Aa which reduces J ( a ) is given by Aa = -hH(c~o)-'VJ(a0) = -XS(LYO)
(9)
where X is a positive scalar which defines a point on the Newton's search direction S(ao). Equation 9 is useful to tune the process parameter, but it requires evaluation of the Hessian matrix and its inverse. To solve this problem, Marshall (1980) developed an algorithm which makes use of a constant approximation of the Hessian matrix evaluated by a previous simulation of the control system. The present algorithm evaluates a good approximation of the Hessian matrix in real time whenever an estimation is needed, thus improving the convergence characteristics. Finally the inversion problem can be solved by applying a special method (Haley, 1980) to find H-' at the difference points of the parameter space from an initial one. Consider the following notation Aaii = [0, ..., 0, Aai, 0, ...,OIT and Aaij = [0,...,0, Aai, 0, ..., 0, Aa;, 0, ..., 0IT
(10)
If n parallel models are added to the SP model with parameters a Aai and if eq 8 is used, the principal diagonal elements of the Hessian matrix are
+
n
hii = LIJ(ao Aa?
+ A q i ) - J(ao)- Aagi]
(11)
To evaluate the remaining entries, consider n(n - 1)/2 extra parallel models with parameters a + Aaij. From eq 8 and with hii evaluated previously, the hij = hji is 1 h.. = [J(a0 + Aaij) - J(a0) - g;Aai - g;Aaj AaiAa; (hiiAa? + hjjAa?)] (12) Equations 11and 12 allow on-line evaluation of the Hessian matrix. The difference between this approximation and the Marshall approximation can be appreciated from what follows. The second derivative of J with respect to ai and a) is 2 J &,+At g ) d t (13) de de aaiaaj aai aaj aaiaaj
d2J
(--+
If e is assumed to be a linear function of ai,the term d2e/(aaidaj)is zero and an approximation of the Hessian matrix can be achieved with evaluation of the gradient vector only. The proposed algorithm computes a Hessian matrix which includes the second derivative of e. The algorithm was stable and had satisfactory dynamics even when the initial model was wrong by as much as 100% as will be shown later. On the other hand, the linear approximation failed to converge when a large mismatch between model and plant was imposed. Convergence Considerations. It is assumed that J ( a ) (R" R ) is a continuous function with continuous first and second derivatives. The minimization algorithm generates a sequence of points in the parameter space, a', ap,..., CYk starting from an initial one, a@ We want to find a condition to assure that this sequence approximates to the desired minimum, a. From eq 9, the parameter variation at each step is a k + l = ak - X k s ( a k ) = pk (14) In the neighborhood of a h , the search direction must be
-
a decent direction for the function J (Stoer and Bulrisch, 1980), Le.,
The partial derivative of eq 15 is given by
a&
-=--
ai
apk
81
(16)
and by use of eq 14
With eq 17 and from the definition of the search direction given in eq 9, condition 15 becomes vJ(cUk)TH(cUk)-lvJ(Ck) >0
(18)
If condition 18 is satisfied and X is sufficiently small, then J(ak+')< J ( a k )and the parameters values will converge to the minimum point a. Evaluation of the Hessian Matrix by Using Pseudosensitivity Models. An alternative implementation of the identification algorithm was performed by using pseudosensitivity models to evaluate the sensitivity coefficients. For example, for derivative of a given function with respect to the time delay is
a ar [G(s,a)e"']
= -sG(~,a)e-~'
(19)
which requires differentiation with respect to time. When pure differentiation is necessary, it is often possible to approximate to the derivative by using "pseudosensitivity" coefficients. In this case, the sensitivity models are approximations to those derived theoretically. Figure 4 shows a flow chart to calculate the derivatives of the error needed in the evaluation of the Hessian matrix of J , for a first-order system plus delay. Simulation Results of the Identification Algorithm. The behavior of the proposed identification algorithm is illustrated in Figure 5. In Figure 5a, the plant operation was assumed to be represented by Gp(s)Tp(s)= k, exp(srp)/(sup+ 1) = 80 exp(-s80)/s80 l),and the initial model was G,(s)T,(s) = k, exp(-sr,)/(su, + 1) = 40 exp(-s120)/(s40 + 1). When the system is at steady state, it is perturbed to allow the identification of all the plant parameters. It can be seen that the mismatch reduces with successive estimations. In Figure 5b, a second-order plant was considered, G,(s)Tp(s)= 80 exp(-s80)/(s80 + l)(s20 + 11,the initial model being the same as before. Obviously, the mismatch does not reduce to zero with the successive estimations but the performance of the control system is effectively improved. Parts c and d of Figure 5 compare the behavior of the identification algorithm with the on-line evaluation of the Hessian matrix and with the linear approximation, respectively. The plant and initial model were the same for = 80 exp(-s80)/(s80 + 1) and G,both cases: Gp(s)Tp(s) (s)T,(s) = 40 exp(-s40)/(s40 + l),respectively. This is obviously not a good initial model due to the underestimation of time delay. However, the proposed algorithm converges in spite of the initial mismatch. On the other hand, with the linear approximation, the control system becomes unstable.
+
IV. Control System Design Controller Design. For the SP control system shown in Figure 2, the transfer function from the reference input
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 777 k c =k L m ( u m ( ~ + ~ ) - l )
(24)
The maximum controller gain for minimum bl and b2 occurs at bl = b2 = b. With this assumption, k , and 7 b are now kc
=L(+) k, k,k,b2
Tc
=Um
In this way, the controller parameters are obtained as functions of the model parameters and an extra one, b, which defines the pole locations of the input-output transfer function. This parameter can be easily adjusted on-line by the operator, and the estimation of the upper and lower bounds will be discussed later. Filter Design. In the previous section, a modified SP was introduced. It makes use of a filter to improve the disturbance attenuation properties. Watanabe et al. (1983) proposed a filter structure which, for a first-order plant, is 1 F ( s ) = kl k2sb + 1 They imposed the conditions (sa, + l)[Gm(s) - F(S)G,(~)T,(~)II~=-~,~, = 0 (29)
+
Figure 4. Sensitivity scheme for the evaluation of the Hessian matrix for a first-order model.
to the output (GI($) is given by eq 1. If model and process match perfectly, it reduces to
lim [G,(s) - F(s)G,(s)T,(s)] = 0
(30)
S-0
Condition 29 removes the pole of G,(s), and condition 30 assures null steady-state error. Finally, the gain factors kl and k2 are where G = G,G,. A strategy to design the controller G, is to impose that the zeros of 1+ G lie in a desired position on the left half of the s plane. However, if model and plant do not match perfectly, GI will have infinite poles. None of the approximations of the delay term can be used to examine the poles location of the closed-loop system. This is mainly because the error introduced by the approximation is very important. In this way, for each approximation we will have an “optimal” controller gain. Since in an adaptive scheme the model approaches the process after successive estimations, the controller parameters can be adjusted accordingly. We can assume no mismatch between process and model and a particular controller structure (PI, PID, etc.). Hence, the controller parameters are functions of the model parameters and the desired poles location. Let us examine the pole location of eq 20 for a first-order system and a PI controller (G,(s) = k , ( l + 1 1 ~ 7 , ) ) .The characteristic polynomial is given by 1 + k,k, k,k, s2
+s
am
+-=
Team
0
where -l/um is the pole of the model. Imposing the roots of eq 21 to be at - l / b l and - l / b z , we have
Solving for k , and
T,,
we have
In this way the disturbance rejection can be improved by shifting to the left the poles of G&) of eq 5. A remark needs to be made at this point regarding the claims of using the filter F(s) as proposed by Watanabe et al. (1983). We are using it as an approximation of the ideal filter without imposing a desired transient response of Gdf(s)to step disturbances. Besides, an alternative filter formulation is still possible, which is simpler to design based on the inverse of the plmt. The disturbance transfer function of the system with the filter in eq 5 can be rearranged to
where (34)
After the disturbance has entered into the process, no action can be performed until time 27, (see Figure 6). An empirical improvement of the disturbance rejection action may be done by shifting to the left the pole of G’. If 7, > a, when an unmeasurable step disturbance enters the system, the output due to G, exp(s7,) is near its steadystate value at time 27,. Thus, acting on G’, it is feasible
778 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
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Figure 5. Performance of the identification algorithm with the proposed approach to the Hessian matrix. (a, top left) Plant, GpTp= 80 exp(-s80)/(s80 + 1); initial model, G,T, = 40 exp(-s120)/(s40 + 1). (b, top right) Plant, GpTp= 80 exp(-s80)/(s80 + 1)(s20+ 1); initial model, G,T, = 40 exp(-s120)/(s40 + 1). (c, bottom left) Plant, GpTp= 80 exp(-s80)/(s80 + 1);initial model, G,T, = 40 exp(-s40)/(s40 + 1). (d, bottom right) Performance of the identification algorithm with linear approximation to the Hessian matrix. Plant, G,T, = 80 exp(-s80)/(s80 + 1);initial model, G,T, = 40 exp(-s40)/(s40 + 1).
to improve the disturbance rejection behavior. The poles of G‘are the zeros of 1 + G,G, and the poles of G, and F. The controller G, was designed to act on the zeros of 1+ G,G,. Consequently,F can be thought of as composed of the inverse of G, and a rational function X(s). Then F is
t
I
(35)
where (36)
and n is the order of the system. The gain k is evaluated from lim sGdf(s) = 0 (37) S-0
and the pi’s are
--_
I
I
Figure 6. Components of the output time response to a step disturbance input: (a) step input, (b) time response of G, exp(-sT,), (c) time response of G‘ exp(-2s~,), and (d) time response of the output (Gdf).
where is the dominant time constant. If the delay term is larger than the dominant time constant, G ’ exp(-2sr,) must respond fast to extinguish the response of G, exp-
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 779
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404
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200
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SO0 Tima (S.C
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Figure 7. (a, left) Performance of the control scheme under load step changes G Tp = G,T, = 80 exp(-s160)/(s80 + 1)and (b, right) under * -) inverse approach. load step changes GpTp= G,T, = 80 exp(-s80)/(s80 + 1): (--) without filter, (-- -) Watanabe's approach, (-a-
Ok
Figure 8. Stability limits for a fist-order plant plus delay with SP compensator and the proposed controller in a gain-delay mismatch plane: a, = 48, and 7, = (1 + 0 , ) ~(b, ~ middle) ; GpTp= 25 exp(-s80)/(s80 + l), k, = (1 (a, left) GpTp= 25 exp(-s80)/(s80 + l), k, = (1 + O&,, + Ok)kp, a, = 80, and T , = (1 + 0 , ) ~and ~ ; (c, right) GpTp= 25 exp(-s80)/(s80 + l),k, = (1 + Ok)kp, a, = 112, and T , = (1
+
(-ST,) in eq 33 which is near steady state at time 27,. But if conversely amd> T, the response of G, exp(-s~,) is far from steady state at time 27,. Then if amd - r, is larger than b, the value of a m d - 7, for p i makes the responses of G, exp(-sT,) and G'exp(-2srm) approach each other near time 7, + amd (see Figure 6). Figure 7 compares the disturbance rejection performance of the SP with and without the filter for a first-order system. In this case, no model-plant mismatch was assumed and GpTp= 80 exp(-s160)/(s80 + 1). In Figure 7a, Watanabe's approach to the disturbance rejection filter has nearly similar performance than with the inverse of the free delay plant approach. It can be seen that both approaches improve the system disturbance attenuation properties, in comparison with the system without filter. In Figure 7b the model and plant delay were decreased to 80 units of time. Now the performances of the two filters deviate considerably from each other. However, in both cases, the disturbance rejection action is effectively improved. Stability and Performance Analysis. The controller and filter parameters were defined previously as functions of the model parameters and an extra parameter b. This restricts the design procedure to the on-line adjustment of only one parameter ( b ) by the operator, leading to a simpler design task. An estimation of the upper bound of the model-plant mismatch allows evaluation of the up-
per bound of b, which can be used as the initial value. In the steady state, when a good estimation of the process parameters is available, b can be decreases to obtain a fast disturbance rejection. A lower bound of b can be evaluated with an estimation of the minimum mismatch between process and model parameters when the system is operating at steady state. It is also helpful to know, from a stability and performance point of view, whether it is preferable to underestimate or overestimate the plant parameters. Stability limits in a model-plant parameters mismatch plane are given in Figure 8. They were developed for the nominal process Gp(s)Tp(s)= 25 exp(-s80)/(80s + 1) by using Bode's criteria and for three different values of b. The figure shows that if the model gain is smaller than the process gain, the control system loses stability robustness. This is due to the fact that the controller gain is invesely proportional to the model gain. Similarly, when the model time constant is greater than the process time constant (Figure 8c), the stability zone becomes narrower. Regarding the delay term, it is preferable that the model time delay be smaller than the plant one. In Figure 9, the performance level curves in a modelplant parameter mismatch plane are given for the same nominal process. The performance index used as described in the Appendix. As expected, in the neighborhood of the origin (small errors), the performance index decreases as
780 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
Ok
Figure 10. Block diagram of the adaptive temperature control system.
1
" = , I 2
56
--
~
-
Figure 9. Performance level curves for a first-order plant plus delay with compensator and the proposed controller in a gain-delay mismatch plane: (a, top) G,T, = 25 exp(-s80)/(s80 + l),k , = (1 + O&,, a, = 80, and 7, = (1 + OJiP; (b, bottom) G,T, = 25 exp(, b = 40. s80)/(s80 + I), k , = (1 + O,)k,, T, = (1 + O ? ) T ~ and
b decreases. For each value of b, three level curves were considered to show the sensitivity of the performance with respect to mismatch. With no error in the time constant, the sensitivity increases as b decreases. Furthermore, as b decreases, it is preferable to overestimate the gain of the process in the same way as for stability limits. Figure 9b shows that from the performance point of view, both the time constant and the gain should be either underestimated or overestimated. The performance is as before very sensitive to the delay mismatch. In conclusion, for the case under study, as expected, we need a good evaluation of the delay term while the gain and the time constant allow larger mismatch. Also, an underestimation or an overestimation of both the gain and the constant and a large value of b (near the time constant of the process) are good initial parameters for the adaptive scheme. In this way, it is possible to localize zones of mismatch with acceptable performance and acceptable degrees of stability, that is, to establish a robust operating zone. V. Illustrative Example On-Line Control of an Electrical Furnace. The adaptive algorithm was applied to control the temperature of an electrical furnace. The block diagram of the control system is shown in Figure 10. A t first, the step response
of the system was studied to evaluate the initial model parameters. A reduced first-order plus delay model was considered. The sampling frequency used was 2 s, and the evaluated model parameters were a, = 80 [samplingunits] , 7, = 93 [sampling units], and k, = 22 at 50 "C to 29 at 30 "C. The identification of such a system during the transition from one operation point to another is difficult due to the continuous variation of the process gain. To solve this problem, only gain estimation was considered in the transition time (set point changes). To estimate the dynamic parameters a t steady state, a square impulse perturbation of variable duration time was introduced in the system. The maximum admissible disturbance in the process and the estimated time constant define this perturbation. Figure l l a shows the response of the adaptive system. After the set point changes, the adaptive system estimates the process parameters by means of perturbations. Four successive estimations are needed to obtain a negligible model-plant mismatch. Figure I l b shows the b variation effects on the disturbance rejection when a load step change is applied. With b = 30 the disturbance rejection is faster than with b = 60. The value of b = 60 defines a robust operating condition for the adaptive system when no adequate estimation of the process parameters is available. When the system is at steady state and with a good estimation of the process parameters, b = 30 can be used to obtain a faster disturbance rejection. This value of b appears to be the minimum, since by use of b = 20 the closed-loop system exhibits an oscillatory behavior, as shown in Figure l l c .
VI. Conclusions A robust adaptive control scheme for processes with time delays has been presented which has the following features: The identification algorithm evaluates a real time approximation of the Hessian matrix. Two different approaches can be considered for the Hessian evaluation based on parallel modeling and pseudosensitivity models. Controller and filter parameters are related in a straight forward manner to the model parameter and one adjustable parameter. This simplifies, considerably, the on-line tuning of the controller. Two different structures for the filter are possible. One is based on an approximation due to Watanabe and Ito (1981),while the other makes use of the inverse of the free delay part of the model. The stability and dynamical performance of the closed-loop system were studied in connection with the model-plant parameter mismatch. The "goodness" of the proposed scheme was illustrated by means of simulation experiments and by the on-line control of an electrical
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 781 I
1
I*npl
rc I
J ( a ) = plant-model integral square error
k = gain
!
pi = time constants of the inverse approach filter S(a)= Newton's search direction of the minimum
SP = Smith Predictor t = time T,(s) = delay transfer function of the model Tp(s)= delay transfer function of the plant Greek Symbols a = model parameter vector Aa = incremental value of the model parameter At = incremental value of time x(s) = rational function in Laplace domain T = dead time T~ = integral component of the PI controller Subscripts c = controller k = identification counter m = model p = plant
Appendix In the construction of the performance level curves, the following index is considered
P.I. = log ( l "0 ( y ( t ) - rf(t))2dt) where y ( t ) is the control system output and r&) (reference) is defined as rf(t) = 0 if 0 C t C T~ rf(t) = s.p. -
1 12CU
2uy)
3MO
Tim
Figure 11. Time response of the adaptive temperature control system. (a, top) Process and model time response during the successive estimations. (b, middle) Process time response to a load step change with b = 30 and 60. (c, bottom) Process time response to a load step change with b = 20.
furnace. The results have shown that the proposed algorithm converges in spite of the large initial errors. It was also shown that the closed-loop system poles can be shifted on the left half of the s plane according to the model-plant mismatch, resulting in improvement of the control system robustness.
Nomenclature a = time constant amd= dominant time constant bi = time constants of the disturbance filter (F(s) e(t,a) = plant-model error F(s) = disturbance filter gi = gradient component of the ith model parameter Gd(s) = disturbance input-plant output transfer function without disturbance filter G&) = disturbance input-plant output transfer function with disturbance filter G,(s) = free delay transfer function of the model GJs) = free delay transfer function of the plant G,(s) = reference input-plant output transfer function H(a) = [hij(a)]Hessian matrix of J
if
T
