2993
Ind. Eng. Chem. Res. 1996,34, 2993-3000
PROCESS DESIGN AND CONTROL An Anti-Wind-UpProportional Integral Structure for Controlling Time-Delayed Multiinput-Multioutput Processes Pablo F. Puleston**+and Ricardo J. Mantz' Laboratory of Industrial Electronics, Control and Instrumentation, Faculty of Engineering, National University of L a Plata, CC91, 1900-La Plata, Argentina
A predictive multiinput-multioutput (MIMO) proportional plus integral controller (PMPI)for time-delayed systems is developed. The proposed structure is able to contend with MIMO delayed processes, simultaneously fulfilling tracking and regulation demands. A practical and simple tuning method for regulation and tracking is proposed, and it is illustrated by a n example. Finally, the performance of the PMPI when dealing with saturation in actuators is treated, analyzing the anti-reset wind-up features of the controller.
I. Introduction It is well-known that the classic proportional integral (derivative) (PI(D)) regulator is the most widely used control strategy in the process industry. This is due to its remarkable effectiveness, abundance of tuning techniques, and broad applicability. However, despite its proven commendable properties, conventional PI(D) controllers are frequently incapable of fulfilling exacting engineering requirements. These limitations used to show up when controlling: (a) processes for which tracking and regulating characteristics are important, (b) multiinput-multioutput (MIMO) coupled systems, and (c) time-delayed systems. Attempting to solve difficulties emerging from case a, some unconventional PI(D) structures have been proposed (Iserman, 1981; Astrom and Wittenmark, 1984; Gawthrop, 1986; Gerry, 1987). Modified PUD) structures stand out among them (Hangglund and Astrom, 1985; Gawthrop, 1986; Eitbelrg, 1987; Mantz and Tacconi, 1989, 1981; Astrom, 1992). Derived from a simple but significant alteration to classical structures, modified PI(D) controllers allow a greater degree of independence in the design and practical tuning of the regulating and tracking features. A new modified P+I structure particularly suited to MIMO systems has been recently published (Puleston and Mantz, 1993). It has been formulated following optimal control concepts. This structure can reduce (even avoid) reset wind-up consequences: significant output overshoots due t o the combination of integral action in the control loop and saturation of the real system actuators (Shinskey, 1979; Astrom and Wittenmark, 1984; Hanus et al., 1987, 1989, 1991; Walgama and Sternby, 1993; Park and Choi, 1993). It can be said that the latter controller is more qualified than classical structures when confronting processes described in cases a and b. However, if the process contains significant dead time, both (classical and modified) structures are in trouble. The present paper introduces a predictive modified MIMO PI structure able to deal simultaneously with +
CONICET
* CICPBA.
systems under conditions enumerated in cases a-c. The contents of the paper are arranged as follows. The next section contains the development of the proposed structure. For its derivation the optimal control approach is followed, but the integrator final state is imposed to be a function of the reference vector. At the end of the section, the main features are discussed and a practical tuning method is presented. In section I11 an example is given, controlling a two-stage chemical reactor with time delay in the control. It shows the controller performance simultaneously facing tracking and regulating requirements (even under reset wind-up conditions). Finally, the last section deals with the conclusions.
11. Derivation of the Predictive Modified PI (PMPI) 11.1. The Control Policy. For the development of this paper we consider the control of processes with transport delay. Let us assume that the input control signal of the system is delayed by T, so that it has an input point which cannot be affected instantaneously but only after an interval of time T. Then, the system is described by
cy
+ Bu(t-T)
i ( t )= h ( t ) ( t )= C d t )
(1)
For the first T s the output of the system is the uncontrolled initial condition response, and hence, in a sense, the system will be not optimal for time T. But, given that the control is delayed, there is nothing that can be done to affect the output instantaneously. For the derivation of the proposed structure let us assume there exists a constant control uo which in steady state will maintain y ( t )at vector of constant set point yo. It means that lim i ( t )= 0 = Ax,
+ Bu,
lim y ( t ) = yo where the subscript 0 refers to the steady state equilibrium value. We wish the resulting controller t o have integral action in an attempt to achieve zero steady state error to set point changes and t o step external disturbances.
0888-588519512634-2993$09.00/0 0 1995 American Chemical Society
2994 Ind. Eng. Chem. Res., Vol. 34,No. 9,1995
Furthermore, the control system sensitivity to slow variations of the process parameters is reduced by forcing the controller to contain an integrator. Then, in order to include integral action to the control, we augment the system by adding a set of integral variables, given by
(3) Now, it is possible to redefine the extended system by shifting its origin to the steady state equilibrium point. The system equations are then given by
Substituting eqs 4 and 8 into eq 7 yields
u(t)= -F+(t+T)
- Ffl(t+T)
+ H& + F,f(yo)
(9)
+
where Hayo = Fp3co uo (Kwakernaak and Sivan, 1972; Puleston and Mantz, 1993),with HO= [C(-A + BFp)Bl-'. An adequate selection of the function f7yd will provide different tracking behaviors without affecting optimal regulation (it is obvious that the fourth term in eq 9 exclusively varies when the reference vector changes. Then, since yo remains constant for regulation, the term FIPyo) has no bearing on the control action in regulation mode). This paper stresses a linear relation between the integral final state and the set-point vector (40 = f i ~= )Ky,-,). In this conditions eq 9 becomes u ( t ) = -F+(t+T)
- Ffl(t+T) + Hyo
(10)
+
with H = Ho K. At this point, it is important to note that the control at time t depends on the future extended state a t time t T. Hence, a completed state predictor must be calculated in order to be included in the PMPI structure. 11.2. The Extended State Predictor. According t o eqs 4 and 5 the predicted extended state at time t T (neglecting disturbances between t and t !f') can be written as (Donoghue, 1977)
+
with
+
+
x_,(t+T) =
%,(t)+ f
T[4,(t - t)B,u(t)l d t
(11)
where
4e(t)= (state transition matrix of the extended system} = Defining u(t-T) = u ( t ) we transform eq 4 into
&,(t)=A&$)
+ B&)
converting the delayed problem to an undelayed problem to which the standard optimal control theory can be applied. Our aim is find a control structure which minimizes the regulation index J , while providing a great degree of independence for tracking tuning. We define J as
4(t) =
{state transition matrix of the original system} = eAt
O ( t ) = Lt4(t) dz = [4(t)- 11A-l = A-'[@(t)- I] II.2.a. Computation of x(t+Z'). From eq 11 x_(t+T)= x(t+T)
- xo
= #(T)[x(t)- xol
+ JTG
@(t-t>B[u(t)- uO1d t (12)
where R1, and R2 are positive definite weighting matrices. It can be shown that the control law which minimizes J is
where F = (Rz-lBTP,), and the matrix P, satisfies the algebraic Riccati equation for the augmented system. F is partitioned in matrices F p and FI, which characterize the feedback of x ( t ) and q(t), respectively. The proposed conIrol struEture (PMPI) comes out by imposing the integrator final state ( q ~as ) a function of the reference vector (yo): 40
=x r o )
(8)
Straightforward calculations give the predicted state vector as
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 2995
P(s)= @(s)B= [SI- A]-lB
H
(15)
and Xi(s) is the transform of the contribution of the initial conditions at time t. Observe that the full description of x(t+T) has two required parts: that due to the input u ( t ) and that due t o the initial conditions T s ahead. From now on, the notation a(t+T/t)w i l l be used rather than x ( t + T ) to indicate the prediction based on mea%urementsup to time t. In correspondence, for eq 14, X(s) will be used instead of X(s)eST. We now turn to a consideration of the integral state predictor. II.2.b. Computation of q(t+T). Equation 11 provides the following expression for the shifted integral state
i ,. . . . . . . . . . . . . . . . . .
I
1
.... ,
1
p Q Predictor
q(t+T) = q(t+n - 40
=
ce(nw- xO1+ q(t)- qo + f T
Ce(t-t) B[u(t)- ~
~.. . . . . .
u +
0 dt 1
(16)
Once again, straightforward calculations yield
’
w-
-
I
Figure 1. Proposed controller for time-delayed MIMO systems.
q(t+n = ce(n~ ( t+)q(t)t
In the expression above the prediction error is integrated over the whole control time, a fact that may result in steady-state offset &r significant model error. Such undesirable aftermath can be avoided by minimizing the prediction-error integration. This is easily attained by using the prediction e-sTX(s)in the place of state X(s)(Hammarstrom, 1980). Then eq 23 turns into
The integral predictor calculated in eq 17 can also be denoted as q(t+T) = q ( t )
+ Aq(t,t+r)
(18)
where Aq(t,t+T) takes into consideration the integrated T. It is natural that the value between t and t integral predictor should have to make use of the predicted state a(t+T/t)to compute Aq(t,t+T). This can be easily demonstrated by transforming eq 17:
+
where
C h )U(s)= Laplace{htCB(t-t) B u ( t )dt} (21) S
In summary, the integral is divided into two parts. The first part integrates the measured output deviation, and only the last part uses the predicted state to compute the integrated error between t and t T. Therefore, in the presence of model errors or biased disturbance in the output, the integral does not blow up since the biased prediction error is integrated just over a period of T units of time. Note that if an inadequate cancellation would have been made in eq 23, it would have led to a? expression for the predictor such as: Q(s)esT= l/s[CX(s) - YO] (Donoghue, 19771, resulting in steady-state offset after external step disturbances or model error. Following the notation previously suggested (section II.2.a) for predictors, we will use $(t+T/t)and Q(s)rather than q(t+T) and Q(s)esT. 11.3. Some Remarks Related to the Proposed Structure. Taking the Laplace transform of the control law introduced in section 11.1 (eq 10) gives
+
U(s)= -F$W
- F,$(s)
+ HYo
(25)
There with, the PMPI structure (Figure 1)is attained by implementing eq 25 using predictors computed in section 11.2 (eqs 14 and 24). The controller in Figure 1 can be interpreted as a Laplace{ht-T Ce(t-t) Bu(t)dt} (22) predictive MIMO PI structure, resembling different predictive SISO PI structures used in industrial proand replacing eq 14 into eq 19. Then, it can be written cesses. Therefore, for those readers well-versed with as: classical PI controllers, this interpretation will permit a better understanding of the proposed structure. Solely 1 1 Q(s)P = Q(S) ~ X ( S- )yo]- ;[cx(s) - e - s T ~ o ~ to take advantage of perception developed in the usage of SISO PI, we introduce a PI-like representation of the (23)
cS~ $ (P(s>e-sWs) T) =
+
2996 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
,... . ....Process .................. .,
I j IErr! ,..........._.:.. ........
~
-
Predictor
Predictor
Figure 2. PMPI interpreted as a predictive modified PI structure.
bance rejection without considering the reference tracking. (11)Improve tracking by adjusting the matrix WP (or H). Step I is not new to control engineers. For step I1 many tuning criteria can be proposed. We suggest a very practical tuning method. In this method, step I1 is accomplished by a tuning technique based on a time domain analysis of the control action vector u(t). For this purpose, we consider the effects of the adjusting matrix Wp over the control action. It can be easily observed that: (a) shortly after starting, the control action is basically constituted by the proportional contribution:
where
PMPI. For this purpose, operating with eq 25 (assuming that C and F p are of the same size and invertible) it becomes (b) when the elements of Y(t+T/t)become greater than the elements of Wpyo @(t+T/t)> Wpyo),the elements of up(t)change their signs, restraining the output rise. According to this, we define Wp as
where
Kp = F P C 1 W p= (FpC-l)-lH
(27)
are gain matrices and
ires, = CX(S, B(s) = {Yo- [CX(s)- Ce-”X(s)
+ Y(s)l}
(28)
are predictors of the output and the output error, respectively. Proportional and integral terms do not operate on the same predictive error signal. The P term has a modified error signal at its input (reference YOis weighted by matrix W p before being compared with the predicted output Y(s)). On the other hand, for steady-state accuracy, the: term integrates a direct prediction of the output error E(s)(in this case vector YOis not weighted). For different values of Wp the control system described by eq 26 (Figure 2) has a structure resemblance t o SISO PI configurations: classical PI (for Wp = identity) (Iserman, 1981;Astrom and Wittenmark, 1984; Gerry, 19871, IP controller (for Wp = 0) (Iserman, 1981; Nandam, 1986; Gerry, 1987), and modified PI (for Wp as a diagonal matrix) (Hangglud and Astrom, 1985; Gawthrop, 1986; Eitbelrg, 1987; Mantz and Tacconi, 1989, 1990; Astrom et al., 1992). Nevertheless the proposed structure is far more versatile and complete than a mere set of SISO PI controllers working in parallel. We have previously exposed (eq 9, section 11.2.b)that the proposed structure admits independent tracking tuning while preserving optimal regulation properties. This can be easily seen now, inspecting the PMPI illustrations: the control matrices Kp and FI (or their corresponding pair Fp and FI in Figure 1)are tuned for optimal disturbance rejection. Since Wp (or H in Figure 1)is out of the feedback loop, an adequate tuning of this matrix enhanced the system tracking behavior without affecting its regulation properties. On the basis of the former concepts we can infer a tuning criterion for the proposed structure: (I) Design matrices Kp and FI (or F p and F I ) for optimal distur-
W p= aWo
(30)
where a is a scalar factor, and WO= (FPC-~)-~HO is the set-point weighting matrix provided by the standard optimal regulator theory (corresponding to the term Flf(Y0) = 0 in eq 9) (Kwakernaak, 1972). A simple design possibility emerges: large values of a lead to stronger immediate controller action, while low values of a soften the controller action after a reference change. Therefore, smaller tracking overshoots can be obtained by choosing 0 < a < 1. Now, we would like to make a brief discussion about the behavior of the PMPI in the presence of nonlinearities in practical systems. One of the most commonly encountered nonlinearities in real world control engineering is the saturation in actuators, and often it is the dominant nonlinearity. When conventional controllers with integral action confront actuator saturation, serious performance degradation (large overshoots and large setting times) may take place. Reset wind-up or integrator saturation (Shinskey, 1979;Astrom and Wittenmark, 1984; Hanus et al., 1987, 1989, 1991; Walgama and Sternby, 1993; Park and Choi, 1993) occurs if the process input saturates and the controller continues integrating the error. The output of the integrator can then assume very large values, taking a long time to get it back to a normal value again. Generally in industrial processes, deviations (about the equilibrium state) due to set-point changes are larger than those caused by disturbances, thus for setpoint changes a stronger control signal will be needed to reinstate the system equilibrium. Therefore, the most critical situation, as far as actuator saturation is concerned, will be with the system acting as a tracker. The proposed structure permits a fast-recovery tuning, not imposing an underdamped tracking response. Thus, the adequate selection of matrix WP (or H ) may reduce tracking overshoots. This reduction is a direct consequence of a smaller tracking control action. We can summarize how the PMPI decreases reset wind-up overshoots with the following idea: smaller control actions in the tracking mode saturate actuators for
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 2997
Figure 3. Schematic diagram of the catalytic reactor.
shorter periods, shirking integrator overload. In the next section the reset wind-up effects are analyzed using a concrete example.
-0.1
o
1000
zow
3000
4o00
sow moo
7000
eo00
9000
ioow
111. Application: Control of a Thermal Process In this section we illustrate the features of the PMPI controlling a MIMO coupled system. The system parameters for this example have been taken from a pilot plant designed to perform catalytic conversion studies on light petroleum distillates. The plant is basically designed around an adiabatic tubular reactor, surrounded by five independent electric furnaces. Each one of these heating stages is monitored by a thermopair. For illustrative purposes, we have simplified the process to be a two-stage reactor (Figure 3). This assumption does not affect the conceptual aspects of the analysis. It is made only to keep the example as uncluttered as possible, without being trivial. The system is described (MKS units) by:
where output y ( t ) = ijl(t)yz(t)IT is the reactor inner temperature, state x ( t ) = [ x l ( t ) x2(t)lT is the temperature of the furnace shell, and control action u ( t ) = [ul(t)u2(t)lTis heating power (which has a 490 s delay). In every vector, the elements 1and 2 correspond to stage 1 and stage 2, respectively. In order to calculate the gain matrices of the propose structure, we follow the tuning criterion suggested in section 11.3. First, matrices K p and FI are computed without considering tracking specifications. Therefore, these matrices arise from optimizing the regulation index J 1100
0 1
10 0 0
101
-‘;@)[O 4000 4000 ] u-( t ) } dt (32) becoming
0.05 0
F ~ = [o
0.051
(33)
Figure 4 depicts the regulating response of the system when a disturbance reaction (a step disturbance of heat flow) occurs in stage 1 of the inner reactor . It can be seen that this design leads to fast recovery from disturbances in the output. Nevertheless, it is usual that tuning for optimal regulation may result in a controller with inappropriate tracking features. In fact, for the set-point weighting matrix that the standard optimal control theory would provide from minimizing J
w,= [la03 -0.023
1.03
(see eq 9), the tracking response of the controlled system would present a 33% overshoot (Figure 5, a = 1). It would be inadmissible for this process. Of course, tracking overshoots could be equalized by annexing an extra term in the performance index J, in order t o ponder tracking behavior. However, this procedure would end up deteriorating the dynamic response to disturbances. On the contrary, in our structure, no trade off is set between tracking and regulation: once the desired regulation has been achieved, the closed loop gain matrices are kept invariant and tracking enhancement is sought by adjusting the matrix Wp. For this purpose, we follow the second step of the proposed tuning criterion (eq 30): Figure 5 displays how the system tracking behavior is affected by variations of the factor
2998 Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995
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BOW
Woo
1OOW
W.'
loo0
0
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'
O
6ot\a
a
a
e4
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.L"
.
~
0
1000
2000
3000
4000
5000
6OW
7000
8000
9000
10000
0
(OW
2000
Tim6 (sscl
Figure 5. System tracking response when a reference step is applied to stage 1. (a, top) Output y(t): curves a, a = 1; curves b,
a = 0.75; curves c, a = 0.5. (b, bottom) Control action u ( t ) : curves a, a = 1; curves b, a = 0.75; curves c, a = 0.5.
a. A remarkable overshoot reduction is obtained by chosing a adequately. These improvements are dramatically manifested when dealing with real systems with saturable actuators. The proposed structure proves to be considerably senseless to reset wind-up if matrix Wp is correctly selected. Figures 6 and 7 permit us to compare the tracking performance of the system for different values of actuator saturation. The output and the control action corresponding to standard tuning (Wp = Wo, a = 1)is shown in Figure 6a and 6b, respectively. In Figure 7a tracking overshoots have been reduced by selecting a = 0.5. An inspection of the control action in Figure 7b reveals that actuators are saturated for shorter periods than in the case of Figure 6. Under these conditions, the control action for the tracking mode is not too demanding, consequently saturation is avoided or a t least diminished. In the previous analysis the advantages of the proposed structure have been illustrated. To bring this to an end, we would like to briefly discuss how model error influences the system behavior. This approach may result in a point of interest because several chemical processes (and particularly heat processes) are subject to significant model error. In due course, modeling errors have been taken into account by assuming that the model of the system from which the predictor has been computed is
3000
4 W
5000 6000 Time [sec]
7000
BOO0
8000
10000
Figure 6. Effects of saturation on the tracking response for a = 1 (Wp = WO).(a, top) Output y(t): curves a, no saturation; curves b, saturation a t 50 W; curves c, saturation a t 30 W. (b, bottom) Control action u ( t ) : curves a, no saturation; curves b, saturation a t 50 W; curves c, saturation a t 30 W.
k ( t ) = [A
+ AAb(t)+ [B + hBl~(t-T+bT) (34)
where AT and the error matrices AA and AB determine the model error. Many sets of values of AA, AB, and AT have been tested in order to infer conclusions about the system behavior in the presence of modeling errors. For each test, AT and the elements of the error matrices have been picked up at random from a Gaussian distribution of zero mean and a given percentage error (e % = 100u/element,where u is the standard deviation). In Tables 1 and 2 and Figure 8 some results can be observed. Table 1 dispjays the means of overshoot (fiJ and settling time (t,) when no actuator saturation exists. Each row has been computed using 20 different sets of error matrices, taken from Gaussian distributions of 10, 20,30, and 40% error, respectively. Table 2 is analogous to Table 1, but actuator saturation has been considered (Usaturation = 30 W). Note that in every case the ratio between the standard tuning (a= 1)and the proposed structure (a= 0.5) is in favor of the latter. Finally, Figure 8 depicts a set of time responses of the system under the most stringent situation: model parameters taken from a Gaussian distribution of 40% modeling error and with actuator saturation at Usaturation = 30 W (in accordance with the fourth row in Table 2). Figure 8a corresponds to standard tuning, while Figure 8b reproduces the proposed structure responses. It can be seen that, in spite of the significant model error, the
Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995 2999
L
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Ul
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Figure 7. Effects of saturation on the tracking response for a = 0.5. (a, top) Output y ( t ) : curves a, no saturation; curves b, saturation a t 50 W; curves c, saturation a t 30 W. (b, bottom) Control action u(t): curves a, no saturation; curves b, saturation a t 50 W; curves c, saturation at 30 W.
Figure 8. Effects of model error: system outputs using 10 different sets of error matrices (and A27 taken from a Gaussian distribution of 40% model error (with actuator saturation at 30 W) (a, top) Tracking responses for a = 1 (Wp = WO).(b, bottom) Tracking responses for a = 0.5.
Table 1. Case with No Actuator Saturation: Comparative Table (a = 1 against a = 0.6) of Means of Overshoot and Settling Time. Each Row Was Computed Using 20 Different Modeling Errors Taken from Gaussian Distributions of 10, 20, 30 and 40%Error, Respectively
Table 2. Case with Actuator Saturation at 30 W Comparative Table (a = 1 against a = 0.6) of Means of Overshoot and Settling Time. Each Row Was Computed Using 20 Different Modeling Errors Taken from Gaussian Distributions of 10, 20, 30, and 40%Error, Respectively Saturation a t 30 W mean of settling time (5%): t s mean of overshoot: f i p error e,
No Saturation ~~
~
error e, % = 1000/
element 10 20 30 40
mean of overshoot:
up
mean of settling time (5%): t s -
Mp,% Mp,% &fPW 2, ts t s w (a = 1) ( a = 0.5) MJO.5) (a = 1) (a = 0.5) t,(0.5) 17.6 19.3 21.1 23.2
4.8 5.8 7.1 7.9
3.66 3.32 2.96 2.93
5021 5073 5286 5304
4461 4517 5128 5200
1.12 1.09 1.03 1.02
PMPI responses are not significantly deteriorated, and they are much better than those of the standard tuning.
IV. Conclusions Time-delayed systems are difficult to control and prone to oscillate. Troubles enlarge when confronting multiinput-multioutput coupled processes. The controller proposed in this paper (a predictive PI control structure for time-delayed MIMO systems) admits optimal regulation according to a performance index, without imposing restrictions for tracking tuning. Therefore, it can satisfy tracking and regulating requirements simultaneously.
% = lOOU/
element 10 20 30 40
Mp,% Mp,% &fPW t, ts td1Y ( a = 1) ( a = 0.5) M,(0.5) (a = 1) ( a = 0.5) &(0.5) 36.8 37.7 38.9 40.9
6.1 7.4 8.8 9.5
6.03 5.09 4.42 4.30
6017 6160 6357 6398
4799 4987 5338 5422
1.25 1.22 1.19 1.18
A practical tuning method has been proposed. Even though better results can be obtained by using more sophisticated tuning, the effectiveness of the proposed method is proven by an example. It is notorious that reset wind-up effects worsen in the presence of time-delayed systems. The PMPI structure demonstrates advantageous anti-wind-up features, eliminating (or considerably reducing) undesired overshoots due to actuator saturation. In the example, the influence of model error has been analyzed. The proposed structure has shown to maintain good properties, even when confronting significant modeling errors.
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Received for review December 29, 1994 Accepted May 11, 1995 @
IE9403450
@
Abstract published in Advance A C S Abstracts, August 1,
1995.
