the identification data set by the closed-loop system, coupled with a control-relevant prefilter, yields a model whose corresponding control system improves its ...
CLOSED-LOOP IDENTIFICATION OF RESTRICTED COMPLEXITY MODELS USING ITERATIVE REFINEMENT Daniel E. Rivera and Saurabh Bhatnagar Dept. of Chemical, Bio, and Materials Engineering and Control Systems Engineering Laboratory Computer-Integrated Manufacturing Systems Research Center Arizona State University Tempe, AZ 85287-6006 Society Affiliation(s): AIChE, IEEE Abstract A novel technique for closed-loop sytem identification of reduced-order models is presented. The method arrives at a process model and its corresponding compensator in an iterative fashion by introducing a series of step changes at the manipulated variable/setpoint. The bias introduced into the identification data set by the closed-loop system, coupled with a control-relevant prefilter, yields a model whose corresponding control system improves its performance at every step. The method is appealing to chemical engineering practitioners because it combines the tasks of system identification with controller commissioning to produce a simple-to-use yet reliable autotuning procedure. 1. Introduction Identification testing in the chemical process industries is commonly conducted while the plant is in normal operation. As a result, the identification test must be designed such that the important process variables do not deviate significantly from setpoint. One way of achieving this requirement is by conducting the identification test while the plant remains in the presence of control. Closed-loop identification is therefore a problem of significant practical and industrial importance. Previous work by Rivera and Tedja [1] analyzed certain important aspects of closed-loop identification (input signal design, model structure selection, and parameter estimation) from the traditional approach to the problem: one attempts to create conditions that make the closed-loop system Strong System Identifiable [2] and then selects design variables in the identification problem such that a useful model for control is obtained. In [1] it was shown that controller detuning is often necessary if detrimental bias caused by the closed-loop system is to be avoided. In this paper we consider an alternate control-relevant approach which takes a different philosophical approach to the problem: instead of avoiding closed-loop bias, why not take advantage of 1
it? By establishing a link between the desired closed-loop objective and the system identification objective, it becomes possible to use the closed-loop system to naturally weight those frequencies that are most important for control design. The result of such an approach is shown here. This work philosophically parallels several iterative schemes for simultaneous modelling and enhanced controller performance which have been recently proposed in the literature [3],[4]. Like [4] we apply control-relevant frequency weighting on prediction-error estimation techniques using prefiltering; however, our method uses the Internal Model Control design procedure [5] to construct the compensator instead of LQG. [3] also uses the IMC design technique, but relies on the closed-loop technique developed by [6] for system identification. In our study we find that the application of a control-relevant prefilter as developed in [7] during the initial open-loop test of the procedure results in a substantial improvement in the rate of convergence of the method. The overall effectiveness of the technique has been illustrated with the help of two examples. 2. Theoretical Development We begin by assuming that the system to be identified and controlled is represented in the form: y(t) = p(z)u(t) + ν(t)
(1)
Here p(z) is the true plant model, assumed to be a linear transfer function in terms of z, the forward shift operator and ν(t) is a stationary noise sequence with power spectrum Φν which is uncorrelated with u. The goal of the identification procedure is to estimate a transfer function model p˜ and noise model p˜e from observations of y and u while the true plant in (1) is under classical feedback control: y(t) = p˜(z)u(t) + p˜e (z)e(t)
(2)
(2) encompasses the entire family of prediction-error model structures which include such wellknown methods as ARMAX, ARX, FIR, Box-Jenkins and Output Error estimation. We demonstrate the identification procedure for excitation injection at two points in the loop 1) at manipulated variable (ud ) 2) setpoint (y). Recommendations are made based on the comparison of the closed loop time series obtained for each case. For signal injection at the manipulated variable output y(t) is described by y(t) =
1 p(z) ud (t) + ν(t) 1 + pc 1 + pc
(3)
while the manipulated variable u(t) satisfies u(t) =
1 c(z) ud (t) − ν(t) 1 + pc 1 + pc
2
(4)
For signal injection at the setpoint the closed loop responses are governed by p(z)c(z) 1 r(t) + ν(t) 1 + pc 1 + pc c(z) c(z) r(t) − ν(t) u(t) = 1 + pc 1 + pc
y(t) =
(5) (6)
If the input/output data is Prefiltered then yF (t) = L(z)y(t)
uF (t) = L(z)u(t)
for signal injection at the manipulated variable the filtered prediction error is eF (t) = L˜ p−1 ˜u(t)) = L˜ p−1 ˜)ǫud (t) + (ǫ + p˜p−1 η)ν(t)) e (y(t) − p e ((p − p
(7)
while for signal injection at the setpoint the corresponding equation is eF (t) = L˜ p−1 ˜)p−1 ηr(t) + (ǫ + p˜p−1 η)ν(t)) e ((p − p
(8)
where
η = pc(1 + pc)−1 Complementary sensitivity function ǫ = (1 + pc)−1 Sensitivity function The equations for the power spectrum of the prediction error for each case are then | L |2 | p˜e |2 | L |2 = (| p − p˜ |2 | p−1 η |2 Φr + | 1 + p˜c |2 | ǫ |2 Φν ) | p˜e |2
ΦeF mani = (| p − p˜ |2 | ǫ |2 Φud + | 1 + p˜c |2 | ǫ |2 Φν ) ΦeF setpoint
(9) (10)
which can be readily shown via Parseval’s Theorem to be related to the 2-norm sum of the timedomain prediction error. (9) and (10) clearly show that prefiltering acts as a frequency-dependent weight on the goodness-of-fit in identification, and can be judiciously chosen by the control designer to influence the estimation procedure in a way that most benefits the ensuing controller design problem. We arrive at an expression for the control-relevant prefilters by defining an identification objective that seeks to minimize the difference between the control error of the nominal control system (based on the identified model) versus the error that is actually seen when the controller is implemented on the true plant. Such an objective function is represented as min || e − emodel ||2 = min || p˜
p˜
3
(r − d) (r − d) − ||2 1 + pc 1 + p˜c
(11)
which can be rearranged to yield min || p˜
p − p˜ p˜c 1 · · · (r − d) ||2 p˜ 1 + p˜c 1 + pc
(12)
Establishing a relationship between (9) and (10) with (12) (and assuming a sufficiently high input signal to noise ratio, which allows us to ignore the term associated with Φν ) leads to the prefilter statements L(z)mani = L(z)setpoint
p˜−1 η˜p˜e (r − d) ud ǫ˜p˜p˜e (r − d) = rident
(13) (14)
While (13) is a low pass filter (Fig 8) the prefilter for the for setpoint signal injection case (14) posses band pass charactersistics (Fig 9). These prefilters are successively updated for each iteration (i.e., each new step change introduced in ud or r), resulting in an improved closed-loop performance at each step. Hence the entire identification procedure can be summarized in steps as 1. Design a controller that stabilizes the initial ARX model estimated from prefiltering the openloop data using the method described in [7] versus no prefiltering on the data. 2. Generate closed loop data from the loop comprising the true plant and the first compensator by introducing a step change at the manipulated variable or setpoint. 3. Based on the choice of the point of signal injection choose (13) or (14) and prefilter the closed loop data. 4. Feed the prefiltered data to the ARX identifier for generation of a second model and hence a second compensator. 5. Set up the closed loop and go to step 2 and keep repeating the procedure until satisfactory responses are obtained. The benefits of this procedure are shown with examples. 3. Examples Example 1:
To illustrate the effectiveness of the proposed closed-loop identification procedure
we consider a first-order plant with deadtime equaling the time constant of the process: p(s) = 4
e−s s+1
(15)
The objective is to use AutoRegressive with eXternal input (ARX) estimation is to fit the plant to the model p˜(z) =
z2
K(z − β) − α1 z + α2
(16)
The sampling time is set at T = 0.1 min. The IMC-derived controller for (16) assuming a step input for the H2 -optimal portion of the design and a second-order filter can be represented parsimoniously as a five-term difference equation in velocity form △uk = Kc [ek − τI ek−1 + τD ek−2 ] + τF1 △uk−1 + τF2 △uk−2
(17)
The desired closed-loop time constant is τcl = 1.0 min. Following the steps of the proposed identification scheme we begin by designing a controller that stabilizes the initial ARX model estimated from prefiltering the open-loop data versus no prefiltering on the data. The true plant and the first compensator now constitute a closed-loop system which will now be used to generate closed loop data by giving a step change at the manipulated variable. In both cases the data is prefiltered using (13) or (14) (as the case may be) and fed to the ARX identifier for generation of a second model and hence a second compensator. This iterative procedure is continued until there is no significant change in the closed loop performance. Fig 2. and Fig 3. depict the closed loop responses at various steps when signal injection point is manipulated variable while Fig 4. and Fig 5. show the closed loop responses when signal injection is performed at the setpoint. Clearly, as noted in both cases the use of the control relevant prefilter on the first open loop step substantially improves the rate of convergence. Example 2:
Consider a case when the closed loop responses of a control system have become
highly oscillatory because the parameters of the plant have undergone changes with time. We will show that application of the proposed identification method on the badly tuned loop can retune the controller to have nice overdamped closed loop responses. We shall assume that the closed loop was originally tuned for e−s (18) s+1 This apriori knowledge will be of great help in designing the closed loop prefilter for the first iterap=
tion. We therefore begin by exciting the badly tuned closed loop by a set-point change at the manipulated variable. We tap the u(t) and y(t) data and prefilter it using the closed loop-prefilter based on the apriori knowledge of the plant. The prefiltered data is fed to the ARX identifier estimating a model of the type (16) which in turn is further utilized to yield a controller of form given by (17). Now that the construction of the closed loop is complete the new set up is again used to 5
generate closed loop data. Like the Example 1 this iterative procedure of signal generation, prefiltering, model estimation and controller commissioning is repeated till the satisfactory tuning of the compensator is achieved. Fig 6 shows the various steps in retuning the badly tuned controller. Fig 7 illustrates the same but for the Prett-Garcia type of a controller with a form similar to (17) but with one lesser move on the manipulated variable. The effectiveness of this technique has also been shown with Output Error estimation in lieu of ARX. Acknowledgments The support of the National Science Foundation (Grant No. CTS-9110528) is gratefully acknowledged.
References [1] Rivera, D.E. and A. Tedja, “A Control-Relevant Methodology for Closed-Loop Identification,” 1991 AIChE Annual Meeting, paper 144b, Los Angeles, November, 1991. [2] Gustavsson, I., L. Ljung, and T. S¨oderstr¨om, “Identification of processes in closed loop identifiability and accuracy aspects,” Automatica, 13, 59, 1977. [3] Lee, W.S., B.D.O. Anderson, R.L. Kosut, I.M.Y. Mareels, “On Adaptive Robust Control and Control Relevant System Identification,” 1992 American Control Conference, Chicago, IL, 1992. [4] Zang, Z., R.R. Bitmead, M. Gevers, “Disturbance Rejection: On-line refinement of controllers by closed loop modelling,”1992 American Control Conference, Chicago, IL, 1992. [5] Morari M., E. Zafiriou, Robust Process Control, Prentice-Hall, 1989. [6] Hansen, F., G. Franklin, and R.L. Kosut, “Closed-loop identification via the fractional representation: experiment design,” 1989 American Control Conference, Pittsburgh, PA. [7] Rivera, D.E., J.F. Pollard, and C.E. Garc´ıa, “Control-Relevant Prefiltering: A Systematic Design Approach and Case Study” IEEE Transactions on Automatic Control, Special Issue on System Identification for Control Design, 37, 964,1992. [8] Prett and Garc´ıa, C.E.,Fundamental Process Control, Butterworths, Stoneham, MA, 1988.
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FIGURES
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Figure 1: Closed-loop feedback structure with signal injection at manipulated variable
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Figure 2: Example 1: Closed-Loop responses without open-loop prefiltering (signal injection at manipulated variable)
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Figure 3: Example 1: Closed loop responses with open-loop prefiltering (signal injection at manipulated variable)
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Figure 4: Example 1: Closed-Loop responses without open-loop prefiltering (signal injection at setpoint)
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Figure 5: Example 1: Closed loop responses with open loop prefiltering (signal injection at setpoint)
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Figure 6: Example 2: Iterative closed loop response refinement with controller given by △uk = Kc [ek − τI ek−1 + τD ek−2 ] + τF1 △uk−1 + τF2 △uk−2
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Figure 7: Example 2: Iterative closed loop response refinement with controller given by △uk = Kc [ek − τI ek−1 + τD ek−2 ] + τF1 △uk−1
Prefilter based on closed loop data 10 0
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Figure 8: Iterative closed loop prefilter for signal injection at manipulated var. p˜−1 η˜p˜e (r−d) ud
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Figure 9: Iterative closed loop prefilter for signal injection at set-point. L(z)setpoint =
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ǫ˜p˜p˜e (r−d) rident
