In recent years there has been an extensive and growing interest in feedback control systems, which automatically adjust their controller settings to compensate for changes in the process or the environments. Such systems are known generically as adaptive control systems. Adaptive control techniques have considerable potential for application where controlling processes are not well understood or they have significant non-linearities or time-varying parameters.
The relation of the input (manipulated variables) U and the output (process variable) Y is modeled by the regression model:
Y(k) = SAi Y(k-i) + SBi U(k-i-td) + e
A = 1 + a1 z^-1 + a2 z^-2 + … + an z^-n
B = b0 + b1 z^-1 + … + bm z^-m
Where e is the “white noise” and unknown polynomial coefficients Ai, Bi are identified in real time by recursive least square method. The model transfer function can be expressed as:
G(z) = B(z) / A(z)
To be able to predict and/or control the output Y the conditional probability density functions with boundary conditions is employed. The theory of the Bayesian system classification has appeared to be a suitable tool to solve this class of problems. This approach also yields stability and robustness with respect to unmodeled terms and it has a good disturbance rejection properties.
A control algorithm is, as a rule, only a small part of software ensuring the digital control. The dominating part is computer activity associated with data management. The choice of sampling rate becomes as integral part of the control design because it influences heavily the discrete output signal to be processed. Very fast sampling makes it possible to observe high-frequency disturbances and the optimal controller tries to eliminate them, sometimes uselessly or even harmfully complicating its function. On the other hand, very slow sampling may bring too little information and the control quality decreases.
The period should be selected as small as possible. However, in practice other factors must also be taken into consideration when choosing the sampling period such as process dynamics (dominant time constants of the controlled process), the spectrum of the disturbances and the computational requirements.
The following remarks point out some cases (directly related to control) when special data processing is of importance.
The adaptive controller/Modeling Algorithm is sensitive to outliers, rare exceptional data occurring due to a failure (e.g. measuring device). Thus a simple test for suitable range of measurements and /or their changes is almost inevitable.
The scaling of data must be algorithm-friendly. Awkward scaling may cause higher ” computation noise”, scaling also has an impact on convergence of algorithms. For example, an (almost) linear dependence of data makes convergence of identification algorithms rather slow.
Extreme care should be taken when creating a model from closed loop data sets if PID control was invoked with rather high sampling rate.
MODEL TYPE ‘Delta’
|PV(t)||= PV(t-1) + [V11,V12,V13,…]||*||PCSOut[t-td1] – PCSOut[t-td1-1]
PCSOut[t-td1-1] – PCSOut[t-td1-2]
PCSOut[t-td1-2] – PCSOut[t-td1-3]
|+ [V21,V22,V23,…]…||*||Disturbance1[t-td2] – Disturbance1[t-td2-1]
Disturbance1[t-td2-1] – Disturbance1[t-td2-2]
Disturbance1[t-td2-2] – Disturbance1[t-td2-3]
|+ [V31,V32,V33,…]…||*||Disturbance2[t-td3] – Disturbance2[t-td3-1]
Disturbance2[t-td3-1] – Disturbance2[t-td3-2]
Disturbance2[t-td3-2] – Disturbance2[t-td3-3]
|+ [Vp1,Vp2,Vp3,…]…||*||PV[t-1] – PV[t-2 ]
PV[t-2] – PV[t-3]
PV[t-3] – PV[t-4]
V1 = [V11, V12, V13,…], etc.
td1, td2, td3, … transportation delays
MODEL TYPE ‘Absolute’
V1 = [V11, V12, V13,…], etc.
td1, td2, td3, … transportation delays
Important Modeling Considerations
The validity and accuracy of the models identified depends on the accuracy of the data being submitted for modeling. The data being used should be qualified data and should not include extreme upset conditions such as a shutdown, unless a model is being created for recovering from extreme upset conditions. Please note that a model created to cater to extreme upset conditions may not be valid for normal operating conditions. Process knowledge is important in order to specify the correct modeling parameters.
Analyze the Process
The first step towards model identification is analyzing the process. This includes identifying all the variables that may have an effect on the desired controlled variable (process variable). This includes as a minimum the manipulated variable. This may also include estimating the transportation delays for the manipulated/disturbance variables and determining the gain sign if possible.
Data Sampling Period
The sampling period determines the span over which models are identified. Ideally the sampling period should be close to 10% of the Time Constant of the process. If the data sampling period is much higher than the control scan period, and the loop is tuned aggressively, the output of the loop will have made several moves in between two data samples. Thus the energy delivered into the process will be significantly different than that represented in the data. This will not be apparent to the modeling algorithm resulting in lower overall gains identified. A balance between sampling period based on 10% of the process time constant and the control scan period will have to be reached in determining the optimum data sampling period. If the loop is not tuned aggressively, a sampling period much greater than the control scan time may be used without affecting the validity of the Model.
Key Model Parameters
If a change in the manipulated variable or disturbance variable results in an increase in the process variable, the gain sign between those variables is considered to be positive. Conversely if the change results in a decrease in the process variable, the gain sign is considered to be negative. If the gain signs between the manipulated variable and disturbance variables to the controlled variable are known, the model identified may be improved by constraining the gain signs.
Time Constant is defined as the time it takes for the process to reach 63.2% of it’s final value once it starts to respond to a change in the manipulated or disturbance variable. This does not include the transportation delay (deadtime).
Transportation delay is the amount of time it takes for the process variable to start responding to a change in the manipulated or disturbance variable. The transportation delay for different variables for the same process variable may be different. This value is specified in units of sampling period. Transportation delay is the most important parameter used by the modeling algorithm in identifying models. The modeling algorithm may suggest the transportation delays based on the coefficient threshold. Starting with the transportation delay specified by the user, the modeling algorithm computes the polynomial equation for each variable and discards the smaller coefficients considering these as transportation delays.
Open loop simulation is the single most important mechanism for testing the validity of the model. When analyzing the simulation plot, the following key parameters should be noted to determine validity of the model.
The phase of the Predicted PV should be in phase with the phase of the actual process. Which means the rise and fall of the Predicted PV should occur at the same time as the actual PV.
The peak-to-peak gain of the Predicted PV should be close to the peak-to-peak gain of the actual PV. There may be a DC Offset between the Predicted PV and the actual PV, which could be a result of external unmeasured disturbances and may be ignored.
The process direction (i.e. rise and fall) of the Predicted PV should be the same as the actual PV.
It should be noted that the prediction may start degrading over longer horizons due to drifts in the actual process and due to influence of external unmeasured disturbances. If the prediction is good for a reasonable number of sampling periods from the start of open-loop simulation, the prediction may be deemed sufficiently accurate and the model may be considered to be valid.