... the main ideas of PIDlab Java applets ...

What do you know about your process (A priori information) ?

Physical behavior

If you are controlling temperature, pressure, flow or concentration you can suppose that your process has a monotone step response. You don't know anything about the mathematical model (e.g. the process order or transfer function parameters). In this case you need to obtain some experimental data (see below) and then you can use the Java applet 'Fractional PID laboratory' which offers the model set approach based on experimental data and a priori information about the transfer function form. This method is very robust because all typical process models in the arbitrary order lag/deadtime form are included in the class of a priori admissible processes.

Transfer function

If you are sure that you know the mathematical model of the process, you can use the 'Fractional PID laboratory' or the 'PID Controller Designer'. In both applets, you can define your custom transfer function and make a controller design for your nominal model. But this way is not recommended for above mentioned types of processes with monotone step response, because then you can often obtain an unstable closed loop if your model is not precise enough! This is why all traditional heuristic methods based on nominal mathematical model often fail! The transfer function as an enough precise mathematical model can be used, when you control some oscillating electrical or mechanical systems.

Experimental data

If you use the model set approach, you must have some experimental data obtained from an identification experiment. The following two types of experimental data (or their combination) are supported in the Java applet 'Fracional PID laboratory'.

One sample of the frequency response

Using relay identification experiment, you can obtain one sample of the process frequency response. The recommended phase shift of the sample is about -135'. The sample of the frequency response can be e.g. measured using REX PIDAT controller block.

Moments of the impulse response

From the rectangle pulse response, you can compute moments of the impulse response. The moments can be e.g. measured using REX PIDMA controller block.

Design specifications

You must specify some requirements on the standard closed control loop, e.q. the robustness or some frequency limitations. The following types of design specifications are supported in both Java applets 'Fracional PID laboratory' and 'PID Controller Designer'

1) Gain margin (Gm)

Specifies the robustness to the gain changes in the closed loop. Should always be at least 2, which means that the Nyquist plot should cross the negative real axes at the right side from the value -1/2=-0.5. This specification can be used in our applets by mouse click in the Nyquist plot figure.

2) Phase margin (Pm)

Specifies the robustness to the delay changes in the closed loop. Should always be at least 60', which means that the Nyquist plot should cross the unit circle at the right side from the point with phase shift 60'. This specification can be also used in our applets by mouse click in the Nyquist plot figure.

3) Sensitivity function (SF) upper limit

Specifies a maximum gain of disturbances that are affecting the measurement of the process variable (PV). Should always be less than some given value Ms (e.g. 1.5), which means that the Nyquist plot cannot enter the protected area given by dark blue 'M-circle'. This specification can be added to the controller design by toggling the corresponding M-circle checkbox in our applets.

4) Complementary sensitivity (CSF) function upper limit

Specifies the quality of tracking the reference setpoint (SP) and the damping of disturbances appearing in the closed loop. Should always be less than some given value Mt (e.g. 1.5), which means that the Nyquist plot cannot enter the protected area given by light blue 'M-circle'. This specification can be added to the controller design by toggling the corresponding M-circle checkbox in our applets.

Summarizing the design specifications 1)-4) : The Nyquist plot should always pass at the right side from control points Gm and Pm and should not enter the protected area given by M-circles as it is shown on the previous picture.

5) Closed loop bandwidth

You always need to have a proper closed loop bandwidth to dump high frequency disturbances and to eliminate unmodeled dynamic at higher frequencies. It means that the complementary sensitivity function must be from a given frequency close to zero in some tolerance band. This design specification can be used in our applets by mouse click in the CSF figure.

6) Low frequency disturbance dumping

You always want do dump disturbances in the whole frequency range, but it is not possible thanks to the 'water bad effect' of sensitivity function shown in the previous right figure. You only can dump disturbances at low frequencies, which means that the sensitivity function must be close to zero in some tolerance band at low frequencies. This design specification can be used in our applets by mouse click in the SF figure.

Summarizing the design specifications 2)-6) : The design specifications can be formed into 6 numbers (Mt, et, wt, Ms, es, ws) as shown in following figures.

How to find controller parameters using applets ?

Now it is clear which shape of the Nyquist plot and sensitivity functions is sufficient according to above 6 design specifications. Firstly, one need to choose the filter parameter N in controller derivative part. This in fact means the limitation of derivative output, when a step appears on the input. The default value is 10 and usually is not necessary to change it. The recommended interval is from 2 to 20. Then one must choose the ratio between integral and derivative time constant. The default value is 0.25 and usually is again not necessary to change it. After that, it is possible to start with shaping of Nyquist plot and sensitivity functions. The design specifications can be added by mouse click in Nyquist plot, SF, or CSF figure. Each design specification creates one region in the Robustness Regions figure where the plane of two free parameters K, Ki is depicted. Each region splits this plate into two parts and only one part is the suitable area of controller parameters. If all specifications have to be fulfilled simultaneously, the controller parameters must be chosen in the intersection of all suitable areas (e.q. intersection of Gm and Pm area). If there is no intersection, the specifications can't be fulfilled simultaneously. From the admissible area is the best choice the point 'R' with maximum Ki coordinate (the fastest closed loop). Finally, it is possible to decrease the feed-forward parameter 'b' to obtain closed loop step response without any overshot, while the robustness is not disturbed. In this way one obtains a complete 2DOF PID Controller in ISA form.

What is a value set?

Usually, the process frequency response is known precisely only at a few frequencies. One sample of the process frequency response at frequency w1 can be obtained using relay identification experiment. The process model is uncertain at all frequencies different from w1. The uncertainty is infinite. If we add some a priori information about the process - e.g. that the step response is monotone (it is true 95% of typical industrial processes - temperature, pressure, concentration, flow) then the uncertainty will became finite. For each frequency different from w1 one obtains so called value set - the closed area in frequency domain. The value set represents 'generalized' point of process frequency response and is created by all possible monotone processes satisfying experimental data. Also value sets in Nyquist plot plane can be computed easily by multiplying process value sets by controller C(jw).

If we want to create robust controller, the Nyquist plot limitations must be fulfilled for all value sets at critical frequencies. This technique is implemented in the Java applet 'Fractional PID laboratory'.

How to ensure the fulfillment of design specifications for the whole model set?

If you have only experimental data from the process (samples of frequency response or moments of the impulse response) the process frequency response is uncertain. Now, lets explain the uncertainty on the 'one frequency response sample case'. At frequency w1 different from experimental w, all processes from the model set (in the admissible form and consistent with measured point) create the generalized point of frequency response - value set. The value set is a closed area in frequency domain and it quantifies the model uncertainty at frequency w1. Using 'Fractional PID laboratory', you can easily compute the value set boundary (it is for all frequencies generated by the same extremal transfer functions). By mouse dragging on the value set boundary in 'Process frequency response' window, you can choose ultimate bounding processes of the whole model set (usually only two). Then you can move to the Nyquist plot plane and make a robust design for these two processes. You can check, that the design specifications are fulfilled for all value sets at critical frequencies as shown in the right picture below. The same idea can be used if the value sets are computed from the moments of the impulse response.

How to use computed parameters in industrial practice ?

Note, that the controller parameters computed with 'Fractional PID laboratory' or 'PID Controller Designer' can be directly used in blocks PIDU, PIDAT, PIDMA which are a part of a real-time control system REX. In this way, your results can be directly implemented at some target platform in industry. Note that the controllers included in REX blockset provides autotuning function which will tune your controller very reliably according to the above ideas! more info...

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