Open loop tuning methods

Open loop tuning methods are where the feedback controller is disconnected and the experi-menter excites the plant and measures the response. They key point here is that since the con-troller is now disconnected the plant is clearly now no longer strictly under control. If the loop is critical, then this test could be hazardous. Indeed if the process is open-loop unstable, then you will be in trouble before you begin. Notwithstanding for many process control applications, open loop type experiments are usually quick to perform, and deliver informative results.

To obtain any information about a dynamic process, one must excite it in some manner. If the system is steady at setpoint, and remains so, then you have no information about how the process behaves. (However you do have good control so why not quit while you are ahead?) The type of excitation is again a matter of choice. For the time domain analysis, there are two common types of excitation signal;– the step change, and the impulse test, and for more sophisticated analysis, one can try a random input test. Each of the three basic alternatives has advantages and disadvantages associated with them, and the choice is a matter for the practitioner.

Step change The step change method is where the experimenter abruptly changes the input to the process. For example, if you wanted to tune a level control of a buffer tank, you could sharply increase (or decrease) the flow into the tank. The controlled variable then slowly rises (or falls) to the new operating level. When I want a quick feel for a new process, I like to perform a step test and this quickly gives me a graphical indication of the degree of damping, overshoot, rise time and time constants better than any other technique.

Impulse test The impulse test method is where the input signal is abruptly changed to a new value, then immediately equally abruptly changed back to the old value. Essentially you are trying to physically alter the input such that it approximates a Dirac delta function.

Technically both types of inputs are impossible to perform perfectly, although the step test is probably easier to approximate experimentally.

The impulse test has some advantages over the step test. First, since the input after the ex-periment is the same as before the exex-periment, the process should return to the same pro-cess value. This means that the time spent producing off-specification (off-spec) product is minimised. If the process does not return to the same operating point, then this indi-cates that the process probably contains an integrator. Secondly the impulse test (if done perfectly) contains a wider range of excitation frequencies than the step test. An excitation signal with a wide frequency range gives more information about the process. However the impulse test requires slightly more complicated analysis.

Random input The random input technique assumes that the input is a random variable approx-imating white noise. Pure white noise has a wide (theoretically infinite) frequency range, and can therefore excite the process over a similarly wide frequency range. The step test, even a perfect step test, has a limited frequency range. The subsequent analysis of this type of data is now much more tedious, though not really any more difficult, but it does require a data logger (rather than just a chart recorder) and a computer with simple regression software. Building up on this type of process identification where the input is assumed, within reason, arbitrary, are methods referred to as Time Series Analysis (TSA), or spectral analysis;– both of which are dealt with in more detail in chapter6.

Open-loop or process reaction curve tuning methods

There are various tuning strategies based on an open-loop step response. While they all follow the same basic idea, they differ in slightly in how they extract the model parameters from the

Plant

-

-output input

6

-...

...

...

...

...

K

T

-

- L

time tangent at inflection pt

Figure 4.17: The parameters T and L to be graphically estimated for the openloop tuning method relations given in Table4.2.

recorded response, and also differ slightly as to relate appropriate tuning constants to the model parameters. This section describes three alternatives, the classic Ziegler-Nichols open loop test, the Cohen-Coon test, and the ˚Astr ¨om-H¨agglund suggestion.

The classic way of open loop time domain tuning was first published in the early 40s by Ziegler and Nichols³, and is further described in [150, pp596–597] and in [179, chap 13]. Their scheme re-quires you to apply a unit step to the open-loop process and record the output. From the response, you graphically estimate the two parameters T and L as shown in Fig.4.17. Naturally if your response is not sigmoidal or ‘S’ shaped such as that sketched in Fig.4.17and exhibits overshoot, or an integrator, then this tuning method is not applicable.

This method implicitly assumes the plant can be adequately approximated by a first order trans-fer function with time delay,

Gp≈Ke^−θs

τ s + 1 (4.25)

where L is approximately the dead time θ, and T is the open loop process time constant τ . Once you have recorded the openloop input/output data, and subsequently measured the times T and L, the PID tuning parameters can be obtained directly from Table4.2.

A similar open loop step tuning strategy due to Cohen and Coon published in the early 1950s is where you record the time taken to reach 50% of the final output value, t2, and the time taken to reach 63% of the final value, t3. You then calculate the effective deadtime with

θ = t2− ln(2)t³ 1− ln(2) and time constant,

τ = t3− t¹

The open loop gain can be calculated by dividing the final change in output over the change in the input step.

3Ziegler & Nichols actually published two methods, one open loop, and one closed loop, [204]. However it is only the second closed loop method that is generally remembered today as the “Ziegler–Nichols” tuning method.

Once again, now that you have a model of the plant to be controlled in the form of Eqn.4.25, you can use one the alternative heuristics given in Table4.2. The recommended range of values for the deadtime ratio for the Cohen-Coon values is 0.1 < θ/τ < 1. Also listed in Table4.2are the empirical suggestions from [16] known as AMIGO, or approximate M -constrained integral gain optimisation. These values have the same form as the Cohen-Coon suggestions but perform slightly better.

Table 4.2: The PID tuning parameters as a function of the openloop model parameters K, τ and θ from Eqn.4.25as derived by Ziegler-Nichols (open loop method), Cohen and Coon, or alterna-tively theAMIGOrules from [16].

Controller Kc τi τd

Fig.4.18illustrates the approximate first-order plus deadtime model fitted to a higher-order over-damped process using the two points at 50% and 63%. The subsequent controlled response using the values derived from in Table4.2is given in Fig.4.19.

Figure 4.18: Fitting a first-order model with deadtime using the Cohen-Coon scheme.

Note how the fitted model is a reason-able approximation to the actual response just using the two data points and gain.

See Fig.4.19for the subsequent controlled

response. ^{0 5 10 15 20 25 30 35}

Conventional thought now considers that both the Zeigler-Nichols scheme in Table4.2and the Cohen-Coon scheme gives controller constants that are too oscillatory and hence other modified tuning parameters exist, [178, p329]. Problem4.1demonstrates this tuning method.

−2 0 2

P−only

y & setpoint

0 100 200

−5 0 5

u

time

−2 0 2

PI−control

0 100 200

−5 0 5

time

0 100 200

−5 0 5

time

−2 0 2

PID control

Figure 4.19: The closed loop response for a P, PI and PID controlled system using the Cohen-Coon strategy from Fig.4.18.

Problem 4.1 Suppose you have a process that can be described by the transfer function

Gp= K

(3s + 1)(6s + 1)(0.2s + 1)

Evaluate the time domain response to a unit step change in input and graphically estimate the parametersLandT. Design a PI and PID controller for this process using Table4.2.

Controller settings based on the open loop model

If we have gone to all the trouble of estimating a model of the process, then we could in principle use this model for controller design in a more formal method than just rely on the suggestions given in Table 4.2. This is the thinking behind the Internal Model Control or IMC controller design strategy. The IMC controller is a very general controller, but if we restrict our attention to just controllers of the PID form, we can derive simple relations between the model parameters and appropriate controller settings.

The nice feature of the IMC strategy is that it provides the scope to adjust the tuning with a single parameter, the desired closed loop time constant, τc, something that is missing from the strategies given previously in Table4.2. A suitable starting guess for the desired closed loop time constant is to set it equal to the dominant open loop time constant.

Table4.3gives the PID controller settings based on various common process models. For a more complete table containing a larger selection of transfer functions, consult [179, p308].

An simplification of the this IMC idea in an effort to make the tuning as effortless as possible is given in [186].

Perhaps the easiest way to tune a plant when the transfer function is known is to use the MATLAB

functionpidtune, or the GUI,pidtoolas shown in Fig.4.20.

Table 4.3: PID controller settings based on IMC for a small selection of common plants where the control engineer gets to chose a desired closed loop time constant, τc.

Plant PID constants

KcK τi τd

K/(τ s + 1) τ

τc

τ –

K

(τ1s + 1)(τ2s + 1)

τ1+ τ2

τc

τ1+ τ2 –

K/s 2

τc 2τc –

K s(τ s + 1)

2τc+ τ

τ_c² 2τc+ τ 2τcτ 2τc+ τ Ke^−θs

τ s + 1

τ

τc+ θ τ –

τ + θ/2

τc+ θ/2 τ + θ/2 τ θ 2τ + θ

Figure 4.20: PID tuning of an arbitrary transfer function using the MATLABGUI.