Homework answers / question archive / ELEE4220/6220: Quiz 7 Homework (Quiz) 7: PID Control for Systems with Deadtime The goal of this homework is two-fold

ELEE4220/6220: Quiz 7 Homework (Quiz) 7: PID Control for Systems with Deadtime The goal of this homework is two-fold

Electrical Engineering

Share With

---

ELEE4220/6220: Quiz 7

Homework (Quiz) 7: PID Control for Systems with Deadtime

The goal of this homework is two-fold. First, you will analyze the effects of deadtime delays in linear systems. Second, this homework serves as an introduction to simulations with xcos. Furthermore, there are some minor design elements in this homework.

Deadtime delays appear in many plants or processes where transportation from one location to another incurs a delay. Echos in a room with a PA system are a good example, but conveyor belts or extruder systems may incur delays, too. We can describe a delay as a signal x(t) that arrives at the sensor without change in magnitude, but with a lag τ, meaning, the sensor receives the signal x(t−τ). We will now consider a simple first-order process with deadtime (Figure 1) and attempt to control the output variable with

 

Figure 1: Feedback control loop for a system with deadtime. The process can be split into two parts: a conventional, first-order system and a delay e^−sτ, which represents the time τ that the process output needs to reach the sensor. The optimum control strategy – primarily with respect to stability – needs to be found.

It is possible to describe the process as two consecutive parts. The first is the first-order system with a 0.5s time constant 2/(s + 2). Its output, in the time domain y_o(t) needs a certain time τ to reach the sensor. The sensor signal, still in the time domain, is therefore y(t) = y_o(t − τ). Since the Laplace transform of a deadtime component is e^−sτ, the process transfer function becomes

 

          (1)

 

We need a control system that tracks the input variable x(t). The controller function is H(s), which leads to the closed-loop transfer function of

 

)            (2)

 

It is evident from Equation 2 that conventional Laplace-domain methods fail, because the transfer function contains the irrational term e^−sτ. Only when τ → 0 does this system behave like a conventional linear system – despite the fact that the deadtime function is linear and time-invariant.

Part of this homework is a simulation with Scilab/xcos, but some initial theoretical questions will allow us to better appreciate the problem.

1. For the delay τ = 0.2s, plot the Bode diagram (magnitude and phase) from 0.1Hz to 10Hz. Note that ω = 2πf = 1/τ. Tip: It may be useful to plot the phase alone over the frequency in linear scale and then transfer the curve into the Bode diagram and its logarithmic frequency scale.        (10 points)
2. In a separate plot, indicate the Bode asymptotes of 2/(s+2) and then plot the combined frequency response, i.e., the Bode diagram for G(s). Determine the frequency at which the phase crosses the critical value of -180°, and then determine the gain margin of G(s). (5 points)
3. Next, examine the closed-loop system and its stability when the deadtime component is approximated with the first-order Pad´e approximation,

e−sτ ≈

2 − τs          (3)

 

2 + τs

You may use the open-loop pole-zero configuration of the approximated G(s) in a root-locus approach or examine the gain margin similar to the previous question. (5 points)

4. For pure P-control H(s) = k_p, what is the largest value of k_p that keeps the closed-loop system stable? (5 points)
5. Set up a simulation with Scilab/xcos. Use a unit step function as input and show five seconds of simulated output. If you use the PID control block, set the D- and I-components to zero – but see note below: try not to use it; instead, combine gain, integral, and derivative into a DIY PID, then set k_D and k_I initially to zero. Demonstrate (e.g., by including suitable plots in your homework submission) that the value of k_p that you found above leads to strong oscillatory behavior and that a larger k_p turns the closed-loop system unstable. (15 points)
6. Next, attempt to use PID control. To determine the three coefficients k_p, k_D, and k_I, use the closed-loop Ziegler-Nichols method ( described below). Present the coefficients that you determined and the dynamic response of the closed-loop system. What overshoot behavior and settling time (defined as 2% of the final value) do you see? (5 points)
7. The Cohen-Coon method (also described below) promises successful tuning of a system in the presence of a deadtime. Use the CohenCoon method to determine the three coefficients k_p, k_D, and k_I, and compare the dynamic response to that of the Ziegler-Nichols tuning method.    (5 points)
8. BONUS: Can you beat the Ziegler-Nichols and Cohen-Coon methods? A lower overshoot with similar settling time qualifies as ”better”, a faster settling time with comparable overshoot also qualifies. Ideally, you would improve both the settling time and reduce the overshoot. You may use all tools at your disposal, including trial-and-error. You may use any controller with or without additional compensator(s). Document all control elements in your loop and show the step response with its overshoot and settling time metrics. Provide a reason why you use those – even if it is trial-and-error. Depending on the reasoning and the final outcome, you may be awarded up to 20 bonus points.

A note on the xcos setup: The numerical treatment of this simulation is especially difficult, and the xcos default settings will likely abort the simulation or at least appear to develop chaotic behavior. Selection of a suitable solver algorithm has the strongest influence on the simulation, and the Runge-Kutta 4(5) solver and the Crank-Nicholson 2(3) solver appear to perform best. Relaxing the tolerances is also recommended if the simulation gets stuck (Figure 2). In addition, the delay block requires enough buffer to locally store the delayed data for the small time steps that the solver uses.

In addition, the PID block is less robust than an explicit PID function assembled from gain blocks, integral, derivative, and summation.

 

Figure 2: Specific settings for the xcos simulation. (A) Simulation setup window. Here, you already set the duration of the simulation (small green arrow). Loosening the tolerances from 10⁻⁶ to 10⁻⁴ makes the simulation more robust, but the most important setting is the solver selection. The optimum solver may differ between operating systems (compiler, math implementation), but it appears that RK45 and

CRANI work best on Linux systems. (B) Setup for the delay. The dead time of 0.2s is set here, but it is also necessary to significantly increase the buffer size.

Ziegler-Nichols PID Tuning

The original method by Ziegler and Nichols^[1] calls for a measurement of the closed-loop response near the stability boundary. The obtain the PID

coefficients,

1. Set k_D = 0 and k_I = 0 to obtain pure P-control.
2. Ideally, provide a narrow pulse at the input. However, a step function at the input generally yields the same results.
3. Start with low k_p and slowly increase it until you get an oscillatory response.
4. Continue increasing k_p until the closed-loop system shows an almost undampened response.
5. Record the gain for this response (designated k_u) and the period of the oscillation. T_u.
6. Set the PID controller to k_p = k_u/1.7, k_D = k_p·T_u/8, and k_I = 2k_p/T_u. The controller is now tuned.

 

 

Cohen-Coon PID Tuning

Cohen and Coon^[2] proposed an open-loop based tuning method that is more tolerant to deadtime delays. The disadvantage of this method is that it assumes a first-order system or – at least – a second-order system with one very fast pole. The obtain the PID coefficients, apply a step input to G(s) at t = 0, then

1. Measure the step response of the process, i.e., G(s) in this case. In the simulated step response, the dead time and the onset of the exponential equilibration are clearly distinguishable. This is not always the case in real-world systems where the transition can be blurred. Therefore:
2. Record the time at which the step response reaches 50% of its final value (T_1/2) and the time at which the step response reaches 63% of its final value (T63).
3. Estimate the dead time τ,

 

      (4)

 

4. estimate the time constant of the first-order component T1,

T1 = T63 ⁻ τ        (5)

and finally compute the dead time as fraction of the time constant, R_T = τ/T1. Note that both τ and T1 should be familiar (see Figure 1), but the method described here is statistically more robust.

5. Record the steady-state gain of the process g as the ratio between equilibrium output and amplitude of the step input. Note that for our process, g = 1, and this step is provided for reasons of completeness only.
6. Set the PID controller to

 

 

and finally k_D = k_pτ_D and k_I = k_p/τ_I.