CHEN90032 · Process Simulation and Control
Closed-Loop Behaviour & PID Controllers
This chapter of the University of Melbourne CHEN90032 Process Simulation and Control guide closes the feedback loop round the process. It builds the ideal PID controller transfer function Gc(s) = Kc(1 + 1/(TIs) + TDs), derives the servo (set-point) and regulator (disturbance) closed-loop transfer functions, and reduces the loop to the characteristic equation 1 + GOL = 0. You will see how the closed-loop poles set speed, shape and stability, why proportional-only control leaves an offset, and how integral action removes it — the spine of the process-control half of the subject.
What this chapter covers
- 01Write the ideal parallel PID law m(t) and its transfer function Gc(s) = Kc(1 + 1/(TIs) + TDs)
- 02State the job and units of each term: gain Kc, integral (reset) time TI [time], derivative time TD [time]
- 03Draw the standard negative-feedback loop and identify the five blocks Gc, Gv, Gp, Gd, Gm
- 04Derive the servo transfer function C/R = GcGvGp / (1 + GcGvGpGm) by block algebra
- 05Derive the regulator transfer function C/QL = Gd / (1 + GcGvGpGm) for disturbance rejection
- 06Form the characteristic equation 1 + GOL = 0 with GOL = GcGvGpGm, and read the closed-loop poles
- 07Show that proportional-only control leaves a steady-state offset of 1/(1 + KcKp)
- 08Show that integral action removes the offset but raises the closed-loop order by one
- 09Check that GOL is dimensionless and every closed-loop pole has a negative real part (stability)
Proportional control of a first-order process: closed-loop TF and offset
- +1Open-loop transfer function: GOL = KcGp = (4 × 5)/(10s + 1) = 20/(10s + 1). The loop gain KcKp = 20 is dimensionless.
- +1Servo transfer function: C/R = GOL/(1 + GOL) = [20/(10s + 1)] / [1 + 20/(10s + 1)] = 20/(10s + 21).
- +1Standard first-order form (divide top and bottom by 21): C/R = (20/21)/((10/21)s + 1) = 0.952/(0.476 s + 1). Closed-loop gain Kcl = 20/21 = 0.952; closed-loop time constant τcl = 10/21 = 0.476 s — faster than the open-loop 10 s.
- +1Closed-loop pole from 1 + GOL = 0: 10s + 21 = 0, so s = −2.1 s−1. It is real and negative, so the loop is stable (and equals −1/τcl).
- +1Offset to a unit set-point step, by the final-value theorem: c(∞) = C/R at s = 0 = 20/21 = 0.952.
- +1Offset = R(∞) − c(∞) = 1 − 0.952 = 0.048 = 1/(1 + KcKp) = 1/21. Proportional-only control leaves a small residual offset; raising Kc shrinks it but never removes it.
Key terms
- PID controller
- A controller combining proportional, integral and derivative action on the error. Ideal parallel transfer function Gc(s) = Kc(1 + 1/(TIs) + TDs); P uses only the first term and PI drops the derivative (TD = 0).
- Controller gain Kc
- The proportional gain acting on the present error e = r − c. Larger Kc gives a stronger, faster correction and a smaller offset, but reduces the stability margin. Dimensionless when the loop signals are scaled.
- Integral (reset) time TI
- The tuning constant of the integral term, with units of time. A smaller TI means stronger integral action. Integral action keeps acting while any error remains, so it removes the steady-state offset.
- Derivative time TD
- The tuning constant of the derivative term, with units of time. Derivative action anticipates the trend of the error and adds damping, allowing a higher Kc, but it amplifies measurement noise, so it is used sparingly.
- Servo (set-point) transfer function
- The closed-loop response of the output to a set-point change, C/R = GcGvGp/(1 + GcGvGpGm). The numerator is the forward path; the denominator is 1 plus the loop gain.
- Regulator (disturbance) transfer function
- The closed-loop response of the output to a load disturbance, C/QL = Gd/(1 + GcGvGpGm). It shares the same denominator as the servo case; a larger loop gain gives better disturbance rejection.
- Open-loop transfer function GOL
- The product of every element once round the loop, GOL = GcGvGpGm. It must be dimensionless, and it is the quantity that appears in the characteristic equation.
- Characteristic equation
- 1 + GOL = 0, the shared denominator of the closed-loop transfer functions set to zero. Its roots are the closed-loop poles; the loop is stable if and only if every root has a negative real part. Proportional-only control leaves a steady-state offset of 1/(1 + KcKp), which integral action drives to zero at the cost of raising the closed-loop order by one.
Closed-Loop Behaviour & PID Controllers FAQ
Why does proportional-only control leave an offset while PI does not?
With P-only control the manipulated output is proportional to the error, so it needs a non-zero error to hold the valve away from its bias — the loop settles with a residual offset of 1/(1 + KcKp). Integral action keeps changing the output for as long as any error exists, so it only stops when the error is exactly zero. The cost is that the integrator adds a pole at s = 0, raising the closed-loop order by one, which can reduce the stability margin.
What is the characteristic equation and why does it matter?
It is 1 + GOL = 0, where GOL = GcGvGpGm is the product of every block round the loop. Both the servo and the regulator transfer functions share this denominator, so its roots are the closed-loop poles that govern speed, shape and stability of the whole loop. Clearing fractions turns it into a polynomial in s, and the loop is stable only if every root has a negative real part — the starting point for the Routh and Bode methods that follow.
Can AI help me with closed-loop transfer functions and PID controllers in CHEN90032?
Yes, used as a study aid rather than an answer service. Sia can explain, step by step, how to reduce a block diagram to C/R and C/QL, how the PID terms map onto Kc, TI and TD, and how to form and read 1 + GOL = 0, and it can check your reasoning against your own worked attempt. It will not sit your closed-book exam or lab for you, and no tool can promise a specific grade; the aim is to help you master the method so you can reproduce it under exam conditions.
Studying with AI? Sia — free AI chemical engineering tutor works through CHEN90032 step by step.
Exam move
Treat this chapter as block algebra with physical meaning. Drill the single-loop shortcut — closed-loop = (forward path) / (1 + loop gain) — until you can write C/R and C/QL from a diagram without hesitation, always with the full loop product GcGvGpGm underneath. Practise both directions: given a plant and a controller, form the characteristic equation 1 + GOL = 0 and read the poles; and given a control aim, decide whether P, PI or PID is needed and why. Fix the two key facts in memory: P-only control leaves an offset of 1/(1 + KcKp), and integral action removes it while adding a pole at s = 0. The written final is closed-book with a provided formula sheet, so rehearse the method and always check that GOL is dimensionless and every pole sign matches your stability claim. Pace at about 1.8 minutes per mark (3 hours over 100 marks), and confirm the exam date for the ~June 2027 end-of-Semester-1 exam period on Canvas.
