Homework problems

All problems by chapter, linked to the slide where they are posed

1. Mathematical Preliminaries & Dynamic Response

  • Problem #1 — For the CSTR of Example 1.3, find all the approximate transfer functions relating the outputs \(c_A\) and \(T\) to the inputs \(c_{A,i}\), \(T_i\) and \(T_c\). Show that all six transfer functions share one denominator, and explain why. …
  • Problem #2 — (a) Find \(f(t)\) for \(F(s)=\frac{s+\beta}{(s+a)^2+\omega^2}\) (b) Find \(f(t)\) for \(F(s)=\frac{s\sin\phi+\omega\cos\phi}{s^2+\omega^2}\) Check your answers with sympy.inverse_laplace_transform or MATLAB ilaplace.
  • Problem #3 — For Example 1.8, carry out the inverse transforms: (a) the impulse response \(\bar c_2(t)\) for \(C_i=5/4\); (b) the pulse response \(\bar c_2(t)\), using the dead-time property for the factor \(e^{-s/4}\). Plot both together with …
  • Problem #4 — (a) Show that the overdamped step response can be written with two exponentials \(e^{-t/\tau_1}\) and \(e^{-t/\tau_2}\), and find \(\tau_1\), \(\tau_2\) in terms of \(\tau\) and \(\zeta\). (b) Starting from the underdamped solution, derive …
  • Problem #5 — (a) Show that the 1/1 Padé approximation is more accurate than the first-order Taylor expansion \(e^{-t_ds}\approx1-t_ds\). (Compare the Taylor series of both.) (b) Derive the 2/2 Padé approximation and find its poles and zeros for …
  • Problem #6 — Blending tank and transfer pipe — A nearly perfectly mixed blending tank (first order) feeds a downstream unit through a long transfer pipe. A test shows that the pipe behaves as if its volume were split into five equal perfectly stirred tanks in series. …
  • Problem #7 — \(\frac YX=\frac{K(\tau_zs+1)}{(\tau_1s+1)(\tau_2s+1)},\qquad \tau_1>\tau_2\) Calculate the response to a step of magnitude \(M\) and show that (a) \(y(t)\) has an extremum only if \(\dfrac{1-\tau_z/\tau_2}{1-\tau_z/\tau_1}>1\); (b) …

2. Analysis of Single-Loop Control Systems

  • Problem #1 — The temperature \(T\) of a tray near the top of a distillation column is controlled by adjusting the reflux flow \(R\). Draw a block diagram for this feedback loop. Treat both the feed flow rate \(F\) and the feed composition \(x_F\) as …
  • Problem #2 — Proportional-only control of the stirred-tank heater above. The transmitter has a span of 50 °F and a zero of 55 °F. Nominal conditions are \(\bar T=80\) °F and \(\bar T_i=65\) °F. \(K_c=5\), \(K_v=1.2\) (dimensionless), \(K_{IP}=0.75\) …
  • Problem #3 — Blending tank — The exit concentration \(c_3\) is controlled with stream 2. (a) Draw the block diagram of the composition loop. (b) Derive each transfer function and substitute numbers. (c) The PI controller was tuned at the nominal conditions. …

3. Tuning of Single-Loop Control Systems

  • Problem #1 — A process stream is heated in a shell-and-tube exchanger, and \(T_2\) is controlled with the steam valve. In an open-loop test the steam pressure \(P_s\) was stepped from 18 to 20 psig. Valve \(K_v=0.9\) psi/psi, I/P transducer …

4. Stability of Closed-Loop Systems

  • Problem #1 — \(G_p=\frac{2(s+1)}{s(s+2)(s^2+1)},\qquad 1+G_cG_p=0\) (a) Show with an analytical technique of your choice that the closed loop cannot be stabilized by a proportional controller. (b) With a PI or a PD controller (positive gain) …

5. Frequency-Response Analysis & Design

  • Problem #1 — \(G_p=\frac{2e^{-s}}{10s+1},\qquad G_c=K_c\left(1+\frac{1}{\tau_Is}\right),\quad \tau_I=10\) (a) Show that \(\tau_I=10\) makes \(G_{OL}\) a pure integrator with dead time. Sketch its Bode plot by hand. (b) Find the phase crossover …

6. Multi-loop Systems

  • Problem #1 — (a) Compute the RGA. Which pairing do you choose, and how strong is the interaction? (b) Design the two decouplers. Are they realizable? Compare the static and dynamic versions. (c) Tune two PI controllers with the ITAE …

Starter code for each problem is at the end of the chapter’s Colab notebook.