Control Systems Theory and Design
Control Systems Theory and Design tests how you model a process, choose a control structure, demonstrate stability and tune a loop that remains robust when the plant changes. Questions often combine block-diagram algebra with commissioning reality: a correct formula is not enough if the answer ignores dead time, valve dynamics or sampling.
Core rules and concepts
- Represent loop elements as transfer functions. For negative feedback with unity feedback, Gcl = GcGp/(1 + GcGp); the characteristic equation 1 + GcGp = 0 determines stability.
- Use Routh-Hurwitz for absolute stability and Bode or Nyquist for relative stability. Gain margin is the additional gain in dB before the Nyquist plot reaches -1; phase margin is the phase angle above -180 degrees at gain crossover. Typical targets are 6-10 dB and 30-45 degrees, but dead time and valve nonlinearity may require more.
- Model a self-regulating process as FOPDT: G = K e^(-θs)/(τs + 1). Dead time θ limits achievable bandwidth; a large θ/τ ratio makes the loop difficult. Integrating and inverse-response processes need different structures.
- PID implementations differ in gain convention, derivative filtering and whether derivative acts on error or PV. Use derivative on PV, anti-windup and bumpless transfer. Sample at least 5-10 times faster than the dominant time constant; otherwise use discrete design and consider aliasing.
- Lambda/IMC tuning for FOPDT: Kc = τ/[K(λ + θ)], Ti = τ. Choose λ for robustness: λ ≈ θ for dead-time-dominant loops, larger for noisy or nonlinear loops.
Worked example
A flow loop has K = 2, τ = 10 s and θ = 2 s. With λ = 2 s, Kc = 10/[2(2 + 2)] = 1.25 and Ti = 10 s. The gain is dimensionless only when controller, transmitter and valve use consistent engineering units. If the transmitter is 0-100 m3/h and the valve is 0-100%, convert both to fractions before applying Kc. Choosing λ = 10 s gives Kc = 0.417: slower, but more tolerant of dead-time error.
Traps
Confusing gain crossover with gain margin; tuning in manual; derivative kick on setpoint changes; quoting a PID gain without its unit convention; and treating a stable simulation as proof of site stability.