ChE 182 Chemical Process Dynamics and Control Lecture 11 Bode and Nyquist Stability Criterion

General Stability Criterion The feedback control system is stable if and only if all roots of the characteristic equation are negative or have negative real parts. Otherwise, the system is unstable.

Characteristic equation Characteristic equation of feedback loop: 1+GOL = 0 GOL = -1 |GOL(jω)| = 1 ∠GOL (jω) = -180°

Root-Locus Diagram

Marginal stability or conditional stability Complex conjugate root pair on the imaginary axis s = ±jb -sustained oscillation with ωc = b -roots: s = ±jωc |GOL(jωc)| = 1 ∠GOL (jωc) = -180°

Critical frequency or phase crossover frequency, ωc Definitions Critical frequency or phase crossover frequency, ωc value of ω for which φOL(ω) = -180° Gain crossover frequency, ωg value of ω for which AROL(ω) = 1 Marginally stable system: ωc = ωg

Bode Stability criterion Consider an open-loop transfer function GOL = GcGvGpGm that: 1. is strictly proper (n ≥ m), 2. has no poles on or to the right of imaginary axis (with the possible exception of a single pole at the origin), 3. has a single ωc and a single ωg. Then the closed-loop system is stable if AROL(ωc) < 1. Otherwise, it is unstable.

On-line Tuning: Continuous cycling method 1. After the process has reached steady-state, use only P-control by eliminating the integral and derivative action (set τD to zero and set τI to the largest possible value). 2. Set Kc equal to a small value and place the controller in automatic mode. 3. Introduce a small, momentary set-point change. Gradually increase KC in small increments until continuous cycling occurs. This value of Kc is called the critical or ultimate gain, Kcu. The peak-to-peak period is called the critical or ultimate period, Pu.

On-line Tuning: Continuous cycling method

On-line Tuning: Continuous cycling method 4. Calculate the PI or PID controller settings using the Ziegler-Nichols tuning relations or the more conservative Tyreus-Luyben settings.

Controller Tuning: Frequency response analysis For proportional only control: GOL = KcG; where G = GvGpGm AROL(ω) = KcARG(ω) At the stability limit: ω = ωc AROL = 1 Kc = Kcu AROL(ωc) = Kcu ARG(ωc) = 1 Kcu = 1/ ARG(ωc) Pu = 2π/ ωc

Nyquist Stability Criterion Consider an open-loop transfer function GOL = GcGvGpGm that: 1. is strictly proper (n ≥ m), 2. has no unstable pole-zero cancellation Let: N = number of times (-1,0) is encircled in the clockwise direction P = number of poles that lie to the right of imaginary axis Z = N + P Then the closed-loop system is stable if and only if Z = 0.

Gain and Phase Margins (Bode plot) -measures of relative stability Gain Margin: GM ≡ 1/AROL(ωc) -for a stable system: GM > 1 Phase Margin: PM ≡ 180° + φOL(ωg)

Gain and Phase Margins (Nyquist plot) -measures of relative stability