1. Course Review Part 3

2. Manual stability control Manual servo control

3. 6.6 Reset windup The condition where the integral action drives the controller output to one of its limits. Usually because of –Poor configuration –Undersized (or oversized) valve The controller cannot reach setpoint - offset. This offset causes the integral term to wind up The controller may be slow in responding to a change in the controlled variable because of the dominant integral term

4. Reset Windup SP PV Output limit Controller Output

5. Real World Temp Desired temp in C Temp Error in C - + Actual Valve Position Temperature disturbances y GvGp1 Gm1 Gd1 d1 Gc1Gp2 Gd2 Steam supply disturbances d2 Desired Steam flow valve position Steam flow 7. Cascade Motivating Example Valve may be nonlinear -Steam supply disturbances are fast, but can only be mitigated once they affect temperature -Steam dynamics may be complicated or even unstable

6. Real World Temp Desired temp in C Temp Error in C - + Valve Position Temperature disturbances y GvGp1 Gm1 Gd1 d1 Gc1Gc2Gp2 Gd2 Gm2 Steam supply disturbances d2 Desired Steam flow Steam flow - + Cascade mitigates: –Nonlinearities in the valve –Fast disturbances in d2 –Dynamics in Gd2 –Instability or dynamics in Gp2 Split the control problem into two parts: –Steam flow –Temperature control, given steam flow If inner loop  < outer loop  /5, then the inner loop can be ignored

7. 8. Feedforward Feedback is mostly about the poles of the transfer function: long term behaviour and stability Feedforward is about the zeros of the transfer function: short term dynamics Feedback cannot affect the zeros of a transfer function. Feedforward can. Can be very useful if there is a long time delay in Gd and Gp.

8. Zeros in Gd are modified by adding Gff * Gp. Perfect feedforward is not possible if: -delay in Gp is greater than delay in Gd, (requires future values of disturbance) -Gp has RHP zeros (Gff would be unstable) In such cases, use static feedforward, or leave out the unstable part. Level y Gp Change in Pressure Gd d Boiler Feed Water flow Feedforward Boiler steam drum level control: Process shows inverse response: confuses feedback control. RHP zero can be mitigated by feedforward. Gff For perfect cancellation

9. 9. Routh Stability Criterion Useful to find limiting values of Kc and  I Can only be used on polynomials –If characteristic equation contains exponentials, use a Pade approximation 1.Write characteristic equation as a polynomial: Make coefficient of highest power of s (a n ) positive If any coefficient is negative or zero, system is not stable 2. If all coefficients are positive, construct Routh array:

10. Routh Array will have n+1 rows Row 1anan a n-2 a n-4 … (pad with 0 if necessary) Row 2a n-1 a n-3 a n-5 … Row 3b1b1 b2b2 b3b3 … Row 4c1c1 … ….… Row n+1z1z1 Each row has one less column than the row above. Row n+1 will have only one column. The number of unstable poles is equal to the number of sign changes going down the first column of the array

11. Calculation

12. 10. Frequency Response

13. Amplitude Ratio and Phase Shift using Transfer Functions 1.Replace S with j  in the transfer function: G(s)  G(j  ) 2.Rationalize G: make it equal to a + jb, where a and b may be functions of  (G is now a complex number that is a function of  ) 3.AR = |G| = sqrt(a 2 + b 2 ) 4.  = tan -1 (b/a)

14. Amplitude Ratio and Phase Shift using Transfer Functions For systems in series: GpGcGv InputOutput Transfer functions multiply: G = Gc * Gv * Gp Amplitude ratios multiply and Phase angles add: AR = AR(Gc) * AR(Gv) * AR(Gp)  =  (Gc) +  (Gv) +  (Gp) Logarithms of Amplitude Ratios add: Log(AR) = log(AR(Gc)) + log(AR(Gv)) + log(AR(Gp))

15. 10.4 Bode Stability Criterion Input amplitude Output amplitude Process  c is the frequency at which the phase shift of the forward path = -180 o If the AR at  c < 1, then the system is stable. The “-” sign in the negative feedback gives another 180 o At phase lag of 180 o for the forward path, input = previous input*AR ControllerValve - InputOutput

16. 10.4 Bode Stability Criterion, Gain and Phase Margin Phase Margin Gain Margin AR < 1 at  c, so system is stable AR = 1  = -180  cc gg

17. 11. Multivariable Systems

18. Break

19. Feedback Control of unstable system