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## Course Review

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**Course Review**Part 3**Manual stability control**• Manual servo control**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**Reset Windup**SP PV Output limit Controller Output**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 Steam supply disturbances Temperature disturbances d2 d1 Desired Steam flow valve position Real World Gd2 Gd1 Actual Valve Position Temp Error in C Desired temp in C Steam flow Temp Gc1 Gv Gp2 Gp1 y + - Gm1**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 Steam supply disturbances Temperature disturbances d2 d1 Real World Gd2 Gd1 Desired Steam flow Temp Error in C Valve Position Desired temp in C Steam flow Temp Gc1 Gc2 Gv Gp2 Gp1 y + + - - Gm2 Gm1**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.**Change in Pressure**d Gff Gd Feedforward Boiler steam drum level control: Process shows inverse response: confuses feedback control. RHP zero can be mitigated by feedforward. For perfect cancellation Boiler Feed Water flow Level • 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. Gp y**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 • Write characteristic equation as a polynomial: Make coefficient of highest power of s (an) positive If any coefficient is negative or zero, system is not stable 2. If all coefficients are positive, construct Routh array:**Routh Array will have n+1 rows**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**Amplitude Ratio and Phase Shift using Transfer Functions**• Replace S with j in the transfer function: G(s) G(j) • 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 ) • AR = |G| = sqrt(a2 + b2) • = tan-1(b/a)**Amplitude Ratio and Phase Shift using Transfer Functions**For systems in series: Gc Gv Gp Input Output 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))**10.4 Bode Stability Criterion**Output amplitude Input amplitude Input Output Controller Valve Process - c is the frequency at which the phase shift of the forward path = -180o If the AR at c < 1, then the system is stable. The “-” sign in the negative feedback gives another 180o At phase lag of 180o for the forward path, input = previous input*AR**AR = 1**10.4 Bode Stability Criterion, Gain and Phase Margin Gain Margin AR < 1 at c, so system is stable g c Phase Margin = -180**Feedback**• Control of unstable system