Proportional & Integral Controllers

1. Set TD = 0 & TI = ∞ 2. Increase KP until the system just starts to oscillate (KP = KPO). The frequency of oscillation here is ωc and the period is TO=2π/ωc. Set controller gains as:
NB: this transfer function is non-proper and is therefore difficult to realise in practice. Proper T.F.: Strictly proper T.F.: Order N(s) ≤ Order D(s) Order N(s) < Order D(s)
Error {e(t)} t
The PID regulator is given by: Here:
KI = KP TI
1 de(t ) u (t ) = K P e(t ) + ∫ e(t )dt + TD TI dt
K D = K P TD
KI K D s 2 + K P s + K I N ( s)U (s) = KP + + KDs = = E (s) s s D( s)
Proportional & Integral Controllers Proportional + Integral (PI) controllers were developed because of the desirable property that systems with open loop transfer functions of type 1 or above have zero steady state error with respect to a step input. The PI regulator is:
R2 C2 R3 R1 _ + C1
e(t)
1 R1 + R1 + sC U (s) 1 = − + E ( S ) R u (t ) 3 R2 +
1 sC1 1 sC 2

R1 1 C2 (1 + sC1R1 ) = − + + R sC R C (1 + sC R ) 1 3 1 2 2 3
Practical PID controllers
1 U ( s) 1 = KP + (1 + sTD ) E ( s) (1 + sαTD ) sTI
1. The 1/(1+sαTD) term acts as an effective low-pass filter on the P+D regulator to attenuate noise in the derivative block. 2. If α = 0 the original PID form is obtained 3. Typically α = 0.1 to place the filter as far away from the derivative action as possible. It is generally impractical to move the filter further away as the control action (effort) becomes too much and can-not be realised (supply rail or limiters saturate the control effort) 4. PD controllers could be realised on their own if the plant does not require an integrator. Circuit realisation:
Type P PI PID KP TI TD
Created by Dr Grant Covic for paper: Systems and Control 1998: Document 2 of 3
Notes: 1. Even though first and second order systems cannot in principle fully oscillate, practical systems always contain transport delays and non-linearities that make them oscillate if the loop gain is high. 2. A disadvantage with the Z-N technique is that it really needs to be performed in real-time in the actual plant. In some cases it may be undesirable to have the plant/process oscillate even for tuning purposes.
The Ziegler-Nichols tuning Technique for P, PI & PID controllers (Golten & Verwer §7.7) Proportional: PI: PID:
Gc ( s ) = K P
1 Gc ( s) = K p 1 + sTI 1 1 + sTD Gc ( s ) = K P + sTI 1 + s0.1TD
These are generally easier to realise, and also reduce the susceptibility of the derivative action to noise.
Created by Dr Grant Covic for paper: Systems and Control 1998: Document 2 of 3
CLTF =
K P ( s + 1 TI ) C (s) = 2 R( s) s( s + 3s + 2) + sK P + K P TI
Created by Dr Grant Covic for paper: Systems and Control 1998: Document 2 of 3
Tuning PI Controllers General approach to tuning: 1. Initially have no integral gain (TI large) 2. Increase KP until get satisfactory response 3. Start to add in integral (decreasing TI) until the steady state error is removed in satisfactory time (may need to reduce KP if the combination becomes oscillatory) Anti-windup in I & PI controllers Under some operating conditions non-linearities in the plant or controller can stop an Integral controller from removing the steady state error. If the Integrator output is not limited, then during this time the total of the integrated (summed) error {≈ΣKIe(t)dt} will continue to build. Once the restrictions are finally removed, problems can arise because this built up “energy” must be removed before the integral control can act normally… this can take a long time. To avoid this, anti-windup circuits are added that place ± limits on the integral total. These limits are usually placed on the summed output of the P&I controller as well.
Ex: assume we wish to apply PI regulator to a type 0 plant:
G p (s) =
Gc(s) R(s) + E(s)
K P (1 + 1 ) sTI
1 s 2 + 3s + 2
Gp(s)
U(s)
−
1 s + 3s + 2
2
C(s)
Thus:
OLTF = K P ( s + 1 G p (s) ) TI s
K U (s) = KP + I E (s) s
But can be realised easily in the following form:
1 sTI
C1 R2
e(t) R1
+ U(s)
1 + R2 sC1 U ( s) =− E (S ) R1