P, PI, Lead, Lag compensators

Control engineering and signal processing

Michala Srnová2009/2010

Content Compensator P compensator PI compensator Lead compensator Lag compensator PID controller

Compensator Additional component Compensate a deficient performance Alter of root locus or frequency response Suitable place

Compensator II Classical compensators

P PI PID Lead Lag PD/Lead Lag/Lead

P compensator Proportional control Transfer function

Frequency independent Steady state problems Stability problems Error multiplication → small fluctuation of error signal

→ great changes of control signal

P compensator Step response

Steady state error or oscillations Not often used alone With connection with integral part (PI) or in PID

PI compensator Proportional + Integral compensator

Ki – integral gain Changes type of the system No steady state error to step Can make system slower

PI compensator II Frequency response

Ti = Kp/Ki

High-frequency attenuator PI introduce a phase lag !

Lead compensator Lead phase → phase of compensator +

Lead: |z| < |p|

Lag compensator |z| > |p|

PI

Lead and Lag Compensators Alter of frequency response Calculations

Definition of α, ωm ,τ

Transfer function

Calculation for maximum phase angle ( ω=ωm)

System gain = -10 log α → frequency = ωm

System with comp.: gain = 0 db by frequency ωm

PID controller Proportional + integral + derivative

Transfer function:

KD – derivative gain

Used in majority of closed-loop industrial processes

PID controller Step response

Rise time Overshoot Settling Time Steady state Error

Effect of increasing the parameters:

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