Lec 11. Common Controllers Some commonly used controllers –Proportional Controller –Integration Controller –Derivative Controller Reading: 5-8. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

Design of Unity-Feedback Systems Objectives in designing controller C(s) –Stability (first priority) –Steady state error (static) –Time specifications (dynamic) + plant controller

Some Commonly Used Controllers Proportional (P) controller: Integration (I) controller: Derivative (D) controller: Proportional plus Integration (PI) controller: Or combination of them, such as Controllers using root locus & frequency-response design Phase-lead compensator Phase-lag compensator Lag-lead compensator

Proportional Controller With a proportional controller, the closed loop transfer function has a time constant: Uncompensated plant is a first order system with time constant T and steady state tracking error for unit step input: + plant controller

Step Responses (T=1) Without controller Larger K will increase the response speed A steady state error is present no matter how large K is

Remarks on P Controllers A proportional controller does not change the system type For a type 0 system G(s), increasing K can only decrease the steady state position error, not eliminate it Stability may be an issue for higher order systems

Integration Controller With an integration controller, the unity feedback system has closed loop transfer function that is a standard form of second order system: + plant controller

Step Responses (T=1) uncompensated system

Remarks on I Controllers A integration controller can increase the system type by 1 –For a type 0 system, the compensated system is of type 1 –Hence, steady state error for step input is completely eliminated Compared with proportional controller, instability can be more easily caused by increasing K

PI Controller Proportional controller: increase system response without sacrificing too much on stability Proportional plus Integration (PI) controller: Integration controller: increase system type (thus eliminate tracking error), but may lead to instabilty + plant controller

Step Responses (T=1) No steady state position error Very fast response by increasing K p uncompensated system

Derivative Controller Derivative controller C(s)=K d s –Anticipate and correct error before it becomes too large –Add damping to the system –Highly sensitive to the error e(t) –Tend to increase the stability of the system Almost always used along with proportional or PI controller + plant controller

Effect on Stability Using the proportional controller Using the integration controller Using the derivative controller Ranges of K for stability: + plant controller

Example Plant is a double integrator Two choices for the controller C(s) –A proportional controller –A proportional plus derivative (PD) controller + plant controller

Proportional Controller Proportional controller Closed loop transfer function Step Response (J=10, K p =1): Changing K p can only change the frequency of the undamped oscillation

PD Controller Proportional plus derivative controller Closed loop transfer function is a (non-standard) second order system Step Response (J=10, K p =1): Increasing the derivative component will lead to more damping in the step response, and improve the stability of the system

Step Response (J=10, K d =10) The improved stability caused by the derivative component allows us to choose larger gain for the proportional component, thus reducing steady state error (indirectly)

PID Controller A combination of proportional, integration and derivative controllers: Three parameters to adjust (more flexibility) Widely used for controlling practical systems