Lec 11. Common Controllers

• Some commonly used controllers– Proportional Controller– Integration Controller– Derivative Controller

• Reading: 5-8.

Design of Unity-Feedback Systems

• Objectives in designing controller C(s)– Stability (first priority)– Steady state error (static)– Time specifications (dynamic)

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plantcontroller

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:

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Step Responses (T=1)Step Response

Time (sec)

Am

plit

ud

e

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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:

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plantcontroller

Step Responses (T=1)Step Response

Time (sec)

Am

plit

ud

e

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

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 ControllerProportional 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

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plantcontroller

Step Response

Time (sec)

Am

plit

ud

e

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Responses (T=1)

No steady state position error

Very fast response by increasing Kp

uncompensated system

Derivative Controller • Derivative controller C(s)=Kds

– 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

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Effect on Stability

Using the proportional controller

Using the integration controller

Using the derivative controller

Ranges of K for stability:

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Example• Plant is a double integrator• Two choices for the controller C(s)

– A proportional controller

– A proportional plus derivative (PD) controller

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plantcontroller

Proportional ControllerProportional controller

Closed loop transfer function

Step Response (J=10, Kp=1):

Step Response

Time (sec)

Am

plit

ud

e

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Changing Kp can only change the frequency of the undamped oscillation

PD ControllerProportional plus derivative controller

Closed loop transfer function is a (non-standard) second order system

Step Response (J=10, Kp=1):

Increasing the derivative component will lead to more damping in the step response, and improve the stability of the system

Step Response

Time (sec)

Am

plit

ud

e

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Step Response (J=10, Kd=10)Step Response

Time (sec)

Am

plit

ud

e

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

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