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Tracking Performance

Tracking A Reference Signal

Type-N System

Tracking Performance of Type-0, Type-1and Type-2 System

Tracking Performance of Systems with

Disturbance

Tracking Performance of Non-unity

Feedback Systems

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Tracking A Reference SignalConsider a feedback system as described in the following figure:

Normally, the system is required to regulate the output signal c(t) according

to the instructional input signal r(t), that is, to make sure c(t) to track r(t). If

sensor dynamics is considered in the feedback channel, this tracking regulationperformance is well described by the following transfer function:

And the tracking performance is defined as:

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Tracking A Reference SignalIn practice, we are specifically interested in the tracking performance of

a feedback system with respect to the following instructional input signals:

Correspondingly, we have: R(s)=1/s ------ Unit-Step

R(s)=1/s2------ Unit-RampR(s)=1/s3 ----- Unit-Acceleration

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Tracking A Reference Signal

If the feedback system is stable, that is, 1+G(s)H(s)=0has only roots in the

left-half complex plane, then the tracking performance can be obtained as:

Specifically, for the special instructional input signals:

Unit-Step Signal

Unit-Ramp Signal

Unit-Acceleration Signal

Fact: thansfer function G(s)H(s) plays critical role in determining the tracking

performance and it deserves special attention !

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Type-N SystemWithout loss of generality, we assume H(s)=1, that is, the sensor dynamics is

not considered. In this case, the feedback system is

G(s)

A feedback system with a unit gain in its feedback channel is called

Unity Feedback System.

The tracking performance of a unity feedback system will be determined by

the forward transfer function G(s). Note that the expression ofG(s) can be

always characterized as:

R(s) C(s)E(s)

Definition: The type of a unity-feedback system is defined by Nand the unityfeedback system in this form is called a Type-N system.

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Type-N SystemExample 1:

We have

and K=20, N=0, so this is a Type-0 system

Example 2:

We have

and K=100, N=1, so this is a Type-1 system

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Type-N SystemExample 3:

We have

and K=437.5, N=2. So this is a Type-2 system.

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Tracking Performance of Type-0,

Type-1 and Type-2 SystemTracking performance of Type-N system can be obtained as follows:

For unit-step reference signal

where H(s)=1 and

Type-0 system (N=0):



Kp=K is called position error

constant

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Type-1 and Type-2 System

For unit-ramp reference signal

where H(s)=1 and




Kv=Kis called velocity error constant

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For unit-acceleration reference signal

where H(s)=1 and



Type-2 system (N=2): Ka=Kis called acceleration error

constant

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Type-1 and Type-2 SystemIn summary:

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Type-1 and Type-2 SystemExamples:

Note that

This is a Type-0 system. Therefore, Kp=K=20 and,

=1/21Forr(t)=1(t)

Forr(t)=t

Forr(t)=t2/2

Forr(t)=t

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Note that

This is a Type-1 system: Kv=K=100, and

Forr(t)=1(t)

Forr(t)=t

For r(t)=t2/2

=1/100

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Note that

So this is a Type-2 system and Ka=K=437.5

Forr(t)=1(t)

Forr(t)=t

For r(t)=t2/2 =1/437.5

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Tracking Performance of Systems

with DisturbanceConsider a feedback system with disturbance

Transfer function:

So the steady-state tracking error is:

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with DisturbanceFor a unit-step disturbance, D(s)=1/s, we have

and

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with DisturbanceExample:

For this example, we have

=-1/(0+1000)=-0.001

It is noted that ifG1(s) includes an integrator 1/s, then the error caused by

the disturbance will be completely eliminated.

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Feedback SystemsIf sensor and/or transducer

dynamics are considered, then a

non-unity feedback system willbe presented.

We can convert it into an

equivalent unity-feedbacksystem first, then apply the

results established before.

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Feedback Systems

In general, the system could

include both disturbance as

shown above. The tracking error

can be obtained as:

Consider unit-step input and disturbance: R(s)=D(s)=1/s

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Feedback SystemsObviously, if

=1 =0

then, the tracking error can be achieved.

In general, if the following four conditions are satisfied, then the equations

above to guarantee the zero error tracking will be held:

1) 1+G1(s)G2(s)H(s) is stable,

2) G1(s) is a Type-1 element,3) G2(s) is a Type-0 element,

4) H(s) is a Type-0 element with a dc gain of unity: H(0)=1.

Note that the Conditions 1)-4) are only SUFFICIENT conditions for achievingzero-error tracking.

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88324 - X.Chen -Note 7