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Necessary of Compensation

In order to obtain the desired performance of the system, we use compensating networks.

Compensating networks are applied to the system in the form of feed forward path gain

adjustment.

Compensate a unstable system to make it stable.

A compensating network is used to minimize overshoot.

These compensating networks increase the steady state accuracy of the system. An

important point to be noted here is that the increase in the steady state accuracy brings

instability to the system.

Compensating networks also introduces poles and zeros in the system thereby causes

changes in the transfer function of the system. Due to this, performance specifications ofthe system change.

Effects of Lead Compensator

Lead compensator adds a dominant pole and zero. This increases the damping of the

closed loop system.

This increased damping means less overshoot, less rise time, less settling time. Thus there

is an improvement in transient response.

It improves phase margin of the closed loop system

It increases bandwidth of the closed loop system. More bandwidth faster is the response.

Steady state error doesnt get affected

Limitations of Lead compensation

Maximum lead angle available from a lead network is 60o.Thus if a lead of 70 to 90 is

required multiple lead compensators are required.

Lead compensation requires additional gain to offset the attenuation in lead network. More bandwidth sometimes makes the system susceptible to noise

Effects of Lag Compensator

It improves steady state performance.

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The attenuation due to lag compensator shifts the gain crossover frequency to a lower

frequency point. Thus bandwidth of the system gets reduced.

The system becomes more sensitive to parameter variations

This makes system less stable

Reduced bandwidth means slower response

Effects of Lag-Lead Compensator

Lag lead compensator is used when both fast response and good accuracy is required

Increases low frequency gain which improves the steady state

It increases bandwidth and thereby speed of response

Since it has larger bandwidth settling and rise time are low

MODULE 5

Drawbacks of transfer function

Transfer function defined under zero initial conditions

Only applicable to LTI system

Restricted to single input single output system

Doesnt provide any information regarding internal state of the system

Advantages of state variable analysis

Applicable to any type of system

Can be applied to MIMO systems

Properties of state transition matrix ((t)=)

(t)=,then (0)==I

-1

(t)= = (-t)

(t1+t2) = (t1)* (t2) = *= (t1)* (t2)

(t1-t2)* (t2-t3) = (t1-t3) = (t1-t3)

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Characteristics equation

Asymptotically stable system

A system which produces zero output for zero input irrespective of initial conditions.

An asymptotically stable system is also BIBO stable, but not vice versa

No. BIBO Asymptotically stable

1 Zero state stabilty Zero input stability

2 External stability Internal stability

3 All poles of G(s) on Left

Half Plane

All eigen values of A are in

LHP

4 Bounded output for

bounded input

Output=0 for zero input

5 Necessary and sufficient

condition

Impulse response should

be bounded

All eigen values of A are in

LHP