Control Systems--The Last Basic Course, Pt II

8/3/2019 Control Systems--The Last Basic Course, Pt II

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Performance of feedback systems

Performance of second order feedback systems

A 2nd order equation may be given as the following; R(s) is used forunit step function

As decreases, the roots approach the imaginary axis and response becomes more

oscillatory.

Standard performance measures are defined in terms of step response.


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Settling time Ts = 4 = 4/.

If is increased in left hand side, it has to be such that denominator does not turn complex.

Tp gives normalized peak time. Why normalized? Because it is free of .


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Effect of a third pole and zero on second order systems

Where the transfer function is

Examples 5.1 and 5.2 are good ones.

S-Plane Root Location and Transient Response the information imparted is graphic


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The two equations above are of utmost importance.

Poles of T(s) decide the response modes, while zeros decide the weightings of those;

moving a zero closer to pole reduces overall contribution of mode function corresponding tothat pole.

Steady state error of Feedback Control Systems

Initial value theorem is for transient response, while final value theorem is for steady

state.

Error is found for closed loop system from function E(s).

Consider the case of a step input of magnitude A belowonly (A/s) is the change.

Position error constant Kp gives the idea of loop transfer function without any storage

element put in circuit. Kp is an extremely important parameter, obviously.

Type Number N gives the idea of integrations performed or, say, need to be performed.


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Further,

The steady state error for a unit step input with one or more integrations is always

zero.


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We need to have integrations more than or equal to 2 to get zero steady state error for ramp

input.Note it here again that integration means incorporating energy storage componentsin the circuit.

The error constant type in terms if which you are going to think would depend over the type


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of inputthe power of s it would generate. One more thing which is implicit here is a

composite function consisting many powers oftcan be analyzed in terms of different errorconstants.

Kp, Kv and Ka give a system's ability to eliminate/reduce steady state errormeasure for

steady state error analysis.

Designer makes the choice of error constants and then keeps monitoring keeping within

requirements of performance.

The Non-unity Feedback Systems

The first obvious change is units of output Y(s)--they're different from output of H(s).

K of a block also provides with means to perform change from one unit to another.

A trnasfer function calculated with limit s--> 0 gives itsDC gain.

For solving non-unity feedback, we rearrange blocks to get a unity feedback system which

must first be stable.


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Performance Indices

Needed in cases of both improving the design and very designing of the system.

This is about quantitatively measuring the performance of the system so as particular

specification can be emphasized.

An optimum control system is one with the index having reached an extremum, which

often is a minimum. The index is, therefore, tried to keep the lowest possible, which can be

zero by the way.

1 suitable index is integral of square of error, ISE

Although ISE is the most popular, following is an account of few more indices developed to

suit specific purposes and computations.


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The Simplification of Linear Systems

A very pleasent fact is that a pole of a transfer function with its real part much more

negative than rest of them affects the transient response insignificantly.

The whole purpose of this simplification is to be able to approximate as much as possiblethe response of higher order tranfer fuctions with transfer functions of lower order. One

algebraic method is associated approximation.

The gain constant K remains same for both the actual fucntion and the lower order

approximation to ensure that the steady state response of the both is same(well,

almost).

Gh(s) is the transfer function of higher order and Gl(s) is that of lower order. The degree of denominator is kept larger than the numerator to ensure there is no DC offset

which renders the linearity futile by violation of homogeneity.

Values of ci and di is given by the equation M2q = 2q, q being 1, 2, ... depending over the

many equations ve require.

When the dominant pole of Gh(s) is to be retained, the denominator is not approximated,

only the numerator is approximated.


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Control Systems--The Last Basic Course, Pt II

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