stability is the most important system specification

7/28/2019 Stability is the Most Important System Specification

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* STABILITYISTHEMOSTIMPORTANTSYSTEM

SPECIFICATION. IFASYSTEMISUNSTABLE,

TRANSIENTRESPONSEANDSTEADY-STATEERRORS

AREMOOTPOINTS.* ANUNSTABLESYSTEMCANNOTBEDESIGNEDFOR

ASPECIFICTRANSIENTRESPONSEORSTEADY-STATE

ERRORREQUIREMENT.


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TOTALRESPONSEOFASYSTEM

is the sum of the forced and natural responses, or

C(t) = cforced (t)+ cnatural(t)


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STABILITYFORLINEAR, TIME-INVARIANTSYSTEMS.

Using the natural response:

1. A system is stable if the natural response approaches zero

as time approaches infinity.

2. A system is unstable if the natural response approaches

infinity as time approaches infinity.

3. Asystem is marginally stable if the natural response

neither decays nor grows but remains constant or

oscillates.

Using the total response (BIBO):

1. A system is stable if every bounded input yields a bounded

output.

2. A system is unstable if any bounded input yields an

unbounded output.


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SECOND-ORDER SYSTEMS


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OVERDAMPED RESPONSE

* has a pole at the origin that comes from the unit step input and two

real poles that come from the system* The input pole at the origin generates the constant forced response; each of

the two system poles on the real axis generates an exponential

natural response whose exponential frequency is equal tothe ole location.


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UNDERDAMPED RESPONSE

This function has a pole at the origin that comes from the unit step input and two complex poles that comefrom the system.

the poles that generate the natural response are at s= -1 j(sq2)

the real part of the pole matches the exponential decay frequency of the sinusoids amplitude, while the

imaginary part of the pole matches the frequency of the sinusoidal oscillation

The time constant of the exponential decay is equal to the reciprocal

of the real part of the system pole. The value of the imaginary part

is the actual frequency of the sinusoid, as depicted in Figure 4.8.

* This sinusoidal frequency is given the name damped frequency of

oscillation, vd. Finally, the steady-state response (unit step)

was generated by the input pole located at the origin


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UNDAMPED RESPONSE

*has a pole at the origin that comes from the unit step inputand two imaginary poles that come from the system.

The input pole at the origin generates the constant forced

response, and the two system poles on the imaginary axis at

j3 generate a sinusoidal natural response whose frequency

is equal to the location of the imaginary poles.

c(t) = K1 + k4 cos(3t )

Note that the absence of a real part in the pole pair

corresponds to an exponential that does not decay.

Mathematically, the exponential is e -0t =1.


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CRITICALLY DAMPED RESPONSE

* This function has a pole at the origin that comes from the unit step input and two

multiple real poles that come from the system.The input pole at the origin generates the constant forced response, and the two

poles on the real axis at 3 generate a natural response consisting of an

exponential and an exponential multiplied by time, where the exponential

frequency is equal to the location of the real poles

* Critically damped responses are the fastest possible without the overshoot that

is characteristic of the underdamped response.


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Natural Frequency, Wn - The natural frequency of a

second-order system is the frequency of oscillation of the

system without damping.

- For example, the frequency of oscillation of a series RLC

circuit with the resistance shorted would be the natural

frequency.


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DAMPING RATIO,

Aviable definition for this quantity is one that compares the

exponential decay

frequency of the envelope to the natural frequency.

the ratio is constant regardless of the time scale of the response

the reciprocal, which is proportional to the ratio of the natural periodto the exponential time constant, remains the same regardless of the

time base.


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stability is the most important system specification

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