Transient Response Definition
Damping Oscillation: A typical Transient Response Example
For a system with transfer function G(s), whether open loop or closed loop and input R(s), the output is
\[C\left( s \right)=\text{ }G\text{ }\left( s \right)\text{ }R\left( s \right)\]
For distinct poles, whether real or complex, the partial fraction expansion of C(s) and the corresponding solution c(t) are, from
\[C\left( s \right)=\frac{{{K}_{1}}}{\left( s+{{p}_{1}} \right)}+\frac{{{K}_{2}}}{\left( s+{{p}_{2}} \right)}+\ldots +\frac{{{K}_{n}}}{\left( s+{{p}_{n}} \right)}\text{ (1)}\]
$c\left( t \right)={{K}_{1}}\exp \left( {{p}_{1}}t \right)+{{K}_{2}}\exp \left( {{p}_{2}}t \right)+\ldots +{{K}_{n}}\exp \left( {{p}_{n}}t \right)$
The denominator of C(s)= G(s)R(s) and its partial faction expansion contain terms due to the poles of input R(s) and those of the system G(s). The terms due to R(s) yield the forced solution whereas the system poles give the transient solution, and this is the part of the response into which more insight is needed.
System Stability
This is the most important characteristic of the transient response. For a system to be useful, the transient solution must decay to zero. This leads to the following definition,
The fundamental stability theorem can be formulated by examination of (1). If any system pole –p_{i} is positive or has a positive real part, then the corresponding exponential grows, so the system is unstable. A positive real part means that the pole lies in the right half of the splane. Hence;
A system is stable if and only if all the system poles lie in the left half of the s plane.
Transient Response First Order System (Simple Lag)
The first order system shown in the following figure is very common for analysis purposes in control system.
Fig. 1: First Order System
For a step input R(s) =1/s,
\[C(s)=\frac{{}^{1}/{}_{T}}{s(1+{}^{1}/{}_{T})}=\frac{{{K}_{1}}}{s}+\frac{{{K}_{2}}}{s+{}^{1}/{}_{T}}\]
${{K}_{1}}={{\left. \frac{{}^{1}/{}_{T}}{s+{}^{1}/{}_{T}} \right}_{s=0}}=1$
${{K}_{2}}={{\left. \frac{{}^{1}/{}_{T}}{s} \right}_{s={}^{1}/{}_{T}}}=1$
Hence the transient response is,
The first term is the forced solution due to the input and the second the transient solution, due to the system pole. Figure 2 shows this transient as well as c(t). The transient is seen to be a decaying exponential if it takes long to decay, the system response is slow, so the speed of decay is of key importance. the commonly used measure of this speed of decay is the time constant.
Fig. 2(a): Step Response of Simple Lag Network
Fig. 2(b): Step Response of Simple Lag Network
Time Constant of Second Order System
Since e^{t/T}= e^{1} when t=T, it’s seen that:
 The time constant for a simple lag (1/Ts+1) is T seconds.
 This is, in fact, the reason why a simple lag transfer function is often written in this form. The coefficient of s then immediately indicates the speed of decay.
 It takes 4T seconds for the transient to decay to 1.8% of its initial value.
 At t= T,
c (T)=10.368=0.632
The values at t=T provide one point for sketching the curves in figures (2a,2b). Also the curves are initially tangent to the dashed lines, since
\[\frac{d}{dt}\left( {{e}^{{}^{t}/{}_{\tau }}} \right){{}_{t=0}}=\frac{1}{T}{{e}^{{}^{t}/{}_{T}}}{{}_{t=0}}=\frac{1}{T}\]
These two facts provide a good sketch for the response.
Now consider the correlation between this response and the pole position at s=1/T in Fig .1. The purpose of developing such insight is that it will permit the nature of the transient response of a system to be judged by inspection of the polezero pattern.
For the simplelag, two features are important:

Stability
As discussed, for stability, the system pole 1/T must lie the left half of the splane, since otherwise the transient e^{t/T} grows instead of decays as t increases.

Speed Of Response
To speed up the response of the system (that is to reduce its time constant T), the pole 1/T must be moved left.
Transient Response of Second Order System (Quadratic Lag)
This very common transfer function can be reduced to the standard form
\[G\left( s \right)=\frac{\omega _{n}^{2}}{{{s}^{2}}+2\zeta {{\omega }_{n}}s+\omega _{n}^{2}}\]
Where
ω_{n }= undraped natural frequency
ζ = damping ratio
For a unit step input R(s) =1/s, the transform of the output is
\[C\left( s \right)=\frac{\omega _{n}^{2}}{s({{s}^{2}}+2\zeta {{\omega }_{n}}s+\omega _{n}^{2})}\]
The characteristic equation would be
${{s}^{2}}+2\zeta {{\omega }_{n}}s+\omega _{n}^{2}=0$
These system poles depend on ζ:
$\zeta >1:overdamped:{{s}_{1,2}}=\zeta {{\omega }_{n}}\pm {{\omega }_{n}}\sqrt{{{\zeta }^{2}}1}$
$\zeta =1:critically~damped:{{s}_{1,2}}={{\omega }_{n}}$
$\zeta <1:underdamped:{{s}_{1,2}}=\zeta {{\omega }_{n}}\pm j{{\omega }_{n}}\sqrt{1{{\zeta }^{2}}}$
Fig.3: System Poles Quadratic Lag
Figure 3 shows the splane for plotting the pole positions.
 For ζ>1, these are on the negative realaxis, on both sides of ω_{n . }
 For ζ=1, both poles coincide at ω_{n. }
 For ζ<1, the poles move along a circle of radius ω_{n} centered at the origin, as may be seen from the following expression for the distance of the poles to the origin:
$\left {{s}_{1,2}} \right={{\left[ {{\left( \zeta {{\omega }_{n}} \right)}^{2}}+{{\left( {{\omega }_{n}}\sqrt{1{{\zeta }^{2}}} \right)}^{2}} \right]}^{{}^{1}/{}_{2}}}={{\omega }_{n}}$
From the geometry in figure 3, it is seen also that
$Cos\left( \theta \right)=\frac{\zeta {{\omega }_{n}}}{{{\omega }_{n}}}=\zeta $ Hence,
The damping ratio $\zeta =\cos \phi $ , where ∅ is the position angle of the poles with the negative real axis.
Time constant
This is the time constant in seconds for the amplitude of oscillation to decay to e^{1} of its initial value:${{e}^{\zeta {{\omega }_{n}}t}}={{e}^{1}}$, Hence
\[T=\frac{1}{\zeta {{\omega }_{n}}}\]
Analogous to the simple lag, the amplitude decays to 2% of its initial value in 4T seconds. It is again important to determine the correlation between dynamic behavior and the pole positions in the splane in Figure 3:

Absolute Stability
The real part $\zeta {{\omega }_{n}}$ of the poles must be negative for the transient to decay; that is, the poles must lie in the left half of the splane.

Relative Stability
To avoid excessive overshoot and unduly oscillatory behavior, damping ratio ζ must be adequate. Since ζ = Cosϕ, the angle ϕ may not be close is 90^{o}.

Time Constant
The time constant is reduced (that is, the speed of decay of the transient is increased) by increasing by increasing the negative real part of the pole position.

Speed of Response
The speed of response of the system is increased by increasing the distance ω_{n} of the poles to the origin.

Undammed Natural Frequency
This equals the distance of the poles to the origin. Moving the poles out radially (with ζ constant) increases the speed of response while the percentage overshoot remains constant.

Frequency of Transient Oscillations ${{\omega }_{n}}\sqrt{1{{\zeta }^{2}}}$
This frequency also called the resonance frequency or damped natural frequency equals the imaginary part of the pole positions.
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