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Transient Response | First and Second Order System Transient Response

Transient Response Definition 

Transient Response
In control systems, a transient response (which is also known as a natural response) is the system response to any variation from a steady state or an equilibrium position. The examples of transient responses are step and impulse responses which occur due to a step and an impulse input respectively.

 A typical Transient Response Example

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,

System Stability Condition
A system is stable if transient solution decays to zero and is unstable if this solution grows.

The fundamental stability theorem can be formulated by examination of (1). If any system pole –pi 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 s-plane. 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.

first order 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,

First Order System Transient Response
\[c\left( t \right)=1-{{e}^{-{}^{t}/{}_{T}}}\]

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.

Step Response of First Order System

Fig. 2(a): Step Response of Simple Lag Network

Step Response of First Order System

Fig. 2(b): Step Response of Simple Lag Network

Time Constant of Second Order System

Definition
This is the time in seconds for the decaying exponential transient to be reduced to    e-1 =0.368 of its initial value. 
                                                                    

Since e-t/T= e-1 when t=T, it’s seen that:

  1. The time constant for a simple lag (1/Ts+1) is T seconds.
  2. 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.
  3. It takes 4T seconds for the transient to decay to 1.8% of its initial value.
  4. At t= T,

c (T)=1-0.368=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 pole-zero pattern.

For the simple-lag, two features are important:

  1. Stability

 As discussed, for stability, the system pole -1/T must lie the left half of the s-plane, since otherwise the transient e-t/T grows instead of decays as t increases.

  1. 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

 ωn =  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:

$\zeta >1:overdamped:{{s}_{1,2}}=-\zeta {{\omega }_{n}}\pm {{\omega }_{n}}\sqrt{{{\zeta }^{2}}-1}$

$\zeta =1:critically~damped:{{s}_{1,2}}=-{{\omega }_{n}}$

$\zeta <1:underdamped:{{s}_{1,2}}=-\zeta {{\omega }_{n}}\pm j{{\omega }_{n}}\sqrt{1-{{\zeta }^{2}}}$

System Poles Quadratic Lag

Fig.3: System Poles Quadratic Lag

Figure 3 shows the s-plane for plotting the pole positions.

  • For ζ>1, these are on the negative real-axis, on both sides of -ωn .
  • For ζ=1, both poles coincide at -ωn.
  • For ζ<1, the poles move along a circle of radius ωn 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 s-plane in Figure 3:

  1. 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 s-plane.

  1. 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 90o.

  1. 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.

  1. Speed of Response

                        The speed of response of the system is increased by increasing the distance ωn of the poles to the origin.

  1. 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.

  1. 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.

About Ahmad Faizan

Mr. Ahmed Faizan Sheikh, M.Sc. (USA), Research Fellow (USA), a member of IEEE & CIGRE, is a Fulbright Alumnus and earned his Master’s Degree in Electrical and Power Engineering from Kansas State University, USA.

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