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Frequency response Bode diagrams

Motivation

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 Frequency domain analysis and design are instrumental in achieving appropriate performance and robustness of a closed loop system.
 Frequency response can be experimentally obtained.
 Nyquist plots and Bode plots are essential tools in control systems design.

Steady state periodic response
System performance often characterised in terms of response to important input types.
 Impulse response= output when a short, sharp shock is applied
 Step response= output when input is suddenly changed
 Frequency response: characterises steady- state output of system when input is a sinusoid of frequency ω

Sinusoidal steady-state response
Causal, stable, LTI system G(s)
Gain: Phase:
Claim: Gain = 𝐺𝐺 𝑗𝑗𝜔𝜔 ,Phase = ∠𝐺𝐺 𝑗𝑗𝜔𝜔
Taken from Astrom and Murray, Feedback systems

Proof of Claim (assume BIBO stable)
Let𝑢𝑢𝑢𝑡𝑡 =𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡
∫0𝑡𝑡 𝑔𝑔 𝑣𝑣 𝑒𝑒𝑗𝑗𝜔𝜔 𝑡𝑡−𝑣𝑣 𝑑𝑑𝑣𝑣 = 𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡 ∫0𝑡𝑡 𝑔𝑔(𝑣𝑣)𝑒𝑒−𝑗𝑗𝜔𝜔𝑣𝑣𝑑𝑑𝑣𝑣
 𝑦𝑦𝑢(𝑡𝑡)𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡= ∫0𝑡𝑡 𝑔𝑔(𝑣𝑣)𝑒𝑒−𝑗𝑗𝜔𝜔𝑣𝑣𝑑𝑑𝑣𝑣
 𝑦𝑦𝑢 𝑡𝑡 𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡 − 𝐺𝐺(𝑗𝑗𝜔𝜔) = − ∫𝑡𝑡∞ 𝑔𝑔(𝑣𝑣)𝑒𝑒−𝑗𝑗𝜔𝜔𝑣𝑣𝑑𝑑𝑣𝑣
≤ ∫𝑡𝑡∞ |𝑔𝑔 𝑣𝑣 |𝑑𝑑𝑣𝑣 → 0
𝑦𝑦𝑢 𝑡𝑡 = 𝐺𝐺(𝑗𝑗𝜔𝜔) 𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡+ (transient0 as t∞)

Proof (continued)
Soif𝑢𝑢𝑡𝑡 =cos𝜔𝜔𝑡𝑡=R𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡, 𝑦𝑦𝑡𝑡 =∫0𝑡𝑡𝑔𝑔𝑣𝑣R𝑒𝑒𝑗𝑗𝜔𝜔(𝑡𝑡−𝑣𝑣) 𝑑𝑑𝑣𝑣
= R ∫0𝑡𝑡 𝑔𝑔 𝑣𝑣 𝑒𝑒𝑗𝑗𝜔𝜔 𝑡𝑡−𝑣𝑣 𝑑𝑑𝑣𝑣 = R 𝑦𝑦𝑢(𝑡𝑡)
= R 𝐺𝐺(𝑗𝑗𝜔𝜔) 𝑒𝑒𝑗𝑗𝜔𝜔𝑡𝑡 + transient
= |𝐺𝐺 𝑗𝑗𝜔𝜔 | cos 𝜔𝜔𝑡𝑡 + ∠𝐺𝐺 𝑗𝑗𝜔𝜔 + transient

Steady state response:
 Note that in steady state we obtain
 The steady state output is periodic of the same frequency but

Frequency Response for Some Unstable Systems
 We can also formally define a frequency response for non-BIBO systems, provided there are no poles with positive real part.
 Just set 𝑠𝑠 = 𝑗𝑗𝜔𝜔 in the transfer function. E.g.  𝐻𝐻(𝑠𝑠)= 𝑠𝑠±𝑛𝑛 has freq. response (𝑗𝑗𝜔𝜔)±𝑛𝑛
 𝐻𝐻 𝑠𝑠 = 1 has freq. response 1
 (Made rigorous by adding a small real part -ε to any poles on imaginary axis; multiplying H(s) by (𝑠𝑠+𝜖𝜖)𝑛𝑛 if relative degree n>0; then
𝑠𝑠 2 + 𝜔𝜔 02 1
( 𝑗𝑗 𝜔𝜔 ) 2 + 𝜔𝜔 02
letting ε0.)

Nyquist plot
• Plot of G(jω) in complex plane
• Curve is parameterised by frequency.
• Bode plots are more convenient.

Link to Fourier Transforms
Fourier transform
Fourier transform
 Any input signal can be regarded as a “sum” of terms containing for

Bode diagrams

 We represent the frequency response in a “convenient” form using two plots.
gives additive property.
Multiplication with gives .
gives more compact diagrams.

Why? (additive property)
 Note if we have then,

Bode diagrams
 Consider a transfer function:  Then, we can write

Bode diagrams
 If we obtain Bode diagrams (gain and phase) for some basic functions, then Bode diagrams of an arbitrary transfer function is obtained by “adding” gain plots and phase plots of the individual building blocks.
 Bode diagrams are invaluable in “loop shaping” design techniques.
 We will plot Bode diagrams for a number of simple transfer functions.

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