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System analysis based on the frequency response

Relationship of the performance specifications between the frequency and time domain

Relationship of the performance specifications between the frequency and time domain

Relationship of the performance specifications between the frequency and time domain

Relationship of the performance specifications between the frequency and time domain

System analysis based on the frequency response Performance specifications in the frequency domain 1. For the closed loop systems The general frequency response of a closed loop systems is shown (1) Resonance frequency ωr: ω A(ω) A(0) 0.707A(0) ωr ωb Mr (2) Resonance peak Mr : (3) Bandwidth ωb:

For the open loop systems (1) Gain crossover frequency ωc: For the unity feedback systems, ωc≈ ωb , because: (2) Gain margin Kg: (3) Phase margin γc:

System analysis based on the frequency response Generally Kg and γc could be concerned with the resonance peak Mr : Kg and γc ↑ —— Mr ↓. ωc could be concerned with the resonance frequency ωr and bandwidth ωb : ωc↑ —— ωr and ωb ↓.

Relationship of the performance specifications between the frequency and time domain The relationship between the frequency response and the time response of a system can be expressed by following formula: But it is difficult to apply the formula .

Relationship of the performance specifications between the frequency and the time domain: for the typical 2th-order system For the typical 2th-order system: We have:

(1) Bandwidth ωb(or Resonance frequency ωr)  Rise time tr Relationship of the performance specifications between the frequency and time domain (1) Bandwidth ωb(or Resonance frequency ωr)  Rise time tr Generally ωb(or ωr )↑—— tr ↓ because of the “time scale” theorem: So ωb(or ωr )↑—— tr ↓ alike : ωc↑—— tr ↓ because of ωc≈ ωb . For the large ωb , there are more high-frequency portions in c(t), which make the time response to be faster.

(2) Resonance peak Mr  overshoot σp% Relationship of the performance specifications between the frequency and time domain (2) Resonance peak Mr  overshoot σp% Normally Mr ↑ —σp% ↑ because of the large unbalance of the frequency signals passing to c(t) . Kg and γc ↓ —σp% ↑is alike because of Kg and γc ↓—Mr ↑. Some experiential formulas:

Relationship of the performance specifications between the frequency and time domain (3) A(0) → Steady state error ess So for the unity feedback systems:

System analysis based on the frequency response “three frequency band” theorem The performance analysis of the closed loop systems according to the open loop frequency response. 1. For the low frequency band the low frequency band is mainly concerned with the control accuracy of the systems. The more negative the slope of L(ω) is , the higher the control accuracy of the systems. The bigger the magnitude of L(ω) is, the smaller the steady state error ess is. 2. For the middle frequency band The middle frequency band is mainly concerned with the transient performance of the systems. ωc↑—tr ↓; Kg and γc ↓—σp% ↑

“three frequency band” theorem The slope of L(ω) in the middle frequency band should be the –20dB/dec and with a certain width . 3. For the high frequency band The high frequency band is mainly concerned with the ability of the systems restraining the high frequency noise. The smaller the magnitude of L(ω) is, the stronger the ability of the systems restraining the high frequency noise is.

System analysis based on the frequency response Example: ω 0dB Ⅰ －40 －20 Compare the performances between the system Ⅰand system Ⅱ Solution : essⅠ> essⅡ trⅠ > trⅡ Ⅱ The ability of the system Ⅰrestraining the high frequency noise is stronger than systemⅡ