Question

Determine the transfer function of the phase-lead compensator that can be used with the follow open-loop transfer function in order to give a phase margin of 45° 30 G(s) = s(s+3) Include the Bode plots for the original system (G(s)) and the compensated system (Gc(s)G(s), where Gc(s) is the transfer function of the phase-lead compensator). The Bode plots must show t phase margin (use the margin command or Itiview in MATLAB).

Answer 1

Using Matlab:

Gs=tf(30,[1 3 0]);%Define the system
sisotool(Gs)%open SISO system analysis and design tool

Control System Designer - Bode Editor for Loop Transfer_C CONTROL SYSTEM BODE EDITOR VIEW -ox do E È 5 CEO Tuning Methods New Plot Store Retrieve Compare Export Preferences Open Save Edit Multimodel Session Session Architecture Configuration FILE ARCHITECTURE Data Browser Controllers and Fixed Blocks ULTS PREFERENCES GRAPHICAL TUNING Bode Editor Edit feedback loop using Bode plot Root Locus Editor for Loop Transfer_ C X Root Locus Editor for Loop Transfer_C for Loop Transfer_C LOUI Closed-Loop Bode Editor Edit closed loop using Bode plot Root Locus Editor Edit compensators using root locus plot Imag Axis Designs Nichols Editor Edit feedback loop using Nichols plot -3 -2.5 -2 -1.5 -1 Real Axis -0. 5 0 0.5 1 AUTOMATED TUNING PID 10 Transfer_r2y: step xl PID Tuning Tune PID compensator using robust response time or classical methods Responses Loop Transfer_C 10 Transfer_r2y IO Transfer_r2u 10 Transfer_du2y 10 Transfer_dy2y Step Response From:r To: y Optimization Based Tuning Optimize compensator parameters to satisfy design requirements Preview TOzdt LQG Synthesis Obtain feedback compensator using Linear-Quadratic-Gaussian design Amplitude Loop Shaping Find feedback compensator to match specified open-loop shape 101 102 IMC quency (rad/s) Internal Model Control (IMC) Tuning Obtain feedback compensator usina IMC desian 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ^ a 40 6:20 AM 5/6/2020 W

Now Select Controller Type as PD, Design mode as Frequency. Don't forget to select Design With First Order Derivative Filter.

PID Tuning Compensator CM = 0.49522 x (1 +0.25s) (1 + 0.083) Select Loop to Tune LoopTransfer_C Add new loop ... Specifications Tuning method: Robust response time Controller Type: POTPI O PD besign with first order derivative filter Design mode: Frequency PID 0.0971 Bandwidth (rad/s) 9.71 Reset rameters Phase Margin (deg! 00 Update Compensator Help

Assign Phase Margin as 45 and relevant Bandwidth, Click on Update compensator.

You will see the Compensator Transfer function at the top.

Now in Matlab Command window:

Gs=tf(30,[1 3 0]);
Gcs=zpk(-0.25,-0.08,0.49522)%Compensator Transfer Function
bode(Gs)%Bode Plot
[ Gm1 , Pm1 , Wcg1 , Wcp1 ] = margin( Gs )%determin all the margins.
figure();
bode(Gs*Gcs)
[ Gm2 , Pm2 , Wcg2 , Wcp2 ] = margin( Gs*Gcs )

Output:

Bode Diagram Magnitude (dB) Phase (deg) 10-1 100 101 Frequency (rad/s)

Bode Diagram Magnitude (dB) -90 Phase (deg) -180 10-3 10-2 R 101 10² 10-1 10° Frequency (rad/s)

Gm1 =

Inf

Pm1 =

30.5572

Wcg1 =

Inf

Wcp1 =

5.0814

Gm2 =

Inf

Pm2 =

39.1329

Wcg2 =

Inf

Wcp2 =

3.3255