The block diagram shown in the figure above models the CL behavior of a 1- DOF...

  

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The block diagram shown in the figure above models the CL behavior of a 1- DOF SEA with impedance controller. Derive the expression of the impedance reflected at the input port Z(s) = using either a graphical method (block diagram reduction) or an algebraic method. In either case, please include the complete procedure. -415 [3 POINTS] Assuming that the impedance to be rendered is a virtual stiffness P (i.e., sZ (s) =P), prove that the CL system satisfies the passivity condition if P ≤ak, a = 0.8333. [5 POINTS] Determine the value of P resulting in a system bandwidth of 10 Hz. Then, using Matlab, trace the Bode diagrams (magnitude in dB, phase in deg) of Z(jw) for that P, and estimate the CL asymptotic behavior of the SEA at low, high, and intermediate frequencies (i.e., derive the equations of 3 asymptotes and plot them in the Bode diagrams). HINTS: For (1), the resulting Z(s) should look exactly as the one discussed in Sec. 6.7 (Modelling). For (2), verify the conditions for coupled stability described in Sec. 6.8 (Coupled Stability) using the provided numerical values for the control gains and physical parameters. For (3), use the approximate expression of the system bandwidth described in Sec. 6.9 (Bandwidth), then verify your result using the MATLAB bode function. Controller SZ (s) Ko+ OPA T₂ ]] K SEA JA=0.13 kg m² K= 400 Nm/rad K₂=1 1 = 0.4 K₁ = 10000 1₁ = 4000 The block diagram shown in the figure above models the CL behavior of a 1- DOF SEA with impedance controller. Derive the expression of the impedance reflected at the input port Z(s) = using either a graphical method (block diagram reduction) or an algebraic method. In either case, please include the complete procedure. -415 [3 POINTS] Assuming that the impedance to be rendered is a virtual stiffness P (i.e., sZ (s) =P), prove that the CL system satisfies the passivity condition if P ≤ak, a = 0.8333. [5 POINTS] Determine the value of P resulting in a system bandwidth of 10 Hz. Then, using Matlab, trace the Bode diagrams (magnitude in dB, phase in deg) of Z(jw) for that P, and estimate the CL asymptotic behavior of the SEA at low, high, and intermediate frequencies (i.e., derive the equations of 3 asymptotes and plot them in the Bode diagrams). HINTS: For (1), the resulting Z(s) should look exactly as the one discussed in Sec. 6.7 (Modelling). For (2), verify the conditions for coupled stability described in Sec. 6.8 (Coupled Stability) using the provided numerical values for the control gains and physical parameters. For (3), use the approximate expression of the system bandwidth described in Sec. 6.9 (Bandwidth), then verify your result using the MATLAB bode function. Controller SZ (s) Ko+ OPA T₂ ]] K SEA JA=0.13 kg m² K= 400 Nm/rad K₂=1 1 = 0.4 K₁ = 10000 1₁ = 4000