Position Control by means of a Flexible Transmission

Position Control by means of a Flexible Transmission

The flexible transmission has been described in Chapter 7, Section 7.5.4. In this Section we will discuss the design of the controller for the position control of the third pulley (see Figure 8.33), based on the identified model. The control input is the reference for the axis position of the first pulley. It was shown in Section 7.5.4 that the system has two very low damped vibration modes. It is operated with a sampling period of 50 ms. The identified model (open loop) is

A(q ⁻¹ ) = 1 − 1.609555q ⁻¹ + 1.87644q ⁻² − 1.49879q ⁻³ + 0.88574q ⁻⁴

B(q ⁻¹ ) = 0.3053q ⁻¹ + 0.3943q ⁻²

4. 
   = 2
   

Figure 8.33. Digital control of a flexible transmission

The frequency characteristic of this model has been presented in Figure 7.24. The two vibration modes of the model are characterized by ω₀ = 11.949 rad/s, ζ =0.042 and ω₀ = 31.462 rad/s, ζ = 0.023. The model has an unstable zero (because |b₂|>|b₁|).

Since the model has an unstable zero the pole placement strategy will be used for controller design. It is desired to get a regulation and a tracking behavior characterized by a pair of well damped dominant poles, corresponding to the first vibration mode of the flexible transmission. The controller should also assure certain robustness specifications in terms of modulus margin, delay margin and maximum value of the modulus of the input sensitivity function at high frequencies. The performance and robustness specifications are summarized next:

• 
  Tracking dynamics: discretization of a second-order continuous time
  system with ω₀ = 11.94 rad/s and ζ = 0.9 (Rise time t_R ≈ 0,285s which
  corresponds to t_R ≈ 6T_s)
  

• 
  Zero steady state error (controller should include an integrator)
  

• 
  Dominant poles of the closed loop corresponding to the discretization of a second-order continuous time system with ω₀ = 11.94 rad/s and ζ = 0.8;
  
• 
  Modulus margin: ∆M ≥ 0.5 ; delay margin: ∆τ ≥ 0.05s
  

• 
  |S_up (q^-1)|_max ≤ 10 dB for f ≥ 0.35 f_s
  

A first controller design, for which only the dominant poles have been specified, leads to the sensitivity functions S_yp and S_up shown in Figures 8.34 and 8.35

(curves A).