Position Control by means of a Flexible Transmission

F igure 8.42. Frequency characteristics of the identified model for the 360° flexible robot arm

Figure 8.43. Pole – zero map of the identified model for the 360° flexible robot arm
The design of the controller will be done using the iterative pole placement
design with shaping of the sensitivity functions, by simultaneous tuning of H_S (respectively H_R) and of auxiliary poles (see Chapter 3, Section 3.6 and (Prochazka
and Landau 2003)). The effective computations have been carried on with
ppmaster (MATLAB^®)⁹ (Prochazka and Landau 2001). The specifications are the same as in (Langer and Landau 1999):

• 
  Tracking dynamics: discretization of a second-order continuous time
  system with ₀=2.6173rad/s and  = 0.9
  
• 
  Zero steady state error (controller should include an integrator)
  
• 
  Dominant poles of the closed loop system corresponding to the
  discretization of a second-order continuous time system with ₀=2.6173
  rad/s and  = 0.8
  
• 
  ^{Modulus margin:} M  0.5 ^{(-6 dB); delay margin:   0.05s (1Ts )}
  
• 
  Constraints on the input sensitivity function |S_up (q^-1)|: < 15 dB at low frequencies (< 4 Hz); <0 dB from 4.5 Hz to 6.5 Hz; < 15 dB from 6.5 Hz
  to 8 Hz; <10 dB from 8 to 10 Hz (f_s=20Hz)
  

The low value imposed on |S_up| between 4.5 Hz and 6.5 Hz (0.225 to 0.325 f/f_s) is a constraint resulting from the low value of the open loop gain and the uncertainties
upon the model in this frequency region. The bound |S_up| at high frequencies will limit the effect of the measurement noise upon the control signal.
The desired dominant closed loop poles are chosen as indicated in the

S₁
specifications, and an integrator is introduced in the controller ( H  1 q ^1 ).

Note that it is not necessary to damp the second and third (high frequency)

vibration mode because closed loop band pass (defined by the dominant closed
loop poles), the disturbances and the tracking model dynamics are all are at low frequencies. Therefore these poles will be kept unchanged by specifying them as
poles of the closed loop (see the partial internal model design, Section 3.5.5).
The result of this first design is a controller for which |S_yp| and |S_up| are far
outside the imposed templates at high frequencies (Figures 8.44 and 8.45 – curves
A). In such situations auxiliary poles have to be added. The total number of poles which can be specified (without increasing the size of the controller) is

S₁
n _P  n _A  n _B  d  n _H  1  12

Six poles have been have been already assigned. It is therefore possible to add auxiliary poles of the form

P_F ( z ^1 )  (1  p₁ z ^1 ) ⁶ ;  0 .5  p₁  0.05

Taking p₁   0.5 ^{one gets a controller leading to sensitivity functions which are}

slightly above the templates in two frequency regions (Figures 8.44 and 8.45 –
curves B). |S_yp| is above the template around 1 Hz and |S_up| is above the template
between 4 and 6 Hz. First, to improve the design, we will consider the introduction
serve for the computation of the discrete time filter, is chosen with a resonance
frequency f₀ =1Hz (6.28 rad/s). The damping for the denominator is chosen as
_den =0.8 (in order that the auxiliary poles which will be introduced be well
damped). The desired attenuation is M_t = -5.5 dB, leading to _num =0.424. The
characteristics of the discrete time filters H and P₂ are given in Table 8.9¹⁰. The

sensitivity functions obtained with the new controller are illustrated in Figures 8.44 and 8.45 – curves C. It remains to correct now the frequency characteristics of |S_up|

S₂