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This part deleted on 04 Jan. 2022:
   I have often wondered why the "lambda literature" makes a coupling between tauc and tau, see equation (***) above.
   For example, an integrating process has tau=\infty, but tauc can be small (e.g., tauc=theta from the SIMC-rule)
   However, note the similarity between equations (***) with the smooth tuning formula (**). 
   This leads to the following special case where the "lambda literature"-tuning approach is reasonable: 
   
   Start Special case: 
       Consider a first-order process with no delay and assume we have a disturbance at the input 
       which should be rejected with as little control effort as possible.  
       
       Then taud=tau and from (**) (Rule A) we get that for acceptable disturbance rejection ("smooth tuning") we must require:
	      tauc = 1/wd = tau/kd'
       This implies that we have the following tuning rule in terms of kappa,
	      kappa = 1/kd'.
       Here kd' = kd*dmax/ymax, so we have that
	      kappa = 1/kd' = ymax/(kd*dmax) = ratio between max. allowed output change (ymax) 
					       and output change without control (kd*umax) (steady-state)
       Alterntively, in terms of lambda=tauc:
             lambda=tauc = tau*kappa =  tau*ymax/(kd*dmax)
       For k'd>1, we need to "speed up" the response by a factor k'd to get acceptable disturbance rejection.

       Note: the above analysis used the approximation |1+L|\approx |L|. Here is a bit more accurate analysis. 
       The disturbance response is 
                 y = Gd*d/(1+L). 
       Here Gd = kd/(tau*s+1) and with the SIMC tunings we get, L = GC = 1/(tauc*s) so S=1/(1+L) = tauc*s/(tauc*s+1).
       For tauc < tau, the peak value of the transfer function |SG_d| is approximately (asymptotic value)
		kd*tauc/tau = kd*kappa
       which holds well for kappa<1. 
       However, for kappa=1 the asymptotic analysis does not quite hold, and the peak is only half (0.5*kd). 
                       This follows from 1/sqrt(2)*sqrt(2)=1/2=0.5
       Note that we have here considered sinusoidal disturbances. For comparison, Forsman ("Reglerteknik", p. 49, 2005, in Swedish) 
       finds a factor 0.368 (rather than 0.5) for a step disturbance.
              
       Thus, selecting kappa=1 reduces the disturbance sensitivity by a factor about 2, and this explains why kappa=1 may be a 
       reasonable starting point for the tuning. As mentioned, a common tuning rule is to select kappa even larger, say kappa=3, which must
       mean that, strictly speaking, we do not need control for disturbance rejection, but rather we use control to stabilze and avoid drift.
       
       Conclusion: From my derivation with sinusoidal input disturbances, the recommended value for kappa (the "lambda-factor") is:  
               kappa = Ratio between allowed output variation and output variation without control (steady-state)

BUT: There is also the issue of input saturation due to noise if Kc is too large, and this may a reason for not speeding up the response too much, see approach 3.

    End Special Case.