Dahlin’s algorithm

There is a variety of specifications that can be imposed on the system closed-loop response for a given change in set point. These lead to a number of alternative discrete-time control algorithms—the best known of which are the Deadbeat and Dahlin s algorithms. 

From eq. (30.14) it is clear that Dahlin s algorithm is physically realizable if the dead time in HGp(z) is not larger than (k + )T. 

With Dahlin s algorithm we can avoid the excessive control action produced by the deadbeat algorithms, thus reducing significantly the undesired large overshoots or highly oscillatory closed-loop response. 

Unfortunately, the deadbeat and Dahlin s algorithms usually contain poles that cause severe ringing of the controller output. This may be the... 

Discuss the construction of the deadbeat and Dahlin s algorithms. Which one imposes more stringent specifications on the closed-loop response What are the consequences of such stringent requirements ... 

To compute Dahlin s algorithm we assume that the closed-loop deadtime is equal to the process dead time plus one sampling period to account for the delay in the sampling and holding operations (i.e., 6 = 2+1=3 seconds). Also, we assume that the desired response has a time constant ji = 2. Then from the design eq. (30.14) we take... 

Dahlin s algorithm, 331 modified version, 333 damping coefficient, 81,83 data fitting, 115,117,119,122,127 data reconciliation, 370,414 data validation, 116,370 DGS (distributed control system), 483 deadband, 158... 

Fig. 7.96. Comparison of response of the controlled variable using deadbeat and Dahlin s response specification algorithm...

Therefore, the design eq. (30.6) yields Dahlin s control algorithm ... 

The digital control algorithms discussed in Sections 30.2 and 30.3 were designed for set point changes (servo problem). Therefore, the question arises as to how well they perform for load (disturbance) changes. It is a fortuitous coincidence that algorithms such as the deadbeat or Dahlin s perform well for both set point and load changes. 

For a discussion of the deadbeat algorithms the reader can consult Ref. 10, while Dahlin s method can be found in the original paper ... 

An alternative to discrete PI or PID algorithms is one that is determined by sampled-data techniques using the z-transformation. This algorithm does not have parameters K Tr, and td, but is expressed as a ratio of polynomials in powers of Z whose coefficients are specified to achieve a certain response. Some criteria for selecting coefficients are like the methods described in the previous section while others select the coefficients to obtain a specified type of closed-loop response. For example, Dahlin s method specifies the response to a step change in set point to be a first-order lag with dead time. The minimal prototype sampled-data algorithm is a type of optimal control in which the output is specified to reach set point, that is, zero error, in the fastest time without exlubiting oscillations. 

Next two Direct Synthesis algorithms for discrete-time application are considered Dahlin s method and the Vogel-Edgar method. The discrete-time version of a related method considered in Chapter 12, Internal Model Control, is also presented. 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

---