Dahlin algorithm

This is a generalised version of the control algorithm. For example, by setting 

This is the equation for the proportional-on-error, derivative-on-error noninteractive controller. We can however choose coefficients to produce almost any control algorithm. For a first order plus deadtime processes we use 

Generally 6 will not be an exact multiple of ts and so the value of MV (jv+ d is linearly interpolated between adjacent values of MV. 

The tuning constant (2) is the required time constant for the trajectory of the approach to SP, as it is in the Lambda tuning method. 

The performance of the Dahlin algorithm is similar to that of the Smith predictor and IMC. It is equally sensitive to the accuracy of the deadtime (6) used in deriving N and hence the value of j). It too can be extended to higher order models. 

---

As our final minimal-prototype controller, let us consider the Dahlin algorithm. The basic notion is to specify a desired step setpoint response that looks like a deadtime followed by an exponential rise up to the setpoint. See Fig. 20.4. 

VII.42 (a) Design the Dahlin algorithm for the process of Problem VII. 39. The algorithm must be physically realizable and should provide no slower response than that of the open-loop process. 

Figure 30.5 Closed-loop responses to unit step change in set point using deadbeat and Dahlin algorithms.

How can you eliminate the ringing from a deadbeat or Dahlin 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. 

DAHLIN<44) suggested that, in order to avoid the large overshoots and oscillatory behaviour which are characteristic of the deadbeat algorithm, the specification of the system closed-loop response to a step change in set point should be the same as that for a first-order system with dead time. The first-order time constant can then be employed as a design parameter which can be adjusted to give the desired closed-loop response. Hence ... 

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

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. 

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

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... 

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. 

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 ... 

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

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... 

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. 

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... 

---