Online Identification of Multivariable Discrete Time Delay Systems Using a Recursive Least Square Algorithm

Algorithm 1

Step 1. Data acquisition {𝑈(𝑘),𝑌(𝑘)} and initialization:

   set Θ_𝐺=Θ_𝐺0 and 𝑃=𝛽𝐼 where 𝛽 is a scalar and

   𝐼 is the identity matrix of size

   (𝑝,𝑝(𝑛_𝑎+𝑛_𝑏+𝑛_𝑐+1)) and 𝑘=0,

Step 2. Increment 𝑘 and construct the observation vector

   𝜑(𝑘,𝐷), the generalized observation vector

   Φ^𝑇(𝑘,Θ_𝐺) using (9) and (16),

Step 3. Estimate Θ_𝐺 using the developed identification

   method:

   Υ(𝑘)=𝑌(𝑘)−Θ^𝑇(𝑘−1)𝜑(𝑘,𝐷)

   Θ𝑃(𝑘)=𝑃(𝑘−1)−𝑃(𝑘−1)Φ(𝑘,_𝐺)Φ^𝑇(𝑘,Θ_𝐺)𝑃(𝑘−1)1+Φ^𝑇(𝑘,Θ_𝐺)𝑃(𝑘−1)Φ(𝑘,Θ_𝐺)

   Θ_𝐺(𝑘)=Θ_𝐺(𝑘−1)+𝑃(𝑘)Φ(𝑘,Θ_𝐺)Υ(𝑘)

Step 4. Return to Step 2 until 𝑘=𝑁 where 𝑁 is the

   number of input/output data.