Table 5 Comparative results of the K-MHFLMS estimates and errors for the example with \(p = 1,2,3,4\),\(\sigma = 0.2\),\(\delta { = 0}{\text{.4}}\).

From: Design of multi-innovation hierarchical fractional adaptive algorithm for generalized bilinear-in-parameter system using the key term separation principle

\(p\)

\(t\)

\(a_{1}\)

\(a_{2}\)

\(\lambda_{1}\)

\(\lambda_{1}\)

\(\rho_{1}\)

\(\rho_{2}\)

\(\theta_{1}\)

\(\theta_{2}\)

Fitness

\(1\)

2000

 − 0.23658

0.61316

 − 0.00448

0.15714

0.30575

 − 0.23658

 − 0.35185

0.35185

0.93785

4000

 − 0.42722

1.09984

0.01403

0.62382

0.57568

 − 0.42722

 − 0.29423

0.29423

0.83627

6000

 − 0.51701

1.42078

0.03594

1.07872

0.79555

 − 0.51701

 − 0.21698

0.21698

0.76298

8000

 − 0.57753

1.62690

0.09322

1.23263

0.98202

 − 0.57753

 − 0.15734

0.15735

0.72233

10,000

 − 0.58809

1.72885

0.11933

1.26087

1.14354

 − 0.58809

 − 0.12694

0.12694

0.69739

\(2\)

2000

 − 0.39408

1.12456

 − 0.00295

0.60998

0.57982

 − 0.39408

 − 0.28452

0.28452

0.83606

4000

 − 0.57323

1.57245

0.10338

1.17575

0.98155

 − 0.57323

 − 0.16139

0.16139

0.72465

6000

 − 0.59516

1.71535

0.15399

1.17450

1.28820

 − 0.59516

 − 0.13271

0.13271

0.68371

8000

 − 0.61984

1.79492

0.21404

1.10776

1.57618

 − 0.61984

 − 0.09710

0.09710

0.65131

10,000

 − 0.61435

1.79320

0.18870

1.16114

1.84180

 − 0.61435

 − 0.08043

0.08043

0.62069

\(3\)

2000

 − 0.49436

1.44851

0.02263

1.04872

0.80803

 − 0.49436

 − 0.20660

0.20661

0.76201

4000

 − 0.61201

1.69921

0.17696

1.10997

1.30093

 − 0.61201

 − 0.12166

0.12166

0.68328

6000

 − 0.60138

1.73957

0.18838

1.11955

1.71431

 − 0.60138

 − 0.09983

0.09984

0.63621

8000

 − 0.62422

1.76291

0.24024

1.05794

2.08552

 − 0.62422

 − 0.07345

0.07345

0.60489

10,000

 − 0.62485

1.72034

0.19788

1.15025

2.41461

 − 0.62485

 − 0.05747

0.05747

0.57628

\(4\)

2000

 − 0.56175

1.62421

0.06612

1.19771

0.99333

 − 0.56175

 − 0.15775

0.15776

0.72306

4000

 − 0.62738

1.71589

0.20410

1.04499

1.59757

 − 0.62738

 − 0.10407

0.10408

0.65203

6000

 − 0.60661

1.70596

0.19943

1.10367

2.10275

 − 0.60661

 − 0.07609

0.07610

0.60020

8000

 − 0.63110

1.70567

0.25147

1.02722

2.52066

 − 0.63110

 − 0.05490

0.05490

0.57829

10,000

 − 0.64054

1.64660

0.20143

1.13502

2.87212

 − 0.64054

 − 0.03635

0.03636

0.55858

\({{\varvec{\Sigma}}}\)

 

0.80000

1.40000

0.40000

1.90000

3.00000

 − 2.00000

0.65000

 − 1.20000

 
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