Table 4 Comparative results of the K-MHFLMS \((p = 4)\) estimates and errors for the example with \(\delta { = 0}{\text{.4,0}}{.7,1}{\text{.0,1}}{.2,1}{\text{.4}}\),\(\sigma = 0.8\).

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

\(\delta\)

\(t\)

\(a_{1}\)

\(a_{2}\)

\(\lambda_{1}\)

\(\lambda_{1}\)

\(\rho_{1}\)

\(\rho_{2}\)

\(\theta_{1}\)

\(\theta_{2}\)

Fitness

\(0.4\)

2000

 − 0.59509

1.90210

0.44453

1.24634

1.60054

 − 0.59509

 − 0.02255

0.02255

0.63297

4000

 − 0.62935

1.84584

0.51378

1.12167

2.60722

 − 0.62935

0.03071

 − 0.03071

0.55940

6000

 − 0.58289

1.75407

0.54912

1.18331

3.35389

 − 0.58289

0.05550

 − 0.05550

0.54775

8000

 − 0.64046

1.67079

0.56839

1.05076

3.89695

 − 0.64046

0.07643

 − 0.07643

0.57998

10,000

 − 0.63706

1.62265

0.52575

1.22441

4.40759

 − 0.63706

0.13087

 − 0.13087

0.60759

\(0.7\)

2000

 − 0.61293

1.88239

0.46899

1.25927

1.64766

 − 0.61293

 − 0.02396

0.02396

0.62709

4000

 − 0.63366

1.85215

0.51657

1.12413

2.53532

 − 0.63366

0.02973

 − 0.02973

0.56226

6000

 − 0.58442

1.77845

0.56580

1.19081

3.16825

 − 0.58442

0.05701

 − 0.05701

0.54387

8000

 − 0.64342

1.71719

0.58121

1.05989

3.65086

 − 0.64342

0.06913

 − 0.06913

0.56604

10,000

 − 0.63974

1.67374

0.53983

1.23639

4.10152

 − 0.63974

0.12903

 − 0.12903

0.57785

\(1.0\)

2000

 − 0.62765

1.84413

0.50178

1.29780

1.64785

 − 0.62765

 − 0.03881

0.03881

0.62633

4000

 − 0.64135

1.86129

0.53570

1.13971

2.42009

 − 0.64135

0.02912

 − 0.02912

0.56698

6000

 − 0.59150

1.81358

0.59512

1.20439

2.96076

 − 0.59150

0.06755

 − 0.06754

0.54196

8000

 − 0.64653

1.77706

0.61084

1.08204

3.39544

 − 0.64653

0.06416

 − 0.06415

0.55626

10,000

 − 0.64381

1.73829

0.56641

1.25201

3.79619

 − 0.64381

0.13080

 − 0.13080

0.55468

\(1.2\)

2000

 − 0.63509

1.80564

0.52436

1.33201

1.62110

 − 0.63509

 − 0.08029

0.04504

0.62935

4000

 − 0.64703

1.86194

0.54538

1.15388

2.31533

 − 0.64703

 − 0.15472

 − 0.20334

0.56697

6000

 − 0.59480

1.83321

0.60006

1.20974

2.80187

 − 0.59480

 − 0.17708

 − 0.30667

0.53800

8000

 − 0.64711

1.81077

0.61534

1.09579

3.20222

 − 0.64711

 − 0.23893

 − 0.35477

0.54609

10,000

 − 0.64241

1.77714

0.56999

1.25834

3.57081

 − 0.64241

 − 0.20927

 − 0.46204

0.53667

\(1.4\)

2000

 − 0.64419

1.79625

0.52871

1.31568

1.69941

 − 0.64419

 − 0.23781

 − 0.14471

0.61506

4000

 − 0.64625

1.85899

0.54744

1.16329

2.30089

 − 0.64625

 − 0.28915

 − 0.33947

0.56636

6000

 − 0.59496

1.83977

0.59348

1.21077

2.72546

 − 0.59496

 − 0.31486

 − 0.43319

0.54126

8000

 − 0.64531

1.82839

0.61115

1.11019

3.08361

 − 0.64531

 − 0.37431

 − 0.48570

0.54655

10,000

 − 0.63820

1.80227

0.56535

1.26125

3.41559

 − 0.63820

 − 0.35001

 − 0.59261

0.53432

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

 

0.80000

1.40000

0.40000

1.90000

3.00000

 − 2.00000

0.65000

 − 1.20000

 
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