[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Next Issue
Volume 11, July
Previous Issue
Volume 11, May
You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.
 
 

Algorithms, Volume 11, Issue 6 (June 2018) – 11 articles

  • Issues are regarded as officially published after their release is announced to the table of contents alert mailing list.
  • You may sign up for e-mail alerts to receive table of contents of newly released issues.
  • PDF is the official format for papers published in both, html and pdf forms. To view the papers in pdf format, click on the "PDF Full-text" link, and use the free Adobe Reader to open them.
Order results
Result details
Section
Select all
Export citation of selected articles as:
12 pages, 2676 KiB  
Article
Efficient Deep Learning-Based Automated Pathology Identification in Retinal Optical Coherence Tomography Images
by Qingge Ji, Wenjie He, Jie Huang and Yankui Sun
Algorithms 2018, 11(6), 88; https://doi.org/10.3390/a11060088 - 20 Jun 2018
Cited by 24 | Viewed by 5615
Abstract
We present an automatic method based on transfer learning for the identification of dry age-related macular degeneration (AMD) and diabetic macular edema (DME) from retinal optical coherence tomography (OCT) images. The algorithm aims to improve the classification performance of retinal OCT images and [...] Read more.
We present an automatic method based on transfer learning for the identification of dry age-related macular degeneration (AMD) and diabetic macular edema (DME) from retinal optical coherence tomography (OCT) images. The algorithm aims to improve the classification performance of retinal OCT images and shorten the training time. Firstly, we remove the last several layers from the pre-trained Inception V3 model and regard the remaining part as a fixed feature extractor. Then, the features are used as input of a convolutional neural network (CNN) designed to learn the feature space shifts. The experimental results on two different retinal OCT images datasets demonstrate the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Machine Learning for Medical Image Analysis)
Show Figures

Figure 1

Figure 1
<p>Outline of the proposed algorithm.</p>
Full article ">Figure 2
<p>Retinal dataset examples. (<b>a</b>) Original B-scans. (<b>b</b>) Corresponding preprocessed B-scans.</p>
Full article ">Figure 2 Cont.
<p>Retinal dataset examples. (<b>a</b>) Original B-scans. (<b>b</b>) Corresponding preprocessed B-scans.</p>
Full article ">Figure 3
<p>Inception V3 with filter dimensions and illustration of the inception layer.</p>
Full article ">Figure 4
<p>The classification sensitivity of different methods.</p>
Full article ">Figure 5
<p>The mean accuracy of different architectures.</p>
Full article ">
4 pages, 157 KiB  
Editorial
Special Issue on Algorithms for Scheduling Problems
by Frank Werner, Larysa Burtseva and Yuri N. Sotskov
Algorithms 2018, 11(6), 87; https://doi.org/10.3390/a11060087 - 20 Jun 2018
Cited by 4 | Viewed by 4623
Abstract
This special issue of Algorithms is devoted to the development of scheduling algorithms based on innovative approaches for solving hard scheduling problems either exactly or approximately. Submissions were welcome both for traditional scheduling problems as well as for new practical applications. The main [...] Read more.
This special issue of Algorithms is devoted to the development of scheduling algorithms based on innovative approaches for solving hard scheduling problems either exactly or approximately. Submissions were welcome both for traditional scheduling problems as well as for new practical applications. The main topics include sequencing and scheduling with additional constraints (setup times or costs, precedence constraints, resource constraints, and batch production environment) and production planning and scheduling problems arising in real-world applications. Full article
(This article belongs to the Special Issue Algorithms for Scheduling Problems)
23 pages, 2958 KiB  
Article
Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models
by Christer Dalen and David Di Ruscio
Algorithms 2018, 11(6), 86; https://doi.org/10.3390/a11060086 - 17 Jun 2018
Cited by 13 | Viewed by 5225
Abstract
A method for tuning PI controller parameters, a prescribed maximum time delay error or a relative time delay error is presented. The method is based on integrator plus time delay models. The integral time constant is linear in the relative time delay error, [...] Read more.
A method for tuning PI controller parameters, a prescribed maximum time delay error or a relative time delay error is presented. The method is based on integrator plus time delay models. The integral time constant is linear in the relative time delay error, and the proportional constant is seen inversely proportional to the relative time delay error. The keystone in the method is the method product parameter, i.e., the product of the PI controller proportional constant, the integral time constant, and the integrator plus time delay model, velocity gain. The method product parameter is found to be constant for various PI controller tuning methods. Optimal suggestions are given for choosing the method product parameter, i.e., optimal such that the integrated absolute error or, more interestingly, the Pareto performance objective (i.e., integrated absolute error for combined step changes in output and input disturbances) is minimised. Variants of the presented tuning method are demonstrated for tuning PI controllers for motivated (possible) higher order process model examples, i.e., the presented method is combined with the model reduction step (process–reaction curve) in Ziegler–Nichols. Full article
(This article belongs to the Special Issue Algorithms for PID Controller)
Show Figures

Figure 1

Figure 1
<p>Consider PI control of the FOPTD process model, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mi>s</mi> </mfrac> </mrow> </semantics> </math>. The figure shows the robustness <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math> as a function of the MP parameter <math display="inline"> <semantics> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> </semantics> </math>, given constant robustness, for the interval <math display="inline"> <semantics> <mrow> <mn>1.5</mn> <mo>≤</mo> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>≤</mo> <mn>4.0</mn> </mrow> </semantics> </math>.</p>
Full article ">Figure 2
<p>Consider a control feedback system where the plant model is described by the process model, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math>, PI controller, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>K</mi> <mi>p</mi> </msub> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mi>s</mi> </mrow> <mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> <mi>s</mi> </mrow> </mfrac> </mrow> </semantics> </math>, and the disturbance model, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math>, where disturbance <span class="html-italic">v</span> at the input when, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math>, and at the output when, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>. Input <span class="html-italic">u</span>, output <span class="html-italic">y</span> and reference <span class="html-italic">r</span>.</p>
Full article ">Figure 3
<p>Reference example (Example 2). Consider PI control of the IPTD model, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mi>s</mi> </mfrac> </mrow> </semantics> </math>. The figure illustrates the trade-off between the Pareto performance objective <span class="html-italic">J</span> (Equation (<a href="#FD41-algorithms-11-00086" class="html-disp-formula">41</a>)) and robustness <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math> (Equation (<a href="#FD10-algorithms-11-00086" class="html-disp-formula">10</a>)). It illustrates the MP parameters <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.7</mn> </mrow> </semantics> </math> for Algorithm 1 proposed in <a href="#sec4-algorithms-11-00086" class="html-sec">Section 4</a>. SIMC is added for comparison.</p>
Full article ">Figure 4
<p>Consider PI control of an IPTD process, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> <mfrac> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>τ</mi> <mi>s</mi> </mrow> </msup> <mi>s</mi> </mfrac> </mrow> </semantics> </math> with process parameters <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>. PI controller <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>K</mi> <mi>p</mi> </msub> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mi>s</mi> </mrow> <mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> <mi>s</mi> </mrow> </mfrac> </mrow> </semantics> </math> with settings as in Algorithm 1. The figure shows the indices <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math>, <math display="inline"> <semantics> <msub> <mrow> <mi>ITAE</mi> </mrow> <mrow> <mi>v</mi> <mi>u</mi> </mrow> </msub> </semantics> </math>, <math display="inline"> <semantics> <msub> <mrow> <mi>IAE</mi> </mrow> <mi>r</mi> </msub> </semantics> </math>, <math display="inline"> <semantics> <msub> <mrow> <mi>ITAE</mi> </mrow> <mi>r</mi> </msub> </semantics> </math>, <math display="inline"> <semantics> <msub> <mrow> <mi>IAE</mi> </mrow> <mi>r</mi> </msub> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>TV</mi> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>ISE</mi> </mrow> </semantics> </math>, <math display="inline"> <semantics> <msub> <mrow> <mi>ITAE</mi> </mrow> <mrow> <mi>v</mi> <mi>u</mi> </mrow> </msub> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>IAE</mi> </mrow> </semantics> </math> as a function of varying the MP parameter <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>∈</mo> <mrow> <mo>[</mo> <mn>1.5</mn> <mo>,</mo> <mn>4.0</mn> <mo>]</mo> </mrow> </mrow> </semantics> </math> and with prescribed RTDE <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1.6</mn> </mrow> </semantics> </math>.</p>
Full article ">Figure 5
<p>Example 2 (Reference example). Consider PI control of an IPTD process model, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mi>s</mi> </mfrac> </mrow> </semantics> </math>. The figure illustrates the time-domain responses, given a prescribed robustness <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1.59</mn> </mrow> </semantics> </math>, of the following methods: the PO PI, SIMC with prescribed closed loop time constant <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.24</mn> <mspace width="0.166667em"/> <mi>τ</mi> </mrow> </semantics> </math> and Algorithm 1 where the MP parameter <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math> (proposed in <a href="#sec4-algorithms-11-00086" class="html-sec">Section 4</a>) and RTDE <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1.79</mn> </mrow> </semantics> </math>. An output disturbance unit step is presented at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> and an input disturbance unit step at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics> </math>.</p>
Full article ">Figure 6
<p>Example 3. Consider PI control of the FOPTD process model, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>K</mi> <mfrac> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>τ</mi> <mi>s</mi> </mrow> </msup> <mrow> <mi>T</mi> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </semantics> </math>, where <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>5.7</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics> </math>. The figure shows the trade-off curves with the Pareto performance objective J (Equation (<a href="#FD41-algorithms-11-00086" class="html-disp-formula">41</a>)) and robustness <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math> (Equation (<a href="#FD10-algorithms-11-00086" class="html-disp-formula">10</a>)). It illustrates the MP parameters <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.7</mn> </mrow> </semantics> </math> for Algorithm 1 (proposed in <a href="#sec4-algorithms-11-00086" class="html-sec">Section 4</a>). SIMC with set-point time constant <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics> </math> is added for comparison.</p>
Full article ">Figure 7
<p>Example 3. Consider PI control of the FOPTD process model, <math display="inline"> <semantics> <mrow> <msub> <mi>H</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>K</mi> <mfrac> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>τ</mi> <mi>s</mi> </mrow> </msup> <mrow> <mi>T</mi> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </semantics> </math>, where <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>5.7</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics> </math>. The figure illustrates the time-domain responses, given a prescribed robustness <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1.59</mn> </mrow> </semantics> </math>, of the following methods: the PO PI, SIMC with prescribed closed loop time constant <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.10</mn> <mspace width="0.166667em"/> <mi>τ</mi> </mrow> </semantics> </math>, and Algorithm 1 where the MP parameter <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math> (proposed in <a href="#sec4-algorithms-11-00086" class="html-sec">Section 4</a>) and RTDE <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1.56</mn> </mrow> </semantics> </math>. An output disturbance unit step is presented at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> and an input disturbance unit step at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>140</mn> </mrow> </semantics> </math>.</p>
Full article ">Figure 8
<p>Example 4. Consider PI control of the higher order process model (Equation (<a href="#FD47-algorithms-11-00086" class="html-disp-formula">47</a>)). The figure illustrates the trade-off curves with the Pareto performance objective J (Equation (<a href="#FD41-algorithms-11-00086" class="html-disp-formula">41</a>)) and robustness <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math> (Equation (<a href="#FD10-algorithms-11-00086" class="html-disp-formula">10</a>)). It shows the MP parameter settings <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.7</mn> </mrow> </semantics> </math> for Algorithm 1 proposed in <a href="#sec4-algorithms-11-00086" class="html-sec">Section 4</a>. SIMC is added for comparison.</p>
Full article ">Figure 9
<p>Example 4. Consider PI control of the higher order process model (Equation (<a href="#FD47-algorithms-11-00086" class="html-disp-formula">47</a>)). The figure illustrates the time-domain responses, given a prescribed robustness <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1.59</mn> </mrow> </semantics> </math>, of the following methods: the PO PI controller vs. SIMC with closed loop time constant <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>1.33</mn> <mspace width="0.166667em"/> <mi>τ</mi> </mrow> </semantics> </math>, and PRC + Algorithm 1 where the MP parameter setting <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.7</mn> </mrow> </semantics> </math> (proposed in <a href="#sec4-algorithms-11-00086" class="html-sec">Section 4</a>) and RTDE <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1.63</mn> </mrow> </semantics> </math>. An output disturbance unit step is presented at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> and an input disturbance unit step at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>35</mn> </mrow> </semantics> </math>.</p>
Full article ">Figure 10
<p>Example 5. Consider PI control of the higher order underdamped process model (Equation (<a href="#FD48-algorithms-11-00086" class="html-disp-formula">48</a>)). The figure shows the trade-off curves with the Pareto performance objective J (Equation (<a href="#FD41-algorithms-11-00086" class="html-disp-formula">41</a>)) and robustness <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math> (Equation (<a href="#FD10-algorithms-11-00086" class="html-disp-formula">10</a>)). It illustrates the PO PI controllers and PRC + Algorithm 1 variants with MP parameter settings <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>4.0</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.7</mn> </mrow> </semantics> </math>.</p>
Full article ">Figure 11
<p>Example 5. Consider PI control of the higher order underdamped process model (Equation (<a href="#FD48-algorithms-11-00086" class="html-disp-formula">48</a>)). The figure illustrates the time-domain responses, given a prescribed robustness <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1.59</mn> </mrow> </semantics> </math>, of following methods: the PO PI and the PRC + Algorithm 1 where the MP parameter setting <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics> </math> (proposed in <a href="#sec4-algorithms-11-00086" class="html-sec">Section 4</a>) and RTDE <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>2.20</mn> </mrow> </semantics> </math>. An output disturbance unit step is presented at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> and an input disturbance unit step at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>80</mn> </mrow> </semantics> </math>.</p>
Full article ">Figure 12
<p>Example 6. PI control of the higher order underdamped process model (Equation (<a href="#FD48-algorithms-11-00086" class="html-disp-formula">48</a>)). The figure illustrates the trade-off curves with the Pareto performance objective J (Equation (<a href="#FD41-algorithms-11-00086" class="html-disp-formula">41</a>)) and robustness <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math> (Equation (<a href="#FD10-algorithms-11-00086" class="html-disp-formula">10</a>)). It shows the PO PI controllers with robustness <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>s</mi> </msub> </semantics> </math> and the ζ-PRC + Algorithm 1 variant where the RTDE <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>2.7</mn> </mrow> </semantics> </math> is fixed and the main tuning parameter is ζ.</p>
Full article ">Figure 13
<p>Example 6. PI control of the higher order underdamped process model (Equation (<a href="#FD48-algorithms-11-00086" class="html-disp-formula">48</a>)). The figure illustrates the time-domain responses, given a prescribed robustness <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1.59</mn> </mrow> </semantics> </math>, for the following methods: the PO PI and the ζ-PRC + Algorithm 1 variant with MP parameter and MTDE settings <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mo>=</mo> <mi>δ</mi> <mo>=</mo> <mn>2.7</mn> </mrow> </semantics> </math>, and tuning parameter <math display="inline"> <semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>0.74</mn> </mrow> </semantics> </math>. An output disturbance unit step is presented at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> and an input disturbance unit step at time <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>80</mn> </mrow> </semantics> </math>.</p>
Full article ">
20 pages, 2157 KiB  
Article
ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems
by Yong-Hong Lan and Zhe-Min Cui
Algorithms 2018, 11(6), 85; https://doi.org/10.3390/a11060085 - 14 Jun 2018
Cited by 2 | Viewed by 4627
Abstract
This paper presents a second order P-type iterative learning control (ILC) scheme with initial state learning for a class of fractional order linear distributed parameter systems. First, by analyzing the control and learning processes, a discrete system for P-type ILC is established, and [...] Read more.
This paper presents a second order P-type iterative learning control (ILC) scheme with initial state learning for a class of fractional order linear distributed parameter systems. First, by analyzing the control and learning processes, a discrete system for P-type ILC is established, and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by introducing a new norm and using the generalized Gronwall inequality, which is less conservative than the existing one. Finally, the validity of the proposed method is verified by a numerical example. Full article
Show Figures

Figure 1

Figure 1
<p>The desired output <math display="inline"><semantics> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> </semantics></math>.</p>
Full article ">Figure 2
<p>The desired output <math display="inline"><semantics> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> </semantics></math>.</p>
Full article ">Figure 3
<p>The ninth iterative output <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>=</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>The ninth iterative output <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>=</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>The error surface <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>=</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>The error surface <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>=</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>The iterative number-max tracking error.</p>
Full article ">Figure 8
<p>The iterative number-max tracking error with different <math display="inline"><semantics> <mi>α</mi> </semantics></math>.</p>
Full article ">
19 pages, 333 KiB  
Article
Efficient Approximation for Restricted Biclique Cover Problems
by Alessandro Epasto and Eli Upfal
Algorithms 2018, 11(6), 84; https://doi.org/10.3390/a11060084 - 12 Jun 2018
Cited by 3 | Viewed by 5157
Abstract
Covering the edges of a bipartite graph by a minimum set of bipartite complete graphs (bicliques) is a basic graph theoretic problem, with numerous applications. In particular, it is used to characterize parsimonious models of a set of observations (each biclique corresponds to [...] Read more.
Covering the edges of a bipartite graph by a minimum set of bipartite complete graphs (bicliques) is a basic graph theoretic problem, with numerous applications. In particular, it is used to characterize parsimonious models of a set of observations (each biclique corresponds to a factor or feature that relates the observations in the two sets of nodes connected by the biclique). The decision version of the minimum biclique cover problem is NP-Complete, and unless P=NP, the cover size cannot be approximated in general within less than a sub-linear factor of the number of nodes (or edges) in the graph. In this work, we consider two natural restrictions to the problem, motivated by practical applications. In the first case, we restrict the number of bicliques a node can belong to. We show that when this number is at least 5, the problem is still NP-hard. In contrast, we show that when nodes belong to no more than two bicliques, the problem has efficient approximations. The second model we consider corresponds to observing a set of independent samples from an unknown model, governed by a possibly large number of factors. The model is defined by a bipartite graph G=(L,R,E), where each node in L is assigned to an arbitrary subset of up to a constant f factors, while the nodes in R (the independent observations) are assigned to random subsets of the set of k factors where k can grow with size of the graph. We show that this practical version of the biclique cover problem is amenable to efficient approximations. Full article
(This article belongs to the Special Issue Algorithms for Hard Problems: Approximation and Parameterization)
12 pages, 2733 KiB  
Article
A Combined Syntactical and Statistical Approach for R Peak Detection in Real-Time Long-Term Heart Rate Variability Analysis
by David Pang and Tomohiko Igasaki
Algorithms 2018, 11(6), 83; https://doi.org/10.3390/a11060083 - 7 Jun 2018
Cited by 2 | Viewed by 5003
Abstract
Long-term heart rate variability (HRV) analysis is useful as a noninvasive technique for autonomic nervous system activity assessment. It provides a method for assessing many physiological and pathological factors that modulate the normal heartbeat. The performance of HRV analysis systems heavily depends on [...] Read more.
Long-term heart rate variability (HRV) analysis is useful as a noninvasive technique for autonomic nervous system activity assessment. It provides a method for assessing many physiological and pathological factors that modulate the normal heartbeat. The performance of HRV analysis systems heavily depends on a reliable and accurate detection of the R peak of the QRS complex. Ectopic beats caused by misdetection or arrhythmic events can introduce bias into HRV results, resulting in significant problems in their interpretation. This study presents a novel method for long-term detection of normal R peaks (which represent the normal heartbeat in electrocardiographic signals), intended specifically for HRV analysis. The very low computational complexity of the proposed method, which combines and exploits the advantages of syntactical and statistical approaches, enables real-time applications. The approach was validated using the Massachusetts Institute of Technology–Beth Israel Hospital Normal Sinus Rhythm and the Fantasia database, and has a sensitivity, positive predictivity, detection error rate, and accuracy of 99.998, 99.999, 0.003, and 99.996%, respectively. Full article
Show Figures

Figure 1

Figure 1
<p>A typical electrocardiography (ECG) waveform and RR interval.</p>
Full article ">Figure 2
<p>Block diagram of ECG signal processing.</p>
Full article ">Figure 3
<p>Vectorization process.</p>
Full article ">Figure 4
<p>Final result of the vectorization and noise-removal processes.</p>
Full article ">Figure 5
<p>(<b>a</b>) Relation of the ECG signal sequence to the automata states and other ECG parameters and (<b>b</b>) the automata’s state transition diagram.</p>
Full article ">Figure 6
<p>Occurrence of the signal noted as an “isolated QRS-like artefact” in record #16272 [<a href="#B12-algorithms-11-00083" class="html-bibr">12</a>,<a href="#B13-algorithms-11-00083" class="html-bibr">13</a>,<a href="#B15-algorithms-11-00083" class="html-bibr">15</a>].</p>
Full article ">Figure 7
<p>(a, b, and c) Typical waveforms, which sometimes occurred and were annotated as beats in the Fantasia database, but cannot be recognized by the automata as heartbeats [<a href="#B12-algorithms-11-00083" class="html-bibr">12</a>,<a href="#B13-algorithms-11-00083" class="html-bibr">13</a>,<a href="#B14-algorithms-11-00083" class="html-bibr">14</a>,<a href="#B15-algorithms-11-00083" class="html-bibr">15</a>].</p>
Full article ">Figure 8
<p>A single misdetection of the R peak results in wrong NN intervals, which can greatly modify the result of the frequency analysis of the HRV as compared to the actual analysis without misdetection.</p>
Full article ">
18 pages, 4389 KiB  
Article
Research on Fault Diagnosis of a Marine Fuel System Based on the SaDE-ELM Algorithm
by Yi Wei and Yaokun Yue
Algorithms 2018, 11(6), 82; https://doi.org/10.3390/a11060082 - 7 Jun 2018
Cited by 8 | Viewed by 4839
Abstract
Since the traditional fault diagnosis method of the marine fuel system has a low accuracy of identification, the algorithm solution can easily fall into local optimum, and they are not fit for the research on the fault diagnosis of a marine fuel system. [...] Read more.
Since the traditional fault diagnosis method of the marine fuel system has a low accuracy of identification, the algorithm solution can easily fall into local optimum, and they are not fit for the research on the fault diagnosis of a marine fuel system. Hence, a fault diagnosis method for a marine fuel system based on the SaDE-ELM algorithm is proposed. First, the parameters of initializing extreme learning machine are adopted by a differential evolution algorithm. Second, the fault diagnosis of the marine fuel system is realized by the fault diagnosis model corresponding to the state training of marine fuel system. Based on the obtained fault data of a marine fuel system, the proposed method is verified. The experimental results show that this method produces higher recognition accuracy and faster recognition speed that are superior to the traditional BP neural network, SVM support vector machine diagnosis algorithm, and the un-optimized extreme learning machine algorithm. The results have important significance relevant to fault diagnosis for a marine fuel system. The algorithm based on SaDE-ELM is an effective and practical method of fault diagnosis for a marine fuel system. Full article
Show Figures

Figure 1

Figure 1
<p>Research framework.</p>
Full article ">Figure 2
<p>Single hidden layer neural network framework.</p>
Full article ">Figure 3
<p>Structure of Extreme Learning Machine (ELM) algorithm.</p>
Full article ">Figure 4
<p>Flow chart of the Self-adaptive Different Evolution Optimized Extreme Learning Machine (SaDE-ELM) algorithm.</p>
Full article ">Figure 5
<p>Pseudocode of the proposed SaDE-ELM algorithm.</p>
Full article ">Figure 6
<p>Marine fuel oil system. 1—tank, 2—strainer, 3—circulating pump, 4—delivery valve, 5—fuel system filter, 6—high pressure pump, 7—high-pressure pipe, 8—injector, and 9—return pipe.</p>
Full article ">Figure 7
<p>Pressure waveform characteristic parameters.</p>
Full article ">Figure 8
<p>Pressure wave characteristics of training data.</p>
Full article ">Figure 9
<p>Pressure waveforms for the different states of the marine fuel system.</p>
Full article ">Figure 10
<p>Comparison of the convergence effect of the algorithm.</p>
Full article ">Figure 11
<p>BP neural network algorithm for marine fuel system fault diagnosis.</p>
Full article ">Figure 12
<p>Support vector machine algorithm for marine fuel system fault diagnosis.</p>
Full article ">Figure 13
<p>Extreme learning machine algorithm for marine fuel system fault diagnosis.</p>
Full article ">Figure 14
<p>SaDE-ELM algorithm for marine fuel system fault diagnosis.</p>
Full article ">
15 pages, 262 KiB  
Article
A Randomized Algorithm for Optimal PID Controllers
by Yossi Peretz
Algorithms 2018, 11(6), 81; https://doi.org/10.3390/a11060081 - 5 Jun 2018
Cited by 9 | Viewed by 4021
Abstract
A randomized algorithm is suggested for the syntheses of optimal PID controllers for MIMO coupled systems, where the optimality is with respect to the H -norm, the H 2 -norm and the LQR functional, with possible system-performance specifications defined by regional pole-placement. [...] Read more.
A randomized algorithm is suggested for the syntheses of optimal PID controllers for MIMO coupled systems, where the optimality is with respect to the H -norm, the H 2 -norm and the LQR functional, with possible system-performance specifications defined by regional pole-placement. Other notions of optimality (e.g., mixed H 2 / H design, controller norm or controller sparsity) can be handled similarly with the suggested algorithm. The suggested method is direct and thus can be applied to continuous-time systems as well as to discrete-time systems with the obvious minor changes. The presented algorithm is a randomized algorithm, which has a proof of convergence (in probability) to a global optimum. Full article
(This article belongs to the Special Issue Algorithms for PID Controller)
16 pages, 293 KiB  
Article
Scheduling a Single Machine with Primary and Secondary Objectives
by Nodari Vakhania
Algorithms 2018, 11(6), 80; https://doi.org/10.3390/a11060080 - 5 Jun 2018
Cited by 6 | Viewed by 4040
Abstract
We study a scheduling problem in which jobs with release times and due dates are to be processed on a single machine. With the primary objective to minimize the maximum job lateness, the problem is strongly NP-hard. We describe a general algorithmic scheme [...] Read more.
We study a scheduling problem in which jobs with release times and due dates are to be processed on a single machine. With the primary objective to minimize the maximum job lateness, the problem is strongly NP-hard. We describe a general algorithmic scheme to minimize the maximum job lateness, with the secondary objective to minimize the maximum job completion time. The problem of finding the Pareto-optimal set of feasible solutions with these two objective criteria is strongly NP-hard. We give the dominance properties and conditions when the Pareto-optimal set can be formed in polynomial time. These properties, together with our general framework, provide the theoretical background, so that the basic framework can be expanded to (exponential-time) implicit enumeration algorithms and polynomial-time approximation algorithms (generating the Pareto sub-optimal frontier with a fair balance between the two objectives). Some available in the literature experimental results confirm the practical efficiency of the proposed framework. Full article
(This article belongs to the Special Issue Algorithms for Scheduling Problems)
Show Figures

Figure 1

Figure 1
<p>The initial Earliest Due date (ED)-schedule <span class="html-italic">S</span> for the problem instance of Example 1.</p>
Full article ">Figure 2
<p>Complementary ED-schedule <math display="inline"><semantics> <msub> <mi>S</mi> <mn>1</mn> </msub> </semantics></math> for Example 1.</p>
Full article ">Figure 3
<p>A complementary ED-schedule <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>1</mn> </msub> </semantics></math>.</p>
Full article ">
19 pages, 1961 KiB  
Article
A Fire Detection Algorithm Based on Tchebichef Moment Invariants and PSO-SVM
by Yongming Bian, Meng Yang, Xuying Fan and Yuchao Liu
Algorithms 2018, 11(6), 79; https://doi.org/10.3390/a11060079 - 25 May 2018
Cited by 16 | Viewed by 5061
Abstract
Automatic fire detection, which can detect and raise the alarm for fire early, is expected to help reduce the loss of life and property as much as possible. Due to its advantages over traditional methods, image processing technology has been applied gradually in [...] Read more.
Automatic fire detection, which can detect and raise the alarm for fire early, is expected to help reduce the loss of life and property as much as possible. Due to its advantages over traditional methods, image processing technology has been applied gradually in fire detection. In this paper, a novel algorithm is proposed to achieve fire image detection, combined with Tchebichef (sometimes referred to as Chebyshev) moment invariants (TMIs) and particle swarm optimization-support vector machine (PSO-SVM). According to the correlation between geometric moments and Tchebichef moments, the translation, rotation, and scaling (TRS) invariants of Tchebichef moments are obtained first. Then, the TMIs of candidate images are calculated to construct feature vectors. To gain the best detection performance, a PSO-SVM model is proposed, where the kernel parameter and penalty factor of support vector machine (SVM) are optimized by particle swarm optimization (PSO). Then, the PSO-SVM model is utilized to identify the fire images. Compared with algorithms based on Hu moment invariants (HMIs) and Zernike moment invariants (ZMIs), the experimental results show that the proposed algorithm can improve the detection accuracy, achieving the highest detection rate of 98.18%. Moreover, it still exhibits the best performance even if the size of the training sample set is small and the images are transformed by TRS. Full article
Show Figures

Figure 1

Figure 1
<p>The schematic diagram of support vector machine (SVM) classification.</p>
Full article ">Figure 2
<p>The flow diagram of particle swarm optimization-support vector machine (PSO-SVM) model construction.</p>
Full article ">Figure 3
<p>The flow diagram of fire image detection based on the PSO-SVM model.</p>
Full article ">Figure 4
<p>Candidate images: (<b>a</b>–<b>c</b>) stable burning of fire; (<b>d</b>–<b>f</b>) initial extinguishing of fire; (<b>g</b>–<b>i</b>) final extinguishing of fire.</p>
Full article ">Figure 5
<p>The fitness curve of particle swarm optimization.</p>
Full article ">Figure 6
<p>Effect of training dataset size on recognition rate.</p>
Full article ">Figure 7
<p>Images transformed by translation, rotation, and scaling (TRS). (<b>a</b>) Original image; (<b>b</b>) image translated with [−70, −50]; (<b>c</b>) image translated with [50, 70]; (<b>d</b>) image rotated by <math display="inline"><semantics> <mrow> <msup> <mrow> <mn>15</mn> </mrow> <mo>∘</mo> </msup> </mrow> </semantics></math>; (<b>e</b>) image rotated by <math display="inline"><semantics> <mrow> <mo>−</mo> <msup> <mrow> <mn>15</mn> </mrow> <mo>∘</mo> </msup> </mrow> </semantics></math>; (<b>f</b>) image scaled with 0.8; (<b>g</b>) image scaled with 1.2.</p>
Full article ">
21 pages, 1142 KiB  
Article
A Modified Artificial Bee Colony Algorithm Based on the Self-Learning Mechanism
by Bao Pang, Yong Song, Chengjin Zhang, Hongling Wang and Runtao Yang
Algorithms 2018, 11(6), 78; https://doi.org/10.3390/a11060078 - 24 May 2018
Cited by 9 | Viewed by 4815
Abstract
Artificial bee colony (ABC) algorithm, a novel category of bionic intelligent optimization algorithm, was achieved for solving complex nonlinear optimization problems. Previous studies have shown that ABC algorithm is competitive to other biological-inspired optimization algorithms, but there still exist several insufficiencies due to [...] Read more.
Artificial bee colony (ABC) algorithm, a novel category of bionic intelligent optimization algorithm, was achieved for solving complex nonlinear optimization problems. Previous studies have shown that ABC algorithm is competitive to other biological-inspired optimization algorithms, but there still exist several insufficiencies due to the inefficient solution search equation (SSE), which does well in exploration but poorly in exploitation. To improve accuracy of the solutions, this paper proposes a modified ABC algorithm based on the self-learning mechanism (SLABC) with five SSEs as the candidate operator pool; among them, one is good at exploration and two of them are good at exploitation; another SSE intends to balance exploration and exploitation; moreover, the last SSE with Lévy flight step-size which can generate smaller step-size with high frequency and bigger step-size occasionally not only can balance exploration and exploitation but also possesses the ability to escape from the local optimum. This paper proposes a simple self-learning mechanism, wherein the SSE is selected according to the previous success ratio in generating promising solutions at each iteration. Experiments on a set of 9 benchmark functions are carried out with the purpose of evaluating the performance of the proposed method. The experimental results illustrated that the SLABC algorithm achieves significant improvement compared with other competitive algorithms. Full article
Show Figures

Figure 1

Figure 1
<p>An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at [0,0], the step-size is generated according to Equation (<a href="#FD5-algorithms-11-00078" class="html-disp-formula">5</a>) with <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> and the angular direction is uniformly distributed.</p>
Full article ">Figure 2
<p>Comparison of the convergence results of the average best solution with 30 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 3
<p>Comparison of the convergence results of the average best solution with 60 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 4
<p>Comparison of the convergence results of the average best solution with 100 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 4 Cont.
<p>Comparison of the convergence results of the average best solution with 100 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 5
<p>The success ratio <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>r</mi> <mi>a</mi> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </semantics></math> of each SSE (operator). The horizontal ordinate denotes success ratio of each solution search equation at the midpoint of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) and the end of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> </mrow> </semantics></math>) with 30 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 6
<p>The success ratio <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>r</mi> <mi>a</mi> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </semantics></math> of each SSE. The horizontal ordinate denotes success ratio of each solution search equation (operator) at the midpoint of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) and the end of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> </mrow> </semantics></math>) with 60 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 6 Cont.
<p>The success ratio <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>r</mi> <mi>a</mi> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </semantics></math> of each SSE. The horizontal ordinate denotes success ratio of each solution search equation (operator) at the midpoint of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) and the end of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> </mrow> </semantics></math>) with 60 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 7
<p>The success ratio <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>r</mi> <mi>a</mi> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </semantics></math> of each SSE. The horizontal ordinate denotes success ratio of each solution search equation (operator) at the midpoint of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) and the end of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> </mrow> </semantics></math>) with 100 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">Figure 7 Cont.
<p>The success ratio <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>r</mi> <mi>a</mi> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </semantics></math> of each SSE. The horizontal ordinate denotes success ratio of each solution search equation (operator) at the midpoint of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) and the end of the experiments (<math display="inline"><semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mi>E</mi> </mrow> </semantics></math>) with 100 dimensions. (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>1</mn> </msub> </semantics></math>) Sphere Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics></math>) Quartic Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>3</mn> </msub> </semantics></math>) Schwefel’s Problem 2.22; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>4</mn> </msub> </semantics></math>) Rosenbrock Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>5</mn> </msub> </semantics></math>) Rastrigin Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>6</mn> </msub> </semantics></math>) Griewank Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>7</mn> </msub> </semantics></math>) Ackley Function; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>8</mn> </msub> </semantics></math>) Schwefel’s Problem 2.26; (<math display="inline"><semantics> <msub> <mi>f</mi> <mn>9</mn> </msub> </semantics></math>) Penalized Function.</p>
Full article ">
Previous Issue
Next Issue
Back to TopTop