{"id":49062,"date":"2026-09-14T13:36:50","date_gmt":"2026-09-14T13:36:50","guid":{"rendered":"https:\/\/imperix.com\/doc\/?p=49062"},"modified":"2026-09-14T13:31:16","modified_gmt":"2026-09-14T13:31:16","slug":"parametric-system-identification-using-the-imperix-sys-id-toolbox","status":"publish","type":"post","link":"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox","title":{"rendered":"Parametric system identification using the imperix Sys. Id. toolbox"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 ez-toc-wrap-right-text counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Hardware-requirements\" >Hardware requirements<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Identification-procedure\" >Identification procedure<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Simulink-and-PLECS-models\" >Simulink and PLECS models<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Identification-in-Cockpit\" >Identification in Cockpit<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Parametrization-of-the-transfer-function\" >Parametrization of the transfer function<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Simulation-validation\" >Simulation validation<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Visualizing-simulation-data-in-PLECS\" >Visualizing simulation data in PLECS<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Conclusions\" >Conclusions<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#Further-reading\" >Further reading<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/imperix.com\/doc\/help\/parametric-system-identification-using-the-imperix-sys-id-toolbox\/#References\" >References<\/a><\/li><\/ul><\/nav><\/div>\n\n<p class=\"wp-block-paragraph\">This note illustrates a system identification example that derives a parametric model using the imperix System Identification toolbox. It presents a step-by-step procedure, starting by preparing the Simulink\/PLECS model, performing the frequency response measurement, and finally deriving the parameters. The example considers a <a href=\"https:\/\/imperix.com\/doc\/implementation\/step-down-buck-converter\">buck converter<\/a> with an inductor and a resistor connected at its output, and the goal is to identify their values.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"618\" height=\"436\" src=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/buck.png\" alt=\"Buck converter considered in the parametric system identification.\" class=\"wp-image-49215\" style=\"width:309px;height:auto\" srcset=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/buck.png 618w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/buck-300x212.png 300w\" sizes=\"auto, (max-width: 618px) 100vw, 618px\" \/><figcaption class=\"wp-element-caption\">The buck converter considered in this example.<\/figcaption><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Hardware-requirements\"><\/span>Hardware requirements<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The following list describes the elements used to build the buck converter:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A <a href=\"https:\/\/imperix.com\/products\/power\/sic-mosfet-module\/\">phase leg module<\/a> (PEB-800-40, PEB8038, PEB8024, or PEB4050);<\/li>\n\n\n\n<li>A <a href=\"https:\/\/imperix.com\/products\/control\/rapid-prototyping-controller\/\">programmable controller<\/a> (B-Box 4, B-Box 3, or B-Box 3 Micro);<\/li>\n\n\n\n<li><a href=\"https:\/\/imperix.com\/software\/acg-sdk\/\">Control development tools for Simulink\/PLECS<\/a> (ACG SDK), with a valid license;<\/li>\n\n\n\n<li>A DC power supply;<\/li>\n\n\n\n<li>An inductor and a resistor;<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Identification-procedure\"><\/span>Identification procedure<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The procedure to identify the system parameters divides into two main steps:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Identify the frequency response from the duty cycle to the inductor current;<\/li>\n\n\n\n<li>Derive its analytical expression and parameterize it using the identified response.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Since the system is known, it&#8217;s possible to make assumptions about the roles of the various components using the laws of physics. In this case, the RL series dynamics will dominate the frequency response mentioned in step 1. After finding an appropriate analytical expression, parameterizing it yields the inductance and resistance values by finding the numerical values that fit the measured frequency response.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The imperix System Identification toolbox implements the <em>Empirical Transfer Function Estimation<\/em> (ETFE) technique to estimate the frequency response between two signals. This requires deciding where to perturb the system and which two signals to record to compute the frequency response from one to the other. For more details regarding the ETFE implementation, see the <a href=\"https:\/\/imperix.com\/doc\/software\/system-identification-injector\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/software\/system-identification-injector\">Sys. Id. Injector<\/a> block documentation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As the frequency response of interest is from the duty cycle to the inductor current, it is reasonable to inject the perturbation on the duty cycle and to measure the perturbed duty cycle signal <code>d<\/code> and inductor current <code>i_L<\/code>. The following schematic depicts the identification setup, where the grey area shows the identification target.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"618\" height=\"516\" src=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/buck_prbs.png\" alt=\"Schematic showing the injection point used in the parametric identification procedure.\" class=\"wp-image-49217\" style=\"width:309px;height:auto\" srcset=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/buck_prbs.png 618w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/buck_prbs-300x250.png 300w\" sizes=\"auto, (max-width: 618px) 100vw, 618px\" \/><figcaption class=\"wp-element-caption\">Identification setup for the <em>duty-cycle to inductor current<\/em> transfer function.<\/figcaption><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Simulink-and-PLECS-models\"><\/span>Simulink and PLECS models<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The following Simulink and PLECS models implement the setup shown above. The perturbation signal is injected using the <a href=\"https:\/\/imperix.com\/doc\/software\/system-identification-injector\">Sys. Id. injector<\/a> block, connected between the duty-cycle reference and the PWM modulator input, matching the configuration above. The <a href=\"https:\/\/imperix.com\/doc\/software\/probe-variable\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/software\/probe-variable\">probe<\/a> blocks are used to acquire in <a href=\"https:\/\/imperix.com\/doc\/help\/cockpit-user-guide\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/help\/cockpit-user-guide\">Cockpit<\/a> the duty cycle <code>d<\/code> and inductor current <code>i_L<\/code>.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1442\" height=\"577\" src=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/simulink-1.png\" alt=\"Simulink model used to carry out the parametric identifcation.\" class=\"wp-image-49481\" srcset=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/simulink-1.png 1442w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/simulink-1-300x120.png 300w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/simulink-1-767x307.png 767w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/simulink-1-1024x410.png 1024w\" sizes=\"auto, (max-width: 1442px) 100vw, 1442px\" \/><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"999\" height=\"351\" src=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs-1.png\" alt=\"PLECS model used to carry out the parametric identification.\" class=\"wp-image-49673\" srcset=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs-1.png 999w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs-1-300x105.png 300w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs-1-766x269.png 766w\" sizes=\"auto, (max-width: 999px) 100vw, 999px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-a6f16ceb-4783-45a2-80cb-e74bdc2153b8\" href=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/PN215_param_sysId_Simulink.zip\">PN215_param_sysId_Simulink<\/a><a href=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/PN215_param_sysId_Simulink.zip\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-a6f16ceb-4783-45a2-80cb-e74bdc2153b8\">Download<\/a><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-file\"><a id=\"wp-block-file--media-ba2e039c-f962-44e1-9754-cac7a0ef2729\" href=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/PN215_param_sysId_PLECS.zip\">PN215_param_sysId_PLECS<\/a><a href=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/PN215_param_sysId_PLECS.zip\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-ba2e039c-f962-44e1-9754-cac7a0ef2729\">Download<\/a><\/div>\n<\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Identification-in-Cockpit\"><\/span>Identification in Cockpit<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<div class=\"wp-block-simple-alerts-for-gutenberg-alert-boxes sab-alert sab-alert-danger\" role=\"alert\">This note assumes inductance and resistance are unknown for didactic purposes; however, their nominal ratings are generally known. The voltage and duty cycle must be adjusted to <a href=\"https:\/\/imperix.com\/doc\/implementation\/safety-and-protection-in-the-lab\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/implementation\/safety-and-protection-in-the-lab\">safely perform<\/a> the procedure without damaging your specific equipment.<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">To build and deploy the models to an imperix controller by pressing <code>Ctrl + B<\/code> in Simulink or <code>Ctrl + Alt + B<\/code> in PLECS. More information is available in <a href=\"https:\/\/imperix.com\/doc\/help\/programming-imperix-controllers\">PN138<\/a>. After building the model,&nbsp;Cockpit&nbsp;will open automatically. It will also automatically detect the <a href=\"https:\/\/imperix.com\/doc\/help\/working-with-cockpit-projects\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/help\/working-with-cockpit-projects\">Cockpit project<\/a> file (<code>.ixproj<\/code>) provided with the models. This pre-configures the project workspace by adding a <a href=\"https:\/\/imperix.com\/doc\/help\/scope-module\">Scope module<\/a> and a <a href=\"https:\/\/imperix.com\/doc\/help\/system-identification-module\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/help\/system-identification-module\">System Identification module<\/a>, and adds the <code>i_L<\/code> and <code>d<\/code> variables to the Scope, which are needed by the Sys. Id. module.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Identification procedure<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Once Cockpit is open, the following steps detail how to perform the identification process:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Turn on the DC voltage source and make sure the variable <code>V_dc<\/code> settles around the expected value. In this case, it&#8217;s set to 200 V. If not, check the sensor&#8217;s sensitivity and whether it is connected to the correct channel. The <code>PWM enable<\/code> switch should be off, so <code>i_L<\/code> should be zero.<\/li>\n\n\n\n<li>The tunable reference signals should bring the system to the desired operating point. In this case, as the duty cycle ranges in [0,1], <code>d_ref<\/code> = 0.5 provides equal headroom in both directions for the PRBS signal.<\/li>\n\n\n\n<li> The options on the right panel (under the <em>Sys. Id.<\/em> section) configure the injection parameters:\n<ul class=\"wp-block-list\">\n<li>Since the system has no control loop, setting&nbsp;Injection mode&nbsp;to open-loop or closed-loop makes no difference.<\/li>\n\n\n\n<li>The perturbation amplitude should yield a good&nbsp;signal-to-noise&nbsp;ratio without introducing harmonics that would reduce estimation accuracy&nbsp;(i.e., avoid <em>non-linear operation<\/em>). In this case, 0.1 is a good starting point.<\/li>\n\n\n\n<li>The advanced parameters are left at the default value in this case. The <a href=\"https:\/\/docs.google.com\/spreadsheets\/d\/1VkCzmW4CWLQpLCXn4ZP4M8psfZOmAocB3WJWr-bkP4Q\" data-type=\"link\" data-id=\"https:\/\/docs.google.com\/spreadsheets\/d\/1VkCzmW4CWLQpLCXn4ZP4M8psfZOmAocB3WJWr-bkP4Q\">PN306<\/a> provides more information on these.<\/li>\n\n\n\n<li>The numerator and denominator signals should be chosen according to the frequency response of interest; in this case, from the duty cycle to the inductor current. Then, <code>d<\/code> is the denominator and <code>i_L<\/code> is the numerator.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Enable the PWM and ensure the resulting current is constant.<\/li>\n\n\n\n<li>Press the start injection button, wait for the injection to end, and the Bode plot of the identified transfer function will appear. The picture below illustrates the obtained result.<\/li>\n\n\n\n<li>The waveform can be exported for later processing with the right-click menu (<code>.csv<\/code> or <code>.mat<\/code>). Alternatively, <a href=\"https:\/\/imperix.com\/doc\/help\/cockpit-user-guide?currentThread=b-box-4#Snapshots\">snapshots<\/a> can help in comparing multiple acquisitions.<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1587\" height=\"873\" src=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/cockpit_cursors-1.png\" alt=\"Bode plot of the resulting frequency response.\" class=\"wp-image-49760\" style=\"width:780px;height:auto\" srcset=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/cockpit_cursors-1.png 1587w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/cockpit_cursors-1-300x165.png 300w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/cockpit_cursors-1-767x422.png 767w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/cockpit_cursors-1-1024x563.png 1024w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/cockpit_cursors-1-1536x845.png 1536w\" sizes=\"auto, (max-width: 1587px) 100vw, 1587px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Parametrization-of-the-transfer-function\"><\/span>Parametrization of the transfer function<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The previous section yielded the frequency response from the duty cycle to the inductor current, and it is possible to derive an analytical expression for it. For the buck converter under consideration, it can be a first-order system [1]:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>G<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msub><mi>i<\/mi><mi>L<\/mi><\/msub><mi>d<\/mi><\/mfrac><\/mstyle><mo>\u2248<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msub><mi>V<\/mi><mrow><mtext><\/mtext><mi>dc<\/mi><\/mrow><\/msub><mrow><mi>R<\/mi><mo>+<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mi>L<\/mi><\/mrow><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">G(j\\omega) = \\dfrac{i_L}{d} \\approx \\dfrac{V_\\mathrm{dc}}{R + j\\omega L}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The above transfer function neglects the delay due to PWM and sampling, as it is not needed to derive the inductance and resistance values. The expression can be parametrized using the low-frequency and high-frequency regions of the identified frequency response. For small and large values of <math data-latex=\"\\omega\"><semantics><mi>\u03c9<\/mi><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math>:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mtable displaystyle=\"true\" columnalign=\"left right left right\" class=\"tml-jot\" style=\"width:100%;\"><mtr><mtd class=\"tml-right\" style=\"padding:0;width:50%;padding-left:0em;padding-right:0em;\"><\/mtd><mtd class=\"tml-right\" style=\"padding-left:1em;padding-right:0em;\"><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">\u2192<\/mo><mn>0<\/mn><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><mrow><mi>|<\/mi><mi>G<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>|<\/mi><mo>\u2248<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msub><mi>V<\/mi><mrow><mtext><\/mtext><mi>dc<\/mi><\/mrow><\/msub><mi>R<\/mi><\/mfrac><\/mstyle><mspace width=\"0.2778em\"><\/mspace><mo stretchy=\"false\">\u27f9<\/mo><mspace width=\"0.2778em\"><\/mspace><mi>R<\/mi><mo>\u2248<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msub><mi>V<\/mi><mrow><mtext><\/mtext><mi>dc<\/mi><\/mrow><\/msub><mrow><mi>|<\/mi><mi>G<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>|<\/mi><\/mrow><\/mfrac><\/mstyle><\/mrow><\/mtd><mtd class=\"tml-right\" style=\"padding:0;width:50%;padding-left:1em;padding-right:0em;\"><mtext><span class=\"tml-eqn\"><\/span><\/mtext><\/mtd><\/mtr><mtr><mtd class=\"tml-right\" style=\"padding:0;width:50%;padding-left:0em;padding-right:0em;\"><\/mtd><mtd class=\"tml-right\" style=\"padding-left:1em;padding-right:0em;\"><mrow><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><\/mtd><mtd class=\"tml-right\" style=\"padding:0;width:50%;padding-left:1em;padding-right:0em;\"><mtext><span class=\"tml-eqn\"><\/span><\/mtext><\/mtd><\/mtr><mtr><mtd class=\"tml-right\" style=\"padding:0;width:50%;padding-left:0em;padding-right:0em;\"><\/mtd><mtd class=\"tml-right\" style=\"padding-left:1em;padding-right:0em;\"><mrow><mi>\u03c9<\/mi><mo stretchy=\"false\">\u2192<\/mo><mo form=\"prefix\" stretchy=\"false\">+<\/mo><mi>\u221e<\/mi><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mspace width=\"1em\"><\/mspace><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><mrow><mi>|<\/mi><mi>G<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>|<\/mi><mo>\u2248<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msub><mi>V<\/mi><mrow><mtext><\/mtext><mi>dc<\/mi><\/mrow><\/msub><mrow><mi>\u03c9<\/mi><mi>L<\/mi><\/mrow><\/mfrac><\/mstyle><mspace width=\"0.2778em\"><\/mspace><mo stretchy=\"false\">\u27f9<\/mo><mspace width=\"0.2778em\"><\/mspace><mi>L<\/mi><mo>\u2248<\/mo><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mfrac><msub><mi>V<\/mi><mrow><mtext><\/mtext><mi>dc<\/mi><\/mrow><\/msub><mrow><mi>\u03c9<\/mi><mi>|<\/mi><mi>G<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>|<\/mi><\/mrow><\/mfrac><\/mstyle><\/mrow><\/mtd><mtd class=\"tml-right\" style=\"padding:0;width:50%;padding-left:1em;padding-right:0em;\"><mtext><span class=\"tml-eqn\"><\/span><\/mtext><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{align}\n\\omega\\rightarrow0:&amp;\n|G(j\\omega)|\\approx \\dfrac{V_\\mathrm{dc}}{R}\\;\\Longrightarrow\\; R\\approx \\dfrac{V_\\mathrm{dc}}{|G(j\\omega)|}\\\\\\\\\n\n\\omega\\rightarrow+\\infty:\\quad&amp;\n|G(j\\omega)| \\approx \\dfrac{V_\\mathrm{dc}}{\\omega L}\\;\\Longrightarrow\\;\nL \\approx \\dfrac{V_\\mathrm{dc}}{\\omega|G(j\\omega)|}\n\\end{align}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n\n\n<p class=\"wp-block-paragraph\">To compute R, the appropriate frequency value is the lowest one available. Choosing <math data-latex=\"\\omega\"><semantics><mi>\u03c9<\/mi><annotation encoding=\"application\/x-tex\">\\omega<\/annotation><\/semantics><\/math> to derive <math data-latex=\"L\"><semantics><mi>L<\/mi><annotation encoding=\"application\/x-tex\">L<\/annotation><\/semantics><\/math> is a more delicate task: due to unmodeled parasitic elements, the above approximation holds only up to a certain frequency, that is, where the slope of the magnitude Bode plot is \u221220 dB\/dec. For this system, 3 kHz falls almost in the middle of that region. The user can easily read the required values using the cursors, as described in the <a href=\"https:\/\/imperix.com\/doc\/help\/system-identification-module\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/help\/system-identification-module\">Sys. Id. module<\/a> documentation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><math data-latex=\"\\omega\\approx2\\pi\\cdot12.2\\,\\mathrm{Hz}\"><semantics><mrow><mi>\u03c9<\/mi><mo>\u2248<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo>\u22c5<\/mo><mn>12.2<\/mn><mspace width=\"0.1667em\"><\/mspace><mrow><mtext><\/mtext><mi>Hz<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\omega\\approx2\\pi\\cdot12.2\\,\\mathrm{Hz}<\/annotation><\/semantics><\/math> yields <math data-latex=\"|G(j\\omega)| \\approx 24.49\\,\\mathrm{dB}\\approx16.77\"><semantics><mrow><mi>|<\/mi><mi>G<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>|<\/mi><mo>\u2248<\/mo><mn>24.49<\/mn><mspace width=\"0.1667em\"><\/mspace><mrow><mtext><\/mtext><mi>dB<\/mi><\/mrow><mo>\u2248<\/mo><mn>16.77<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">|G(j\\omega)| \\approx 24.49\\,\\mathrm{dB}\\approx16.77<\/annotation><\/semantics><\/math>, which yields <math data-latex=\"R\\approx11.9\\,\\Omega\"><semantics><mrow><mi>R<\/mi><mo>\u2248<\/mo><mn>11.9<\/mn><mspace width=\"0.1667em\"><\/mspace><mrow><mi mathvariant=\"normal\">\u03a9<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">R\\approx11.9\\,\\Omega<\/annotation><\/semantics><\/math>;<\/li>\n\n\n\n<li><math data-latex=\"\\omega\\approx2\\pi\\cdot3\\,\\mathrm{kHz}\"><semantics><mrow><mi>\u03c9<\/mi><mo>\u2248<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><mo>\u22c5<\/mo><mn>3<\/mn><mspace width=\"0.1667em\"><\/mspace><mrow><mtext><\/mtext><mi>kHz<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\omega\\approx2\\pi\\cdot3\\,\\mathrm{kHz}<\/annotation><\/semantics><\/math> yields <math data-latex=\"|G(j\\omega)| \\approx 12.05\\,\\mathrm{dB}\\approx4\"><semantics><mrow><mi>|<\/mi><mi>G<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mi>\u03c9<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>|<\/mi><mo>\u2248<\/mo><mn>12.05<\/mn><mspace width=\"0.1667em\"><\/mspace><mrow><mtext><\/mtext><mi>dB<\/mi><\/mrow><mo>\u2248<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">|G(j\\omega)| \\approx 12.05\\,\\mathrm{dB}\\approx4<\/annotation><\/semantics><\/math>, which yields <math data-latex=\"L\\approx2.65\\,\\mathrm{mH}\"><semantics><mrow><mi>L<\/mi><mo>\u2248<\/mo><mn>2.65<\/mn><mspace width=\"0.1667em\"><\/mspace><mrow><mtext><\/mtext><mi>mH<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">L\\approx2.65\\,\\mathrm{mH}<\/annotation><\/semantics><\/math>.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Simulation-validation\"><\/span>Simulation validation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">This section compares the results of two different simulations, one with nominal L and R values (2.2 mH and 11.5 \u03a9) and one with identified ones, with the experimental data previously obtained. To run the identification in simulation, users should also enable the block for simulation, as discussed in <a href=\"https:\/\/imperix.com\/doc\/software\/system-identification-injector\">SD041<\/a>. The same settings can be used as in the experimental identification procedure. These Simulink and PLECS models are already configured to identify the system in simulation, and the following plot reports the obtained data:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"815\" src=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/bode_simValidation-1024x815.png\" alt=\"Comparison of the experimental frequency response versus two simulations: one with nominal parameters, one with parameters derived from the parametric identification.\" class=\"wp-image-49480\" srcset=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/bode_simValidation-1024x815.png 1024w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/bode_simValidation-300x239.png 300w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/bode_simValidation-768x611.png 768w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/08\/bode_simValidation.png 1375w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Comparison of the identified system in two simulations against the experimental identification. The previously identified parameters yield the green curve, while the nominal values give the red one.<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The identified values better represent the system than the nominal values, at least within the frequency range typically considered for controller design [1].<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Visualizing-simulation-data-in-PLECS\"><\/span>Visualizing simulation data in PLECS<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Simulink automatically plots the system&#8217;s frequency response at the end of the simulation. In PLECS, the injector block saves data to two .csv files in a folder named after the specified injector. To reduce the friction of writing custom code to plot the data, the PLECS model provides a GNU Octave script. Users can manually run it at the end of the simulation as follows:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1374\" height=\"816\" src=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs_script.png\" alt=\"Screenshot of the PLECS script available in the Simulation menu.\" class=\"wp-image-49670\" style=\"width:780px;height:auto\" srcset=\"https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs_script.png 1374w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs_script-300x178.png 300w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs_script-768x456.png 768w, https:\/\/imperix.com\/doc\/wp-content\/uploads\/2026\/09\/plecs_script-1024x608.png 1024w\" sizes=\"auto, (max-width: 1374px) 100vw, 1374px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Conclusions\"><\/span>Conclusions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The note demonstrated a procedure to identify the frequency response of the duty cycle to inductor current transfer function for a buck converter. Then, it presented a simple way to parameterize it as a first-order system, yielding inductance and resistance values that better fit the actual system behavior than the nominal values in the typical frequency range of interest (20 to 10 times below the switching frequency). A higher-order simulation model would also allow characterization of parasitic elements, improving matching in the high-frequency range. This requires determining which elements to use and how to connect them.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Further-reading\"><\/span>Further reading<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">It is also possible to use the system identification data to design or validate a controller in the frequency domain, as described in <a href=\"https:\/\/imperix.com\/doc\/uncategorized\/controller-tuning-in-the-frequency-domain-using-imperix-sys-id-toolbox\" data-type=\"link\" data-id=\"https:\/\/imperix.com\/doc\/uncategorized\/controller-tuning-in-the-frequency-domain-using-imperix-sys-id-toolbox\">PN216<\/a>. This note demonstrates how to derive the parameters necessary for the tuning procedure from the Bode plot, as well as how to fine-tune the controller directly in the experimental setup.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"References\"><\/span>References <span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/doi.org\/10.1007\/978-3-030-43881-4\">[1]<\/a> R. W. Erickson, D. Maksimovi\u0107, <em>Fundamentals of Power Electronics<\/em>, Springer, 2020<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This note illustrates a system identification example that derives a parametric model using the imperix System Identification toolbox. It presents a step-by-step procedure, starting by&#8230;<\/p>\n","protected":false},"author":34,"featured_media":50077,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","footnotes":""},"categories":[3],"tags":[],"software-environments":[103,104],"provided-results":[108,107],"related-products":[50,31,32,92,166,110],"guidedreadings":[],"tutorials":[],"user-manuals":[],"coauthors":[181],"class_list":["post-49062","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-help","software-environments-matlab","software-environments-plecs","provided-results-experimental","provided-results-simulation","related-products-acg-sdk","related-products-b-board-pro","related-products-b-box-rcp","related-products-b-box-micro","related-products-b-box-rcp-3-0","related-products-tpi"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Parametric system identification using the imperix Sys. 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