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the MapleSim Control Design Toolbox to Design a PID Controller
1. Introduction2. Pole Placement Method3. Exact Pole Placement Method4. Gain-Phase Margin Method5. Automatic Tuning
<Text-field bookmark="Introduction" style="Heading 1" layout="Heading 1">1. Introduction</Text-field>This worksheet illustrates how the MapleSim Control Design Toolbox can be used to design PID controllers using several methods. This worksheet is divided into 3 sections. In the first section, we will use the Pole Placement method to design a PI controller for a second-order system so that we can confine the closed-loop poles to a desired region. In the second section, we will use the Exact Pole Placement method to design a PID controller so that we can specify the exact location of the dominant poles. Finally, in the last section, we will use the Gain-Phase Margin method to design a PID controller for a fifth-order system.restart; libname;
<Text-field bookmark="Pole Placement Design Method" style="Heading 1" layout="Heading 1">2. Pole Placement Method</Text-field>The Pole Placement method is a unique feature of the MapleSim ControlDesign toolbox. It allows you to specify a desired region for the closed-loop poles based on damping ratio, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, and the natural frequency, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM5Njk7bkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. In this section, we will show how the Pole Placement method can be used to design a PI controller for an arbitrary second-order system.We can create a second-order system using the DynamicSystems[NewSystem] 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRJXN5czFGJ0YvRjJGPQ==print(); # input placeholderAs mentioned above, the desired region for the closed-loop poles is defined by the damping ratio, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, and the natural frequency, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM5Njk7bkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. For this example, the values of 0.7 and 5/2 have been chosen for the damping ratio and natural frequency, respectively.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJnpldGE7MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMC43RidGOQ==print(); # input 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print(); # input placeholderBased on the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiw2JVEoJnpldGE7MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjQvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0Y2USdub3JtYWxGJw== and LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEqJm9tZWdhO24xRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn, the command ControlDesign[FeasibleGain] provides you with the option to output a list of 3 possible gain values for the PI controller as sample designs for which all the closed-loop poles will lie in the desired region. Alternatively, the command gives you the option to plot a region of feasible gain values.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print(); # input 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%;For this example, we will use the gain values for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEja2NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEja2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn as provided in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEka3AxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEkcmhzRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiUtRlM2Ji1GIzYjLUYsNiVRKnNvbHV0aW9uMUYnRi9GMkY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZTNiYtRiM2Iy1JI21uR0YkNiRRIjJGJ0Y5RjlGaG5GW28tRlM2Ji1GIzYjLUZjbzYkUSIxRidGOUY5RmhuRltvRjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJ0ZNLUYsNiVRJGtpMUYnRi9GMkY1Rk8tRlM2JC1GIzYlRldGXm9GXm9GOQ==print(); # input placeholderUsing these gain values, the transfer function for the PI controller can be determined as: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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRJEdjMUYnRi9GMkY9print(); # input placeholderAt this point, we can automatically obtain the transfer function for the closed-loop system using the command ControlDesign[PIDClosedLoop].LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEvQ2xvc2VkTG9vcFN5czFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUS5Db250cm9sRGVzaWduRidGL0YyLUkobWZlbmNlZEdGJDYmLUYjNiMtRiw2JVEuUElEQ2xvc2VkTG9vcEYnRi9GMkY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZZNiQtRiM2JS1GLDYlUSVzeXMxRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkhGSi9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSRHYzFGJ0YvRjJGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRlJGVA==print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRL0Nsb3NlZExvb3BTeXMxRidGL0YyRj0=print(); # input placeholderAs expected, the poles of the closed-loop reside in the region defined by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiw2JVEoJnpldGE7MUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjQvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0Y2USdub3JtYWxGJw== and LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVEqJm9tZWdhO24xRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn.
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%;The pole locations can be determined using the following command: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
The step response of the open and closed-loop system can be determined using the DynamicSystems[ResponsePlot] command. From the plots, we can see how the including of a PI controller was able to stabilize the system.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjcDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUS9EeW5hbWljU3lzdGVtc0YnRi9GMi1JKG1mZW5jZWRHRiQ2Ji1GIzYjLUYsNiVRLVJlc3BvbnNlUGxvdEYnRi9GMkY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZTNiQtRiM2Ti1GLDYlUS9DbG9zZWRMb29wU3lzMUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJVN0ZXBGJ0YvRjItRlM2JC1GIzYjLUYsNiNRIUYnRjlGYW8tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZlby9GTkZmby1GNjYtUSInRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRmZwLUYsNiVRKWR1cmF0aW9uRidGL0YyRmdwRmNwLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkjbW5HRiQ2JFEiNUYnRjlGYW9GY3BGZ3AtRiw2JVEqbnVtcG9pbnRzRidGL0YyRmdwRmNwRl9xLUZjcTYkUSQzMDBGJ0Y5RmFvRmNwRmdwLUYsNiVRJmNvbG9yRidGL0YyRmdwRmNwRl9xLUYsNiVRJWJsdWVGJ0YvRjJGYW9GY3BGZ3AtRiw2JVEnbGVnZW5kRidGL0YyRmdwRmNwRl9xLUkjbXNHRiQ2I1E1Q2xvc2Vkfkxvb3B+UmVzcG9uc2VGJ0Zhb0ZjcEZncC1GLDYlUSdsYWJlbHNGJ0YvRjJGZ3BGY3BGX3EtRlM2Ji1GIzYnRmBwLUZmcjYjUSV0aW1lRidGYW8tRmZyNiNRKkFtcGxpdHVkZUYnRmBwRjlGWkZnbkY5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZN%;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%;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%;JSFH
<Text-field bookmark="Exact Pole Placement Design" style="Heading 1" layout="Heading 1">3. Exact Pole Placement Method</Text-field>In this section, we will show how the Exact Pole Placement method is used to design a PID controller for a 2nd order system.As in the previous section, the transfer function of the open loop system can be defined as follows: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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRJXN5czJGJ0YvRjJGPQ==print(); # input placeholderLet us assume that the desired closed-loop pole locations are:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEncG9sZXMyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtZmVuY2VkR0YkNiYtRiM2Mi1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GWC1JI21uR0YkNiRRIjRGJ0Y5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGV0ZZLUZlbjYkUSIyRidGOS1GNjYtUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GYm8tRiw2JVEiSUYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRmFvL0ZOUSwwLjMzMzMzMzNlbUYnRlRGWi1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRldGWUZbb0Zeb0Zkb0Znb0ZULUkmbWZyYWNHRiQ2KC1GZW42JFEiNUYnRjktRiM2I0Zbby8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXEvJSliZXZlbGxlZEdGPUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUYsNiNRIUYnprint(); # input placeholderUsing the ControlDesign[DominantPole] command we can obtain the PID controller transfer 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(); # input placeholderAt this point, we can automatically obtain the transfer function for the closed-loop system using the command ControlDesign[PIDClosedLoop].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print(); # input placeholderprint(); # input placeholderThe step response of the open and closed-loop system is shown in the following plots. The closed-loop response in addition to tracking the input signal is much faster than then open-loop response.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%;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%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmcGxvdHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUShkaXNwbGF5RidGL0YyL0YzUSdub3JtYWxGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GNjYkLUYjNigtRjY2Ji1GIzYlLUYsNiVRI3AzRidGL0YyLUkjbW9HRiQ2LVEiLEYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGVi8lKnN5bW1ldHJpY0dGVi8lKGxhcmdlb3BHRlYvJS5tb3ZhYmxlbGltaXRzR0ZWLyUnYWNjZW50R0ZWLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEjcDRGJ0YvRjJGPUY/RkJGUC1GUTYtUSJ+RidGPUZUL0ZYRlZGWUZlbkZnbkZpbkZbb0Zdby9GYW9GX28tRiw2JVEnbGFiZWxzRidGL0YyLUZRNi1RIj1GJ0Y9RlRGaW9GWUZlbkZnbkZpbkZbby9GXm9RLDAuMjc3Nzc3OGVtRicvRmFvRmJwLUY2NiYtRiM2Jy1GLDYjUSFGJy1JI21zR0YkNiNRJXRpbWVGJ0ZQLUZccTYjUSpBbXBsaXR1ZGVGJ0ZocEY9Rj9GQkY9%;JSFH
<Text-field bookmark="Gain-Phase Margin Design Method" style="Heading 1" layout="Heading 1">4. Gain-Phase Margin Method</Text-field>In this section, we will show how the Gain-Phase Margin method is used to design a PID controller for the following fifth-order system.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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRJXN5czNGJ0YvRjJGPQ==print(); # input placeholderThe system gain margin (in decibels) and associated crossover frequency (in rad/sec) is: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print(); # input placeholderSimilarly, the system phase margin (in degrees) and crossover frequency (in rad/sec) is: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print(); # input placeholderUsing the ControlDesign[GainPhaseMargin] command we can obtain the PID gain values so that the desired gain and phase margin for the closed-loop system is at least, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEjMTVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSJ+RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEjZEJGJy8lJ2l0YWxpY0dGOEYvRi8= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEjNjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSYmZGVnO0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGR0Yv, respectively.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print(); # input placeholderAt this point, we can automatically obtain the transfer function for the closed-loop system using the command ControlDesign[PIDClosedLoop].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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRL0Nsb3NlZExvb3BTeXMzRidGL0YyRj0=print(); # input placeholderThe gain and phase margin and crossover frequencies for the loop transfer function are shown below. These values exceed the minimum requirements specified above.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(); # input 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(); # input placeholderThe step response of the open and closed-loop system is shown in the following plots. Unlike the open-loop response which was more than 100% off in the steady state, the closed-loop response now tracks the input signal.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%;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%;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%;The bode plots of the open-loop system and the loop transfer function are shown below.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%;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%;
<Text-field bookmark="Automatic Tuning" style="Heading 1" layout="Heading 1">5. Automatic Tuning</Text-field>In this section, we will show how to tune the parameters of a PID controller using a single tuning parameter \317\204, which is equivalent to the desired time constant of the closed-loop system.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRJXN5czRGJ0YvRjJGPQ==print(); # input placeholderThe desired closed-loop time constant \317\204 is 1.0 second (for a settling time of approaximately LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiNkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJyZzZG90O0YnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JI21pR0YkNiVRJyYjOTY0O0YnLyUnaXRhbGljR0Y4Ri8tRjM2LVEiPUYnRi9GNkY5RjtGPUY/RkFGQy9GRlEsMC4yNzc3Nzc4ZW1GJy9GSUZURitGLw== seconds).LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmJnRhdTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSNtbkdGJDYkUSQxLjBGJ0Yyprint(); # input 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(); # input 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEvRHluYW1pY1N5c3RlbXNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYlUSxQcmludFN5c3RlbUYnRi9GMi9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYjLUYsNiVRL0Nsb3NlZExvb3BTeXM0RidGL0YyRj0=print(); # input 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%;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%;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%;Verify the resulting closed-loop settling time 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print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
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