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System Definition 2. State-Feedback Control Design 2.1 Determining Controllability 2.2 Designing a State-Feedback Controller 3. Observer-Based Control Design 3.1 Determining Observability 3.2 Designing a Observer-Based Controller 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OnivdwgiZ_oOgnaifZQn`qnRgb?acDQ^[ifTfpS_iK^aqI`=xf@>rdw\\?ouMnet>inOu]ynMnpCXoe`wwqhHHsJ@uiNoLPsHytYIuiVa`WxsOskQdEir<qooQkKndPfoKQ`GHdJGoZNdYF]s^c]_Z^@j:B:bZWGbuWHmu;QheMWiISxAd=yI:sTymDHycRmxOqFwcUeOgXqcREWXwFEsCsQCtobreuo[fN]hEGUgeYWsB`YWnaeqOcq]CFMv>wcJeEJ;B:;:Aew?XPcddmx^EWdgR_EhLsYlOcTWhD_G>MWpUDACIfODMWSpCvdCfRMXDcff;YtegV_UcwXl_tNuC[KxJIDJWCaqxTGwA=rvIXm[YiUcq;ifYU=qfgYwMKyNWDeeUtIb_cuTeEhITTWxiOV[[v^kUcWt`yR[WvhCE_=s\\=iy]ELaDSGCq;Z:J<Lkt<sniV^uKJiw?QTkewjUwWUv?IxM<moiuiIsRTowMNLQMEuoJIJmXW=TPiiR=PRkTPEXvJdsOiOOhnjhXeaMZhXJLO^Ix?hljMjo=b:>:bZWx]uYHmw;?XquYoiTY_YIwyh_fgaydifVqc]GgiqGgShPkvggXeSyPwsmed\\_fQwUekd_UVXSbg_hWcX]SRO]TKcb\\cdf_dwCYroCR<:>^MatCVtIQlwfcrYdCWvAW]`gpva^SHi[nopo^EQdyAZdOmdHn]^odgleXxs>_tPwKi`UX_QoxIOtc?wa@k\\^ybXphY`sNpyW\\C>jU@ybqxKG`fQf]Xe`psxF`U@gpi[KnrjWlP_cKY^`goLqrm_m[a`J^c_x_hWhhAfA>`:>Z:::::::::::::::::::::::::::::::::XjyaG:3:\"\{\}LSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlP0tGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiUhZiNGKi0lKUNISUxEUkVORzYi This application makes use of the MapleSim Control Design Toolbox.

The image at right is of a cart of mass LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= supporting two inverted pendulums of mass LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= with lengths LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYGxfXzFgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYGxfXzJgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn, respectively. The variables and parameters of the system are summarized in the following table: System Parameters Mass of cart LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiQtSSNtaUdGJDYlUSJNRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJy8lJXNpemVHUSMxMEYnRjpGOg==JSFH Mass of pendulums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiQtSSNtaUdGJDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJy8lJXNpemVHUSMxMEYnRjpGOg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Length of pendulum 1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJsRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIxRidGN0Y6RjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy8lJXNpemVHUSMxMEYnRkZGRg== Length of pendulum 2 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJsRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIyRidGN0Y6RjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy8lJXNpemVHUSMxMEYnRkZGRg== Angle of pendulum 1 from vertical LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUScmIzk1MjtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYkUSIxRidGOi9GOFEldHJ1ZUYnL0Y7USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjovJSVzaXplR1EjMTBGJ0Y6Rjo= Angle of pendulum 2 from vertical 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For small LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYoLUYjNiUtSSNtaUdGJDYjUS1gJnRoZXRhO19fMWBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0ZBRjxGOg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYoLUYjNiUtSSNtaUdGJDYjUS1gJnRoZXRhO19fMmBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjhRJ25vcm1hbEYnL0krbXNlbWFudGljc0dGJFEkYWJzRicvJSVvcGVuR1EpJnZlcmJhcjtGJy8lJmNsb3NlR0ZBRjxGOg==, the equations of motion for this system are: 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

LyomSSJNRzYiIiIiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JInZHRiU2I0kidEdGJUYuRiYsKCooSSJtR0YlRiZJImdHRiVGJklSI21zdWIobWkoIiZ0aGV0YTsiLGZvbnRzdHlsZX49fiJub3JtYWwiKSxtbigiMSIpKUdGJUYmISIiKihGMUYmRjJGJklSI21zdWIobWkoIiZ0aGV0YTsiLGZvbnRzdHlsZX49fiJub3JtYWwiKSxtbigiMiIpKUdGJUYmRjRJInVHRiVGJg==

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

LyomSSJtRzYiIiIiLCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkidkdGJTYjSSJ0R0YlRi9GJiomSTcjbXN1YihtaSgibCIpLG1uKCIxIikpR0YlRiYtRik2JC1GKTYkLUlSI21zdWIobWkoIiZ0aGV0YTsiLGZvbnRzdHlsZX49fiJub3JtYWwiKSxtbigiMSIpKUdGJUYuRi9GL0YmRiZGJiooRiRGJkkiZ0dGJUYmRjdGJg==

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

LyomSSJtRzYiIiIiLCYtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkidkdGJTYjSSJ0R0YlRi9GJiomSTcjbXN1YihtaSgibCIpLG1uKCIyIikpR0YlRiYtRik2JC1GKTYkLUlSI21zdWIobWkoIiZ0aGV0YTsiLGZvbnRzdHlsZX49fiJub3JtYWwiKSxtbigiMiIpKUdGJUYuRi9GL0YmRiZGJiooRiRGJkkiZ0dGJUYmRjdGJg==

With some simple term re-writing and by letting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EnYHhfXzFgRictSSNtb0dGJDYtUSI9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctRiw2I1EtYCZ0aGV0YTtfXzFgRidGMw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EnYHhfXzJgRictSSNtb0dGJDYtUSI9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctRiw2I1EtYCZ0aGV0YTtfXzJgRidGMw==, 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, 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 the equations of motion can be converted into state space form, that is 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. 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 print(); # input placeholder JSFH print(); # input placeholder JSFH where:JSFH 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 print(); # input placeholder Since the states of the system, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzFgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzJgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn, act as our outputs and since this design assumes that both state variables are measured, the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiQ0YnLyUlc2l6ZUdRIzEwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== matrix for our system can be defined as: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Using the DynamicSystems[StateSpace] command we can create the state space representation for our system. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

JSFH

The system defined in the previous section can be controlled so that the inverted pendulums remain vertical on top of the cart, that is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EtYCZ0aGV0YTtfXzFgRictSSNtb0dGJDYtUSI9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctRiw2I1EtYCZ0aGV0YTtfXzJgRidGLy1JI21uR0YkNiRRIjBGJ0YzRjM=, using a state-feedback control strategy provided the system is: (1) controllable by the input, and (2) the states LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EnYHhfXzFgRicvJSVzaXplR1EjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EnYHhfXzJgRicvJSVzaXplR1EjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= can be measured directly

The controllability matrix for the above system can be determined using the DynamicSystems[ControllabilityMatrix] command. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEsQ250cmxNYXRyaXhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSomY29sb25lcTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvRjBRJ25vcm1hbEYnLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1GLDYkUTZDb250cm9sbGFiaWxpdHlNYXRyaXhGJ0YvLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JFEkc3lzRidGL0YvRjZGOUYv

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

Rank of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEsQ250cmxNYXRyaXhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGMi9GNVEnbm9ybWFsRic=: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEuTGluZWFyQWxnZWJyYUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiRRJVJhbmtGJ0YvRi9GLy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GMzYkLUYjNiQtRiw2JFEsQ250cmxNYXRyaXhGJ0YvRi9GLy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZJL0YwUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIl

JSFH Determinant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEsQ250cmxNYXRyaXhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGMi9GNVEnbm9ybWFsRic=: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEuTGluZWFyQWxnZWJyYUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiRRLERldGVybWluYW50RidGL0YvRi8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjM2JS1GIzYmLUYsNiRRLENudHJsTWF0cml4RidGLy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZJL0YwUSdub3JtYWxGJ0ZHRkxGR0ZKRkw=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLClJImdHNiIiIiMiIiIpLCZJJWxfXzFHRiZGKEklbF9fMkdGJiEiIkYnRigpSSJNR0YmIiIlRi0pRitGMEYtKUYsRjBGLUYt

Even though the generic rank of the above matrix is 4 (i.e. matrix is generically full rank), we cannot say the system is controllable without verifying the conditions upon which the determinant of the controllability matrix becomes 0. For this system, the determinant becomes 0 when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EnYGxfXzFgRictSSNtb0dGJDYtUSI9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctRiw2I1EnYGxfXzJgRictRiw2I1EhRidGMw==. From this we can conclude that the system is controllable if the lengths of the inverted pendulums differ from each other.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Assuming we have prior knowledge of the desired location of the closed-loop poles for our system, we can use the ControlDesign[StateFeedback][PolePlacement] command to calculate the state feedback gain for a single-input system. For this design, let us assume that the desired location of the closed-loop poles are: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEmcG9sZXNGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSShtZmVuY2VkR0YkNictRiM2Ny1GMzYuUSomdW1pbnVzMDtGJy8lJ2l0YWxpY0dGOC9GMFEnbm9ybWFsRidGNkY5RjtGPUY/RkFGQy9GRlEsMC4yMjIyMjIyZW1GJy9GSUZXLUkjbW5HRiQ2JVEiM0YnRlJGVC1GMzYuUSIsRidGUkZURjYvRjpRJXRydWVGJ0Y7Rj1GP0ZBRkMvRkZRJjAuMGVtRicvRklRLDAuMzMzMzMzM2VtRictRjM2LlEifkYnRlJGVEY2RjlGO0Y9Rj9GQUZDRlxvL0ZJRl1vRk8tSSZtZnJhY0dGJDYoLUZaNiVRIjVGJ0ZSRlQtRiM2JC1GWjYlUSIyRidGUkZURi8vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmRwLyUpYmV2ZWxsZWRHRjgtRjM2LlEiK0YnRlJGVEY2RjlGO0Y9Rj9GQUZDRlZGWEZccC1GMzYuUScmc2RvdDtGJ0ZSRlRGNkY5RjtGPUY/RkFGQ0Zcb0Zjby1GLDYlUSJJRidGUkZURmduRk9GZG8tRjM2LlEoJm1pbnVzO0YnRlJGVEY2RjlGO0Y9Rj9GQUZDRlZGWEZccEZccUZfcUZnbkZPLUZaNiVRIjdGJ0ZSRlRGL0ZSRlQvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjM2LlEiOkYnRlJGVEY2RjlGO0Y9Rj9GQUZDRkVGSEYv The state-feedback gain, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEjS2NGJy8lJXNpemVHUSMxMEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GNlEnbm9ybWFsRic=, is then: 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

We can obtain the closed-loop state-space matrices using the ControlDesign[StateFeedbackClosedLoop] command. Then we can verify that the closed-loop system has its poles located at the desired pole locations. 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 print(); # input placeholder 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

print(); # input placeholder At this point, we can simulate the closed-loop system to verify if the controller that we designed is able to stabilize the inverted pendulums on the cart. Since the controller was developed symbolically we can perturb any number of the system parameters. Doing so, will give us a sense of the controller's robustness to parameter variations. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Investigating the Closed-Loop Response Simulation Parameters Value Mass of cart LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJNRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= Mass of pendulums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Length of pendulum 1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYjUSdgbF9fMWBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNEY0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Length of pendulum 2LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LlEifkYnLyUlc2l6ZUdRIzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkZGMg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYjUSdgbF9fMmBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNEY0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Gravity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElOS44MUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv DocumentTools:-SetProperty(TextArea0,value,10); DocumentTools:-SetProperty(TextArea1,value, 1); DocumentTools:-SetProperty(TextArea2,value, 1.2); DocumentTools:-SetProperty(TextArea3,value, 2); M:=10; m:=1; `l__1`:=1.2; `l__2`:=2; g:=9.81; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=use DocumentTools in M := convert(DocumentTools:-Do(%TextArea0, value), float); m := convert(DocumentTools:-Do(%TextArea1, value), float); `l__1` := convert(DocumentTools:-Do(%TextArea2, value), float); `l__2` := convert(DocumentTools:-Do(%TextArea3, value), float); g:=9.81; end use; use DocumentTools in if g<> 9.81 then M:=10; m:=1; `l__1`:=1.2; `l__2`:=2; g:=9.81; end if; sol, statevars, outputvars := DynamicSystems[Simulate](clsys, [0,0], initialconditions = Vector([0, 0, -.1, .1]), output = [solution, statevariable, outputvariable], parameters = ['M'=M, 'm'=m, 'g'=g, 'l__1'= `l__1`, 'l__2'= `l__2` ]): plt := Array(1 .. 2, 1 .. 2); plt[1, 1] := plots[odeplot](sol, [t, statevars[1]], 0 .. 7, title = typeset(x[1], " response"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt[1, 2] := plots[odeplot](sol, [t, statevars[2]], 0 .. 7, title = typeset(x[2], " response"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt[2, 1] := plots[odeplot](sol, [t, statevars[3]], 0 .. 7, title = typeset(x[3], " response"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt[2, 2] := plots[odeplot](sol, [t, statevars[4]], 0 .. 7, title = typeset(x[4], " response"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); Do(%Plot0=plots[display](plt)); unassign('M'); unassign('m'); unassign('`l__1`'); unassign('`l__2`'); unassign('g'); end use; JSFH 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The state-feedback controller which was designed in the previous section assumed that the states LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EnYHhfXzFgRicvJSVzaXplR1EjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EnYHhfXzJgRicvJSVzaXplR1EjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= are measured directly. This is not practical in many situations, and consequently control designers must turn into observer-based control design to control their systems. Observer-based control design makes use of an observer module to estimate the states. It requires the system to be observable in addition to being controllable.

This section will examine the observability of the system under the following conditions: (1) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzFgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is measured and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzJgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is not, (2) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzJgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is measured and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzFgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is not, and (3) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzFgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzJgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn are measured.

We get a subsystem using DynamicSystems[Subsystem] command where only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzFgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is measurable: 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 print(); # input placeholder print(); # input placeholder The observability matrix can be determined by using the DynamicSystems[ObservabilityMatrix] command. 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

print(); # input placeholder

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

Rank of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EzYE9ic3ZNYXRyaXhfX3hfXzFgRicvJSp1bmRlcmxpbmVHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEuTGluZWFyQWxnZWJyYUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiRRJVJhbmtGJ0YvRi9GLy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GMzYkLUYjNiQtRiw2I1EzYE9ic3ZNYXRyaXhfX3hfXzFgRidGL0YvLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRkkvRjBRJ25vcm1hbEYn

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIl

Determinant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EzYE9ic3ZNYXRyaXhfX3hfXzFgRicvJSp1bmRlcmxpbmVHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEuTGluZWFyQWxnZWJyYUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiRRLERldGVybWluYW50RidGL0YvRi8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjM2JC1GIzYkLUYsNiNRM2BPYnN2TWF0cml4X194X18xYEYnRi9GLy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZJL0YwUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKilJIm1HNiIiIiMiIiIpSSJnR0YmRidGKClJIk1HRiZGJyEiIilJJWxfXzFHRiZGJ0YtRi0=

Since the observability matrix is calculated symbolically, knowing that the matrix is generically full rank does not provide us with enough information to say that the system is observable for all possible values of parameters. We must determine for what parameter values the determinant of the observability matrix becomes 0. For this example, the system is observable for all values of the parameters.

We get a subsystem using DynamicSystems[Subsystem] command where only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EnYHhfXzJgRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn is measurable: 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 print(); # input placeholder The observability matrix can be determined by using the DynamicSystems[ObservabilityMatrix] command. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

Rank of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EzYE9ic3ZNYXRyaXhfX3hfXzJgRicvJSp1bmRlcmxpbmVHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEuTGluZWFyQWxnZWJyYUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiRRJVJhbmtGJ0YvRi9GLy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GMzYkLUYjNiQtRiw2I1EzYE9ic3ZNYXRyaXhfX3hfXzJgRidGL0YvLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRkkvRjBRJ25vcm1hbEYn

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIl

Determinant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EzYE9ic3ZNYXRyaXhfX3hfXzJgRicvJSp1bmRlcmxpbmVHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEuTGluZWFyQWxnZWJyYUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiRRLERldGVybWluYW50RidGL0YvRi8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjM2JC1GIzYkLUYsNiNRM2BPYnN2TWF0cml4X194X18yYEYnRi9GLy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZJL0YwUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqKilJIm1HNiIiIiMiIiIpSSJnR0YmRidGKClJIk1HRiZGJyEiIilJJWxfXzJHRiZGJ0YtRi0=

As in section 3.1.1, the system is observable for all parameter valuesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=. This means that the system is observable when either of the angles are measured.

If the both states are measurable, the observability matrix of the system is: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1E0YE9ic3ZNYXRyaXhfX3hfXzEyYEYnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0Y1LyUqc2VwYXJhdG9yR0Y1LyUpc3RyZXRjaHlHRjUvJSpzeW1tZXRyaWNHRjUvJShsYXJnZW9wR0Y1LyUubW92YWJsZWxpbWl0c0dGNS8lJ2FjY2VudEdGNS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktRiw2JFEvRHluYW1pY1N5c3RlbXNGJy9GN1EnaXRhbGljRictSShtZmVuY2VkR0YkNictRiM2JC1GLDYkUTRPYnNlcnZhYmlsaXR5TWF0cml4RidGT0ZPRjNGNi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GUjYkLUYjNiQtRiw2JFEkc3lzRidGT0ZPRk9GTw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

Rank of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1E0YE9ic3ZNYXRyaXhfX3hfXzEyYEYnLyUqdW5kZXJsaW5lR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEuTGluZWFyQWxnZWJyYUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiRRJVJhbmtGJ0YvRi9GLy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GMzYkLUYjNiQtRiw2I1E0YE9ic3ZNYXRyaXhfX3hfXzEyYEYnRi9GLy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZJL0YwUSdub3JtYWxGJw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIl

The observability matrix is full rank. Clearly, the system is observable when both angles are measured, as well.

JSFHJSFH In section 2.2, we showed how the ControlDesign toolbox could be used to design a state-feedback controller when both angles are measured. In this section, we will show how the ControlDesign toolbox can be used to design an observer-based control system when only one state, let us say LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+, is measured. According to the separation principle, for linear time invariant systems, the state feedback and state observer can be designed independently. We select the desired poles for the observer error dynamic to be about 5-10 times further away from the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoaiYjOTY5O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis than those of the state feedback gain design. This ensures that the state feedback poles are the dominant poles of the system. For this example, the following values for the state-feedback poles and the observer poles were chosen. If you will recall, the state feedback poles that were chosen here are the same as those used in the state-feedback control design section. 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print(); # input placeholder

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print(); # input placeholder

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Using the ControlDesign[ControllerObserver] command, the closed-loop system of the state-feedback controller and observer can be obtained. We can verify that closed-loop system poles match the desired pole locations. 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 print(); # input placeholder 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print(); # input placeholder

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We modify the state-space representation of the closed-loop system so that there are 8 outputs corresponding to all the states of the closed-loop system. The first four outputs represent the state outputs, while the last four outputs represent the observer error. 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JSFH Investigating the Closed-Loop Response Simulation to Observer-Based Control Design Parameters Value Mass of cart LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJNRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= Mass of pendulums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= JSFH Length of pendulum 1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYjUSdgbF9fMWBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNEY0 JSFH Length of pendulum 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DocumentTools:-SetProperty(TextArea7,value, 2); M:=10; m:=1; `l__1`:=1.2; `l__2`:=2; g:=9.81; JSFHuse DocumentTools in M := convert(DocumentTools:-Do(%TextArea4, value), float); m := convert(DocumentTools:-Do(%TextArea5, value), float); `l__1` := convert(DocumentTools:-Do(%TextArea6, value), float); `l__2` := convert(DocumentTools:-Do(%TextArea7, value), float); g:=9.81; end use; use DocumentTools in if g <> 9.81 then M:=10; m:=1; `l__1`:=1.2; `l__2`:=2; g:=9.81; end if; sol1, statevars1, outputvars1 := DynamicSystems[Simulate](closedloop1, 0, initialconditions = Vector([0, 0, -.1, .1, 0.1e-2, 0.5e-2, -0.1e-2, 0.1e-1]), output = [solution, statevariable, outputvariable], parameters = ['M'=M, 'm'=m, 'g'=g, 'l__1'= `l__1`, 'l__2'= `l__2` ]): plt1 := Array(1 .. 4, 1 .. 2); plt1[1, 1] := plots[odeplot](sol1, [t, statevars1[1]], 0 .. 4, title = typeset('x[1]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[1, 2] := plots[odeplot](sol1, [t, statevars1[2]], 0 .. 4, title = typeset('x[2]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[2, 1] := plots[odeplot](sol1, [t, statevars1[3]], 0 .. 4, title = typeset('x[3]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[2, 2] := plots[odeplot](sol1, [t, statevars1[4]], 0 .. 4, title = typeset('x[4]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[3, 1] := plots[odeplot](sol1, [t, statevars1[5]], 0 .. 1, title = typeset('x[1]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[3, 2] := plots[odeplot](sol1, [t, statevars1[6]], 0 .. 1, title = typeset('x[2]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[4, 1] := plots[odeplot](sol1, [t, statevars1[7]], 0 .. 1, title = typeset('x[3]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[4, 2] := plots[odeplot](sol1, [t, statevars1[8]], 0 .. 1, title = typeset('x[4]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); Do(%Plot1=plots[display](plt1)); unassign('M'); unassign('m'); unassign('`l__1`'); unassign('`l__2`'); unassign('g'); end use; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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We design the LQR controller using the ControlDesign[LQR] command. First, using the ControlDesign[ComputeQR] command, we compute the values of the weighting matrices Q and R based on a desired closed-loop time constant. 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print(); # input placeholder

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We design the Kalman observer using the ControlDesign[Kalman] command. 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 print(); # input placeholder 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print(); # input placeholder

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

We get the LQG controller equations using the ControlDesign[ControllerObserver] command.

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print(); # input placeholder

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We get the closed-loop system using the ControlDesign[ControllerObserver] command with the 'closedloop' option 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 %; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JFEqZWlnY2xfbHFnRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GMzYtUSomY29sb25lcTtGJ0YvRjZGOUY7Rj1GP0ZBRkMvRkZRLDAuMjc3Nzc3OGVtRicvRklGTi1GLDYkUS5MaW5lYXJBbGdlYnJhRidGLy1JKG1mZW5jZWRHRiQ2Jy1GIzYmLUYsNiRRLEVpZ2VudmFsdWVzRidGLy8lJ2l0YWxpY0dGOC8lK2V4ZWN1dGFibGVHRjgvRjBRJ25vcm1hbEYnRmVuRmluLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZUNiUtRiM2Ji1GLDYkUSljbHN5c2xxZ0YnRi8tRjM2LlEjOi1GJ0ZlbkZpbkY2RjlGO0Y9Rj9GQUZDRkVGSC1GLDYkUSJhRidGL0YvRmVuRmluLUYzNi5RIjtGJ0ZlbkZpbkY2L0Y6USV0cnVlRidGO0Y9Rj9GQUZDRkVGTy1GMzYuRjVGZW5GaW5GNkY5RjtGPUY/RkFGQ0ZFRkhGZW5GZ25GaW4=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

We modify the state-space representation of the closed-loop system so that there are 8 outputs corresponding to all the states of the closed-loop system. The first four outputs represent the state outputs, while the last four outputs represent the observer error. 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 print(); # input placeholder print(); # input placeholder Investigating the Closed-Loop Response Simulation to LQG Control Design Parameters Value Mass of cart LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJNRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= Mass of pendulums LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Length of pendulum 1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYjUSdgbF9fMWBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNEY0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Length of pendulum 2LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LlEifkYnLyUlc2l6ZUdRIzEwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkZGMg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYjUSdgbF9fMmBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGNEY0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Gravity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtaUdGJDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6Rjo= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFElOS44MUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv DocumentTools:-SetProperty(TextArea8,value,10); DocumentTools:-SetProperty(TextArea9,value, 1); DocumentTools:-SetProperty(TextArea10,value, 1.2); DocumentTools:-SetProperty(TextArea11,value, 2); M:=10; m:=1; `l__1`:=1.2; `l__2`:=2; g:=9.81; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=use DocumentTools in M := convert(DocumentTools:-Do(%TextArea8, value), float); m := convert(DocumentTools:-Do(%TextArea9, value), float); `l__1` := convert(DocumentTools:-Do(%TextArea10, value), float); `l__2` := convert(DocumentTools:-Do(%TextArea11, value), float); g:=9.81; end use; use DocumentTools in if g <> 9.81 then M:=10; m:=1; `l__1`:=1.2; `l__2`:=2; g:=9.81; end if; sol1, statevars1, outputvars1 := DynamicSystems[Simulate](clsyslqg1, 0, initialconditions = Vector([0, 0, -.1, .1, 0.1e-2, 0.5e-2, -0.1e-2, 0.1e-1]), output = [solution, statevariable, outputvariable], parameters = ['M'=M, 'm'=m, 'g'=g, 'l__1'= `l__1`, 'l__2'= `l__2` ]): plt1 := Array(1 .. 4, 1 .. 2); plt1[1, 1] := plots[odeplot](sol1, [t, statevars1[1]], 0 .. 4, title = typeset('x[1]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[1, 2] := plots[odeplot](sol1, [t, statevars1[2]], 0 .. 4, title = typeset('x[2]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[2, 1] := plots[odeplot](sol1, [t, statevars1[3]], 0 .. 4, title = typeset('x[3]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[2, 2] := plots[odeplot](sol1, [t, statevars1[4]], 0 .. 4, title = typeset('x[4]'), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[3, 1] := plots[odeplot](sol1, [t, statevars1[5]], 0 .. 1, title = typeset('x[1]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[3, 2] := plots[odeplot](sol1, [t, statevars1[6]], 0 .. 1, title = typeset('x[2]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[4, 1] := plots[odeplot](sol1, [t, statevars1[7]], 0 .. 1, title = typeset('x[3]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); plt1[4, 2] := plots[odeplot](sol1, [t, statevars1[8]], 0 .. 1, title = typeset('x[4]', "- observer error"), titlefont = [Helvetica, 9], axesfont = [Helvetica, 9], labelfont = [Helvetica, 9], numpoints = 200); Do(%Plot2=plots[display](plt1)); unassign('M'); unassign('m'); unassign('`l__1`'); unassign('`l__2`'); unassign('g'); end use; JSFH 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