Algebraic Riccati Equation Solvers The Continuous and Discrete time Algebraic Riccati Equations arise frequently in the discipline of optimal control. The routines CARE and DARE in the LinearAlgebra package solve the underlying linear systems of those equations. These two routines compute solutions in the real, floating-point domain. Below, the behaviour of these two computational routines are described in more detail.
<Text-field style="Heading 1" layout="Heading 1">The continuous time case</Text-field> In its general form, the linear system of the Continuous time Algebraic Riccati Equation (CARE) takes the form of the following Matrix equation: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEocmVzdGFydEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkcvJSdpdGFsaWNHRjgvJStleGVjdXRhYmxlR0Y4L0YwUSdub3JtYWxGJw== 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKUkiQUdGKUkjK35HRilJIlhHRikiIiItRiU2JEYuRixGLy1GJTYlLCZJIlNHRilGLy1GJTYkRi5JIkJHRilGLylJIlJHRilJIy0xR0YpKUY0Ri0hIiJJIlFHRilGLyIiIQ== In practice, the Matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiU0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkkvJSdpdGFsaWNHRjFGLy9GM1Enbm9ybWFsRic=and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn are the zero Matrix and the identity Matrix respectively. And so a common form is, 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKUkiQUdGKUkjK35HRilJIlhHRikiIiItRiU2JEYuRixGLy1GJTYlRi5JIkJHRikpLUYlNiRGLkY0Ri0hIiJJIlFHRilGLyIiIQ== or, 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKUkiQUdGKUkjK35HRilJIlhHRikiIiItRiU2JEYuRixGLy1GJTYlRi4qJkkiQkdGKUYvKUY1Ri1GLylGLkYtISIiSSJRR0YpRi8iIiE= These equations follow naturally from the following system of differential equation problem that arises frequently in problems of optimal control.In optimal control, the problem is to optimize some objective cost while running a dynamic system. A frequent case is one in which the objective is quadratic while the dynamic system is governed by linear differential equations. The state feedback controller is then called a LQR, or Linear Quadratic Regulator.
<Text-field style="Heading 2" layout="Heading 2">Linear Quadratic Regulator</Text-field> Let the (Matrix) system of linear differential equations which govern the dynamic system be given as, 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 Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHRilGKywmKiZJIkFHRikiIiJGJ0YvRi8qJkkiQkdGKUYvLUkidUdGKUYqRi9GLw== Let the objective cost for the continuous time (infinite horizon) problem be specified as,JSFH 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 LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJi1JIi5HRiQ2JSlJInhHRidJIyt+R0YnSSJRR0YnRi4iIiItRis2JSlJInVHRidGL0kiUkdGJ0Y1RjEvSSJ0R0YnOyIiIUkpaW5maW5pdHlHRiU= JSFH JSFH Then the minimizing condition is , 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 L0kidUc2IiwkLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYmKUkiUkdGJEkjLTFHRiRJIkJHRiRJIlhHRiRJInhHRiQhIiI= JSFH The solution LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbW9HRiQ2LlEqJnVtaW51czA7RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZFLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiUkYnRi9GMi1GIzYnRistSSNtbkdGJDYlUSIxRidGL0YyLyUnaXRhbGljR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy9GM1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNi5RKSZtaWRkb3Q7RidGL0YyRjVGN0Y5RjtGPUY/RkEvRkRRLDAuMTY2NjY2N2VtRicvRkdGXW8tRkw2JVEiQkYnRi9GMkZpbi1GTDYlUSJYRidGL0YyLUYsNi5RIn5GJ0YvRjJGNUY3RjlGO0Y9Rj9GQS9GRFEmMC4wZW1GJy9GR0Zpb0ZVRldGWg==contains the unknown LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn which is the solution to the continuous time algebraic Riccati equation of .
<Text-field style="Heading 1" layout="Heading 1">The discrete time case</Text-field> In its general form, the linear system of the Discrete time Algebraic Riccati Equation (DARE) takes the form of the following Matrix equation: 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKUkiQUdGKUkjK35HRilJIlhHRilGLCIiIkYuISIiLUYlNiUsJkkiU0dGKUYvLUYlNiVGK0YuSSJCR0YpRi8pLCZJIlJHRilGLy1GJTYlKUY3Ri1GLkY3Ri9JIy0xR0YpKUYzRi1GMEkiUUdGKUYvIiIh In practice, the Matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiU0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkkvJSdpdGFsaWNHRjFGLy9GM1Enbm9ybWFsRic=is the zero Matrix and so a common form is, 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKUkiQUdGKUkjK35HRilJIlhHRilGLCIiIkYuISIiLUYlNidGK0YuSSJCR0YpLCZJIlJHRilGLy1GJTYlKUYzRi1GLkYzRi8pLUYlNiVGK0YuRjNGLUYwSSJRR0YpRi8iIiE= or, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYkUSJYRidGLy1GLDYtUSJ+RidGL0YyRjVGN0Y5RjtGPUY/L0ZCUSYwLjBlbUYnL0ZFRk4tRiw2LVEiK0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkQtSSVtc3VwR0YkNiUtRkc2JFEiQUYnRi8tRiM2JC1GRzYkUSVgK35gRidGL0YvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GLDYtUSkmbWlkZG90O0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4xNjY2NjY3ZW1GJy9GRUZfby1JKG1mZW5jZWRHRiQ2JC1GIzYwRkYtRiw2LVEoJm1pbnVzO0YnRi9GMkY1RjdGOUY7Rj1GP0ZBRkRGRkZbby1GRzYkUSJCRidGL0Zbby1GVDYlLUZibzYkLUYjNistRkc2JFEiUkYnRi9GUC1GVDYlRmlvLUYjNiVGZW4vJSdpdGFsaWNHRjQvRjBRJ25vcm1hbEYnRmhuRltvRkZGW29GaW9GaXBGW3FGLy1GIzYkLUZHNiRRJWAtMWBGJ0YvRi9GaG5GW29GZXBGW29GRkZpcC8lK2V4ZWN1dGFibGVHRjRGW3FGL0Zbb0ZWRlAtRkc2JFEiUUYnRi8tRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZbci1JI21uR0YkNiRGam5GL0ZpcEZicUZbcQ== LywoSSJYRzYiISIiLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlKUkiQUdGJUkjK35HRiUsJkYkIiIiLUYoNidGJEkiQkdGJSksJkkiUkdGJUYxLUYoNiUpRjRGL0YkRjRGMUkjLTFHRiVGOkYkRiZGLkYxSSJRR0YlRjEiIiE= In the case if the discrete time problem, the LQR, or Linear Quadratic Regulator can be described as follows.
<Text-field style="Heading 2" layout="Heading 2">Linear Quadratic Regulator</Text-field> Let the (Matrix) system of linear differential equations which govern the dynamic system be given as, 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 Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHRilGKywmKiZJIkFHRikiIiJGJ0YvRi8qJkkiQkdGKUYvLUkidUdGKUYqRi9GLw== Let the objective cost (or performance index) for the discrete time (infinite horizon) problem be specified as,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJi1JIi5HRiQ2JSkmSSJ4R0YnNiNJImlHRidJIyt+R0YnSSJRR0YnRi4iIiItRis2JSkmSSJ1R0YnRjBGMkkiUkdGJ0Y4RjQvRjE7IiIhSSlpbmZpbml0eUdGJQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Then the minimizing condition is , 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 LyZJInVHNiI2I0kiaUdGJSwkLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYnKSwmSSJSR0YlIiIiLUYqNiUpSSJCR0YlSSMrfkdGJUkiWEdGJUY2RjJJIy0xR0YlRjVGOEkiQUdGJSZJInhHRiVGJiEiIg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The solution 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contains the unknown LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn which is the solution to the discrete time algebraic Riccati equation of .
<Text-field style="Heading 1" layout="Heading 1">Usage</Text-field> The calling sequences and options of the CARE and DARE routines are described on their respective help pages. For the CARE routine, the Matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiU0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn are supplied only as optional arguments. For the DARE routine, the Matrix S is an optional argument. Hence the default behaviour is to solve the simpler equations of and . The individual help pages contain more examples and description of all the options. Below are illustrative examples.
<Text-field style="Heading 2" layout="Heading 2">Examples</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZDLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEld2l0aEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiRRLkxpbmVhckFsZ2VicmFGJ0YvLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjwvRjBRJ25vcm1hbEYnRi8tSSNtb0dGJDYtUSI6RidGLy8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlVGOkY9Rj8= 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 JCIzJDRdKyl5T3JfTiEjSg== The inverse of the condition number can be optionally returned, 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkobWFjdGlvbkdGJDYlLUkobWZlbmNlZEdGJDYoLUYjNiYtSSdtdGFibGVHRiQ2Ny1JJG10ckdGJDYoLUkkbXRkR0YkNigtSSNtbkdGJDYkUTEzLjQxMzUwOTEyNDMxMDM5RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0ZFLyUrZ3JvdXBhbGlnbkdGRS8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZMLUY6NigtRj02JFExMS44NTgxMDE3MDUyMDEyMUYnRkBGQ0ZGRkhGSkZNLUY6NigtRj02JFExMi40MTgyOTg1ODU3ODU5MkYnRkBGQ0ZGRkhGSkZNRkNGRkZILUY3NihGTy1GOjYoLUY9NiRRMTkuNzYzMjQwOTQyNzQ4NThGJ0ZARkNGRkZIRkpGTS1GOjYoLUY9NiRRMTQuNjUyMzA2NzIzOTk2MzhGJ0ZARkNGRkZIRkpGTUZDRkZGSC1GNzYoRlRGam4tRjo2KC1GPTYkUTEzLjcyMTg4MDUwMzQ4MzY5RidGQEZDRkZGSEZKRk1GQ0ZGRkgvJSZhbGlnbkdRJWF4aXNGJy9GRFEpYmFzZWxpbmVGJy9GR1EnY2VudGVyRicvRklRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR1EldHJ1ZUYnLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGZHAvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGX3EvJSZmcmFtZUdGX3EvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR1EmZmFsc2VGJy8lLWVxdWFsY29sdW1uc0dGaXEvJS1kaXNwbGF5c3R5bGVHRmlxLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRlxxLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0ZpcUZARkAvSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zoci8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRicvJSlydGFibGVpZEdRKjEzNzQwNTgwNEYnLUkjbW9HRiQ2LVEiLEYnRkAvJSZmZW5jZUdGaXEvJSpzZXBhcmF0b3JHRmFwLyUpc3RyZXRjaHlHRmlxLyUqc3ltbWV0cmljR0ZpcS8lKGxhcmdlb3BHRmlxLyUubW92YWJsZWxpbWl0c0dGaXEvJSdhY2NlbnRHRmlxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRj02JFE2MC4wNzgwMDI2Mzk4MjQyNTgyOTkxRidGQEZjckZmckZA The state-space output of various systems of the DynamicSystems package can also be used as input. 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 JCIzKkdAIVw4TCEqNDohI0o=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1JI21vR0YkNi9RIn5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZDLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0ZDLyUqc2VwYXJhdG9yR0ZDLyUpc3RyZXRjaHlHRkMvJSpzeW1tZXRyaWNHRkMvJShsYXJnZW9wR0ZDLyUubW92YWJsZWxpbWl0c0dGQy8lJ2FjY2VudEdGQy8lJ2xzcGFjZUdGNy8lJ3JzcGFjZUdGN0YvLUYwNiZGMkY1RjgvRjtRJWF1dG9GJ0YrRkFGREZG Solucionadores de equa\303\247\303\265es alg\303\251bricas de Riccati As equa\303\247\303\265es alg\303\251bricas de Riccati em tempo cont\303\255nuo e discreto surgem freq\303\274entemente na disciplina de controle \303\263timo. As rotinas CARE e DARE no pacote LinearAlgebra resolver os sistemas lineares subjacentes dessas equa\303\247\303\265es. Essas duas rotinas computam solu\303\247\303\265es no dom\303\255nio real de ponto flutuante. Abaixo, o comportamento dessas duas rotinas computacionais s\303\243o descritos em mais detalhes.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">O caso do tempo cont\303\255nuo</Font></Text-field> Em sua forma geral, o sistema linear da Equa\303\247\303\243o Alg\303\251brica de Riccati em tempo cont\303\255nuo (CARE) assume a forma da seguinte equa\303\247\303\243o de matriz: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JFEocmVzdGFydEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkcvJSdpdGFsaWNHRjgvJStleGVjdXRhYmxlR0Y4L0YwUSdub3JtYWxGJw== 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKUkiQUdGKUkjK35HRilJIlhHRikiIiItRiU2JEYuRixGLy1GJTYlLCZJIlNHRilGLy1GJTYkRi5JIkJHRilGLylJIlJHRilJIy0xR0YpKUY0Ri0hIiJJIlFHRilGLyIiIQ== Na pr\303\241tica, as Matrizes "S" e "R" s\303\243o a matriz zero e a matriz de identidade, respectivamente. E ent\303\243o uma forma comum \303\251, 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKUkiQUdGKUkjK35HRilJIlhHRikiIiItRiU2JEYuRixGLy1GJTYlRi5JIkJHRikpLUYlNiRGLkY0Ri0hIiJJIlFHRilGLyIiIQ== ou, 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKUkiQUdGKUkjK35HRilJIlhHRikiIiItRiU2JEYuRixGLy1GJTYlRi4qJkkiQkdGKUYvKUY1Ri1GLylGLkYtISIiSSJRR0YpRi8iIiE= Essas equa\303\247\303\265es seguem naturalmente o seguinte sistema de problema de equa\303\247\303\243o diferencial que surge freq\303\274entemente em problemas de controle \303\263timo. No controle \303\263timo, o problema \303\251 otimizar algum custo objetivo durante a execu\303\247\303\243o de um sistema din\303\242mico. Um caso frequente \303\251 aquele em que o objetivo \303\251 quadr\303\241tico, enquanto o sistema din\303\242mico \303\251 governado por equa\303\247\303\265es diferenciais lineares. O controlador de realimenta\303\247\303\243o de estado \303\251 ent\303\243o chamado de LQR ou Regulador Quadr\303\241tico Linear.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Regulador Linear Quadr\303\241tico</Font></Text-field> Deixe o sistema (Matriz) de equa\303\247\303\265es diferenciais lineares que governam o sistema din\303\242mico ser dado como, 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 Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHRilGKywmKiZJIkFHRikiIiJGJ0YvRi8qJkkiQkdGKUYvLUkidUdGKUYqRi9GLw== Deixe o custo objetivo para o problema de tempo cont\303\255nuo (horizonte infinito) ser especificado como,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJi1JIi5HRiQ2JSlJInhHRidJIyt+R0YnSSJRR0YnRi4iIiItRis2JSlJInVHRidGL0kiUkdGJ0Y1RjEvSSJ0R0YnOyIiIUkpaW5maW5pdHlHRiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Ent\303\243o, a condi\303\247\303\243o de minimiza\303\247\303\243o \303\251 , 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 L0kidUc2IiwkLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYmKUkiUkdGJEkjLTFHRiRJIkJHRiRJIlhHRiRJInhHRiQhIiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= A solu\303\247\303\243o de 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cont\303\251m o desconhecidoLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn que \303\251 a solu\303\247\303\243o para a equa\303\247\303\243o alg\303\251brica de Riccati em tempo cont\303\255nuo de .
<Text-field style="Heading 1" layout="Heading 1">O caso de tempo discreto</Text-field> Em sua forma geral, o sistema linear da Equa\303\247\303\243o Alg\303\251brica de Riccati em tempo discreto (DARE) assume a forma da seguinte equa\303\247\303\243o de matriz: 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKUkiQUdGKUkjK35HRilJIlhHRilGLCIiIkYuISIiLUYlNiUsJkkiU0dGKUYvLUYlNiVGK0YuSSJCR0YpRi8pLCZJIlJHRilGLy1GJTYlKUY3Ri1GLkY3Ri9JIy0xR0YpKUYzRi1GMEkiUUdGKUYvIiIh Na pr\303\241tica, as Matrizes "S" s\303\243o a Matriz zero e, portanto, uma forma comum \303\251, 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 LywqLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKUkiQUdGKUkjK35HRilJIlhHRilGLCIiIkYuISIiLUYlNidGK0YuSSJCR0YpLCZJIlJHRilGLy1GJTYlKUYzRi1GLkYzRi8pLUYlNiVGK0YuRjNGLUYwSSJRR0YpRi8iIiE= ou, 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 LywoSSJYRzYiISIiLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYlKUkiQUdGJUkjK35HRiUsJkYkIiIiLUYoNidGJEkiQkdGJSksJkkiUkdGJUYxLUYoNiUpRjRGL0YkRjRGMUkjLTFHRiVGOkYkRiZGLkYxSSJRR0YlRjEiIiE= No caso do problema de tempo discreto, o LQR, ou Regulador Linear Quadr\303\241tico, pode ser descrito como segue.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Regulador Linear Quadr\303\241tico</Font></Text-field> Deixe o sistema (Matriz) de equa\303\247\303\265es diferenciais lineares que governam o sistema din\303\242mico ser dado como, 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 Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHRilGKywmKiZJIkFHRikiIiJGJ0YvRi8qJkkiQkdGKUYvLUkidUdGKUYqRi9GLw== Deixe o custo objetivo (ou \303\255ndice de desempenho) para o problema de tempo discreto (horizonte infinito) ser especificado como,JSFH 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 LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJi1JIi5HRiQ2JSkmSSJ4R0YnNiNJImlHRidJIyt+R0YnSSJRR0YnRi4iIiItRis2JSkmSSJ1R0YnRjBGMkkiUkdGJ0Y4RjQvRjE7IiIhSSlpbmZpbml0eUdGJQ== JSFH JSFH Ent\303\243o, a condi\303\247\303\243o de minimiza\303\247\303\243o \303\251, 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 LyZJInVHNiI2I0kiaUdGJSwkLUkiLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYnKSwmSSJSR0YlIiIiLUYqNiUpSSJCR0YlSSMrfkdGJUkiWEdGJUY2RjJJIy0xR0YlRjVGOEkiQUdGJSZJInhHRiVGJiEiIg== JSFH A solu\303\247\303\243o 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cont\303\251m o desconhecido LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn que \303\251 a solu\303\247\303\243o para a equa\303\247\303\243o alg\303\251brica de Riccati em tempo discreto de .
<Text-field style="Heading 1" layout="Heading 1">Uso</Text-field> As sequ\303\252ncias de chamada e op\303\247\303\265es das rotinas CARE e DARE s\303\243o descritas em suas respectivas p\303\241ginas de ajuda. Para a rotina CARE, as Matrizes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiU0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUnaXRhbGljR0YxRi8vRjNRJ25vcm1hbEYn s\303\243o fornecidos apenas como argumentos opcionais. Para a rotina DARE, o Matrix S \303\251 um argumento opcional. Portanto, o comportamento padr\303\243o \303\251 resolver as equa\303\247\303\265es mais simples de e . As p\303\241ginas de ajuda individuais cont\303\252m mais exemplos e descri\303\247\303\243o de todas as op\303\247\303\265es. Abaixo est\303\243o exemplos ilustrativos.
<Text-field style="Heading 2" layout="Heading 2">Exemplos</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0ZDLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JFEld2l0aEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiRRLkxpbmVhckFsZ2VicmFGJ0YvLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjwvRjBRJ25vcm1hbEYnRi8tSSNtb0dGJDYtUSI6RidGLy8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlVGOkY9Rj8= 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JCIzJDRdKyl5T3JfTiEjSg== O inverso do n\303\272mero da condi\303\247\303\243o pode ser opcionalmente retornado, 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A sa\303\255da do espa\303\247o de estado de v\303\241rios sistemas do pacote DynamicSystems tamb\303\251m pode ser usada como entrada . 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