ControlDesign[LQROutput]  design linear quadratic state feedback regulator (LQR) with output weighting

Calling Sequence


LQROutput(sys, Q, R, opts)


Parameters


sys



System; system object

Q



Matrix; output weighting matrix

R



Matrix; input weighting matrix

opts



(optional) equation(s) of the form option = value; specify options for the LQROutput command





Options


Weighting on the outputinput multiplication term in the cost function. If omitted, a zero matrix with appropriate dimensions will be considered.
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poles = true or false True means the eigenvalues of ABK are returned. See LQR for details. The default value is false.

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riccati = true or false True means the solution of the associated Riccati equation is returned. The default value is false.

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return_Kr = true or false True means the direct gain Kr is returned. The default value is false.

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parameters = {list, set}(name = complexcons)


Specifies numeric values for the parameters of sys. These values override any parameters previously specified for sys. The numeric value on the righthand side of each equation is substituted for the name on the lefthand side in the sys equations. The default is the value of sys given by DynamicSystems:SystemOptions(parameters).



Solvability Conditions


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The data must satisfy the solvability conditions of the standard LQR problem described in the LQR help page.



Description


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The LQROutput command calculates the LQR state feedback gain for a system with output weighting.

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The system sys is a continuous or discretetime linear system object created using the DynamicSystems package. The system object must be in statespace (SS) form.

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In continuoustime domain, the optimal state feedback gain, , is calculated such that the quadratic costfunction

is minimized by the feedback law u = Kx subject to the following statespace
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In discretetime domain, the optimal state feedback gain, , is calculated such that the quadratic costfunction

is minimized by the feedback law u[n] = Kx[n] subject to the following statespace
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Q and R are expected to be symmetric. If the input Q and/or R are not symmetric, their symmetric part will be considered since their antisymmetric (skewsymmetric) part has no role in the quadratic cost function.

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In LQR with output weighting, the weighting matrix Q acts on the outputs rather than the states as in the standard LQR. Also, the weighing matrix N acts on the inner product of the outputs and inputs (creating a bilinear form) rather than the inner product of the states and inputs.

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In addition to the state feedback gain, depending on the corresponding option values, the command returns the closedloop eigenvalues and the solution of the associated Riccati equation.

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The output weighting LQR problem is equivalent to the standard LQR problem with the following weighting matrices:

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The direct gain Kr is computed as follows:

(discrete time)
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If sys contains structured uncontrollable or unobservable states, they are removed using ReduceSystem before computing the LQR state feedback. The resulting gain is then filled with zeros at positions corresponding to the removed states; however, the other outputs are not filled and, consequently, they may have lower dimensions as expected.



Examples


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We can also use LQROuput with discrete models:
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Download Help Document
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