N = Matrix or Vector

poles  = true or false True means the eigenvalues of A-BK are returned. See LQR for details. The default value is false.    

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

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

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 right-hand side of each equation is substituted for the name on the left-hand side in the sys equations. The default is the value of sys given by DynamicSystems:-SystemOptions(parameters).

The data $A,B,\stackrel{\_}{Q},\stackrel{\_}{R},\stackrel{\_}{N}$ must satisfy the solvability conditions of the standard LQR problem described in the LQR help page.

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

The system sys is a continuous or discrete-time linear system object created using the DynamicSystems package. The system object must be in state-space (SS) form.

In continuous-time domain, the optimal state feedback gain, $K$, is calculated such that the quadratic cost-function

In discrete-time domain, the optimal state feedback gain, $K$, is calculated such that the quadratic cost-function

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 (skew-symmetric) part has no role in the quadratic cost function.

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.  

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

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

The direct gain Kr is computed as follows:

If sys contains structured uncontrollable or unobservable states, they are removed using ReduceSystem before computing the LQR state feedback.  The resulting gain $K$ 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.

$\mathrm{with}\left(\mathrm{ControlDesign}\right)\:$

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\:$

$\mathrm{csys}≔\mathrm{NewSystem}\left(\mathrm{Matrix}\left(\left[\left[\frac{3}{\left(s+2\right)\left(s+7\right)}\,\frac{1}{s+1}\right]\,\left[\frac{3s-2}{s^2+5}\,\frac{1}{s^2+5s+1}\right]\right]\right)\right)$

$\mathrm{csys}≔\left[\begin{array}{c}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 2\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\,\mathrm{u2}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\,\mathrm{y2}\left(s\right)\right]\end{array}\right.$

$\mathrm{csys}:-\mathrm{tf}$

$\left[\begin{array}{cc}\frac{3}{s^2+9s+14}& \frac{1}{s+1}\\ \frac{3s-2}{s^2+5}& \frac{1}{s^2+5s+1}\end{array}\right]$

$\mathrm{sys}≔\mathrm{StateSpace}\left(\mathrm{csys}\right)$

$\mathrm{sys}≔\left[\begin{array}{c}\mathbf{State\; Space}\\ \mathrm{continuous}\\ \mathrm{2\; output(s);\; 2\; input(s);\; 7\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(t\right)\,\mathrm{u2}\left(t\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(t\right)\,\mathrm{y2}\left(t\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(t\right)\,\mathrm{x2}\left(t\right)\,\mathrm{x3}\left(t\right)\,\mathrm{x4}\left(t\right)\,\mathrm{x5}\left(t\right)\,\mathrm{x6}\left(t\right)\,\mathrm{x7}\left(t\right)\right]\end{array}\right.$

$\mathrm{sys}:-a\;$$\mathrm{sys}:-b\;$$\mathrm{sys}:-c\;$$\mathrm{sys}:-d$

$\left[\begin{array}{ccccccc}0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ \mathrm{-70}& \mathrm{-45}& \mathrm{-19}& \mathrm{-9}& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& \mathrm{-1}& \mathrm{-6}& \mathrm{-6}\end{array}\right]$

$\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 1& 0\\ 0& 0\\ 0& 0\\ 0& 1\end{array}\right]$

$\left[\begin{array}{ccccccc}15& 0& 3& 0& 1& 5& 1\\ \mathrm{-28}& 24& 25& 3& 1& 1& 0\end{array}\right]$

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$Q≔\frac{1}{5}\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(2\right)\;$$R≔2\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(2\right)$

$Q≔\left[\begin{array}{cc}\frac{1}{5}& 0\\ 0& \frac{1}{5}\end{array}\right]$

$R≔\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

$K≔\mathrm{LQROutput}\left(\mathrm{sys}\,Q\,R\right)$

$K≔\left[\begin{array}{ccccccc}0.717034668157443& 14.2423392627583& 8.97325040212257& 0.992321630768280& \mathrm{-0.0178914427159702}& 0.0116054056366964& \mathrm{-0.00303834255501862}\\ \mathrm{-0.0317449620940457}& \mathrm{-0.565427777675089}& \mathrm{-0.0603711482280991}& \mathrm{-0.00303834255501862}& 0.0952989985741524& 0.298425245845240& 0.0577917808525738\end{array}\right]$

$\mathrm{Kpr}≔\mathrm{LQROutput}\left(\mathrm{sys}\,Q\,R\,\mathrm{poles}=\mathrm{true}\,\mathrm{riccati}=\mathrm{true}\,\mathrm{return\_Kr}=\mathrm{true}\right)\:$$\mathrm{Kpr}\left[1\right]\;$$\mathrm{Kpr}\left[2\right]\;$$\mathrm{Kpr}\left[3\right]\;$$\mathrm{Kpr}\left[4\right]$

$\left[\begin{array}{ccccccc}0.717034668157443& 14.2423392627583& 8.97325040212257& 0.992321630768280& \mathrm{-0.0178914427159702}& 0.0116054056366964& \mathrm{-0.00303834255501862}\\ \mathrm{-0.0317449620940457}& \mathrm{-0.565427777675089}& \mathrm{-0.0603711482280991}& \mathrm{-0.00303834255501862}& 0.0952989985741524& 0.298425245845240& 0.0577917808525738\end{array}\right]$

$\left[\begin{array}{c}\mathrm{-6.99856378303368}+0.\mathrm{I}\\ \mathrm{-4.79073410146030}+0.\mathrm{I}\\ \mathrm{-0.494228759193816}+2.18968411573883\mathrm{I}\\ \mathrm{-0.494228759193816}-2.18968411573883\mathrm{I}\\ \mathrm{-2.00557177848029}+0.\mathrm{I}\\ \mathrm{-1.04883795460928}+0.\mathrm{I}\\ \mathrm{-0.217948275649673}+0.\mathrm{I}\end{array}\right]$

$\left[\begin{array}{ccccccc}2213.3210179011925105661490359729988635623523105777& 1427.3744698955260840335662780084477714201028438168& 170.05490326097842788563325766245079711056354077698& 1.4340693363148856210209140521689160116840515800739& \mathrm{-8.1584867958664927190611477199891984007760389352564}& \mathrm{-3.8143338925075523792756023597033096286517352171425}& \mathrm{-0.063489924188091382049176469030954150915142789638243}\\ 1427.3744698955260840335662780084477714201028438168& 1314.4149489966517127605774990095472393845261805749& 358.10638273538894791100970655990292973259306704383& 28.484678525516698881224832846966240322944663907424& \mathrm{-6.7332125306921498463443365378844628598517488954127}& \mathrm{-7.1469946052254660167289986202242491609016370025761}& \mathrm{-1.1308555553501777075201057104844140218420156581446}\\ 170.05490326097842788563325766245079711056354077698& 358.10638273538894791100970655990292973259306704383& 173.55111848299452019907894155637818079428896173084& 17.946500804245141593919755909088874368587080074509& \mathrm{-0.96420988722855540330309373855482549187031417311249}& \mathrm{-0.37056076992136628982081706414090368560463357756919}& \mathrm{-0.12074229645619825350879543233467955529836974169035}\\ 1.4340693363148856210209140521689160116840515800739& 28.484678525516698881224832846966240322944663907424& 17.946500804245141593919755909088874368587080074509& 1.9846432615365601333495110097986200526978925133139& \mathrm{-0.035782885431940392539024834988131723694174424070744}& 0.023210811273392858470564369426083998560166901460873& \mathrm{-0.0060766851100372365759906089897292226102991059271909}\\ \mathrm{-8.1584867958664927190611477199891984007760389352564}& \mathrm{-6.7332125306921498463443365378844628598517488954127}& \mathrm{-0.96420988722855540330309373855482549187031417311249}& \mathrm{-0.035782885431940392539024834988131723694174424070744}& 0.59690245383661254959345476298732008112992241926137& 1.0702952629406602870546063132728061843987429911545& 0.19059799714830485098021287040917562854135390550258\\ \mathrm{-3.8143338925075523792756023597033096286517352171425}& \mathrm{-7.1469946052254660167289986202242491609016370025761}& \mathrm{-0.37056076992136628982081706414090368560463357756919}& 0.023210811273392858470564369426083998560166901460873& 1.0702952629406602870546063132728061843987429911545& 3.1184288536473559060855917389389997225292005817800& 0.59685049169047937887974968854007127189693335562395\\ \mathrm{-0.063489924188091382049176469030954150915142789638243}& \mathrm{-1.1308555553501777075201057104844140218420156581446}& \mathrm{-0.12074229645619825350879543233467955529836974169035}& \mathrm{-0.0060766851100372365759906089897292226102991059271909}& 0.19059799714830485098021287040917562854135390550258& 0.59685049169047937887974968854007127189693335562395& 0.11558356170514763732723892618907256020381429737325\end{array}\right]$

$\left[\begin{array}{cc}1.63293195981652& \mathrm{-1.65082340253249}\\ 0.712479697627494& 0.382819300946659\end{array}\right]$

$\mathrm{dsys}≔\mathrm{ToDiscrete}\left(\mathrm{csys}\,1\,\mathrm{method}=\mathrm{bilinear}\right)$

$\mathrm{dsys}≔\left[\begin{array}{c}\mathbf{Transfer\; Function}\\ \mathrm{discrete;\; sampletime\; =\; 1}\\ \mathrm{2\; output(s);\; 2\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(z\right)\,\mathrm{u2}\left(z\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(z\right)\,\mathrm{y2}\left(z\right)\right]\end{array}\right.$

$\mathrm{dsys}:-\mathrm{tf}$

$\left[\begin{array}{cc}\frac{3{\left(z+1\right)}^2}{4\left(9z+5\right)z}& \frac{z+1}{3\left(z-\frac{1}{3}\right)}\\ \frac{4\left(z+1\right)\left(z-2\right)}{9\left(z^2+\frac{2}{9}z+1\right)}& \frac{{\left(z+1\right)}^2}{15\left(z^2-\frac{2}{5}z-\frac{1}{3}\right)}\end{array}\right]$

$\mathrm{sys}≔\mathrm{StateSpace}\left(\mathrm{dsys}\right)$

$\mathrm{sys}≔\left[\begin{array}{c}\mathbf{State\; Space}\\ \mathrm{discrete;\; sampletime\; =\; 1}\\ \mathrm{2\; output(s);\; 2\; input(s);\; 7\; state(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(q\right)\,\mathrm{u2}\left(q\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(q\right)\,\mathrm{y2}\left(q\right)\right]\\ \mathrm{statevariable}\=\left[\mathrm{x1}\left(q\right)\,\mathrm{x2}\left(q\right)\,\mathrm{x3}\left(q\right)\,\mathrm{x4}\left(q\right)\,\mathrm{x5}\left(q\right)\,\mathrm{x6}\left(q\right)\,\mathrm{x7}\left(q\right)\right]\end{array}\right.$

$Q≔5\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(2\right)\;$$R≔3\mathrm{LinearAlgebra}:-\mathrm{IdentityMatrix}\left(2\right)$

$Q≔\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

$R≔\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$

$K≔\mathrm{LQROutput}\left(\mathrm{sys}\,Q\,R\right)$

$K≔\left[\begin{array}{ccccccc}0.00356160133647377& \mathrm{-0.494191307766542}& \mathrm{-0.914892355689041}& \mathrm{-0.0264403506039962}& \mathrm{-0.00753844597397812}& \mathrm{-0.0102161879672920}& \mathrm{-0.00237235112808720}\\ 0.0300774217423282& 0.0656631598525158& \mathrm{-0.00165075351937600}& \mathrm{-0.0578915850030273}& \mathrm{-0.0812015931703188}& \mathrm{-0.0812549022768781}& 0.279310675111098\end{array}\right]$

$\mathrm{Kpr}≔\mathrm{LQROutput}\left(\mathrm{sys}\,Q\,R\,\mathrm{poles}=\mathrm{true}\,\mathrm{riccati}=\mathrm{true}\,\mathrm{return\_Kr}=\mathrm{true}\right)\:$$\mathrm{Kpr}\left[1\right]\;$$\mathrm{Kpr}\left[2\right]\;$$\mathrm{Kpr}\left[3\right]\;$$\mathrm{Kpr}\left[4\right]$

$\left[\begin{array}{ccccccc}0.00356160133647377& \mathrm{-0.494191307766542}& \mathrm{-0.914892355689041}& \mathrm{-0.0264403506039962}& \mathrm{-0.00753844597397812}& \mathrm{-0.0102161879672920}& \mathrm{-0.00237235112808720}\\ 0.0300774217423282& 0.0656631598525158& \mathrm{-0.00165075351937600}& \mathrm{-0.0578915850030273}& \mathrm{-0.0812015931703188}& \mathrm{-0.0812549022768781}& 0.279310675111098\end{array}\right]$

$\left[\begin{array}{c}0.767828122439180+0.\mathrm{I}\\ 0.146877156240930+0.\mathrm{I}\\ \mathrm{-0.0598533454582572}+0.309176357637630\mathrm{I}\\ \mathrm{-0.0598533454582572}-0.309176357637630\mathrm{I}\\ \mathrm{-0.132589590854147}+0.\mathrm{I}\\ \mathrm{-0.554251178458362}+0.\mathrm{I}\\ \mathrm{-0.405472587458189}+0.\mathrm{I}\end{array}\right]$

$\left[\begin{array}{ccccccc}0.030421135822715746661483119465619750780079644226519& 0.065909907466081445943651725217081837731677484888381& 0.077863993891928202554012314536566316895231948722372& 0.059112886634046876785210492949616537395421761897085& \mathrm{-0.050188644413976545817420078367331778312895021111404}& \mathrm{-0.062433942231198913457006717445770242913882168709431}& 0.14647440916725792764105957938065613949658264398518\\ 0.065909907466081445943651725217081837731677484888381& 2.2847701972316684633140332644560369698375292274017& 4.7847306516135635638601767892353674998604105167075& 1.5200731282028195434731344834060459690437415027044& \mathrm{-0.051381185070926621076018021910551508907205422423887}& \mathrm{-0.24248117228076022252788740320992280262529833198557}& \mathrm{-0.018026465186631385616928025593331378129935273493821}\\ 0.077863993891928202554012314536566316895231948722372& 4.7847306516135635638601767892353674998604105167075& 12.217596587591310011803119523701421197584453334801& 6.8714533828437224847228249708724585749274573728353& 0.033219117439838470818425867947158666956219639899675& \mathrm{-0.41757515631223804555176954389437678658213715588195}& \mathrm{-0.68989774217826862624169968848499880789368219782819}\\ 0.059112886634046876785210492949616537395421761897085& 1.5200731282028195434731344834060459690437415027044& 6.8714533828437224847228249708724585749274573728353& 7.7290189303281432825264402037496282033620051084464& \mathrm{-0.00089785711277865282387993604941083466168087624032189}& \mathrm{-0.25047345435813408835915911397628042401730449430088}& \mathrm{-0.52414391617243365970614715171054243008109996218594}\\ \mathrm{-0.050188644413976545817420078367331778312895021111404}& \mathrm{-0.051381185070926621076018021910551508907205422423887}& 0.033219117439838470818425867947158666956219639899675& \mathrm{-0.00089785711277865282387993604941083466168087624032189}& 0.096303809125238661441515557835814164660578301981060& 0.10527321517532861157297605229965868726473800562174& \mathrm{-0.30422873354722336756437043493817780506351359200722}\\ \mathrm{-0.062433942231198913457006717445770242913882168709431}& \mathrm{-0.24248117228076022252788740320992280262529833198557}& \mathrm{-0.41757515631223804555176954389437678658213715588195}& \mathrm{-0.25047345435813408835915911397628042401730449430088}& 0.10527321517532861157297605229965868726473800562174& 0.15017244479508106184875357307589186761301102349660& \mathrm{-0.27910155202740406405198625921281687752114303705794}\\ 0.14647440916725792764105957938065613949658264398518& \mathrm{-0.018026465186631385616928025593331378129935273493821}& \mathrm{-0.68989774217826862624169968848499880789368219782819}& \mathrm{-0.52414391617243365970614715171054243008109996218594}& \mathrm{-0.30422873354722336756437043493817780506351359200722}& \mathrm{-0.27910155202740406405198625921281687752114303705794}& 1.0680116872765628785913505609708284332160101080345\end{array}\right]$

$\left[\begin{array}{cc}0.879831651067362& \mathrm{-0.993045941963838}\\ 1.09622204012740& 0.561082720692690\end{array}\right]$