LQRegulatorGains
LQRegulatorGains[ssm,{q,r}]
gives the optimal state feedback gain matrix for the StateSpaceModel ssm and the quadratic cost function, with state and control weighting matrices q and r.
LQRegulatorGains[ssm,{q,r,p}]
includes the state-control cross-coupling matrix p in the cost function.
LQRegulatorGains[{ssm,finputs},{…}]
specifies finputs as the feedback inputs of ssm.
Details and Options
- The standard state-space model ssm can be given as StateSpaceModel[{a,b,…}], where a and b represent the state and input matrices in either a continuous-time or a discrete-time system:
-
continuous-time system 
discrete-time system - The descriptor state-space model ssm can be given as StateSpaceModel[{a,b,c,d,e}] in either continuous time or discrete time:
-
continuous-time system 
discrete-time system - LQRegulatorGains also accepts nonlinear systems specified by AffineStateSpaceModel and NonlinearStateSpaceModel.
- For nonlinear systems, the operating values of state and input variables are taken into consideration, and the gains are computed based on the approximate Taylor linearization and returned as a vector.
- The argument finputs is a list of integers specifying the positions of the feedback inputs
in 
. - LQRegulatorGains[ssm,{…}] is equivalent to LQRegulatorGains[{ssm,All},{…}].
- The cost function is:
-
continuous-time system 
discrete-time system - In LQRegulatorGains[ssm,{q,r}], the cross-coupling matrix p is assumed to be zero.
- The optimal control is given by
, where 
is the computed feedback gain matrix. - For continuous-time systems, the optimal feedback gain is computed as
![kappa=TemplateBox[{r}, Inverse].(b_f.x_r+p)](Files/LQRegulatorGains.en/11.png "kappa=TemplateBox[{r}, Inverse].(b_f.x_r+p)")
, where 
is the solution of the continuous Riccati equation ![a.x_r+x_r.a-(x_r.b_f+p).TemplateBox[{r}, Inverse].(b_f.x_r+p)+q=0](Files/LQRegulatorGains.en/13.png "a.x_r+x_r.a-(x_r.b_f+p).TemplateBox[{r}, Inverse].(b_f.x_r+p)+q=0")
, and 
is the submatrix of 
associated with the feedback inputs 
. - For discrete-time systems, the optimal feedback gain is computed as
![kappa=TemplateBox[{{(, {{{{b, _, f}, }, ., {x, _, r}, ., {b, _, f}}, +, r}, )}}, Inverse].(b_f.x_r.a+p)](Files/LQRegulatorGains.en/17.png "kappa=TemplateBox[{{(, {{{{b, _, f}, }, ., {x, _, r}, ., {b, _, f}}, +, r}, )}}, Inverse].(b_f.x_r.a+p)")
, where 
is the solution of the discrete Riccati equation ![a.x_r.a-x_r-(a.x_r.b_f+p)TemplateBox[{{., {(, {{{{b, _, f}, }, ., {x, _, r}, ., {b, _, f}}, +, r}, )}}}, Inverse].(b_f.x_r.a+p)+q=0](Files/LQRegulatorGains.en/19.png "a.x_r.a-x_r-(a.x_r.b_f+p)TemplateBox[{{., {(, {{{{b, _, f}, }, ., {x, _, r}, ., {b, _, f}}, +, r}, )}}}, Inverse].(b_f.x_r.a+p)+q=0")
. - The optimal control
is unique and stabilizing if 
is stabilizable, 
is detectable, 
, and 
.
Examples
open all close allBasic Examples (5)
Compute the optimum feedback gain matrix for a continuous-time system:
| In[1]:= |
| In[2]:= |
Calculate the optimal control gains for an unstable system:
| In[1]:= |
| In[2]:= |
Compare the open- and closed-loop poles:
Compute the optimal state-feedback gain matrix for a discrete-time system:
| In[1]:= |
Calculate the feedback gains for controlling a two-input system using the first input:
| In[1]:= |
A set of feedback gains for a stabilizable but uncontrollable system:
