BUILT-IN MATHEMATICA SYMBOL
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.
- 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 |
- 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 
.
Compute the optimum feedback gain matrix for a continuous-time system:
| Out[3]= |  |
Calculate the optimal control gains for an unstable system:
Compare the open- and closed-loop poles:
| Out[3]= |  |
Compute the optimal state-feedback gain matrix for a discrete-time system:
| Out[2]= |  |
Calculate the feedback gains for controlling a two-input system using the first input:
| Out[2]= |  |
A set of feedback gains for a stabilizable but uncontrollable system:
| Out[1]= |  |
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