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LQOutputRegulatorGains

LQOutputRegulatorGains
gives the optimal state feedback gain matrix for the StateSpaceModel object ss and the quadratic cost function with output and control weighting matrices q and r.
LQOutputRegulatorGains
includes the output-control cross-coupling matrix p in the cost function.
LQOutputRegulatorGains
specifies sensors as the measured outputs of ss.
LQOutputRegulatorGains
specifies finputs as the feedback inputs of ss.
  • The state-space model ss can be given as StateSpaceModel, where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
continuous-time system
discrete-time system
  • The argument sensors is a list of integers specifying the positions of the measured outputs in .
  • Similarly, finputs is a list of integers specifying the positions of the feedback inputs in .
  • The cost function is:
continuous-time system
discrete-time system
  • The optimal control is given by , where is the computed feedback gain matrix.
  • For continuous-time systems, the optimal feedback gain is computed as , where and . The matrix is the solution of the continuous Riccati equation .
  • For discrete-time systems, the optimal feedback gain is computed as , where and . The matrix is the solution of the discrete Riccati equation .
  • The subscript denotes the submatrix associated with the feedback inputs , and the subscript denotes the submatrix associated with the sensors .
A set of optimal output-weighted state feedback gains for a continuous-time system:
LQ output regulator gains for a discrete-time system:
A set of optimal output-weighted state feedback gains for a continuous-time system:
In[1]:=
Click for copyable input
Out[1]//MatrixForm=
 
LQ output regulator gains for a discrete-time system:
In[1]:=
Click for copyable input
Out[1]=
 
Compute a set of optimal output-weighted state feedback gains of a continuous-time system:
The gains for a cost function with input-output coupling:
The gains for a discrete-time system:
The gains computed using only the first output as the measurement:
Use all outputs as measurements:
The gains using only the first input as the feedback input:
Construct an output-weighted state-feedback gain matrix for an aircraft model:
Plot the closed-loop output response and the control signals:
The output regulator gains can be computed using LQRegulatorGains and equivalent weighting matrices:
LQOutputRegulatorGains gives the same result:
Compute the LQ output regulator gains by solving the underlying Riccati equation:
LQOutputRegulatorGains gives the same result:
Compute the gains for a discrete-time system using DiscreteRiccatiSolve:
LQOutputRegulatorGains gives the same result:
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