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.

## Details and OptionsDetails 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
• 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 , where is the solution of the continuous Riccati equation , and is the submatrix of associated with the feedback inputs .
• For discrete-time systems, the optimal feedback gain is computed as , where is the solution of the discrete Riccati equation .
• The optimal control is unique and stabilizing if is stabilizable, is detectable, , and .

## ExamplesExamplesopen allclose all

### Basic Examples (5)Basic Examples (5)

Compute the optimum feedback gain matrix for a continuous-time system:

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Calculate the optimal control gains for an unstable system:

Compare the open- and closed-loop poles:

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Compute the optimal state-feedback gain matrix for a discrete-time system:

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Calculate the feedback gains for controlling a two-input system using the first input:

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A set of feedback gains for a stabilizable but uncontrollable system:

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