August 2007

370853D-01

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Member of the ssdesign class.

[K, X, eig] = lqr(A, B, Q, R)

[K, X, eig] = lqr(A, B, Q, R, N)

Calculates the optimal steady-state feedback gain matrix **K** that minimizes a quadratic cost function for a linear discrete state-space system model. The cost function weights the model states. The quadratic cost function *J* is defined as the following equation:

integral(**x**'(*t*)**Qx**(*t*)+**u**(*t*)' **Ru**(*t*)+2**x**(*t*)'**Nu**(*t*), *t*, 0, +inf), where *t* is continuous time, **u** is the input vector, and **x** is the state vector.

Name |
Description |

A |
Specifies an n x n state matrix, where n is the number of states. The default is an empty matrix. A is a real matrix. |

B |
Specifies an n x m input matrix, where m is the number of inputs. The default is an empty matrix. B is a real matrix. |

Q |
Specifies a symmetric, positive semi-definite matrix that penalizes the state vector x in the cost function. Q is a real matrix. |

R |
Specifies a symmetric positive definite matrix that penalizes the input vector u in the cost function. The default is the identity matrix. R is a real matrix. |

N |
Specifies a matrix that penalizes the cross product between input and state vectors, such that (Q-N*inv(R)*N') is positive semi-definite. The default is an appropriately sized matrix of zeros. N is a real matrix. |

Name |
Description |

K |
Returns the gain matrix such that K = inv(R)*(B'X+N'). K is a real matrix. |

X |
Returns the symmetric, positive semi-definite (stabilizing) solution to the discrete algebraic Riccati equation. X is a real matrix. |

eig |
Returns the eigenvalues of the matrix (A-BK). These eigenvalues are the closed-loop pole locations. eig is a complex vector. |

A = [-1, -2; 0, -4]

B = [0; 1]

Q = [2, 0; 0, 2]

R = 1

[K, X, eig] = lqr(A, B, Q, R)