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lqr (Control Design Toolkit, MathScript Function)

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)+2x(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)

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