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rec8 - EE221A Discussion 8 Lillian Ratliﬀ 1 LQR Cont time...

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Unformatted text preview: EE221A: Discussion 8 Lillian Ratliﬀ October 22, 2010 1 LQR: Cont. time and ∞-Horizon Problem LTI dynamics: x = Ax + Bu ˙ ∞ Cost function: J = 0 (xT Qx + uT Ru)dt T T Q = Q and R = R (i.e. they are symmetric) and Q is positive semi-deﬁnite and R is positive deﬁnite: (R 0 and Q 0). Then u = −F x, F = R−1 B T P . P = P T > 0 : P A + AT P − P BR−1 B T P + Q = 0. FACT 1.0.1 Self-adjoint linear operators have real eigenvalues. FACT 1.0.2 Positive deﬁnite: eigenvalues strictly greater than 0. i.e. z T M z > 0 for all nonzero vectors z . Positive semi-deﬁnite: no non-negative eigenvalues. Example 1.0.1 x = Ax + Bu, u = −Kx. Then x = Ax − Bkx = (A − Bk )x ˙ ˙ 1 ...
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