lqglqr-example

lqglqr-example - LQG/LQR design example Jo˜ao P. Hespanha...

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Unformatted text preview: LQG/LQR design example Jo˜ao P. Hespanha April 13, 2011 1 LQR state feedback roll-angle roll-rate = = ˙ 2 applied torque 2 + 9 θ ϖ 9 ϖ ¡ . 875 ϖ ¡ 20 τ 9 τ ¡ 50 τ 50 u Figure 1. Aircraft roll angle dynamics Figure 1 shows the roll angle dynamics of an aircraft [1, p. 381]. Defining x θ ϖ τ , we can write the aircraft dynamics as 9 x Ax Bu , where A 1 ¡ . 875 ¡ 20 ¡ 50 , B 50 . 10-2 10-1 10 10 1 10 2 10 3-150-100-50 50 100 magnitude [dB] 10-2 10-1 10 10 1 10 2 10 3-160-140-120-100-80 phase [deg] frequency [rad/s] P(s) from u to θ L(s) for γ = .01, ρ = .01 L(s) for γ = .01, ρ = 1 L(s) for γ = .01, ρ = 100 (a) Open-loop gain for several values of ρ . This parameter al- lows us to move the whole magnitude Bode plot up and down. 10-2 10-1 10 10 1 10 2 10 3-150-100-50 50 100 magnitude [dB] 10-2 10-1 10 10 1 10 2 10 3-160-140-120-100-80 phase [deg] frequency [rad/s] P(s) from u to θ L(s) for ρ = .01, γ = .01 L(s) for ρ = .01, γ = .1 L(s) for ρ = .01, γ = .3 (b) Open-loop gain for several values of γ . Larger values for this parameter result in a larger phase margin. Figure 2. Bode plots for the open-loop gain 1 Open-loop gains. Figure 2 shows Bode plots of the open-loop gain ˆ L Ô s Õ K Ô sI ¡ A Õ ¡ 1 B for several LQR controllers obtained for this system. The controlled output was chosen to be z θ γ 9 θ , which corresponds to G 1 γ , H ....
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This note was uploaded on 12/29/2011 for the course ECE 224b taught by Professor Brewer during the Fall '09 term at UCSB.

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lqglqr-example - LQG/LQR design example Jo˜ao P. Hespanha...

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