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Lecture Signals, systems & inference – Lecture 6: Modal solution of undriven CT LTI state-space models

The following will be discussed in this chapter: Glucose-insulin system, UVA/Padova model (FDA approved!), linearization at an equilibrium yields an LTI model, phase plane trajectories, complex eigenvalue pairs (CT case). | Lecture Signals, systems & inference – Lecture 6: Modal solution of undriven CT LTI state-space models Modal solution of undriven CT LTI state-space models 6.011, Spring 2018 Lec 6 1 Glucose-insulin system From Messori et al., IEEE Control Systems Magazine © IEEE. All rights reserved. This content is excluded from our 2 Creative Commons license. Feb 2018 For more information, see https://ocw.mit.edu/help/faq-fair-use/ UVA/Padova model (FDA approved!) From Messori et al., IEEE Control Systems Magazine © IEEE. All rights reserved. This content is excluded from our Creative Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use/ 3 Feb 2018 Linearization at an equilibrium yields an LTI model e , x(t) = x¯ + x(t) ¯ + q(t) CT case: q(t) = q e , q˙ (t) = f (q(t), x(t)) # h @f i h @f i e˙ q(t) ⇡ e + q(t) e x(t) @q ¯ ,x q ¯ @x ¯ ,x q ¯ e for small perturbations q(t) e and x(t) from equilibrium 4 Phase plane trajectories State trajectories for different initial conditions 5 4 [-6, 3.05] 3 2 1 [-3.2, 1.5] q2(t) 0 [2, -0.9] -1 -2 [4, -2.1] -3 -4 [8, -4] -5 5 -10 -8 -6 -4 -2 0 2 4 6 8 10 q1(t) Complex eigenvalue pairs (CT case) If Ai is a (complex) eigenvalue with eigenvector vi , then its complex conjugate A⇤i is also an eigenvalue, with associated eigenvector vi⇤ . Write i = i + j!i , vi = ui + jvi . Then the contribution of the complex pair to the modal solution is ⇤ ↵ i vi e it + ↵i⇤ vi⇤ e it = h i it Ki e ui cos(!i t + ✓i ) wi sin(!i t + ✓i ) 6 Acoustics and Vibration Animations Have fun exploring the animations created by Prof. Dan Russell, Penn State 7 MIT OpenCourseWare https://ocw.mit.edu 6.011 Signals, Systems and Inference Spring 2018 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. 8