Lecture Signals, systems & inference – Lecture 5: State-space models, equilibrium, linearization. The following will be discussed in this chapter: State variables are (relevant) “memory” variables, defining properties of CT state-space models and the governing equations, linearization at an equilibrium yields an LTI model,. | Lecture Signals, systems & inference – Lecture 5: State-space models, equilibrium, linearization State-Space Models, Equilibrium, Linearization , Spring 2018 Lec 5 1 State variables are (relevant) “memory” variables In physical systems, the natural state variables are typically related to energy storage mechanisms: capacitor voltages or charges, inductor currents or fluxes, positions and velocities of masses, 2 Defining properties of CT state-space models ⇣ ⌘ q˙ (t) = f q(t), x(t), t ⇣ ⌘ y(t) = g q(t), x(t), t • State evolution property • Instantaneous output property 3 Numerical solution of CT state-space model . qi(t0) qi(t0) qi(t) t t0 + ¢ 4 t0 Integrator-adder-gain system x(t) . . q2(t) q2(t) q1(t) q1(t) y(t) -1 + • • -2 8 5 Mechanistic model for capnography 6 and the governing equations V (t) LV¨ (t) + RV˙ (t) + = P C pD (t) + pA ˙ p˙D (t) = V (t) , V˙ (t) > 0 VD pD (t) ˙ p˙D (t) = V (t) , V˙ (t) < 0 VD 7 Equilibrium For a time-invariant nonlinear system with a constant input, an initial state that the system remains at: DT : ¯ = f (q q ¯, x ¯) CT : ¯, x 0 = f (q ¯) 8 Linearization at an equilibrium yields an LTI model e , x[n] = x¯ + x[n] ¯ + q[n] DT case: q[n] = q e , q[n + 1] = f (q[n], x[n]) # h @f i h @f i e + 1] ⇡ q[n e + q[n] e x[n] @q ¯ ¯ q,x @x ¯ ¯ q,x e for small perturbations q[n] e and x[n] from equilibrium 9 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 ¯ ¯ q,x @x ¯ ¯ q,x e for small perturbations q(t) e and x(t) from equilibrium 10 MIT OpenCourseWare Signals, Systems and Inference Spring 2018 For information about citing these materials or our Terms of Use, visit: . 11
