Fall 2001 16.317-3 Basic definitions Linearity- What is a linear dynamical system? A system g is linear with respect to its inputs and output y if superposition holds: G(a11+a22)=a1Gu1+a2G So if y1 Is th he response o of g to u1(91 Gul), and y2 is the response of G to u2(92=Gu2), then the response to a11+a202 IS a191+ a292 A system is said to be time-invariant if the relationship between the input and output is independent of time. So if the response to (t) is yt) then th response to u(t-to is y(t- to (t)is called the state of the system at t because Future output depends only on current state and future input Future output depends on past input only through current state State summarizes effect of past inputs on future output-like the memory of the system Example: Rechargeable fashlight- the state is the current state oj charge of the battery. If you know that state, then you do not need to know how that level of charge was achieved (assuming a perfect battery)to predict the future performance of the fashlightFall 2001 16.31 7—3 Basic Definitions • Linearity — What is a linear dynamical system? A system G is linear with respect to its inputs and output u(t) → G(s) → y(t) if superposition holds: G(α1u1 + α2u2) = α1Gu1 + α2Gu2 So if y1 is the response of G to u1 (y1 = Gu1), and y2 is the response of G to u2 (y2 = Gu2), then the response to α1u1 + α2u2 is α1y1 + α2y2 • A system is said to be time-invariant if the relationship between the input and output is independent of time. So if the response to u(t) is y(t), then the response to u(t − t0) is y(t − t0) • x(t) is called the state of the system at t because: — Future output depends only on current state and future input — Future output depends on past input only through current state — State summarizes effect of past inputs on future output — like the memory of the system • Example: Rechargeable flashlight — the state is the current state of charge of the battery. If you know that state, then you do not need to know how that level of charge was achieved (assuming a perfect battery) to predict the future performance of the flashlight