Behavioral ModelsInput-Output Models6.241 Dynamic Systems and Control Lecture 6: Dynamical Systems Readings: DDV, Chapter 6 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology February 23, 2011 E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 1 / 10Signals Signals: maps from a set T to a set W. Time axis T: topological semigroup1, in practice T = Z, R, N0, or R≥0, and combinations thereof, such as Z × R. Signal space W: vector space, typically Rn, for some fixed n ∈ N. Discrete-time signals �: maps from Z (or N0) to Rn . Continuous-time signals L: maps from R (or R≥0) to Rn . Typically, constraints are imposed on maps to qualify as continuous-time signals: Piecewise-continuity, or Local (square) integrability. DT and CT signals can be given the structure of vector spaces in the obvious way (i.e., time-wise addition and scalar multiplication of signal values). It is possible to mix DT and CT signals (e.g., to describe digital sensing of physical processes, zero-order holds, etc.). 1Semigroup: group without identity and/or inverse. E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 2 / 10Outline 1 2 Behavioral Models Input-Output Models E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 3 / 10Behavioral Models 1/2 A system can be defined as a set of constraints on signals: Behavioral model of a dynamical system Given a time axis T and a signal space W, a behavioral model of a system is a subset B of all possible signals {w : T → W}. A system is linear if its behavioral model is a vector space, i.e., if wa, wb ∈ B ⇒ αwa + βwb ∈ B, ∀α, β ∈ F. A system is time-invariant if its behavioral model is closed with respect to time shift. For any signal w : T W, define the time-shift operator στ as→(στ w )(t ) = w (t − τ ) A system is time-invariant if w ∈ B ⇒ στ w ∈ B, for any τ ∈ T. E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 4 / 10Behavioral Models 2/2 A system is memoryless if, for any v , w ∈ B, and any T ∈ T, the signal e : T W defined as → � e(t) = v(t) if t ≤ T w(t) if t > T is also in B. In other words, a system is memoryless if possible futures are independent of the past. A system is strictly memoryless if there exists a function φ : T × W → {True, False} such that w ∈ B φ(t, w(t)) = True. ⇔In other words, a system is strictly memoryless if the constraints imposed on the signals are purely algebraic, point-wise in time (e.g., no derivatives, integrals, etc.). Note: any notion of regularity imposed on the signals (as a whole), such as piecewise continuity, integrability, etc. requires a system not to be strictly memoryless. (CT systems always have some kind of memory.) E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 5 / 10Example: Memoryless vs. Strictly Memoryless systems Consider a behavioral model B such that w ∈ B if and only if w is piecewise constant, i.e., if there exists a finite partition of T into sets over which w is constant. This system is memoryless, but is not strictly memoryless. E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 6 / 10Outline 1 2 Behavioral Models Input-Output Models E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 7 / 10Input-Output models 1/2 Behavioral models treat all components of signals constrained by the system equally, without any differences in their role or interpretation. In many applications, it is useful to make a distinction between some of the components of the signals (called the input) and the others (called the output). An input-output model is a map S from a set of input signals {u : Tin → Win} and a set of output signals {y : Tout → Wout}. In behavioral terms, an input-output model S is the set B = {(u, y) : y = Su}. Typically we will consider deterministic input-output, i.e., systems that associate a unique output signal to each input signal, where the time axis is Z, R, or combinations thereof. For convenience, we will often assume Tin = Tout = T. E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 8 / 10Input-Output models 2/2 Properties of behavioral models map easily to input-output models. An input-output system S is linear if, for all input signals ua, ub, S(αua + βub) = α(Sua) + β(Sub) = αya + βyb, ∀α, β ∈ F. An input-output system S is time-invariant if it commutes with the time-shift operator, i.e., if Sστ u = στ Su = στ y ∀τ ∈ T. An input-output system S is memoryless (or static) if there exists a function f : Win → Wout such that, for all t0 ∈ T, y(t0) = (Su)(t0) = f (u(t0)). E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 9 / 10� Causality An input-output system S is causal if, for any t ∈ T, the output at time t depends only on the values of the input on (−∞, t]. In other words, define the truncation operator P as (PT u)(t) = u(t) for t ≤ T 0 for t > T . Then an input-output system S is causal if PT SPT = PT S, ∀T ∈ T. An input-output system S is strictly causal if, for any t ∈ T, the output at time t depends only on the values of the input on (−∞, t). E. Frazzoli (MIT) Lecture 6: Dynamical Systems Feb 23, 2011 10 / 10MIT OpenCourseWare http://ocw.mit.edu 6.241J / 16.338J Dynamic Systems and Control Spring 2011 For information about citing these materials or our Terms of Use, visit: