Slides - Automatica - Università degli Studi di Padova

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ne regulation and signal transduction; hormonal, immunological and cardiolar feedback mechanisms; muscular control and locomotion; active sensing, and proprioception; attention and consciousness; and population dynamics idemics. Each of these (and many more) provide opportunities to figure out orks, how it works, and what we can do to affect it. e interesting feature of biological systems is the frequent use of positive feedo shape the dynamics of the system. Positive feedback can be used to create like behavior through autoregulation of a gene, and to create oscillations such se present in the cell cycle, central pattern generators or circadian rhythm. osystems. In contrast to individual cells and organisms, emergent properties regations and ecosystems inherently reflect selection mechanisms that act on le levels, and primarily on scales well below that of the system as a whole. se ecosystems are complex, multiscale dynamical systems, they provide a range of new challenges for the modeling and analysis of feedback systems. t experience in applying tools from control and dynamical systems to bacterial rks suggests that much of the complexity of these networks is due to the ce of multiple layers of feedback loops that provide robust functionality Analisi (1/4): Sistemi Lineari Autonomi a tempo continuo Esponenziale di matrici Forma di Jordan Analisi modale Autovettori Autovettori generalizzati
Analisi (2/4): Teoria della Stabilità Punti di equilibrio Funzioni di Lyapunov Linearizzazione Diagramma delle fasi 100 CHAPTER 4. DYNAMIC BEHAVIOR m l θ x2 0 !2 4.2. QUALITATIVE ANALYSIS u −2π −π 101 0 π 2π x1 1.5 (a) (b) 2 (c) x x 1 2 1 Figure 4.4: Equilibrium points for 1 an inverted pendulum. An inverted pendulum is a model 0.5 x for a class of balance systems in which we wish to keep a system upright, such as a rocket (a). 2 0 Using a simplified model of an inverted 0 pendulum (b), we can develop a phase portrait that !0.5 shows the dynamics of the system (c). The system has multiple equilibrium points, marked !1 by the solid dots along the x2 = !10 line. x 1 , x 2 !1.5 !2 !1 0 1 0 10 20 30 x this can be inferred 1 from the lengths of the Time arrows t in the vector field plot). (a) (b) Figure Equilibrium 4.5: Phase portraitPoints and time domain andsimulation Limit for Cycles a system with a limit cycle. The phase portrait (a) shows the states of the solution plotted for different initial conditions. The limit cycle corresponds to a closed loop trajectory. The simulation (b) shows a single solution An equilibrium point of a dynamical system represents a stationary condition for plotted as a function of time, with the limit cycle corresponding to a steady oscillation of fixedthe amplitude. dynamics. We say that a state xe is an equilibrium point for a dynamical system dx = F(x) We consider the open loop dynamics by settingdt u = 0. The equilibrium points for the system are given by if F(xe) = 0. If a dynamical ⎧ system has an initial condition x(0) = xe, then it will xe stay at the equilibrium point: = ⎪⎩ x(t) = xe for all t ≥ 0, where we have taken t0 = 0. Equilibrium points are one of the most important features of a dynamical system since they define the states corresponding to constant operating conditions. A dynamical system can have zero, one or more equilibrium points. Example 4.4 Inverted pendulum Consider the inverted pendulum in Figure 4.4, which is a part of the balance system we considered in Chapter 2. The inverted pendulum is a simplified version of the ±nπ ⎫ ⎪⎭ , 0 where n = 0, 1, 2,.... The equilibrium points for n even correspond to the pendulum pointing up and those for n odd correspond to the pendulum hanging down. A phase portrait for this system (without corrective inputs) is shown in Figure 4.4c. The phase portrait shows −2π ≤ x1 ≤ 2π, so five of the equilibrium points are shown. ∇ Nonlinear systems can exhibit rich behavior. Apart from equilibria they can also 2 1 !1
Page 1 and 2: Sistemi e Modelli Introduzione al c
Page 3 and 4: Testi Testi di riferimento: Mauro B
Page 5 and 6: Sistemi e ... “Il tutto è maggio
Page 7 and 8: Modelli Matematici Vantaggi: Descri
Page 9 and 10: Sistemi e modelli dinamici Sistemi
Page 11 and 12: Come si costruisce un modello matem
Page 13 and 14: Approccio White-Box (2/2) Vantaggi:
Page 15 and 16: Approccio Gray-Box (2/2) Vantaggi:
Page 17 and 18: Approccio Black-Box (2/2) Vantaggi:
Page 19: Modellizzazione 16 CHAPTER 1. INTRO
Page 23 and 24: Analisi (4/4): Modelli Biologici Co
Page 25 and 26: Identificazione (2/5): Stima ai min
Page 27 and 28: Identificazione (3/5): Stima a mass
Page 29 and 30: Identificazione (5/5): Deconvoluzio
Page 31 and 32: Programma del corso (2/2) Identific

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