Introduction to Modern Control MT 2016

Transcription

1 CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate

2 Outline of this module Instructors: A. Abate, P.J. Goulart, K. Margellos Teaching Assistants: A. Sootla, A. Peruffo Schedule for this week: Lectures (Mon-Fri) 9:30-12:00 Exercises on theory/applications (Mon-Thu) 14:00-17:00 Assessment: exercise attendance and work on case study running through the week and supported by TA Alessandro Abate 1

3 Outline of this module Monday: general introduction, modelling, I/O models (frequency vs. state space), linearisation, simulations in MATLAB, hints at Simulink Tuesday: LTI models, analytic solutions, dynamical stability, controllability and observability Wednesday: optimisation in control; LMIs, duality theory, conic programming, Lyapunov via SOS Thursday: control synthesis; state feedback via pole placement and via LQR, output feedback (filters), notes on MPC Friday: vistas in Control Theory Alessandro Abate 2

4 Textbooks Hespanha, Linear Systems Theory Aström and Murray, Feedback Systems online Boyd and Vandenberghe, Convex Optimization online Alessandro Abate 3

5 A short survey what is your existing background in systems and control theory? Alessandro Abate 4

6 A short history of Control Theory pre-1940s: control in frequency 1940s: Wiener s cybernetics, filtering, applications 50s: Ragazzini and Zadeh, state-space 60s: Kalman filtering, linear systems theory, rocket science 70s: stochastic control 80s: adaptive and robust control 90s: non-linear, geometric control Alessandro Abate 5

7 2000s: optimisation in control, MPC, hybrid systems, networks nowadays: inter-disciplinary research, applications (biology, internet, finance,... ) Alessandro Abate 6

8 Lecture 1 The concept of feedback The concept of model in systems engineering: state-space models The concept of controller : controlling a model via feedback Alessandro Abate 7

9 r forcing. The term feedback refers to a situation in which l systems are connected together such that each system their dynamics are thus strongly coupled. Simple causal ack system is difficult because the first system influences d system influences the first, leading to a circular argument. sed on cause and effect tricky, and it is necessary to analyze onsequence of this is that the behavior of feedback systems, and it is therefore necessary to resort to formal methods The concept of feedback Compare the following two interconnections: in block diagram form the idea of feedback. We often use A dynamical system is a system whose behavior changes over ti to external stimulation or forcing. The term feedback refers to two (or more) dynamical systems are connected together su influences the other and their dynamics are thus strongly cou reasoning about a feedback system is difficult because the fir the second and the second system influences the first, leading to This makes reasoning based on cause and effect tricky, and it is the system as a whole. A consequence of this is that the behavior is often counterintuitive, and it is therefore necessary to resor to understand them. Figure 1.1 illustrates in block diagram form the idea of fee ystem 2 y r System 1 u System 2 y System 1 u System 2 y r System 1 u loop (b) Open loop sed loop systems. (a) The output of system 1 is used as the input of of system 2 becomes the input of system 1, creating a closed loop ection between system 2 and system series 1 isconnection removed, and the system (open loop) vs (a) Closed loop (b) Ope Figure 1.1: Open and closed loop systems. (a) The output of system 1 i system 2, and the output of system 2 becomes the input of system 1, c system. feedback (b) The interconnection between system 2 and system 1 is rem is said to be open loop. (closed loop) dynamical feedback, control feedback Alessandro Abate 8

10 The concept of feedback Watt s Regulator centrifugal governor (flyball governor) for steam engine 2 CHAPTER 1. INTRODUCTION Figure 1.2: The centrifugal governor and the steam engine. The centrifugal governor on the left consists of a set of flyballs that spread apart as the speed of the engine increases. The steam engine on the right uses a centrifugal governor (above and to the left of the flywheel) to regulate its speed. (Credit: Machine a Vapeur Horizontale de Philip Taylor [1828].) Alessandro Abate 9 the terms open loop and closed loop when referring to such systems. A system is said to be a closed loop system if the systems are interconnected in a cycle, as

11 The concept of feedback TCP Protocols global network local network Alessandro Abate 10

12 The concept of feedback Eucariotic Cell 16 CHAPTER 1. INTRODUCTION Figure 1.12: The wiring diagram of the growth-signaling circuitry of the mammalian cell [HW00]. The major pathways that are thought to play a role in cancer are indicated in the diagram. Lines represent interactions between genes and proteins in the cell. Lines Alessandro Abate 11 ending in arrowheads indicate activation of the given gene or pathway; lines ending in a T-shaped head indicate repression. (Used with permission of Elsevier Ltd. and the authors.)

13 The concept of feedback Predator/Prey Ecosystems 2.2. STATE SPACE MODELS TATE SPACE MODELS Har Lyn Figure 2.6: Predator versus prey. The photograph on the left shows a Canadian lynx an a snowshoe hare, the lynx s primary prey. Hare The graph on the right shows the populations o hares and lynxes between 1845 and 1935 Lynx in a section of the Canadian Rockies [Mac37]. Th data were collected on an annual basis over a period of 90 years. (Photograph copyright Tom and Pat Leeson.) discrete-time index (e.g., the month number), we can write H[k ] 1915 = H[k] b r (u)h[k] al[k]h[k], (2 igure 2.6: Predator versus prey. The photograph on the left shows L[k a Canadian + 1] = L[k] lynx and + cl[k]h[k] d f L[k], snowshoe hare, the lynx s primary prey. The graph on the right shows the populations of ares and lynxes where b r (u) is the hare birth rate per unit period and as a function of the Alessandro between Abate 1845 and 1935 in a section of the Canadian Rockies [Mac37]. The 12 ata were collected on an annual basis over a period supply of 90u, years. d f is(photograph the lynx mortality copyrightrate Tomand a and c are the interaction coeffici nd Pat Leeson.) The interaction term al[k]h[k] models the rate of predation, which is assu to be proportional to the rate at which predators and prey meet and is hence g by the product of the population sizes. The interaction term cl[k]h[k] in

14 The use of formal models Objective: abstraction from real system, quantitative description of its underlying dynamics How to build a model? physics-based model (conservation laws, physical geometry) models based on known interactions and properties (e.g.: energy-based models, stoichiometric models) models from experiments (data driven): measurement of model properties, model building via fitting, use of transfer functions use of black box, vs grey box models Alessandro Abate 13

15 State-space Models: a First Example mass-spring-damper system CHAPTER 2. SYSTEM MODELING q c(q) F = m q(t) m q(t) = 1 m ( c( q(t)) kq(t) + u(t)) k ss system with nonlinear damping. The position of the mass is denoted sponding to the rest position of the spring. The forces on the mass are pring with spring constant k and a damper with force dependent on the Alessandro Abate 14

16 State-space Models: a First Example q(t) = 1 m ( c( q(t)) kq(t) + u(t)) State: q(t) Input signal: u(t) Output signal: y(t) = q(t) Alessandro Abate 15

17 State-space Models: a First Example ############################ q(t) = 1 m ( c( q(t)) kq(t) + u(t)) Block diagram for input-output relationship Introduce state variables (integrator outputs):! x 1 (t) = q(t) and x 2! (t) = q(t)! "! "!! Alessandro Abate 16

18 Obtain system of first-order ODE: { ẋ1 (t) = x 2 (t) ẋ 2 (t) = 1 m ( c(x 2(t)) kx 1 (t) + u(t)) To find solution, need two initial conditions Note presence of linear (kx 1 ) & nonlinear parts (c(x 2 )) Given a model, we can analyse it (formal proof, verification) simulate it (testing, validation) control it (synthesis) Alessandro Abate 17

19 State-space Models: a First Example { ẋ1 (t) = x 2 (t) ẋ 2 (t) = 1 m ( c(x 2(t)) kx 1 (t) + u(t)) 1 input u(t) 5 x 1 (t) (cont.), x 2 (t) (dot) Damping c(x 2 ), Elasticity k*x phase plane x 2 (vertical) x 1 (horizontal) Alessandro Abate 18

20 State-space Models: a Second Example Predator-prey dynamics in closed ecosystem (introduced before) State variables: time-dependent population level for the lynxes: l(t), t 0 and for the hares: h(t), t 0 Control Input: hare birth rate b(u), function of food Outputs: population levels l(t), h(t) Model parameters: Mortality rate d. Interaction rates a, c Alessandro Abate 19

21 Model as abstraction of population dynamics: { ḣ(t) = b(u)h(t) a l(t)h(t) l(t) = c l(t)h(t) d l(t) Simulation outputs of developed model: Time dependent Trajectories Phase Plane Time dependent Trajectories Phase Plane no noise noisy Alessandro Abate 20

22 State-space Models: a Third Example Control of inverted pendulum on moving cart (in modern terms, a balance system, e.g. Segway) 36 CHAPTER 2. SYSTEM MODELING m θ l F M p (b) Saturn rocket (c) Cart pendulum system tems. (a) Segway Personal Transporter, (b) Saturn rocket and (c) art. Each of these examples uses forces at the bottom of the system Alessandro Abate a generalization of the spring mass system we saw earlier. ics for a mechanical system in the general form (a) Segway (b) Saturn rock Figure 2.5: Balance systems. (a) Segway Pers inverted pendulum on a cart. Each21 of these exam to keep it upright.

23 State-space Models: a Third Example Dynamics can be derived via Lagrange equations States: position p and angle θ Kinetic energy: T M = 1 2 Mṗ2, T m = 1 2 m(ṗ2 + 2lṗ θ cos θ + l 2 θ2 ) Potential energy: V = mgl cos θ (details discussed in Exercise session) Alessandro Abate 22

24 System Identification data-driven modelling, models from experiments L. Ljung, System Identification: Theory for the User, 2nd edition, Prentice-Hall, 1999 Input data Output data u u Process y y t t u(1), u(2),..., u(n) y(t) = G(s)u(t) y(1), y(2),..., y(n) Alessandro Abate 23

25 The identification procedure Alessandro Abate 24

26 physical models System Identification: a panacea? + general (nonlinear models) + based on physical considerations, allows understanding of structure parameters obtained via nonlinear optimization local minima, time-consuming black-box models + often very effective + intuitive behavior (in particular for 1st & 2nd order models) + identification is fast + many control design methods for linear models black-box relation with physical system? linear not general, only valid locally Alessandro Abate 25