Control Systems Design, SC4026. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft

Transcription

1 Control Systems Design, SC4026 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft

2 Lecture 1 The concept of feedback The role of a controller What is a state? The concept of model in systems engineering: a few examples SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 1

3 reasoning about a feedback system is difficult because the first system influences the second and the second system influences the first, leading to a circular argument. This makes reasoning based on cause and effect tricky, and it is necessary to analyze the system as a whole. A consequence of this is that the behavior of feedback systems is often counterintuitive, and it is therefore necessary to resort to formal methods to understand them. The concept of feedback Figure 1.1 illustrates in block diagram form the idea of feedback. We often use System 1 u System 2 y r System 1 u System 2 y (a) Closed loop (b) Open loop Figure 1.1: Open and closed loop systems. (a) The output of system 1 is used as the input of system 2 2, and the output of system 2 becomes the input of system CHAPTER1, 1. creating INTRODUCTION a closed loop system. (b) The interconnection between system 2 and system 1 is removed, and the system is said to be open loop. 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 SC4026 Fall 2009, dr. A. steam Abate, engine DCSC, on the TU right Delft uses a centrifugal governor (above and to the left of the flywheel) 2 to regulate its speed. (Credit: Machine a Vapeur Horizontale de Philip Taylor [1828].) the terms open loop and closed loop when referring to such systems. A system

4 The concept of feedback 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 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 3 in the diagram. Lines represent interactions between genes and proteins in the cell. Lines 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.)

5 The concept of feedback 2.2. STATE SPACE MODELS Hare Lynx Figure 2.6: Predator versus prey. The photograph on the left shows a Canadian lynx and a snowshoe hare, the lynx s primary prey. The graph on the right shows the populations of hares and lynxes between 1845 and 1935 in a section of the Canadian Rockies [Mac37]. The 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 + 1] = H[k] + b r (u)h[k] al[k]h[k], SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 4 L[k + 1] = L[k] + cl[k]h[k] d f L[k], (2.13) where b r (u) is the hare birth rate per unit period and as a function of the food

6 The concept of feedback SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 5

7 The role of a controller Control: the use of algorithms and feedback in engineered systems 4 CHAPTER 1. INTRODUCTION noise external disturbances noise Actuators System Sensors Output Process Clock D/A Computer A/D Filter Controller operator input Figure 1.3: Components of a computer-controlled system. The upper dashed box represents SC4026 Fall 2009, the process dr. A. Abate, dynamics, DCSC, which TU Delft include the sensors and actuators in addition to the dynamical 6 system being controlled. Noise and external disturbances can perturb the dynamics of the process. The controller is shown in the lower dashed box. It consists of a filter and analog-todigital (A/D) and digital-to-analog (D/A) converters, as well as a computer that implements the control algorithm. A system clock controls the operation of the controller, synchronizing

8 CHAPTER 2. SYSTEM MODELING State-space Models: a First Example q c(q) q(t) = F m m q(t) = 1 m ( c( q) kq + u) 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 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 7 chanics! "

9 State-space Models: a First Example Block diagram for input-output relationship Input signal: u(t) ############################ Output signal: y(t) = q(t)! "!! SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 8

10 ! " State-space Models: a First Example Introduce state variables (integrator! outputs): x 1 (t) = q(t) and! x 2 (t) = q(t)! "!! SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 9

11 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 & nonlinear parts SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 10

12 State-space Models: a Second Example Predator-prey model (introduced before) Now we aim at obtaining a quantitative, abstract simplification of the actual dynamics 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) SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 11

13 cedure calls (RPCs). The server maintains a log of statistics of completed requests. The total number of requests being served, called RIS (RPCs in server), is also measured. The load on the server is controlled by a parameter called MaxUsers, Model parameters: Mortality rate d. Interaction rates a, c Dynamical model: { 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: 40 CHAPTER 2. SYSTEM MODELING Hares Lynxes Population Year Figure 2.7: Discrete-time simulation of the predator prey model (2.13). Using the parameters a = c = 0.014, b r (u) = 0.6 and d = 0.7 in equation (2.13), the period and magnitude of the SC4026 Fall 2009, dr. A. Abate, lynx anddcsc, hare population TU Delft cycles approximately match the data in Figure

14 State-space Models: a Third Example Control of inverted pendulum on moving cart (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 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 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. Each13 of these exam to keep it upright.

15 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 Overall state q = (p, θ), input u = F, output y = (p, θ) SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 14

16 Lagrangian: L = T (q, q) V (q) = (T M + T m ) V Lagrange s Equations: d L dt q L q = [ F 0 ] Obtain [ (M + m) ml cos θ ml cos θ ml 2 ] [ p θ ] + [ ml sin θ θ2 mgl sin θ ] = [ F 0 ] Can synthetically write the dynamics as: M(q) q + K(q, q) = Bu SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 15

17 Notice that we have built a nonlinear model The model will be linearized in Lecture 2 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 16

18 Models from Experiments How to construct state-space models? physics-based model (conservation laws, other physical laws, material properties, physical geometry and dimensions) models based on known interactions and properties (e.g.: energy-based models, stochiometric models) Models from experiments (data driven): use of transfer functions, possibly derived from experiments; measurement of model properties and use of fitting (connection to later part of class) SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 17