Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1. Emilio Frazzoli

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1 Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1, Guzzella Ch. 1 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 22, 2017 E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

2 The hidden technology [Karl Åström] Widely used Very successful Seldom talked about Except when disaster strikes E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

3 E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

4 Outline 1 Overview 2 Logistics 3 Signals and Systems E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

5 Course Objectives 1/3 The course addresses dynamic control systems, i.e., systems that evolve over time, have inputs and outputs. We have three main objectives: Modeling: learn how to represent a dynamic control system in a way that it can be treated effectively using mathematical tools. Analysis: understand the basic characteristics of a system (e.g., stability, controllability, observability), and how the input affects the output. Synthesis: figure out how to change a system in such a way that it behaves in a desirable way. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

6 Course Objectives 2/3 In particular, we will concentrate on systems that can be modeled by Ordinary Differential Equations (ODEs), and that satisfy certain linearity and time-invariance conditions. In this course, we will focus on systems with a single input and a single output (SISO). This will allow us to use classical control tools that are very powerful and easy to use (i.e., mostly graphical), and which are really laying the foundation of any followup work on more challenging control problems. We will analyze the response of these systems to inputs and initial conditions: for example, stability and performance issues will be addressed. It is of particular interest to analyze systems obtained as interconnections (e.g., feedback) of two or more other systems. We will learn how to design (control) systems that ensure desirable properties (e.g., stability, performance) of the interconnection with a given dynamic system. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

7 Your new best friend E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

8 E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

9 Course Objectives 3/3 A large part of the course will require us to work in the Laplace and in the frequency domain and complex numbers, rather than something physical like time and real numbers. This requires a big leap of faith, making the learning curve quite steep for many students. We will make every effort to emphasize the connection between the physical world (and real numbers) and the Laplace/frequency domain (and complex numbers).... if all else fails... E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

10 E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

11 Outline 1 Overview 2 Logistics 3 Signals and Systems E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

12 Course Information Instructor Prof. Emilio Frazzoli Room ML K 32.1 Lead TA Julian Zilly <jzilly@ethz.ch>, Room ML K 42.3 Admin Assistant Ms. Annina Fattor < >, Room ML K32.2 Lectures F 10-12, Room HG F 5, 7. Exercises F 13-15, Various rooms (arranged by groups, refer to Julian). Prof. Office hours TBA. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

13 Reading material Lecture and exercise notes will be posted on the course web site. A nice introductory book on feedback control, available online for free: Feedback Systems: An Introduction for Scientists and Engineers Karl J. Åström and Richard M. Murray Online discussion forum: sign up with your ETH account for L: Control Systems I as a student. Detailed instructions on the course homepage: E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

14 Another suggested (not required) textbook Von Herr Prof. Dr. E. Frazzoli empfohlen Titel Analysis and Synthesis ISBN Autor Lino Guzzella Studentenpreis CHF Preis Normal CHF Erhältlich in den Filialen: ETH Store Polyterrasse Offen Mo-Fr 9-18 Uhr ETH Store Hönggerberg Offen Mo-Fr 8-18 Uhr Sa Uhr Online bestellen: E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

15 Tentative schedule # Date Topic 1 Sept. 22 Introduction, Signals and Systems 2 Sept. 29 Modeling, Linearization 3 Oct. 6 Analysis 1: Time response, Stability 4 Oct. 13 Analysis 2: Diagonalization, Modal coordinates 5 Oct. 20 Transfer functions 1: Definition and properties 6 Oct. 27 Transfer functions 2: Poles and Zeros 7 Nov. 3 Analysis of feedback systems: internal stability, root locus 8 Nov. 10 Frequency response 9 Nov. 17 Analysis of feedback systems 2: the Nyquist condition 10 Nov. 24 Specifications for feedback systems 11 Dec. 1 Loop Shaping 12 Dec. 8 PID control 13 Dec. 15 Implementation issues 14 Dec. 22 Robustness E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

16 Today s learning objectives After today s lecture, you should be able to: Understand control systems in terms of input and output signals. Name examples and describe what states, input and output of a system is Describe the benefits of using control systems to another student Know how to classify signals/systems as linear/nonlinear, causal/acausal, time invariant/variant, memoryless (static) / dynamic Distinguish and calculate different interconnections of systems Distinguish between MIMO and SISO E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

17 Outline 1 Overview 2 Logistics 3 Signals and Systems E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

18 Signals Signals: maps from a set T to a set W. Time axis T, for us this will be the real line, i.e., T = R. One could also consider, e.g., the set of natural numbers (discrete-time systems). Signal space W: for us this will be the real line too, W = R. One could also consider vector-valued signals, for which W = R n for some fixed integer n. y(t) y[k] t k E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

19 Systems: Input-Output models 1/2 In this course, we will consider a system a map between signals, i.e., something that transforms some input signal into some output signal. It is convenient to define an input signal (typically this is the signal that can be manipulated by the designer), and an output signal (which captures how the system performs a certain task). Other signals that are of interest include disturbances and noise. Both are exogenous inputs, but are different in terms of sources and characteristics. More on this later in the course. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

20 Systems: Input-Output models 2/2 An input-output model is a map Σ from an input signal u : t u(t) to an output signal y : t y(t), that is, y = Σu, u Σ y y(t) = (Σu)(t), t T. Block diagram representation E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

21 Your new best friend, revisited E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

22 Memoryless (or static) systems An input-output system Σ is memoryless (or static) if there exists a function S : W W such that, for all t T, y(t) = (Σu)(t) = S(u(t)). Examples: y(t) = 3u(t), y(t) = sin(u(t) 2 ). Not a static system: y(t) = t u(τ) dτ. This system (an integrator) remembers everything that happened in the past. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

23 Time invariance Let the time-shift operator σ τ be defined as follows, for any signal u: (σ τ u)(t) = u(t τ), t T. An input-output system Σ is time-invariant if it commutes with the time-shift operator, i.e., if Σσ τ u = σ τ Σu = σ τ y τ T. Examples: a b E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

24 Linearity An input-output system Σ is linear if, for all input signals u a, u b, and scalars α, β R, Σ(αu a + βu b ) = α(σu a ) + β(σu b ) = αy a + βy b. The key property of linearity is superposition. In other words, if I know that if I apply u a I get y a, and if I apply u b I get y b, then I also know that if I apply u a + u b, I get y a + y b. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

25 Causality An input-output system Σ 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 { u(t) for t T (P T u)(t) = 0 for t > T. Then an input-output system Σ is causal if P T ΣP T = P T Σ, T T. An input-output system Σ 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 (ETH) Lecture 1: Control Systems I 09/22/ / 31

26 Scope of the course In this course, we will consider only LTI SISO systems Single Input, Single Output (i.e., W = R) Linear Time invariant Causal. This is a very restrictive class; in fact, most systems are NOT LTI. On the other hand, many systems are approximated very well by LTI models. This is a key idea. As long as we are mindful of the errors induced by the LTI approximation, the methods discussed in the class are very powerful. Indeed, most control systems in operation are designed according to the principles that will be covered in the course. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

27 Interconnections Control/dynamical systems can be interconnected in various ways: u Σ 1 Σ 2 Serial interconnection: y Σ = Σ 2 Σ 1 u Σ 1 y Parallel interconnection: Σ = Σ 1 + Σ 2 Σ 2 (Negative) Feedback interconnection: u Σ 1 y Σ = (I + Σ 2 Σ 1 ) 1 Σ 1 Σ 2 E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

28 Control objectives Stabilization: make sure the system does not blow up. Regulation: Maintain a desired operating point in spite of disturbances Tracking: follow the reference trajectory as closely as possible. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

29 Basic control architectures r F u P y Feed-forward r e C u P y Feedback F r e C u P y Two degrees of freedom E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

30 benefits/dangers of feedback Feed-forward control relies on a precise knowledge of the plant, and does not change its dynamics. Feedback control allows one to Stabilize an unstable system; Handle uncertainties in the system; Reject external disturbances. However, feedback can introduce instability, even in an otherwise stable system! feed sensor noise into the system. Two degrees of freedom (feedforward + feedback) allow better transient behavior, e.g., can yield good tracking of rapidly-changing reference inputs. E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31

31 Today s learning objectives After today s lecture, you should be able to: Understand control systems in terms of input and output signals. Name examples and describe what states, input and output of a system is Describe the benefits of using control systems to another student Know how to classify signals/systems as linear/nonlinear, causal/acausal, time invariant/variant, memoryless (static) / dynamic Distinguish and calculate different interconnections of systems Distinguish between MIMO and SISO E. Frazzoli (ETH) Lecture 1: Control Systems I 09/22/ / 31