Control Systems I. Lecture 1: Introduction. Suggested Readings: Åström & Murray Ch. 1. Jacopo Tani

Control Systems I Lecture 1: Introduction Suggested Readings: Åström & Murray Ch. 1 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 21, 2018 J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 1 / 30

Figure: Space-X Falcon IX landing Figure: nutonomy self driving car in action J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 2 / 30

Examples of control systems fields of application Thermostat Aerospace Power generation and transmission Robotics Transportation networks J. Tani, E. Frazzoli (ETH) Biological systems Lecture 1: Control Systems I 09/21/2018 3 / 30

Outline 1 Overview and course objectives 2 Logistics 3 Signals and Systems J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 4 / 30

The hidden technology [Karl Åström] Widely used Very successful Seldom talked about Except when disaster strikes J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 5 / 30

Course Objectives 1/3 This course is about control of dynamic systems, i.e., systems that evolve over time, have inputs and outputs. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 6 / 30

Course Objectives 1/3 This course is about control of dynamic systems, i.e., systems that evolve over time, have inputs and outputs. The control problem is finding the right input sequence, over time, such that the system s output follows a reference signal. Or, in other words: make a system behave like the user wants to, and not like it would naturally behave. We learn how to control systems by achieving three objectives: Modeling: learn how to represent a dynamic control system in a way that it can be treated effectively using mathematical tools. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 6 / 30

Course Objectives 1/3 This course is about control of dynamic systems, i.e., systems that evolve over time, have inputs and outputs. The control problem is finding the right input sequence, over time, such that the system s output follows a reference signal. Or, in other words: make a system behave like the user wants to, and not like it would naturally behave. We learn how to control systems by achieving three 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. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 6 / 30

Course Objectives 1/3 This course is about control of dynamic systems, i.e., systems that evolve over time, have inputs and outputs. The control problem is finding the right input sequence, over time, such that the system s output follows a reference signal. Or, in other words: make a system behave like the user wants to, and not like it would naturally behave. We learn how to control systems by achieving three 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. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 6 / 30

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 timeinvariance 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. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 7 / 30

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. Efforts will be made to emphasize the connection between the physical world (and real numbers) and the Laplace/frequency domain (and complex numbers).... if all else fails... J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 8 / 30

J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 9 / 30

Outline 1 Overview and course objectives 2 Logistics 3 Signals and Systems J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 10 / 30

Course Information Instructor Dr. Jacopo Tani <tanij@ethz.ch>, Room ML K 37.3. Lead Teaching Assistant Dr. Shima Mousavi <mousavis@ethz.ch>, Room ML K 37.4. Admin Assistants Julian Zilly <jzilly@ethz.ch>, Annina Fattor <+41 44 632 87 96>, Room ML K32.2. Lectures Friday 10-12, Lecture room HG F 7, with video transmission in HG F 5. Exercises Friday 13-15, Various rooms (see course catalouge). Study center Wednesday 13-15 in room ETZ E 8 (starting from week 3). Instructor office hours Tuesday 14-15 in room ML K 37.3. (Excluding 13.11, 4.12. I will be out of town those weeks. Dr. Mousavi will substitute me.) J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 11 / 30

Smile, you are on camera! All lectures are recorded and will be publicly available online. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 12 / 30

CS I Staff Lead Teaching Assistant Dr. Shima Mousavi <mousavis@ethz.ch> Teaching Assistant Biagosch Carl Philipp Friederich Rockenbauer Giuseppe Rizzi Jasan Zughaibi Luna Meeusen Marc Leibundgut Moritz Reinders Yannik Schnider Email contact <carl.biagosch@juniors.ethz.ch> <rockenbf@student.ethz.ch> <grizzi@student.ethz.ch> <zjasan@student.ethz.ch> <lmeeusen@student.ethz.ch> <leimarc@student.ethz.ch> <rmoritz@student.ethz.ch> <yannicks@student.ethz.ch> J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 13 / 30

Reading material Lecture slides 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 http://www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Online discussion forum: https://piazza.com/, sign up with your ETH account for 151-0591-00L: Control Systems I as a student. Detailed instructions on the course homepage: http://www.idsc.ethz.ch/education/lectures/control-systems-i.html J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 14 / 30

Tentative schedule # Date Topic 1 Sept. 21 Introduction, Signals and Systems 2 Sept. 28 Modeling, Linearization 3 Oct. 5 Analysis 1: Time response, Stability 4 Oct. 12 Analysis 2: Diagonalization, Modal coordinates 5 Oct. 19 Transfer functions 1: Definition and properties 6 Oct. 26 Transfer functions 2: Poles and Zeros 7 Nov. 2 Analysis of feedback systems: internal stability, root locus 8 Nov. 9 Frequency response 9 Nov. 16 Analysis of feedback systems 2: the Nyquist condition 10 Nov. 23 Specifications for feedback systems 11 Nov. 30 Loop Shaping 12 Dec. 7 PID control 13 Dec. 14 State feedback and Luenberger observers 14 Dec. 21 On Robustness and Implementation challenges J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 15 / 30

Today s learning objectives After today s lecture, you should be able to: Understand the approach of control systems in terms of systems with input and output signals Name examples and describe what input, output and states of a system are 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: series, parallel, feedback Distinguish between MIMO and SISO systems J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 16 / 30

Outline 1 Overview and course objectives 2 Logistics 3 Signals and Systems J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 17 / 30

Signals Signals are maps from a set T to a set W. They receive a number as input and produce a number as output. Think of T as the time axis. It will be the real line, i.e., T = R, when talking about continuous-time systems. This is how things work in nature. Or it could be the set of natural numbers: T = N, when talking about discrete-time systems. This is how things work on a computer. 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 J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 18 / 30

Systems: Input-Output models 1/2 Systems: in this course we will consider a system as a map between signals, i.e., something that transforms some input signal into an output signal. Input signals can be thought of as something that can be manipulated by the user. Output signals instead capture how the system responds to a certain input. 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. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 19 / 30

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 Block diagram y(t) = (Σu)(t), t T. representation Depending on how Σ affects u, we classify the system in different ways. For example: Static (memoryless) vs. Dynamic Linear vs. Nonlinear Causal vs. Acausal (not to be confused with casual!) Time invariant vs. Time variant. In this course, we will study how to control dynamic systems that are linear, and time invariant; or LTI systems. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 20 / 30

Memoryless (or static) systems J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 21 / 30

Memoryless (or static) systems An input-output system Σ is memoryless (or static) if for all t T, y(t) is a function of u(t). J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 21 / 30

Memoryless (or static) systems An input-output system Σ is memoryless (or static) if for all t T, y(t) is a function of u(t). In a static (memoryless) system: the output at the present time depends only on the value of input at the present time; not on the value of input in the past or the future time. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 21 / 30

Memoryless (or static) systems: Examples Static systems: y(t) = 3u(t), y(t) = 2 (t+1) u(t), y(t) = sin(u 2 (t)), y(t) = R[u(t)] = u(t)+u (t) 2. Dynamic system: y(t) = t u(τ) dτ, This system (an integrator) remembers everything that happened in the past. y(t) = u(t), y(t) = u(t 2 ), y(t) = u(t a), a 0. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 21 / 30

Causal systems 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]. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Causal systems 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: a system is causal if and only if the future input does not affect the present output. All practically realizable systems are causal. (It is impossible to implement a non causal system in the real world.) J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Causal systems: truncation operator P T We can express causality mathematically by introducing a truncation operator P T : { u(t) for t T (P T u)(t) = 0 for t > T. u P T P T u J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Causal systems 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). J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Causal vs. non-causal systems: Examples Causal systems (all practically/physically realizable systems are causal): y(t) = u(t), y(t) = u(t τ), τ > 0 (systems with delay), y(t) = cos(3t + 1)u(t 1) (don t make the +1 fool you) y(t) = t u(τ) dτ Non-causal system: y(t) = u(t a), a < 0, (actually anti-causal) y(t) = u(t + 1) + u(t) + u(t 1), (non causal) y(t) = u(bt), b > 0, y(t) = t+1 u(τ) dτ. Q: Is y(t) = u(t) causal or non-causal? Furthermore, is it realizable or not? J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Causal vs. non-causal systems: Examples Causal systems (all practically/physically realizable systems are causal): y(t) = u(t), y(t) = u(t τ), τ > 0 (systems with delay), y(t) = cos(3t + 1)u(t 1) (don t make the +1 fool you) y(t) = t u(τ) dτ Non-causal system: y(t) = u(t a), a < 0, (actually anti-causal) y(t) = u(t + 1) + u(t) + u(t 1), (non causal) y(t) = u(bt), b > 0, y(t) = t+1 u(τ) dτ. Q: Is y(t) = u(t) causal or non-causal? Furthermore, is it realizable or not? A: It is causal but not realizable. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 21 / 30

Time-invariant vs. time-variant systems A time invariant system is a time dependant map between input and output signals that is the same at any point in time. In other words, the system manipulates the input in a way that does not depend on when the system is used. More formally: consider the time-shift operator σ τ : (σ τ u)(t) = u(t τ), t T. u σ τ σ τ u J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Time-invariant systems A system is time-invariant if: Σσ τ u = σ τ Σu = σ τ y τ T. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Time-invariant vs. time-variant systems: Examples Time-invariant: y(t) = ku(t) + c, c, k, y(t) = 3sin(u(t)), y(t) = t u(τ)dτ, y(t) = u(t 1) + u(t + 2). Time-variant: y(t) = u(at), a 0 (time scaling), y(t) = 3u(sin(t)) (always time scaling), y(t 2 ) = u(t) (always time scaling, but on the output), y(t) = cos tu(t) (if a coefficient is time dependent), y(t) = u(t) + t (if any summed term - except input/output - is time dependent), y(t) = t u(2τ)dτ. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 22 / 30

Linear systems An input-output system is linear if it is additive and homogeneous. Additivity: Σ(u 1 + u 2) = Σu 1 + Σu 2 Homogeneity: Σ(ku) = kσu. In other words, 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. u a y a u b Σ Σ y b αu a Σ αy a The key idea is superposition: αu a + βu b Σ αy a + βy b J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 23 / 30

Linear vs. nonlinear: Examples Linear: y(t) = u(sin t), y(t) = u(t 2 ) (time scaling doesn t make a system nonlinear), y(t) = au(t), a, y(t) = cos(t)u(t) (coefficients don t influence linearity), y(t) = u(t + a) + u(t b) (time shift does not affect linearity) y(t) = t u(τ)dτ, y(t) = u(t) (differential and integral operators are linear). Nonlinear: y(t) = u 2 (t), y(t) = u(t) + a, (summed terms - except input or output - make a system nonlinear) y(t) = sin(u(t)), y(t) = R(u(t)). J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 24 / 30

Which systems will we treat in this course, and why? In this course, we will consider only LTI SISO systems: Single Input, Single Output, Linear, Time invariant, Causal. This is a very restrictive class of systems; 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. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 25 / 30

Interconnections of systems 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 1 Σ 2 ) 1 Σ 1 Σ 2 J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 26 / 30

What are the objectives of a control system? Stabilization: make sure the system does not blow up. Regulation: maintain a desired operating point in spite of disturbances. Tracking: follow the reference/desired trajectory/behavior as closely as possible. Robustness: the controller that satisfies the above works even if the system is slightly different than we expected. Robustness is a more advanced topic which will be treated in Control Systems II. We will introduce it in the last class of this course. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 27 / 30

Basic control architectures r F u P y Feed-forward J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 28 / 30

Basic control architectures r F u P y Feed-forward r e C u P y Feedback J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 28 / 30

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 J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 28 / 30

Benefits/dangers of feedback Feed-forward control relies on a precise knowledge of the plant, and does not change its dynamics. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 29 / 30

Benefits/dangers of feedback Feed-forward control relies on a precise knowledge of the plant, and does not change its dynamics. Feedback is error based, compensates for unexpected / unmodeled phenomena (disturbances, noise, model uncertainty). 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. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 29 / 30

Today s learning objectives After today s lecture, you should be able to: Understand control systems in terms of input and output signals of systems. Know how to classify signals/systems as linear/nonlinear, causal/acausal, time invariant/variant, memoryless (static) / dynamic. Distinguish and calculate different interconnections of systems. Explain the acronyms MIMO, SISO, LTI. Describe the benefits of using control systems to another student. J. Tani, E. Frazzoli (ETH) Lecture 1: Control Systems I 09/21/2018 30 / 30