Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems. Guillaume Drion Academic year

Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems Guillaume Drion Academic year 2015-2016 1

Outline Systems modeling: input/output approach and LTI systems. Convolution in discrete-time. Convolution in continuous-time: the Dirac delta function. Causality, memory, responsiveness of LTI systems. 2

Systems modeling Modeling and analysis of systems: open loop. Observing and analyzing the environment Input SYSTEM Output Can be used to understand/analyze the behavior of a dynamical system. A good model can predict the future evolution of a system. How can we use systems modeling to predict the future? What is a good model or a good system to model? 4

Systems modeling: state-space representation Last lecture, we saw that the state-space representation of a model can describe its behavior. Example: RLC circuit. i R V v R (t) C v L (t) L v C (t) Such representation can be used to predict the future behavior of the system when subjected to a specific input, providing that we know its current state. 5

Systems modeling: state-space representation Example: the Windkessel model for variations in blood pressure Model Simulation u(t) r P(t) Pr(t) PCa(t) Ca R 6

Systems modeling: state-space representation But what if the system is like this? How many states/equations would you need? 7

or like this? Systems modeling: state-space representation 8

or like this? Systems modeling: state-space representation 9

or like this? Systems modeling: state-space representation Sometimes, you want to know how the system reacts to inputs, but you do not care about all the details of the internal dynamics. Input-output representation! 10

Input-output representation in time domain Any system S can be represented using an input-output representation: u S y Can we mathematically describe the system using the following relationship? What kind of system can be described this way? 11

Input-output representation in time domain Example: think about the evolution of a stock market. Can we accurately predict the evolution of a stock market using a model? 12

Input-output representation in time domain Example: think about the evolution of a stock market. Can we accurately predict the evolution of a stock market using a model? where the evolution depends on the market growth rate, i.e the market growth rate is a parameter of the system. We all know that the market growth rate evolves over time, so the prediction that we will make at a time t using the rate at this time t will not accurately describe the evolution of the stock market! No good prediction! 14

Time-Invariant (TI) systems In a time-invariant system, the input/output relationship does not depend on time: if then or if then Examples: is time-invariant. is time-variant. 16

Input-output representation in time domain Can we mathematically describe a time-invariant system using the following relationship? In particular, can we find a function that will predict the output of the system for any input, whatever the complexity of the input? 20 mv 400 ms 17

Input-output representation in time domain Can we mathematically describe a time-invariant system using the following relationship? It is possible if the system obeys the superposition principle: if and then Superposition principle = additivity + homogeneity. If a system obeys the superposition principle, we can express the (possibly complex) input signal as a sum of simple input signals! 18

Linear, Time-Invariant (LTI) systems Can we mathematically describe a time-invariant system using the following relationship? It is possible if the system obeys the superposition principle: if and then The superposition principle is valid for linear systems. For all these reasons, this course will focus on Linear, Time-Invariant (LTI) systems. 19

Does it make sense to study linear systems? Is a physical/biological system linear? Linearity implies homogeneity:. All physical/biological systems saturate none of them are totally linear. Examples: I/V curve of a diode (left) and force/travel curve of a suspension (right) 20

Does it make sense to study linear systems? Is a physical/biological system linear? However, many systems are almost linear in their functional range, and/or can be decomposed in linear subsystems! Examples: I/V curve of a diode (left) and force/travel curve of a suspension (right) High g High g Low g 21

Input-output representation of LTI systems Can we mathematically describe a LTI system using the following relationship? 22

Input-output representation of LTI systems Can we mathematically describe a LTI system using the following relationship? Using the superposition principle, we can analyze the input/output properties by expressing the input signal into the sum of simple signals: if then 23

Input-output representation of LTI systems Using the superposition principle, we can analyze the input/output properties by expressing the input signal into the sum of simple signals: if then What is the simplest signal: a pulse! The response of a system to a pulse is called the impulse response. Therefore, any LTI systems can be fully characterized by its impulse response. 24

Definition of a pulse in discrete-time The pulse in discrete time is defined by 1 δ[n] -4-3 -2-1 0 1 2 3 4 n 26

Role of a pulse in discrete-time The pulse in discrete time is defined by If we multiply any signal by, we retrieve a signal that only contains the value of the input signal at :. u[0] u[n]δ[n] -4-3 -2-1 0 1 2 3 4 n 27

Role of a pulse in discrete-time The pulse in discrete time is defined by Similarly, if we want to retrieve a signal that only contains the value of the input signal at any value, we need to multiply the signal by : u[n]δ[n-2] u[2] -4-3 -2-1 0 1 2 3 4 n 28

Role of a pulse in discrete-time If we retrieve signals that only contain the value of the input signal at for all and sum them, we retrieve the initial signal: A signal can therefore be decomposed into an infinite sum of unit impulse signals. Example: decomposition of the unit step function: 29

Role of a pulse in discrete-time A signal can therefore be decomposed into an infinite sum of unit impulse signals.... u[-2]δ[n+2] -4-3 -2-1 0 1 2 3 4 + n u[n] u[-1]δ[n+1] -4-3 -2-1 0 1 2 3 4 n = -4-3 -2-1 0 1 2 3 4 + n u[0]δ[n+0] -4-3 -2-1 0 1 2 3 4 n 30...

Impulse response of discrete systems Can we use this decomposition to analyze the input/output properties of a discrete LTI system? Yes, we can use the superposition principle which gives where is the impulse response of the system. 31

Impulse response of discrete systems A LTI system is fully characterized by its impulse response. What does it mean? It means that the response of a LTI system at an instant depends on all the past, present and future values of the input, each of them having a gain equal to. If the impulse response has a finite window, the size of this windows defines the memory of the system. Examples of convolutions. 33

Cascade of systems Show that the impulse response of a cascade of LTI systems is the equal to the convolution of the impulse response of each subsystem. u h 1 [n] x h 2 [n] y 34

Definition of a pulse in continuous-time How can we define a pulse in continuous-time? Similarly to the discrete case, we have to define a signal such that for all signal continuous at the origin, and (step function) is the derivative of the step function, which is discontinuous at the origin! 36

Definition of a pulse in continuous-time: the Dirac delta function The Dirac delta function can be defined as a square of width and height with (defined by its integral equal to 1). ε 0 δ(t) -ε/2 ε/2 0 1/ε t 37

Convolution in continuous-time Any continuous signal can be expressed as a sum (integral) of delta functions: Therefore, the output of a continuous LTI system can be expressed as where is the impulse response of the LTI system. In continuous time, the convolution is 39

Convolution in continuous-time 40

Properties of convolution Convolutions are commutative (show it): Convolutions are associative: Convolutions are distributive (show it): 41

Properties of LTI systems We can extract informations about a LTI system using the shape of the impulse response. Causality: the output only depends on past values of the input. It means that only depends on if. In terms of the impulse response, it means that Indeed, if, causality implies that 43

Memory of LTI systems The memory of a system is defined by the window of its impulse response. The larger the impulse response window, the bigger the memory. h[n] -4-3 -2-1 0 1 2 3 4 n Output depends on the present and the previous input values h[n] -4-3 -2-1 0 1 2 3 4 n Output depends on the present and the 4 previous input values 44

Memory of LTI systems Static systems: depends on only:. This gives where is the static gain of the system. Input Output 0 0 1 K 45

Memory of LTI systems Static systems: depends on only:. This gives where is the static gain of the system. Input Output 0 0 1 K Dynamical systems: the response of the system is limited by the window of its impulse response! (reaches steady-state after some time). Input Output 0 0 1 K 46

Response time of LTI systems The response time of a LTI dynamical system is linked to the time-window of its impulse response. Indeed, if is the length of the impulse response and the length of the input signal, the output signal will have a length of. (can be easily shown graphically). The response-time of a system is defined by its time-constant. 47

Time-constant of LTI systems The general form of a the impulse response of a LTI system is a decaying exponential infinite window. We usually define the time-constant of a system as. 48

Time-constant of LTI systems If the impulse response is the exponential decay: Then and a time- This is the typical response of a first order system. First order systems are characterized by a static gain constant. 49

Time-constant of LTI systems 50

Time-constant of LTI systems Example: high energy photo detector. 51

Time-constant of LTI systems Example: high energy photo detector. 52

Time-constant of LTI systems The time-constant is important for filtering: with = cutoff frequency. Example: the cardiovascular system modeled in lecture #1 (low-pass filter). Low pass filter R H(s) = RC a s +1 + r Atherosclerosis: loss of arterial compliance => C a decreases => τ=rc a decreases τ = RC a 53

Time-constant of LTI systems Other useful response: the step response. 54

Highlights of the day Input/output approach. Delta function in discrete-time. Linear, Time-Invariant systems. Convolution (+properties). Superposition principle. Causality, memory response-time. Impulse response and step response. Time-constant and cutoff frequency. Dirac delta function in continuous-time. 55