Classifications of systems Lecture 2 Systems and Control Theory - - PowerPoint PPT Presentation

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Classifications of systems

Lecture 2

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Overview

• Based on the number of inputs and outputs (SISO, SIMO,

MISO, MIMO, autonomous)

• Continuous vs. Discrete time
• Linear vs. Nonlinear
• Causal vs. Non-causal
• Time-invariant vs. Time-varying
• Lumped vs. Distributed

bold: this course

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Based on the number of inputs and outputs

• SISO: Single Input Single Output
• SIMO: Single Input Multiple Outputs
• MISO: Multiple Inputs Single Output
• MIMO: Multiple Inputs Multiple Outputs
• Autonomous: No inputs (one or more outputs)

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Continuous vs. Discrete time

• During this course we will discuss both simultaneously, in
• rder to emphasize the similarities (and differences)
• A continuous system has continuous input and output signals
• We denote continuous time by 𝑢 ∈ ℝ
• We denote functions of continuous time with round

brackets, e.g.: 𝑦 𝑢

• A discrete system has discrete input and output signals
• We denote discrete time by 𝑙 ∈ ℤ
• We denote functions of continuous time with square

brackets, e.g.: 𝑦 𝑙

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Continuous vs. Discrete time Continuous

For every moment 𝑢 ∈ ℝ, the system has:

• A vector of inputs 𝑣 𝑢
• A vector of outputs 𝑧 𝑢
• A vector of states 𝑦 𝑢

Discrete

For each timestep 𝑙 ∈ ℤ, the system has:

• A vector of inputs 𝑣 𝑙
• A vector of outputs 𝑧[𝑙]
• A vector of states 𝑦[𝑙]

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear: A linear system

• Definition:

A system is linear if 𝑣1 𝑢 → 𝑧1 𝑢 (input 𝑣1 𝑢 results in

• utput 𝑧1 𝑢 ) and 𝑣2 𝑢 → 𝑧2(𝑢) imply that

𝛽 𝑣1 𝑢 + 𝛾 𝑣2 𝑢 → 𝛽 𝑧1 𝑢 + 𝛽 𝑧2(𝑢)

• Properties of a linear system (contained in the definition):
• Superposition
• Homogeneity

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear: A linear system

• Properties of a linear system (contained in the definition):
• Superposition

𝑣𝑏 𝑢 → 𝑧𝑏 𝑢 , 𝑣𝑐 𝑢 → 𝑧𝑐 𝑢 ⇔ 𝑣𝑏 𝑢 + 𝑣𝑐 𝑢 → 𝑧𝑏 𝑢 + 𝑧𝑐 𝑢 This means the output of a system can be found by splitting up the input and solving it separately (analogous to the homogeneous part of an ordinary differential equation)

• Homogeneity

𝛽𝑣 𝑢 → 𝛽𝑧 𝑢

• How to recognize a linear system:
• Linear in all of the variables
• No constant factors

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear: A linear system

• Examples
• 𝑦 = 𝑣

𝑧 = 𝑦 + 2𝑣 Linearity of this system is easily verified, based on the linearity of the derivative: 𝛽 𝑦𝑏 𝑢 + 𝛾 𝑦𝑐 𝑢 = 𝛽𝑣𝑏 𝑢 + 𝛾𝑣𝑐 𝑢 𝛽 𝑧𝑏 𝑢 + 𝛾 𝑧𝑐 𝑢 = 𝛽𝑦𝑏 𝑢 + 𝛾𝑦𝑐 𝑢 + 2𝛽𝑣𝑏 𝑢 + 2𝛾𝑣𝑐 𝑢

• 𝑦1 = 𝑣

𝑦2 =

3 2 𝑦1 + 𝑣

𝑧 = 𝑏𝑦1 − 𝑦2 + 2𝑣

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear: Autonomous linear systems

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear: violating homogeneity

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear: Nonlinear systems

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear: Nonlinear systems

Using a term like nonlinear systems is like referring to the bulk of zoology as the study of non-elephant animals.

• S. Ulam

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear

Predominantly linear

Simple electrical systems

• Circuits with ideal

resistors, capacitors and inductors Simple mechanical systems

• Systems with ideal

springs

Inherently nonlinear

Chemical systems Biological systems Economical systems More involved electrical or mechanical systems …

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Linear vs. Nonlinear

• Reality is nonlinear
• However, this course will only deal with linear systems

(elephants)

• Why we prefer linear systems:

The previously mentioned properties will allow for a thorough study of the system

• Why we are allowed to use linear systems, even in a nonlinear

setting: You can linearize around an equilibrium point (we will do this in the next lecture)

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Causal vs. Non-causal

• A causal system only depends on the present and the past,

not on the future

• An non-causal system (also) depends on the future
• (Almost) all physical systems are causal:
• A telephone:
• It will not ring for future calls
• Any human:
• Is a system that will only react on inputs it has already

received

• If we react because we expect something to happen in

the future, then that expectation arose from past or present inputs

Examples come from: http://www.deekshith.in/2013/03/causal-and-non-causal-systems-better-explained.html

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Causal vs. Non-causal

• How do non-causal systems arise?
• A possibility is by greatly reducing the complexity of a system,

in which some causes of events are taken out of the equations:

• Take for instance an economics model in which we model the

consumption (output)

• An incredible amount of factors determine this, but we only have the

employment numbers (input)

• Current and past employment numbers determine consumption, but

when someone gets fired, they will continue to work for several weeks in most instances, but their consumption will drop immediately  a correct model for this relation would have to be non-causal

• The non-causal model for this input-output relation is not useful if you

want to determine the level of consumption

• You could use the relation to see a drop in employment, before it is

visible in the employment numbers

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Causal vs. Non causal

Examples of non causal systems: expectations

• Modelling housing prices
• People are willing to offer more for houses if they expect

rising prices

• It is hard to measure the expectations of housing prices
• Sometimes economists use their own predictions of

housing prices to replace the expectations

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Causal vs. non causal

Original image Removed details Highlight borders

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Causal vs. Non-causal

The output only depends (directly and indirectly) on the input up to time 𝑙

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Time-invariant vs. Time-varying

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Time-invariant vs. Time-varying

• Examples of time-varying systems:
• The properties of an electrical circuit slowly change over

time

• The human body also has many changing properties
• Systems affected by night and day (Heating of buildings),

when those aspects were ignored in the model

• Examples of (de facto) time-invariant systems:
• A system that describes a physical law, for instance a

system with two masses as its input and their attractive force as an output

• In practice we approximate all systems whose properties

change much slower than the variables as time-invariant

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Lumped vs. Distributed

Reactor 𝑣 𝑢 𝑧 𝑢

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Lumped vs. Distributed

• Now we want to determine the amount of states
• The states should describe the reactor in such a way that we

will be able to determine the reaction rate throughout the reactor

• The rate of the reaction is determined by the local

concentrations of the three compounds, which means we need two states at every point in the reactor (the reactor is not uniform)

• This means we need an infinite amount of states, we say that

this is a distributed system

• The relations in a distributed system are naturally expressed

by PDE’s

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Lumped vs. Distributed

• Do we really need an infinitely large system for this problem?
• Not if we discretize the system, in which we take a finite

amount of points at which we keep track of the concentration

• Physically this would mean we assume that the

concentrations are locally uniform

• For some types of reactors you can go as far as to assume the

reactor is perfectly stirred; with a homogeneous content

• If we have a finite amount of states (for instance by

discretizing), we call the system a lumped system

• It can be expressed in a finite amount of ODE’s

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