State Space Modelling and Control

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State Space Modelling and Control

• Bjørn Langeland

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Practical

• Lecture plan has been provided
• Lesson 1 - 5 by BL
• Lesson 6 -10 by HH
• Literature for lesson 1-5
• Book K. Ogata Modern Control Engineering.
• Various notes will be provided or can be found on
  the homepage.
• Complementary literature
• O. Jannerup og P.H. Sørensen Introduktion til
  reguleringsteknik (in Danish)
• Homepage http//www.iprod.auc.dk/bl/
• Evaluation
• EMSD7 1 module, PE course.
• V7T 2 modules, SE course In groups written
  assignment oral examination

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Why is this course important?

• Because control is in everything. For example in
• production processes welding, painting, bending,
  casting
• mechanical constructions motors, robots,
  hydraulics, pneumatics, ...
• electrical constructions motors (AC, DC), elec.
  circuits , distribution systems.
• information systems navigation, aviation
• thermal systems
• chemical systems
• As an engineer control is an important field by
  be familiar with
• as an expert
• as a interdisciplinary project member
• as project manager
• This course is a piece in your fundamental
  knowledge within control.

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What is the course about?

• Fundamentally, for a given system we want
• to describe the systems dynamical behaviour.
• to investigate the characteristics of the system.
• plus to control the system. I.e. how must the
  system be influenced, so that the system follow
  the reference that we specify.
• Hence, the course is fundamentally about how
• to model a system
• to analyse a system
• and to design a controller for the system

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Lesson 1
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Agenda

• Introduction
• Classical control theory vs. Modern control
  theory
• General State Space representation
• How to represent/describe/model a system?
• Linearization
• Why and how to linearise?

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Modelling and control of what?

• For a given system we want to determine the
  development of a given variable.
• For example
• The height of the water level in a tank
• The temperature in a room, an oven, etc.
• (Fuel) flow to an engine or a tank
• Velocity of a car, airplane, ship or similar
• The position of the sledge on a lathe
• Position and velocity of a robot arm

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Control of motors
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Control of robot (velocity and position)
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Process control Temp., flow, concentration, etc.
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Control of windmill
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Servo control, aircond., autopilot, ect.
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Production equipment
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Control of variables

• Hence, we want, automatically, to control many
  different variables
• Temperature
• Flow
• Velocity
• Position
• Pressure
• Moment
• Concentration
• The theory presented in the course can be used
  regardless of what variables, which we are
  interested in.

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SimuLink
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Response
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Classical and modern control theory

• Classical control theory
• Input-output representation
• Frequent domain (Laplace)
• SISO
• Linear models only
• Time invariant
• Design by graphical techniques
• or by hand
• Trial error
• Engineering. Intuition
• Simple systems

• Modern control theory
• State Space representation
• Time domain
• MIMO
• Linear models only
• Non linear -gt linearization
• Time invariant
• Time variant
• Formal design
• Suitable for computer aided
• Complex systems

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Why use state space?

• We want to represent a systems dynamical
  behaviour
• Input-Output representation (Laplace), Y(s)
  G(s) U(s), insufficiently
• By state space modelling the systems internal
  states are described. Thus, maximum information
  about the system is obtained.
• If we only look at a systems output (ex.
  position of mass), we do not get any information
  about the systems state ex. The velocity of the
  mass.
• The systems we are interested in is often
  described by differential equations, which
  describes changes of systems (often in relation
  to time)

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Why use state space?

• If a systems dynamical behaviour is described by
  more differential equations, these equations can
  be gathered to matrices, which are a suitable
  form for expressing systems when a computer must
  make calculations.
• For MIMO systems the Laplace representation is
  too awkward to use.
• When a system has been described mathematically
  (by state space), a number of formal calculations
  concerning the system can be made.
• I.e. by use of pure mathematics we get
  information concerning the systems
  characteristics.

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Modelling - State space representation

• When representing/describing/modelling a system
  mathematically the
• following two type of equations are used
• State space equation
• Describes the relationship between changes in
  time of the systems state and the system current
  state and the input given to the system
• f (x, u, t)
• 1. order ordinary differential equations
• Output equation
• Describes the relationship between the systems
  output and the current state of the system and
  the input given to the system
• y g (x, u, t)

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Variables in a state space model

• State variables
• describes the systems inner state
• denoted by x
• for example position, speed, temperature,
• often describes a physical characteristic
• Output variables
• variables which can be measured
• denoted by y
• for example often a subset of the state
  variables
• Control variables/Input variables
• the variables which directly can by
  controlled/altered
• denoted by u
• for example current, voltage, open, close,

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State space equation and output equation

• Starting point One nth order differential
  equation
• Objective n 1. order differential equations

• State space equation Output equation

• In the linear case the two set of equations can
  be expressed on matrix form

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How to chose state variables

• As principal rule the outputs of the integrators
  are chosen as the variables to describe the
  system
• Hence
• If a system dynamical behaviour is described by a
  number of variables and its derivatives
• then the state variables (x1, x2, x3, xn) are
  chosen to be
• the system variables
• the system variabless derivatives minus 1.
• In principle other derivatives can be used, but
• by historical reasons x has been chosen
• furthermore a function with x on the left side
  is the simplest form to express a change in
  relation to a given variable.

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State space rep. of linear dynamic system

• Model of system to be controlled

A System matrix (state matrix) B Input
matrix C Output matrix D Direct transmission
matrix
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SISO and MIMO systems

• SISO
• Single Input Single Output
• MIMO
• Multiple Input Multiple Output
• SIMO
• Single Input Multiple Output
• MISO
• Multiple Input Single Output

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Time variant and time invariant systems

• Time variant systems
• Systems which output depends on the time at which
  an input is introduced.
• Time invariant systems
• Systems which behaviour do not change as time
  goes by.
• A system is time variant, when the constants in
  the equations describing the system changes as a
  function of the time at which a input is
  introduced.
• A system with for example a mass which changes is
  called a time variant system (e.g. like a rocket
  consuming fuel)

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Linear and non linear systems

• Linear systems
• fulfil the superposition principle
• systems which can be described by linear
  differential equations
• linear combinations of variables or/and
  derivatives of variables
• Non linear systems
• systems which are described by equations where
  the variables or their derivatives are in the
  power greater than 1.
• systems where the constants in the equations
  depend on the input to the system or by the
  systems state.
• split functions
• sin or cos of variables or derivatives

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Deterministic and stochastic systems

• Deterministic systems
• A system is deterministic, when the same input
  always gives the same output
• Stochastic systems
• A system is stochastic, when the same input gives
  different output
• If the input is from an engine, and the loaded on
  the engine is unknown (but varies stochastically)
  the system is said to be a stochastic system.
• For example when controlling a ship (speed
  or/and position) , the waves, the wind, and the
  current, will make it impossible to know the
  effect of the engines output on the ships
  speed/position.

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Continuous time and discrete time systems

• Continuous time systems
• Systems where all variables are defined/known at
  all instants of time.
• Typically described by differential equations.
• For example Systems where changes to the systems
  happen as time goes by.

• Discrete time systems
• Systems where variables are sampled/measured at
  discrete instants of time, and hence only defined
  at these time instants. Described by difference
  equation.
• For example Systems where changes to the systems
  happen as time goes by, and which we want to
  control by computers.

u(k)
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Type of systems in this course

• In the course both continuous time and discrete
  time systems are treated.
• Systems with the following characteristics are
  treated
• Time invariant
• Deterministic
• Linear
• (Non linear) - linearization
• The course will deal with how such systems
• are modelled
• analytical/mathematical modelling
• (experimental modelling -gt system identification
  (8. semester))
• are analysed
• are designed

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Transfer functions and state space repre.

• A correlation exists between transfer functions
  in the Laplace domain and the state space
  representation in the time domain .
• Laplace
• Transformation of differential equations to an
  arithmetic expression
• can be solved in hand
• Y(s) G(s) U(s)
• State space representation
• Differential equations
• for solving large and complex systems
• x(t) F(x, u, t) y(t) F(x, u, t)
• The purpose of transforming state space
  representation to a Laplace representation is the
  possibility to make a frequency analysis of the
  system. In order to further understand the system
  dynamics.

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Correlation between transfer functions and
state space equations

• Transfer function, G(s)
• State space equations
• Relationship

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Linearization - why?

• In practise almost all dynamic systems are
  non-linear.
• The linear theory, however, gets a relative high
  general relevance, as
• many non-linear systems can be linearized.
• The linear models thus appearing can be used to
  design a control unit for the system around a
  reference state.
• Hence, it must be assumed that the system operate
  in or nearby a certain point (work point).

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Linearization - how?

• A non-linear function is linearized by describing
  the tangent to a given work point on the
  non-linear function.
• Linearization of a function of one variable
• Dx is the change in x
• Dy is the actual change in y
• dy is the approximated change in y
• dy f(x0)Dx
• hence
• f(x0 Dx) f(x0) f(x0)Dx
• Problems
• the curve of the function in the work point
• the work point, x0, must represent norm state

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Linearization in general

• Taylor serie

Equilibrium point f(x0) 0
Higher order joints is not included Dy - dy
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Jacobi matrices
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Today's exercises

• Modelling
• Ogata B-3-9
• Ogata B-3-15
• Transfer function vs. state space representation
• Ogata B-3-10
• Linearization
• To be found on the homepage