State Space Modelling and Control

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State Space Modelling and Control

• Bjørn Langeland

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Lesson 2
3
Lesson 1

• Modelling analysis - design of dynamic systems
  with the purpose of controlling these systems
• Classical vs. modern control theory
• State space representation
• x(t) Ax(t) Bu(t)
• y(t) Cx(t) Du(t)
• Linearization
• Jacobi matrices
• Find the derivatives of each function with
  respect to the state variables

Linear case
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Modelling

• Use physical laws to describe system dynamics
• Derive differential equations
• Arrange differential equations on input-output
  form
• Chose state variables, output variables and input
  variables
• state variables output of integrators
• Set up n 1. order differential equations
• Linearise if necessary
• Jacobi matrices
• Set up state space form
• State equation
• Output equation

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Todays agenda

• Canonical forms
• State transformations
• the choice of state variables is not
  unambiguously
• a system can be represented by a different set of
  state variables
• purpose to bring out certain characteristics of
  the system or to simplify the system
• Eigenvalues
• describes the systems characteristics with
  respect to stability
• Solving the state space equation
• given an initial state, x(t0) and an input, u(t)
• determine the state x(t) and the output y(t)

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What is canonical form

• From the dictionary
• canonical regular, ideal
• By canonical form
• certain characteristics of the systems are
  accentuate
• or the system is expressed on a certain simple or
  appropriate form
• Several canonical forms exists
• controllable canonical form
• observable canonical form
• Jordan canonical form
• diagonal canonical form
• By transformation from a transfer function to a
  state space representation procedures can be
  chosen such that a certain canonical form is
  expressed.

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State transformations

• Given a system described as a linear time
  invariant model
• x(t) Ax(t) Bu(t)
• y(t) Cx(t) Du(t)
• The choice of state variables is not
  unique/unambiguous
• A new state vector, z, can be chosen to describe
  the same dynamic system. The relationship between
  x and z is described by the following state
  transformation
• x Pz
• z(t) P-1APz(t) P-1Bu(t)
• y(t) CPz(t) Du(t)
• P is called a linear transformation matrix, which
  is constant and nonsingular

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State transformation

• It is important to notice that many of the
  systems characteristics remain the
• same under a state transformation
• For example
• The systems transfer function do not change at a
  linear state transformation
• The eigenvalues of the system do not change at a
  linear state transformation
• Hence lI - A lI - P-1AP

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Eigenvalues

• A is a n by n matrix
• Eigenvalues of A is found as the roots in the
  characteristic equation
• lI - A 0
• As A is of dimension n by n there are n
  eigenvalues, l1, , ln
• reel or complex
• different or identical
• Poles of transfer function eigenvalues of A
• Eigenvalues/poles describes the systems
  characteristics with respect to stability.
• For stability eigenvalues must be in the negative
  complex half plan

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Solving the state equation

• Given a system described as a linear model
• x(t) Ax(t) Bu(t)
• y(t) Cx(t) Du(t)
• What is the response/output of the system when
  affected by a given input?
• First of all the initial state of the system must
  be known
• x(t0) x0
• Next, the new state of the system can be
  calculated
• x(t) x(t, x0, u)
• When the new state, x(t), is known, y(t) is
  easily found.
• y(t) Cx(t) Du(t)

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Solving the state equation

• The challenge is to find x(t) x(t, x0, u)
• Two methods are presented in the book
• Time domain method
• Frequency domain method (Laplace)
• In the following only the main characteristic of
  the two methods are presented
• See in the book for a thorough proof .

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Time domain method

• From solving classical differential equations wee
  know that the solution to the linear
  inhomogeneous diff. equation x Ax Bu with
  the initial condition, x0, can be found as
• x xh xp
• where
• xh is the solution to the homogenous equation x
  ax
• and
• xp is the partial solution to the originally
  diff. equation x Ax Bu
• In other words
• x(t,x0,u) x(t,x0,0) x(t,0,u)
• state response zero-input response zero-state
  response

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Solution to the homogenous equation

• Solve x(t,x0,0)
• Hence
• u 0 ? x(t) Ax(t)
• Solution xh eAt x0

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Particular solution

• A particular solution to the originally linear
  inhomogeneous differential
• equation differentialligning x(t) Ax(t)
  Bu(t)
• Solution

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General solution to x(t) Ax(t) Bu(t)

• Solution to the general inhomogeneous linear
  diff. equation x xh xp
• Hence
• If the initial time t0 not equals 0 then the
  general solution is
• (11-43)
• In order to verify the above solution the
  following conditions must be
• fulfilled
• Satisfy the state space equation x(t) Ax(t)
  Bu(t)
• x(t) must convert towards x0 for t ? 0

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Matrix exponential, eAt

• eAt is called the matrix exponential, and can be
  regarded as a linear transformation of the
  initial state, x0 to the vector x(t), and is
  therefore also denoted the state transformation
  matrix. Referred often as F(t) eAt
• Calculation of eAt
• Power series
• Laplace transformation

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Frequency domain method

• Given x(t) Ax(t) Bu(t)
• Laplace transformation gives
• sX(s) - x(0) AX(s) BU(s)
• ?
• sX(s) - AX(s) x(0) BU(s)
• ?
• (sI - A)X(s) x(0) BU(s)
• ?
• X(s) (sI - A)-1x(0) (sI - A)-1BU(s)
• Inverse Laplace transformation gives x(t)
• x(t) L-1X(s)
• x(t) L-1(sI - A)-1x(0) L-1(sI - A)-1BU(s)

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Todays exercises

• Canonical forms
• B-11-1
• Eigenvalues
• B-11-5
• Solving the state equation
• B-11-7
• B-11-8
• Modelling, linearization
• VT7 project assignment (found on the homepage)
• Homepage is now updated www.iprod.auc.dk/bl/Stat
  e_Space_front.htm