State Space Modelling and Control

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State Space Modelling and Control

• Morten Kristiansen
• Bjørn Langeland

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Lesson 3
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Modelling

• Modelling and analysis of dynamic systems with
  regard to design of a control for these systems
• A systems dynamical behaviour can be described by
  differential equations and be expressed at state
  space form formalism by the state space equation
  and output equation
• These systems are modelled graphically as shown
  below (D 0)

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Analysis of system response

• It applies that the response of the system is
  given by
• I.e. Given an initial state, x(0) and an input,
  u(t), can the response characteristic of the
  system be drawn

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Todays agenda

• Design of controllers
• Open and closed loop control systems
• Regulator and servo controllers
• Determination of the input
• System characteristics
• Is it possible to control the system?
  (controllability)?
• Ackermannss formula

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Control of systems

• We want to control the output of a system by
  giving it a certain input, such that
• the system follows a given reference or that
• the systems stays i a certain state (e.g. a
  certain position or velosity)
• Furthermore we want the system to be stable and
  to behave by some given performance measures.
• It is up to the designer of the system to specify
  the performance measures
• Examples of performance measures
• the system reacts fast on a given input
• it is not acceptable that the system has an
  overshoot of more that xx
• steady state must be reached after xx sec.
• Hence, a system can have several performance
  measures

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Open loop system

• If the input of the system can be completely
  planned and without any disturbances or changes
  of the system. Then an open loop control can be
  used.
• Open loop system illustrated graphically
• F is called the reference matrix. It described
  the relation between the reference and the input.
• Does the system have any disturbances, it is not
  possible to compensate for these, because the
  changes of the systems states are not recorded.

Model
Control (Gain)
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Closed loop system

• To compensate for disturbances or if the input is
  dependent on the response of the system. Then a
  closed loop control is used.
• L is called the feedback gain matrix.
• The states of the system are feed back and is
  used to determine the input u.
• u is a function of the system state and a whished
  reference
• u(t) -Lx(t) Fr(t) called the control law

Model
Control
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Servo and regulator controllers

• If we wish that the systems response follows a
  change of the reference then we call it a servo
  system.
• If there are no reference (r(t) 0) then the
  system is called a regulator. Here the wish is to
  control the system so a certain parameter is kept
  constant.
• Graphically it look like
• The control then law looks like
• u(t) -Lx(t)

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Determination of L substitution

• The control law, u(t) -Lx(t), is inserted in
  the state equation. That gives the state equation
  for a closed loop system
• To control the system is u(t) -Lx(t) used to
  decide the input. By this the size of L can be
  found. It is made from the state equation for the
  closed loop equation
• Stability and the systems response characteristic
  is decided from the eigenvalues of the A matrix
  of the system.
• For a closed loop system applies that à (A-BL)

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Pole-placement

• Eigenvalues are found from the characteristic
  polynomial
• lI - (A-BL) 0
• Above is described
• A and B system, also known from modelling of the
  system
• l is the characteristic of the system, also know
  from the designers specification of the system.
• L is the state feedback gain matrix, and is not
  known.
• Eigenvalues also called poles.
• We design and control by choosing L, so the
  placement of the poles are made most appropriate
  - Pole placement.

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Pole-placement

• On one hand we have eigenvalues which gives the
  system the decided characteristic m1, m2, m3, ..
  mn
• From these can a wished characteristic polynomial
  be written
• f(l) (l-m1) (l-m2) (l-m3)... (l-mn) ln
  a1ln-1 an-1l an
• On the other hand are the real eigenvalues for
  the system given by the roots in
• lI - (A-BL) 0
• from this the following polynomial is calculated
  ln b1ln-1 bn-1l bn
• where b-coefficients are the function of l1, l2,
  ln
• By comparing the coefficients, a og b, in the two
  polynomials can
• L l1 l2 ln be decided.

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Design approach

• We can by this way decide L by
• Selecting a row of m1, m2, m3, .. mn which
  describe the systems decided characteristic.
• calculate L
• simulate y(t) for u -Lx(t)
• If y(t) is not as expected/wished then a row of
  new mis are chosen.
• It is an iterative process to decide about L.

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System characteristics

• The three important parameters used to
  characterise the response of a system
• Risetime
• required time for the response to go from 10 of
  steady state to 90 of steady state.
• Peak overshoot
• the response maximum percentage
• exceeding of steady state.
• dependent on the systems damping.
• Settling time
• the time before the response is within
• 1, 2 or 5 of steady state.

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System characteristics

• If it is wished to have a fast reacting system
  (small rise time) then a small damping should be
  chosen. That causes however a very large
  overshoot and a long settling time.
• A compromise between a relative small damping for
  a short rising time and not to big overshoot and
  by then a not too long settling time.
• It is a requirement that the system is stable,
  and settles towards steady state.
• The characteristics of the system is
  given/decided by the systems eigenvalues/poles,
  and they should be in the negative complex half
  part.
• See also chapter 5

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Pole placement vs. System response
2. order system
1. order system
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Can a system be controlled?

• We can also create a control input by deciding L.
  That can be done by placing the eigenvalues
  appropriate.
• BUT, it is not always possible to find a L for a
  given system?
• That a system is controllable means
• the system can be brought from one arbitrary
  condition to another arbitrary condition within
  finite time.
• E.g.
• Can an input signal, u, be found, which can
  influence the system so the system can be
  transferred from one arbitrary start condition,
  x(t0), to an arbitrary final condition, x(t1),
  within finite time.
• If any x(t0) can be transferred to x(t1) is the
  system fully controllable.

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Conditions for system controllability

• If
• start time t0 0 final time t1
• start state x(t0) x(0) final state x(t1)
  0
• For full controllability are applies x(t0) ?
  x(t1)
• consequently x(0) ? 0
• From earlier we have that
• For full controllability are also required

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Controllability matrix

• To prove that a system is fully controllable
  should we be able to find x(0). (Solve the
  integral from the previous slide).
• From the proof in the book
• x(0) -B AB An-1B
• x(0) -QU
• To decide a U, which can affect the system to
  change from x(0) to x(t1) should the following
  equation be solved U x(0)Q-1
• Q should be invert able. (Q-1 should exists).
• The condition for Q-1 is found, is that Q should
  be non-singular which means have a rank n
  (complete rank).
• Q is called the controllable matrix.

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Determination of L - Ackermanns formula

• A general method to determine L is Ackermanns
  formula.
• Given a SISO system
• The wished eigenvalues for the system are m1,
  m2, m3, .. mn
• Therefore f(l) ln a1ln-1 an-1l an
• Ackermanns formula says that L 0 0 0 0
  1 Q-1 f(A)
• where Q B AB A2B An-1B
• and f(A) An a1An-1 an-1A an

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SimuLink example model regulator
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SimuLink example - responses
Eigenvalues 0.1 0.2
Eigenvalues -44i -4-4i
Eigenvalues -1i -1-i
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Todays exercises

• Controllability
• B-11-13 (controllable?)
• Regulator design
• B-12-3
• B-12-5
• B-12-6
• The exercises can be calculated using paper and
  pencil.
• Use Matlab to check the result.