State Space Modelling and Control

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State Space Modelling and Control

• Morten Kristiansen
• Bjørn Langeland

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Lesson 4
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Todays agenda

• Summery of lesson 1 to 3
• Illustrated with an example
• Today's lesson
• design of a servo system
• illustrated with an example

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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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Example Water level in tanks

• Given a plant with 3 tanks containing liquid
  which are coupled together as illustrated in the
  figure (rensningsanlæg).
• We wish to control the flow out of tank number 3,
  by controlling the level of the liquid in tank
  number 3.
• Modelling of the system
• We set up the equations for the liquid balance
  for each tank.
• Liquid in tank liquid input liquid output
• We choose the state variables x1, x2, x3 h1,
  h2, h3,
• y h3 is selected as output variable.
• The flow input into tank 1 is our possibility to
  influence the system. For this reason is the
  following input variable selected u qin

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Example - state space model

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Linearization

• Practical are almost all dynamical systems
  nonlinear.
• The linear theory has however a relative large
  significance, because many nonlinear behaviours
  can be linearised.
• The appearing linear equations from the
  linearization can be used to design of a
  regulator or a control unit around a reference
  condition (work point), which follows a nonlinear
  trajectory.
• That is so to say we have to presume that the
  system is working in or is located around a
  certain point.

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Linearization - how?

• Given a nonlinear model
• It is wished to have a linear model in the work
  point (x0(t), u0(t))
• The partial derived are found from each function
  and the work point is inserted.

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Jacobi matrices
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Example

• Given
• Linearization in the work point x0 1 1 1
• df1/dx1 10x2 df1/dx2 10x1 df1/dx3 5
• df2/dx1 0 df2/dx2 0 df2/dx3 11
• df3/dx1 x3 df3/dx2 x3 df3/dx3 0
• That gives

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Analysis

• Response characteristic
• Solution to state equation,
  is given by
• Stability
• How does the system behave when it is brought out
  of equilibrium?
• A system is stable if it comes close to its
  equilibrium state, xe
• Eigenvalues
• The solution to lI - A 0 gives the system
  eigenvalues.
• For a stable system are the eigenvalues placed in
  the negative complex half plane.
• Controllability
• Is it possible to control the system? I.e. can we
  influence the system to go from one arbitrary
  start state to another arbitrary end state?
• Is investigated it the controllability matrix, Q
  B AB A2B An-1B, can be inverted, so
  Q-1. (Q full rank).

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Example

• Given the tank-example from before with exact
  values inserted
• Controllability? Q B AB A2B Matlab
  ctrb(A,B)
• Rank (Q) 3 So Q-1 is to be found
• I.e. the system is fully controllable.

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Design of controls

• Open vs. closed loop systems
• If the system is not affected by disturbances can
  a open loop system be used
• To be able to compensate for disturbances is a
  closed loop system used.
• For a closed loop system are two types of control
  useful
• Regulator keeps the system in a specified
  wished condition (e.g. equilibrium)
• Servo let the system to follow a reference
  signal (e.g. decided by the user)

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Water tank example again

• No disturbances on the system
• By a planned sequence of the input to the system
  (the water flow into tank 1) can the system be
  brought to a equilibrium state.
• It could e.g. be by x 5 4 3.
• so h1 5, h2 4 and h3 3
• If the system is not affected by any disturbances
  will the system stay in this state.
• The system can be categorised as an open loop
  system.

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Design of a regulator to the example

• It is presumed that the system is in equilibrium
  state. We wish that the system stays in this
  state. But disturbances affect the system.
• To avoid that the output from the system (the
  flow from tank 3) changes because of disturbances
  are a regulator designed.
• It should also be designed how the input to the
  system should be changed when the system is
  affected.

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Regulator

• From the block diagram can be seen that u(t)
  -Lx(t)
• L is calculated by use of Ackermanns formula L
  0 0 1 Q-1 f(A)
• Q controllability matrix
• f(A) An a1An-1 an-1A an
• Matlab acker(A, B, eig)
• The procedure to find a usable u is
• select a row of eigenvalues which describes the
  characteristics of the system.
• calculate L
• simulate y(t) for u(t) -Lx(t)
• The prior condition is that the system is
  controllable.

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Determination of L - 1. attempts

• The following eigenvalues are selected v 0.5
  -0.5 1
• By use of Matlab is L found to be
• L acker(A, B, v) -26 74 -75
• The initial condition is calculated in relation
  to the equilibrium condition so that x0 1 0
  0T. (h1 6, h2 4, h3 3)
• u(t) is known, because L is found.
• The system is brought out of equilibrium (x 0
  0 0T).
• The question is then. How does the system react
  when it is let loose?

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Simulink model of water tank example
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Simulation result 1
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Determination of L - 2. attempts

• A new set of eigenvalues are selected
• v -5-5i -55i -5
• L is found to be -10 30 -25

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Simulation result 2
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Design of a servo today's lesson

• Given a model
• and a reference signal, r(t)
• If we wish that the response of the system
• should follow a change of the reference do
• we call it a servo system.
• Determine the input u(t), so the systems output,
  y(t), is asymptotic and moves towards the
  reference signal.
• I.e. y(t)-r(t) moves towards 0, when t moves
  towards infinite
• Two methods are found to design a servo
• Reference magnification
• Error integration

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Steady state error

• Often is the system not resting in the expected
  state.
• it is called the systems steady state error.
• The reason for this kind of error can be many
  things
• physical friction in the components of the
  systems components
• accuracy of the systems components are
  unsatisfactory
• For some systems it is whatever the properties of
  the physical components, not possible to follow
  certain types of inputs.
• it is caused by the systems structural
  construction.

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Reference magnification

• State- and output equation Control law
  equation
• u(t) -Lx(t) Fr(t)
• Closed loop equation
• (A-BL)x(t)
• Determine the systems transient response. Select
  L so the systems eigenvalues are placed
  appropriate. As by regulator design.
• BFr(t)
• Determine the systems steady state response.
  Select F so

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Servo design for the water tank example

• F -C(A-BL)-1B-1
• calculated by use of Matlab to F 10
• A step reference input
• r 1 is simulated.

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Servo design by error integration

• Error integrator
• z(t) (A-BL) z(t) Wr(t)
• Consequently reduced to a regulator design
  problem.

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The example
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Exercises

• Exercise number 5 (to be found on the homepage).
• Assignment