Discrete time systems - Properties Lecture 6 Systems and Control - - PowerPoint PPT Presentation

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Discrete time systems - Properties Lecture 6 Systems and Control - - PowerPoint PPT Presentation

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STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Discrete time systems - Properties Lecture 6 Systems and Control Theory STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Stability

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Discrete time systems - Properties

Lecture 6

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Stability of Discrete time systems

  • BIBO-Stability (Bounded-Input Bounded-Output)
  • Every bounded input results in a bounded output
  • Internal Stability
  • Stricter than BIBO-Stability
  • All possible internal states return to zero after a finite time in the absence of an input.
  • All Eigenvalues of the matrix A are contained within the a circle of radius 1 around zero in

the complex plane.

  • BIBO-Stability follows from Internal Stability, but the inverse is not necessarily true.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Stability of Discrete time systems

  • A discrete system is BIBO-Stable if all poles of H(z) are within a circle of radius 1 around the
  • rigin.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Can unstable systems exist?

  • According to the mathematical models we have discussed unstable systems need an infinite

amount of energy.

  • What happens in the real world?
  • The system enters a state in which the current linear model is no longer valid.
  • Non linear behavior
  • Smaller unaccounted effects become more prominent
  • ...
  • The system malfunctions and may cause damage to itself or it’s surroundings.
  • Something else bad happens

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Stability: Examples

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Source: http://www.strikepr.net/500px-Stable-unstable1.svg.png

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Airplane stall

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Source: http://i1.wp.com/leavingterrafirma.com/wp-content/uploads/2010/02/Stall1.jpg

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Airplane stall

  • Airplanes generate lift using the Venturi effect.
  • Faster moving air has a lower pressure.
  • Eddy currents may be created due to a too slow airspeed or too

sharp ascent.

  • Turbulent airflow causes a loss of the lift generated by the Venturi

effect.

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  • Without the necessary lift an

airplane becomes an unstable system.

  • https://www.youtube.com/watch?v=WFcW5-1NP60
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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Tilt test for tall vehicles

  • Busses and other tall vehicles have

a tendency to roll when taking turns too quickly.

  • A London bus is loaded with

sandbags and must be able to lean at an angle of at least 28˚ while still returning all tires to the ground.

  • Modern day car manufacturers

have to pass multiple tests for stability while maneuvering.

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Source: http://www.ltmcollection.org/museum/object/link.html?_IXMAXHITS_=1&IXinv=19

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Complex Eigenvalues (DT)

  • As with the roots to the characteristic equation in difference equations, complex and/or

negative Eigenvalues for A create oscillation.

  • The magnitude of the oscillation will grow/decline with .
  • : The oscillation will decrease in magnitude: stable
  • : The oscillation will increase in magnitude: unstable
  • : The oscillation will maintain the same magnitude indefinitely: unstable
  • The smallest achievable period is 2 times the step time, for negative real Eigenvalues.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Complex Eigenvalues (DT)

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Source: http://cnx.org/content/m28650/1.1/

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Controllability and observability

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Not controllable Not observable

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Observability (DT)

  • A system is observable if the current state can be determined in finite time by measuring the
  • utputs.
  • The state space model without inputs gives us:

Now we can determine a set of vector equations in x[0]:

  • If x[k] has n internal states then n equations are needed:

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Observability (CT)

  • Same principles as in discrete time, but now with derivatives.
  • Again a rank n is required for the observability matrix

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Controllability (DT)

  • A system is controllable if it can be brought to a desired state using the inputs in a finite time.
  • Again we start from the state space model:
  • The following equations can de derived:

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Controllability (DT)

  • This last equation can be rewritten as:
  • For a given x[0] and a desired x[n] the required inputs can be found by solving this system.
  • is called the controllability matrix of the system.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Controllability (DT)

  • n is equal to the number of states in the system
  • A system is said to be controllable if the set of equations

can be solved for a given x[0] and any desired x[n].

  • This is the case if the controllability matrix has a rank n.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Detectability and Stabilizability

  • Observability and controllability are important terms in control theory.
  • Detectability and stabilizability are also often used as weaker constraints.
  • A system is detectable if all unstable states are observable.
  • A system is stabilizable if all unstable states are controllable.
  • Detectability and stabilizability are also important terms in control theory.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Example

  • A system with the following state space representation:
  • The observability- and controllability matrixes both have rank 3 and are respectively:
  • The system is both observable and controllable.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Example

  • The second internal state is only indirectly dependent on the input through the other

internal states.

  • Adding 2 more zero’s to A removes this dependence:
  • The resulting controllability matrix now has rank 2. The observability matrix still has rank 3.

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Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Example

  • The first internal state only connected to the output though the 3rd internal state and the

3rd state is only connected to the output directly.

  • If we remove the link from the 3rd internal state to the output, the rank of the observability

matrix drops to 1. The controllability matrix remans unchanged.

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