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

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 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. Systems and Control Theory 2

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 origin. Systems and Control Theory 3

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  The system enters a state in which the current linear model is no longer valid.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 Systems and Control Theory 4

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Stability: Examples Source: http://www.strikepr.net/500px-Stable-unstable1.svg.png Systems and Control Theory 5

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Airplane stall Source: http://i1.wp.com/leavingterrafirma.com/wp-content/uploads/2010/02/Stall1.jpg Systems and Control Theory 6

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.  Without the necessary lift an airplane becomes an unstable system.  https://www.youtube.com/watch?v=WFcW5-1NP60 Systems and Control Theory 7

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. Source: http://www.ltmcollection.org/museum/object/link.html?_IXMAXHITS_=1&IXinv=19 Systems and Control Theory 8

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. Systems and Control Theory 9

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Complex Eigenvalues (DT) Source: http://cnx.org/content/m28650/1.1/ Systems and Control Theory 10

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Controllability and observability 2 Not controllable Not observable Systems and Control Theory 11

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 outputs.  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: Systems and Control Theory 12

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 Systems and Control Theory 13

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: Systems and Control Theory 14

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. Systems and Control Theory 15

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. Systems and Control Theory 16

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. Systems and Control Theory 17

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. Systems and Control Theory 18

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. Systems and Control Theory 19

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. Systems and Control Theory 20