Discrete Time Systems - Representations Lecture 4 Systems and - PowerPoint PPT Presentation

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Discrete Time Systems - Representations Lecture 4 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Discrete Time Systems • For each time step the system has: • A vector of inputs • A vector of outputs • A vector of states Systems and Control Theory 2

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics How to represent a system?  A system can be represented in multiple ways  Block-diagram  State space representation  Difference/differential equations  Impulse response  Transfer functions  In this lecture we will take a look at the different representations Systems and Control Theory 3

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Block-diagram  A block diagram is a visual representation of a system. All LTI’s (Linear Time Invariant) systems can be constructed using these 3 building blocks. Note that every memory element corresponds to one state variable. Systems and Control Theory 4

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  :deposits and withdrawals from of cash from bank account  :current saldo on bank account (before deposit and interest)  :The acquired interest that year  :the saldo on the next year = current saldo + interest + deposits Systems and Control Theory 5

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest 0 50 50 0 Systems and Control Theory 6

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  Simulate the block diagram 2,5 53 0 50 Systems and Control Theory 7

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  Simulate the block diagram 2,6 30 -25 52 Systems and Control Theory 8

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  Simulate the block diagram 1,5 32 0 30 Systems and Control Theory 9

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  Simulate the block diagram 1,6 33 0 32 Systems and Control Theory 10

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  Simulate the block diagram 1,7 65 30 33 Systems and Control Theory 11

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  Simulate the block diagram 3,2 68 0 65 Systems and Control Theory 12

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: compound interest  Simulate the block diagram 3,4 72 0 68 Systems and Control Theory 13

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Bad block diagrams  Delay-free loops  Connecting two outputs without using a sum S 1 S 2 Never use a block diagram that has one of these issues!! Systems and Control Theory 14

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Bad block diagrams  Delay-free loop:  The issue is that this leads to an implicit connection  depends on ,which is not yet known  You can easily rewrite this in an allowd shape Systems and Control Theory 15

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Bad block diagrams  Connecting two outputs  The issue is that this can lead to inconsistencies  According to this block diagram the S 1 output of the systems S 1 and S 2 are equal S 2  There is no way to get around this Systems and Control Theory 16

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics State space representation  Discrete time  This state space representation is again specific to LTI systems:  Linear: it’s easy to see these systems are linear (see lecture about classification of dynamical systems)  Time-invariant: the matrices A,B,C,D do not depend on time, if it were to be a time-variant system the matrices would be replaced by A[k], B[k], C[k] and D[k] Systems and Control Theory 17

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics From block diagram to state space  Block diagram  State space representation  In general  Let the inputs of the memory element be and the outputs .  Trace back to retrieve equations for and  This results in: Systems and Control Theory 18

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics From state space to block diagram (DT)  First Add a delay element for every state x[k] Systems and Control Theory 19

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics From state space to block diagram (DT)  Add a delay element for every state x[k]  Determine the input for every state x[k+1] from the matrixes A and B, as a combination of the states x[k] and inputs u[k] Systems and Control Theory 24 20

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics From state space to block diagram (DT)  Add a delay element for every state x[k]  Determine the input for every state x[k+1] from the matrixes A and B, as a combination of the states x[k] and inputs u[k]  Determine the outputs y[k] in the same way with the matrixes C and D u[k] y[k] 21 Systems and Control Theory 21

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Different state space representations  State space representation is not unique:  Take the following system, which connects u[k] to y[k]:  Now take a non-singular square matrix T and the following system. The relation between u[k] and y[k] will be the same.  With and , we have found a different state space representation for this system.  The same holds for continuous time. Systems and Control Theory 22

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Different state space representations  Input-output behavior is maintained  Internal behavior can be very different  If A has an Eigenvalue decomposition PDP -1 and T = P -1 then the resulting state space model will be greatly simplified  If no Eigenvalue decomposition exists for A then a Jordan form may be used. Systems and Control Theory 23

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Difference equations  Similar to differential equations, but for discrete time  General form:  n is the order of the system  k is usually taken to be larger than zero  Each value y[k+i] represents the output of the system at a moment k+i  Each value u[k+i] represents an external input delivered to the system  Solution in 2 parts:  homogenous: solution from input zero  particular: solution derived as a response from the input Systems and Control Theory 24

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Homogenous difference equations  General form:  Expected form of solution:  Substitution of the expected solution in the difference equation:  Division by r k leads to the characteristic equation:  Solutions of the characteristic equation:  Homogenous solution to the difference equation: Systems and Control Theory 25

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: Fibonacci sequence Systems and Control Theory 26

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: Fibonacci sequence  Homogenous difference equation and starting conditions: ; ,  Characteristic equation:  Roots:  General solution:  Filling in starting conditions:  With solutions: Systems and Control Theory 27

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Complex roots to the characteristic equation  Complex and/or negative roots will result in oscillating behavior.  If the difference equations and starting conditions are both real the complex roots can only be present in conjugate pairs.  This can be converted into a cosine using Euler’s formula: Systems and Control Theory 28

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Non-homogeneous difference equations  General form:  A linear combination of inputs results in the same linear combination of the outputs resulting from each input individually.  The equation can thus be solved for each input individually and the results added together afterwards.  The resulting particular solutions can then be added to the general form of the homogenous solution. Systems and Control Theory 29

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Particular solutions to difference equations Systems and Control Theory 30

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example  Given:  We start by solving the homogenous equation:  We will now fill in the suggested solution: Systems and Control Theory 31