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

Systems and Control Theory

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Discrete Time Systems - Representations

Lecture 4

Systems and Control Theory

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Processing and Data Analytics

Discrete Time Systems

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• For each time step the

system has:

• A vector of inputs
• A vector of outputs
• A vector of states

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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

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Block-diagram

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• 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

• ne state variable.

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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

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Example: compound interest

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50 50

Systems and Control Theory

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Example: compound interest

• Simulate the block diagram

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53 50 2,5

Systems and Control Theory

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Example: compound interest

• Simulate the block diagram

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• 25

30 52 2,6

Systems and Control Theory

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Example: compound interest

• Simulate the block diagram

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32 30 1,5

Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal

Processing and Data Analytics

Example: compound interest

• Simulate the block diagram

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33 32 1,6

Systems and Control Theory

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Processing and Data Analytics

Example: compound interest

• Simulate the block diagram

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30 65 33 1,7

Systems and Control Theory

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Example: compound interest

• Simulate the block diagram

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68 65 3,2

Systems and Control Theory

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Example: compound interest

• Simulate the block diagram

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72 68 3,4

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Bad block diagrams

• Delay-free loops
• Connecting two outputs without using a

sum

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Never use a block diagram that has one of these issues!!

S1 S2

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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

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Bad block diagrams

• Connecting two outputs
• The issue is that this can lead to

inconsistencies

• According to this block diagram the
• utput of the systems S1 and S2 are

equal

• There is no way to get around this

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S1 S2

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State space representation

• 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]

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• Discrete time

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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:

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From state space to block diagram (DT)

• First Add a delay element for every state

x[k]

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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]

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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

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u[k] y[k]

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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.

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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.

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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

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Homogenous difference equations

• General form:
• Expected form of solution:
• Substitution of the expected solution in the difference equation:
• Division by rk leads to the characteristic equation:
• Solutions of the characteristic equation:
• Homogenous solution to the difference equation:

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Example: Fibonacci sequence

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Example: Fibonacci sequence

• Homogenous difference equation and starting conditions:

; ,

• Characteristic equation:
• Roots:
• General solution:
• Filling in starting conditions:
• With solutions:

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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:

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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.

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Particular solutions to difference equations

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Example

• Given:
• We start by solving the homogenous equation:
• We will now fill in the suggested solution:

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Example

• The complete solution has the form:
• Filling in the starting conditions gives us:

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Impulse responses (Discrete time)

• What is an impulse?
• You can decompose any signal in a sum of impulse responses

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Convolution theorem (DT)

• The convolution theorem states:
• Proof:
• Conclusion
• The impulse of a system does describe the input/output behavior

completely.

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Definition of impulse response Time-invariance Superposition Sums are convolutions

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Examples of Dirac-delta’s

• Popping balloons for acoustic measurements

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Example: Leontief model of a planned economy

• Won the nobel prize in 1973
• A simple model that assigns values to

different sectors

• For simplicity we choose a planned
• economy. But today governments all over

the world are using similar models to model their economy.

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Systems and Control Theory

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Example: Leontief model of a planned economy

• Leontief divided the economy in sectors

who buy from eachother.

• To produce one unit of industry 0.40 units
• f agriculture are required

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SALES Agricultu re Industry Services PU R C H AS E Agricultur e 0.40 0.03 0.03 Industry 0.06 0.40 0.10 Services 0.12 0.16 0.20

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Example: Leontief model of a planned economy

• Notation
• :production of sector i in month k
• :the demand to goods from sector i in the next month
• Note: in planned economy demand can be steered, economists can decide how many

rations they give.

• The model
• It is anti-causal: a subdivision of non-causal, for which only future values have to be

known to know the current value

• http://www.unc.edu/~marzuola/Math547_S13/Math547_S13_Projects/M_Kim_Section001_

Leontief_IO_Model.pdf

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