1. Lecture 3: System Representation
Transfer Functions
Graphical Representation
State Space Representation
Reading: Chap
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2. Linear Difference Equation Representation
𝑒(𝑘)
𝑦(𝑘)
Input e(k), and output y(k), k=0,1,2,….
LTI system given by a linear difference equation
Typically assume zero initial conditions:
𝑦 𝑘 + 𝑎 𝑛−1 𝑦 𝑘−1 +⋯+ 𝑎 0 𝑦 𝑘−𝑛 =
𝑏 𝑚 𝑒 𝑘 + 𝑏 𝑚−1 𝑒 𝑘−1 +⋯+ 𝑏 0 𝑦 𝑘−𝑚
𝑦 −1 =𝑦 −2 =⋯=𝑦 −𝑛 =0
𝑒 −1 =𝑒 −2 =⋯=𝑒 −𝑚 =0

3. Transfer Function
Take the z-transform of the linear difference equation to obtain
transfer function of the discrete-time LTI system
Linear difference equation representation for ?
𝑌 𝑧 = 𝑏 𝑚 + 𝑏 𝑚−1 𝑧 −1 +⋯+ 𝑏 0 𝑧 −𝑚 1+ 𝑎 𝑛−1 𝑧 −1 +⋯+ 𝑎 0 𝑧 −𝑛 𝐸(𝑧)
𝐺(𝑧)
𝐸(𝑧)
𝑌(𝑧)
𝐺(𝑧)
𝐺(𝑧)= 𝑧−1 𝑧−2 2

4. Time-Delay Element 𝑒(𝑘) 𝑦 𝑘 =𝑒(𝑘−1) 𝑇 𝑒(𝑘𝑇) 𝑦 𝑘𝑇 =𝑒( 𝑘−1 𝑇) 𝑇 𝐸(𝑧)
Consider a simple LTI discrete-time system whose output y(k) is obtained from the input e(k) by a delay of one time step:
If the input e(k) is obtained by sampling a continuous-time: e(k)=e(kT), then the above operation is a time delay element by time T:
The transfer function of the time-delay element is
𝑒(𝑘)
𝑦 𝑘 =𝑒(𝑘−1) 𝑇
𝑒(𝑘𝑇)
𝑦 𝑘𝑇 =𝑒( 𝑘−1 𝑇) 𝑇
𝐸(𝑧)
𝑌(𝑧)
𝑧 −1

5. Connection of Time Delay Elements
𝑒(𝑘)
𝑦(𝑘) 𝑇 𝑇 𝑇 𝑇
(shift register
using D flip-flops)
A more complicated connection:
𝑒(𝑘) 𝑇 𝑇 +
𝑦(𝑘) − +

6. Simulation Diagram
Simulation diagram is a graphical representation of systems consisting of basic elements of operations:
Time-delay elements
Summation
Multiplication by constant
Example:
can be represented by a simulation diagram:
𝑦 𝑘 =2𝑒 𝑘 −𝑒 𝑘−1 −𝑦(𝑘−1)
𝑒(𝑘)
𝑦(𝑘) 𝑇 − + 𝑇 −

7. Example
𝑦 𝑘 −4𝑦 𝑘−1 +3𝑦(𝑘−2)=𝑒 𝑘−1 −2𝑒 𝑘−2
Simulation diagram:

8. Simulation Diagram for General Linear Difference Equation

9. (Signal) Flow Graph An alternative graphical representation of systems
Basic elements are
Nodes: representing signals
Branches: directed line segment connecting nodes, each with a gain
At each node, signals of all incoming branches are summed and the result is transmitted to all outgoing branches
Example:
𝐸(𝑧)
𝑌(𝑧)
𝐸(𝑧)
𝐺(𝑧)
𝑌(𝑧)
𝐺(𝑧)
𝐸(𝑧)
𝑌(𝑧)
𝐸(𝑧) 1
𝑌(𝑧) + − −1
𝑀(𝑧)
𝑀(𝑧)

10. Previous Example Simulation diagram: 𝑇 𝑇 Flow graph:
𝑦 𝑘 =2𝑒 𝑘 −𝑒 𝑘−1 −𝑦(𝑘−1)
Simulation diagram:
𝑒(𝑘)
𝑦(𝑘) 𝑇 − + 𝑇 −
Flow graph:
𝑌(𝑧)
− 𝑧 −1 1
𝑧 −1
𝐸(𝑧) −1 −1

11. Mason’s Formula
To compute the transfer function from an input node to an output node in an arbitrary flow graph:
Compute the determinant  of the flow graph
Find all forward paths with path gains P1,…,Pk
For each forward path Pi, i=1,…,k, find the determinant (cofactor) i of a (sub) signal flow graph obtained from the original one by removing by branches touching Pi
Then the transfer function from the input node to the output node is

12. Determinant of Flow Graph
=1- (sum of all individual loop gains)
+ (sum of gain products of all two non-touching loops)
- (sum of gain products of all three non-touching loops)
+ …
Example:

13. Application of Mason’s Formula
Forward path
Forward path gain Pi i

14. Example

15. Another Example
Transfer function

16. State Space Representation
Concept of State Variables
State-Variable Model
Relation with Transfer Function Representation
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17. External vs Internal Models
Transfer functions: input-output (I/O, external) representations
Modern perspective: state-variable model (Internal Model)
Systems with identical I/O characteristics may possess drastically different internal structures
Hard constraints on some internal variables
Input and output variables are not enough

18. State-Variable Model Single Input Single Out (SISO) LTI Systems:
input is a scalar:
output is a scalar:
state x is a vector:
A, B, C, D are matrices of proper dimensions
The output y(k), k=0,1,… can be uniquely determined from the input e(k), k=0,1,…, and the initial condition x(0)

19. Numerical Solution
Find y(k) for given input e(k) and initial state x(0)
Recursive solution:

20. Transfer Function of State-Variable Model
Assuming zero initial condition x(0)=0, what is the transfer function from the input signal e(k) to the output signal y(k)?

21. Example Consider the system State variables: In vector form:
System evolution becomes:
Output can be recovered as:
Transfer function:

22. Obtaining State-Variable Model From Transfer Functions
Given a general transfer function
how to obtain equivalent state-variable model?
General procedure
Draw a simulation diagram of the system (many choices)
Assign a state variable to each time delay element’s output
Write the state equation, and the output equation from the diagram

23. One Possible Way
Example:

24. Controller Canonical Form: Simulation Diagram

25. Controller Canonical Form: Flow Graph
Exercise: Check the transfer function from E(z) to Y(z) by Mason’s Formula

26. Observer Canonical Form: Simulation Diagram
Exercise: write the state-variable model

27. Example
State Variables Model 1: controller canonical form

28. Alternative State Variable Model (I)

29. Alternative State Variable Model (II)
Conclusion: A transfer function G(z) can have many different equivalent state-variable models (A,B,C,D), as long as

30. Exercise
State-variable model: