Lecture 3: System Representation
•Transfer Functions
•Graphical Representation
•State Space Representation
•Reading: Chap. 2.7-2.10
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Linear Difference Equation Representation
•Input e(k), and output y(k), k=0,1,2,….
•LTI system given by a linear difference equation
•Typically assume zero initial conditions:
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 ?
•Consider a simple LTI discrete-time system whose output y(k) is obtainedfrom 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
Time-Delay Element
Connection of Time Delay Elements
(shift register
using D flip-flops)
A more complicated connection:
Simulation Diagram
•Simulation diagram is a graphical representation of systemsconsisting of basic elements of operations:
–Time-delay elements
–Summation
–Multiplication by constant
•Example:
can be represented by a simulation diagram:
Example
Simulation diagram:
Simulation Diagram for General Linear Difference Equation
(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 summedand the result is transmitted to all outgoing branches
•Example:
Previous Example
Simulation diagram:
Flow graph:
Mason’s Formula
•To compute the transfer function from an input node to anoutput 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 ofa (sub) signal flow graph obtained from the original one by removingby branches touching Pi
–Then the transfer function from the input node to the output node is
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:
Forward path
Forward path gain Pi
i
Application of Mason’s Formula
Example
Another Example
Transfer function
State Space Representation
•Concept of State Variables
•State-Variable Model
•Relation with Transfer Function Representation
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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 drasticallydifferent internal structures
–Hard constraints on some internal variables
–Input and output variables are not enough
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)
Numerical Solution
•Find y(k) for given input e(k) and initial state x(0)
•Recursive solution:
Transfer Function of State-Variable Model
Assuming zero initial condition x(0)=0, what is the transfer function from theinput signal e(k) to the output signal y(k)?
Example
System evolution becomes:
Output can be recovered as:
Consider the system
In vector form:
State variables:
Transfer function:
Obtaining State-Variable ModelFrom 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
One Possible Way
Example:
Controller Canonical Form:Simulation Diagram
Controller Canonical Form:Flow Graph
Exercise: Check the transfer function from E(z) to Y(z) by Mason’s Formula
Observer Canonical Form:Simulation Diagram
Exercise: write the state-variable model
Example
State Variables Model 1: controller canonical form
Alternative State Variable Model (I)
Alternative State Variable Model (II)
Conclusion: A transfer function G(z) can have many different equivalentstate-variable models (A,B,C,D), as long as
Exercise
State-variable model: