Lecture 3: System Representation Transfer Functions Graphical Representation State Space Representation Reading: Chap. 2.7-2.10 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAA

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

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

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

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) − +

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 −

Example 𝑦 𝑘 −4𝑦 𝑘−1 +3𝑦(𝑘−2)=𝑒 𝑘−1 −2𝑒 𝑘−2 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 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)

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

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

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:

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

Example

Another Example Transfer function

State Space Representation Concept of State Variables State-Variable Model Relation with Transfer Function Representation TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

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

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 the input signal e(k) to the output signal y(k)?

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

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

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 equivalent state-variable models (A,B,C,D), as long as

Exercise State-variable model: