Lecture Guide 4 - Transfer Function And State-space Models.docx

Lecture Guide 4 - Transfer functions and State-Space Models 1 Engr. Caesar Pobre Llapitan

Topics I. II. III. IV. V. VI.

Definition of the transfer function Development of Transfer Functions Poles and zeros of the transfer function Linearization of Nonlinear Models Dynamic behavior of a system with MatLab State-Space and Transfer Function Matrix Models

I. Definition of the Transfer Function (TF)

Transfer functions (TF) are frequently used to characterize the input-output relationships or systems that can be described by Linear Time-Invariant (LTI) differential equations.

The transfer function (TF) of a LTI differential-equation system is defined as the ratio of the Laplace transform (LT) of the output (response function) to the Laplace transform (LT) of the input (driving function) under the assumption that all initial conditions are zero. Consider the LTI system defined by the differential equation a0 y (n)  a1 y (n1)    an1 y' an y  b0 x m   b1 x m1    bm1 x' bm x

where y is the output and x is the input. The TF of this system is the ratio of the Laplace-transformed output to the Laplace-transformed input when all initial conditions are zero, or

Transfer Function (TF)  Gs   

L output  L input  zero initial conditions

Y s  b 0s m  b1s m 1    b m 1s  b m  X s  a 0s n  a 1s n 1    a n 1s  a n

The above equation can be represented by the following graphical representation:

s  X  Input

b0s m  b1s m 1    b m 1s  b m a 0s n  a 1s n 1    a n 1s  a n

s  Y Output

Transfer Function Comments on the Transfer Function (TF) The applicability of the concept of the Transfer Function (TF) is limited to LTI differential equation systems. The following list gives some important comments concerning the TF of a system described by a LTI differential equation: 1. The TF of a system is a mathematical model of that system, in that it is an operational method of expressing the differential equation that relates the output variable to the input variable. 2. The TF is a property of a system itself, unrelated to the magnitude and nature of the input or driving function.

Lecture Guide 4 - Transfer functions and State-Space Models 2 Engr. Caesar Pobre Llapitan

3. The TF includes the units necessary to relate the input to the output; however it does not provide any information concerning the physical structure of the system. (The TF of many physically different systems can be identified). 4. If the TF of a system is known, the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system. 5. If the TF of a system is unknown, it may be established experimentally by introducing known inputs and studying the output of the system. Once established, a TF gives a full description of the dynamic characteristics of the system, as distinct from its physical description II. Development of Transfer Functions

Properties of Transfer Function Models 1. Steady-State Gain The steady-state of a TF can be used to calculate the steady-state change in an output due to a steady-state change in the input. For example, suppose we know two steady states for an input, u, and an output, y. Then we can calculate the steady-state gain, K, from: y  y1 K 2 u 2  u1 For a linear system, K is a constant. But for a nonlinear system, K will depend on the operating condition u, y  . Calculation of K from the TF Model: If a TF model has a steady-state gain, then:

K  lim Gs  s0

This important result is a consequence of the Final Value Theorem Note: Some TF models do not have a steady-state gain (e.g., integrating process) 2. Order of a TF Model

Consider a general nth-order, linear ODE:

an

d ny d n1 y dy d mu d m1u du  a    a  a y  b  b    b1  b0u n1 1 0 m m1 n n1 m m1 dt dt dt dt dt dt

Take Laplace transform, assuming the initial conditions are all zero. Rearranging gives the TF: m

Y s  Gs    Us 

b s

i

a s

i

i 0 n

i 0

1

i

Definition: The order of the TF is defined to be the order of the denominator polynomial. Note: The order of the TF is equal to the order of the ODE.

Lecture Guide 4 - Transfer functions and State-Space Models 3 Engr. Caesar Pobre Llapitan

Physical Realizability For any physical system, n  m. Otherwise, the system response to a step input will be an impulse. This can’t happen.