218023 dynamic system and control lecture 1

Dynamics of systems• Dynamics describes how the states evolves, as a function on the current state and any external inputs • Inputs describe the external excitation of the dynamics •

Mathematical Models of Systems

Dynamics of systems

• Dynamics describes how the states evolves, as a function on the

current state and any external inputs

• Inputs describe the external excitation of the dynamics

• Outputs describe the directly measured variables

 Outputs are a function of the state and inputs ⇒ not

independent variables

 Not all states are outputs; some states can’t be directly

measured

Dynamics of Mechanical Systems

Dynamics of Mechanical Systems

Dynamics of Electrical Systems

Dynamics of Thermal Systems

Derive dynamic equation of the tank

C: Thermal capacitance

h i : Heat rate of input

h o :

: Change of temperature

What is your observation?

• The dynamics of many systems, whether they are mechanical, electrical, thermal, and so on, may be described in terms of

differential equations.

• The differential equations may be obtained by using physical laws governing a particular system (e.g., Newton’s laws for mechanical systems and Kirchhoff’s laws for electrical

systems).

• Deriving reasonable mathematical models is the most

important part of the entire analysis of control systems

System modeling

Models are a mathematical representations

of system dynamics:

• Models allow the dynamics to be

simulated and analyzed, without having

to build the system

• Models are never exact, but they can be

predictive

The model you use depends on the questions

you want to answer

• A single system may have many models

• Time and spatial scale must be chosen to

suit the questions you want to answer

• Always formulate questions before building

a model

The principle of causality

The current output of the system (the output at time t = 0)

depends on the past input (the input for t<0) but does not

depend on the future input (the input for t>0).

Examples of causal systems

- Memoryless system:

- Autoregressive filter:

Examples of noncausal systems

- Central moving average:

- Time reversal:

∝

The coefficients are constants or

functions only of the

independent variable

Linear varying system

Transfer Function and Impulse Response Function

Definition The transfer function of a linear, time-invariant,

differential equation system is defined as the ratio of the Laplace

transform of the output (response function) to the Laplace transform

of the input (driving function) under the assumption that all initial

conditions are zero.

Differential equation

Transfer Function and Impulse Response Function

Differential equation

- The transfer function is a property of a system itself, independent of the magnitude and nature of the input or driving function.

- The transfer function does not provide any information concerning the

physical structure of the system

Laplace Transform

- Laplace transform: A transformation from t (time)

to s (Laplace variable)

- Definition: Laplace transformation is used to map time

domain function into domain function

- This mapping is defined as

: →

Pierre Simon Laplace

List of Common Transforms

Taking a Laplace transform:

Transfer function

Dynamics of Electrical Systems

Taking a Laplace transform

Derive dynamic equation of the tank

↔

; C ;

C: Thermal capacitance

hi: Heat rate of input

ho: Heat rate of output

How to find the output in time domain?

The output Y(s) can be written as the product of G(s) and X(s)

Taking an inverse Laplace transform gives the following convolution integral:

If the input is an impulse

Complete information of the system (the dynamic characteristics of the

system) can be obtained by exciting it with an impulse input and measuring the response.

Matlab for Dynamic Systems and Control

Modelling in State Space Why we need state space model?

- Q: Are transfer function (TF) enough to model systems?

- A: TF is not applicable and convenience for MIMO (multi input

multi output), LTV, and nonlinear system

Frequency domain

Transfer function

SISO-LTI

Time domain

State space model

SISO-LTI

SISO-LTV MIMO-LTI MIMO-LTV Nonlinear system

States and States Variables

that completely determines the behavior of the system.

- State variables of a dynamic system are the variables making up the

smallest set of variables that determine the state of the dynamic system

- State vector: If n state variables are needed to completely describe the

behavior of a given system, then these n state variables can be

considered the n components of a vector x

the x1 axis, x2 axis,…, xn axis, where x1 axis, x2 axis,…, xn axis are state

variables, is called a state space.

- State-space equations include modeling of dynamic systems input

variables, output variables, and state variables.

States Space Model

State equation LTI model

Define state variables outputs of the system state space model

Differential equation

What we can do with the system including derivatives of input?

Consider the differential equation system

define the following n variables as a set of n state variables

with

Rewrite the differential equation:

Compute the coefficients:

Then, we obtain the state equation and output equation

In matrix form

Connection between Transfer Functions and State-Space

Equations

Q?: How

to get an inverse of

a matrix

Connection between Transfer Functions and State-Space

Equations

END OF CHAPTER 1 LINEAR SYSTEM THEORY WILL BE THE NEXT