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# Lecture 02 - PowerPoint PPT Presentation

Lecture 02. State space approach. Control system analysis and design. Step1: Modeling By physical laws By identification methods Step2: Analysis Stability, controllability and observability Step3: Control law design Classical, modern and post-modern control Step4: Analysis

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### Lecture 02

State space approach

• Step1: Modeling

• By physical laws

• By identification methods

• Step2: Analysis

• Stability, controllability and observability

• Step3: Control law design

• Classical, modern and post-modern control

• Step4: Analysis

• Step5: Simulation

• Step6: Implement

Nonlinear Systems by Meiling CHEN 2009

• Differential equation : time-domain approach

• Linear/Nonlinear systems

• Transfer function : frequency-domain approach

• Linear systems

• Dynamic equation: state space approach

• Linear/Nonlinear systems

• Describing function : frequency-domain approach

• Nonlinear systems

Nonlinear Systems by Meiling CHEN 2009

State equation

Dynamic equation

Output equation

State variable

State space

r- input

p- output

Nonlinear Systems by Meiling CHEN 2009

+

+

C

B

+

-

A

Inner state variables

Nonlinear Systems by Meiling CHEN 2009

+

-

+

noise

Example 1

Transfer function

BIBO stable

unstable

Nonlinear Systems by Meiling CHEN 2009

+

+

-2

+

-

+

Example 2

BIBO stable, pole-zero cancellation

Nonlinear Systems by Meiling CHEN 2009

State-space description

Internal behavior description

Nonlinear Systems by Meiling CHEN 2009

Definition: The stateof a system at time is the amount of information at that together with determines uniquely the behavior of the system for

Example

M

Nonlinear Systems by Meiling CHEN 2009

Input 對系統的歷史總結。

Example : Capacitor

electric energy

Example : Inductor

Magnetic energy

Nonlinear Systems by Meiling CHEN 2009

Remark 1:狀態的選擇通常與能量有關， 例如:

Position  potential energy

Velocity  Kinetic energy

Remark 2:狀態的選擇必需是獨立的物理量， 例如:

Nonlinear Systems by Meiling CHEN 2009

M2

M1

B1

B3

B2

Example

Nonlinear Systems by Meiling CHEN 2009

Field circuit

Example

Nonlinear Systems by Meiling CHEN 2009

Transfer function

Laplace transform

matrix

Transfer function

Nonlinear Systems by Meiling CHEN 2009

MIMO system

Transfer function

Nonlinear Systems by Meiling CHEN 2009

+

-

-

Remark : the choice of states is not unique.

exist a mapping

Nonlinear Systems by Meiling CHEN 2009

Non-homogeneous solution

Forced responses

Homogeneous solution

Natural responses

Nonlinear Systems by Meiling CHEN 2009

LTI Dynamic equation

Nonlinear Dynamic equation

Nonlinear Systems by Meiling CHEN 2009

Nonlinear system example:

Pendulum equation

Nonlinear Systems by Meiling CHEN 2009