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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

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

  • Step5: Simulation

    • Matlab, Fortran, simulink etc….

  • Step6: Implement

Nonlinear Systems by Meiling CHEN 2009


Dynamic system descriptions:

  • 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


LTI systems:

State equation

Dynamic equation

Output equation

State variable

State space

r- input

p- output

Nonlinear Systems by Meiling CHEN 2009


D

+

+

C

B

+

-

A

Inner state variables

Nonlinear Systems by Meiling CHEN 2009


Motivation of state space approach

+

-

+

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


system stable

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

單純從 並無法決定x在 以後的運動狀況。除非知道 與 。所以 與 是這個系統過去的歷史總結。故 與 可以作為系統的狀態。

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:

實際上只有一個狀態變數

Nonlinear Systems by Meiling CHEN 2009


K

M2

M1

B1

B3

B2

Example

Nonlinear Systems by Meiling CHEN 2009


Armature circuit

Field circuit

Example

Nonlinear Systems by Meiling CHEN 2009



Dynamical equation

Transfer function

Laplace transform

matrix

Transfer function

Nonlinear Systems by Meiling CHEN 2009


Example

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


The solution of LTI system

Non-homogeneous solution

Forced responses

Homogeneous solution

Natural responses

Nonlinear Systems by Meiling CHEN 2009


Nonlinear systems:

LTI Dynamic equation

Nonlinear Dynamic equation

Nonlinear Systems by Meiling CHEN 2009


m

Nonlinear system example:

Pendulum equation

Nonlinear Systems by Meiling CHEN 2009


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