Lecture #10 State space approach.

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1 Lecture #10 State space approach

2 State equation Dynamic equation Output equation State variable
State space r- input p- output Meiling CHEN

3 Inner state variables C A D B + - Meiling CHEN

4 Motivation of state space approach
Example 1 + - noise Transfer function BIBO stable unstable Meiling CHEN

5 BIBO stable, pole-zero cancellation
Example 2 BIBO stable, pole-zero cancellation -2 + - Meiling CHEN

6 State-space description Internal behavior description
then system stable State-space description Internal behavior description Meiling CHEN

7 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。
Definition: The state of a system at time is the amount of information at that together with determines uniquely the behavior of the system for Example M 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。 Meiling CHEN

8 Example : Capacitor electric energy Input 對系統的歷史總結。 Example : Inductor
Magnetic energy Meiling CHEN

9 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 Meiling CHEN

10 Example K M2 M1 B1 B3 B2 Meiling CHEN

11 Example Armature circuit Field circuit Meiling CHEN

12 Meiling CHEN

13 Dynamical equation Transfer function matrix Transfer function
Laplace transform matrix Transfer function Meiling CHEN

14 Example MIMO system Transfer function Meiling CHEN

15 Remark : the choice of states is not unique.
+ - exist a mapping Meiling CHEN

16 Different state equation description
p is nonsingular Meiling CHEN

17 Definition : Two dynamical systems
with are said to be equivalent. The nonsingular matrix p is called an equivalence transformation. & Theorem: two equivalent dynamical system have the same transfer function. Meiling CHEN