Linear system Lesson 4 State space approach Linear

Linear system Lesson 4 State space approach Linear System by Meiling CHEN

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

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

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

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

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

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

Example : Capacitor electric energy Input 結。 對系統的歷史總 Example : Inductor Magnetic energy Linear System by Meiling CHEN 8

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 Linear System by Meiling CHEN 9

Example K M 2 B 3 M 1 B 2 B 1 Linear System by Meiling CHEN 10

Example Armature circuit Field circuit Linear System by Meiling CHEN 11

Linear System by Meiling CHEN 12

Dynamical equation Transfer function Laplace transform Transfer function matrix Linear System by Meiling CHEN 13

Example MIMO system Transfer function Linear System by Meiling CHEN 14

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

Different state equation description p is nonsingular Linear System by Meiling CHEN 16

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. Linear System by Meiling CHEN 17