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Lecture 3 Modeling (ii) –State Space approach. 3.1 State Variables of a Dynamical System 3.2 State Variable Equation 3.3 Why State space approach 3.4 Derive Transfer Function from State Space Equation 3.5 Time Response and State Transition Matrix. State equation. Dynamic equation.
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Lecture 3 Modeling (ii) –State Space approach 3.1 State Variables of a Dynamical System 3.2 State Variable Equation 3.3 Why State space approach 3.4 Derive Transfer Function from State Space Equation 3.5 Time Response and State Transition Matrix
State equation Dynamic equation Output equation State variable State space r- input p- output Modern Control Systems
D + + C B + - A Inner state variables Modern Control Systems
Motivation of state space approach Example 1 + - + noise Transfer function BIBO stable unstable Modern Control Systems
+ + + -2 + - + Example 2 BIBO stable, pole-zero cancellation Modern Control Systems
then system stable State-space description Internal behavior description Modern Control Systems
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 Modern Control Systems
Input 對系統的歷史總結。 Example : Capacitor electric energy Example : Inductor Magnetic energy Modern Control Systems
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 Modern Control Systems
K M2 M1 B1 B3 B2 Example Modern Control Systems
Armature circuit Field circuit Example Modern Control Systems
Dynamical equation Transfer function Laplace transform matrix Transfer function Modern Control Systems
Example By Newton’s Law Modern Control Systems
State Space Equation Transfer Function Example: Transfer function of the Mass-damper-spring system Modern Control Systems
Example MIMO system Transfer function Modern Control Systems
+ + - - Remark : the choice of states is not unique. exist a mapping Modern Control Systems
Different state equation description p is nonsingular Modern Control Systems
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. Modern Control Systems
