Lecture 02 State space approach NUUEE Nonlinear Systems

Lecture 02 State space approach NUU-EE Nonlinear Systems by Meiling CHEN 2009

Control system analysis and design • Step 1: Modeling – By physical laws – By identification methods • Step 2: Analysis – Stability, controllability and observability • Step 3: Control law design – Classical, modern and post-modern control • Step 4: Analysis • Step 5: Simulation – Matlab, Fortran, simulink etc…. • Step 6: Implement NUU-EE Nonlinear Systems by Meiling CHEN 2009 2

Dynamic system descriptions: 1. Differential equation : time-domain approach Linear/Nonlinear systems 2. Transfer function : frequency-domain approach Linear systems 3. Dynamic equation: state space approach Linear/Nonlinear systems 4. Describing function : frequency-domain approach Nonlinear systems NUU-EE Nonlinear Systems by Meiling CHEN 2009 3

LTI systems: State equation Dynamic equation Output equation State variable State space NUU-EE r- input p- output Nonlinear Systems by Meiling CHEN 2009 4

Inner state variables D B + C - + + A NUU-EE Nonlinear Systems by Meiling CHEN 2009 5

Motivation of state space approach Example 1 + + noise Transfer function BIBO stable unstable NUU-EE Nonlinear Systems by Meiling CHEN 2009 6

Example 2 BIBO stable, pole-zero cancellation -2 NUU-EE + + - + + Nonlinear Systems by Meiling CHEN 2009 + 7

system stable State-space description NUU-EE Internal behavior description Nonlinear Systems by Meiling CHEN 2009 8

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。 NUU-EE Nonlinear Systems by Meiling CHEN 2009 9

Example : Capacitor electric energy Input 結。 對系統的歷史總 Example : Inductor Magnetic energy NUU-EE Nonlinear Systems by Meiling CHEN 2009 10

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 NUU-EE Nonlinear Systems by Meiling CHEN 2009 11

Example K M 2 B 3 NUU-EE M 1 B 2 B 1 Nonlinear Systems by Meiling CHEN 2009 12

Example Armature circuit NUU-EE Field circuit Nonlinear Systems by Meiling CHEN 2009 13

NUU-EE Nonlinear Systems by Meiling CHEN 2009 14

Dynamical equation Transfer function Laplace transform Transfer function NUU-EE matrix Nonlinear Systems by Meiling CHEN 2009 15

Example MIMO system Transfer function NUU-EE Nonlinear Systems by Meiling CHEN 2009 16

Remark : the choice of states is not unique. + + - - exist a mapping NUU-EE Nonlinear Systems by Meiling CHEN 2009 17

The solution of LTI system Homogeneous solution Natural responses NUU-EE Non-homogeneous solution Forced responses Nonlinear Systems by Meiling CHEN 2009 18

Nonlinear systems: LTI Dynamic equation Nonlinear Dynamic equation NUU-EE Nonlinear Systems by Meiling CHEN 2009 19

Nonlinear system example: Pendulum equation m NUU-EE Nonlinear Systems by Meiling CHEN 2009 20