1. Lecture 02 State space approach

2. NUU-EENonlinear Systems by Meiling CHEN 2009 2 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

3. NUU-EENonlinear Systems by Meiling CHEN 2009 3 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

4. NUU-EENonlinear Systems by Meiling CHEN 2009 4 State equation Output equation Dynamic equation State space State variable r- inputp- output LTI systems:

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

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

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

8. NUU-EENonlinear Systems by Meiling CHEN 2009 8 system stable State-space descriptionInternal behavior description

9. NUU-EENonlinear Systems by Meiling CHEN 2009 9 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 M Example 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。

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

11. NUU-EENonlinear Systems by Meiling CHEN 2009 11 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

12. NUU-EENonlinear Systems by Meiling CHEN 2009 12 Example M2M2 M1M1 B3B3 B1B1 B2B2 K

13. NUU-EENonlinear Systems by Meiling CHEN 2009 13 Armature circuitField circuit Example

14. NUU-EENonlinear Systems by Meiling CHEN 2009 14

15. NUU-EENonlinear Systems by Meiling CHEN 2009 15 Dynamical equation Transfer function Laplace transform matrix Transfer function

16. NUU-EENonlinear Systems by Meiling CHEN 2009 16 Example Transfer function MIMO system

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

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

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

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