Viability for a Hybrid Control System Yan Gao( 高 岩 )

1. University of Shanghai for Science and Technology Email: gaoyan@usst.edu.cn University of Shanghai for Science and Technology Viability for a Hybrid ControlSystem Yan Gao(高 岩) The 4th China-Europe Summer School on Complexity Science  12th August,2010 Viability for a Hybrid System Control

2. University of Shanghai for Science and Technology Outlines • Hybrid System • Viability of a Control System • Determining Viability • Multi-Lyapunov ( Nonsmoooth Lyapunov) Function • Population Control for a City Viability for a Hybrid System Control

3. University of Shanghai for Science and Technology 1 Hybrid System Dynamical system with continuous variable (state) and discrete variable (state). Usually, there is an interaction between continuous state and discrete state . In some cases, discrete state is on logic. Discrete state generate a jump. Viability for a Hybrid System Control

4. University of Shanghai for Science and Technology Examples: • Mechanical system: continuous motion interrupted by collisions. • Electrical circuit: charging of capacitor, interrupted by switched opening or closing. • Chemical system: continuous evolution of chemical interaction is control by valves and pumps. • A car with 5 gears, gear shift control generate a jump for velocity. Viability for a Hybrid System Control

5. University of Shanghai for Science and Technology A hybrid system x t Viability for a Hybrid System Control

6. University of Shanghai for Science and Technology Modelling of hybrid control As a complex system, hybrid systems has various forms. A representative model for hybrid control system is as the following. Viability for a Hybrid System Control

7. University of Shanghai for Science and Technology continuous part is discrete part is state is control Viability for a Hybrid System Control

8. University of Shanghai for Science and Technology Swiched control system A hybrid system without jump is said to be switched. Even no jumps in a switched system, but dynamical models changed, it has multi-models. Viability for a Hybrid System Control

9. University of Shanghai for Science and Technology A switched system Viability for a Hybrid System Control

10. University of Shanghai for Science and Technology • Trajectory of switched control system is nonsmooth. • Nonsmoothness appears at switching point. Viability for a Hybrid System Control

11. University of Shanghai for Science and Technology An important switched is linear switched system has of the form: There are m models for the systerm(1).We can choose the switching rule. (1) Viability for a Hybrid System Control

12. University of Shanghai for Science and Technology The earliest publication for hybrid system is : Witsenhausen，IEEE Transactions on Automatic Control,1966. The notion of hybrid system was proposed formally in 80’s ． Viability for a Hybrid System Control

13. University of Shanghai for Science and Technology Some periodicals have some special issue on hybrid system • IEEE Transactions on Automatic Control • Automatica • Lecture Notes in Computer Science Viability for a Hybrid System Control

14. University of Shanghai for Science and Technology 2. Viability Definition of viability: Consider a control system and a set, if for any initial point in this set, there exists a solution of the control system such that it stays in the set for ever, then the set is said to be viable under this control system. Viability for a Hybrid System Control

15. University of Shanghai for Science and Technology x(t) x0 Fig1 Viability for a continuous system Viability for a Hybrid System Control

16. University of Shanghai for Science and Technology x2 2 K x1 1 x3 x0 3 0 ． Fig2 Viability for a hybrid system Viability for a Hybrid System Control

17. University of Shanghai for Science and Technology There are three topics in research on viability: 1.Determining viability; 2.Finding a variable solution; 3.Design of viable region. Viability for a Hybrid System Control

18. University of Shanghai for Science and Technology Applications of viability • Safety region design • Stability and stabilization for a control system • Differential game ( dynamical game) Actually,viable region is a safety region. Viability for a Hybrid System Control

19. University of Shanghai for Science and Technology Viability for a Hybrid System Control

20. Viability for a Hybrid System Control

21. Viability for a Hybrid System Control

22. University of Shanghai for Science and Technology Usually a given a set (region) is not viable. We try to find a subset, particular a bigest one which is viable. Viable region: the bigger, the better. Viability for a Hybrid System Control

23. University of Shanghai for Science and Technology is viable region in Fig3The bigest varible region Viability for a Hybrid System Control

24. University of Shanghai for Science and Technology 3.Viabity Condition Consider a differential inclusion (2) Where is a mapping from to its subset. Viability for a Hybrid System Control

25. Both linear and nonlinear control systems are special cases of the differential inclusion (2). For instance, (2) reduces to a linear control system if That is For instance, (2) reduces to a nonlinear control system if University of Shanghai for Science and Technology , Viability for a Hybrid System Control

26. University of Shanghai for Science and Technology Differential inclusion (2) reduces to a nonlinear control system if that is Viability for a Hybrid System Control

27. University of Shanghai for Science and Technology Definition Let be nonempty. The tangent cone of at is defined by where Viability for a Hybrid System Control

28. Tangent cone of K at K K K Tangent cone of K at Viability for a Hybrid System Control

29. University of Shanghai for Science and Technology Viability condition:The closed set is viable for the differential inclusion (2) if and only if Viability for a Hybrid System Control

30. University of Shanghai for Science and Technology 4.Nonsmooth Lyapunov Function Stability of for a control system is based on Lyapunov function. For switched system, since it is multy-model,switched system has two methods: Common Lyapunov function:There is function, which is a Lyapunov function for each model. Viability for a Hybrid System Control

31. University of Shanghai for Science and Technology In most applications, there are no common Lyapunov function. Multi- Lyapunov function, one is a nonsmooth, Lyapunov function, for instance, a piecewise smooth function. Viability for a Hybrid System Control

32. University of Shanghai for Science and Technology 5. Population Control for a City Continuous model of population is where is population, is a parameter, which depend on birth rate and health level. Viability for a Hybrid System Control

33. University of Shanghai for Science and Technology Migration for the city is taken as discrete. We suppose jump happen once a year Example: Population of Shanghai in 2007 is 18580,000, ．We hope to control the population in 2012 within 21000000． Viability for a Hybrid System Control

34. University of Shanghai for Science and Technology We get imigration to Shanghai: Viability for a Hybrid System Control

35. Thank You Very Much! Viability for a Hybrid System Control