1. Viability for a Hybrid System Control1 Viability for a Hybrid Control System Yan Gao( 高 岩 ) University of Shanghai for Science and Technology Email: gaoyan@usst.edu.cn The 4th China-Europe Summer School on Complexity Science 12th August,2010 University of Shanghai for Science and Technology

2. Viability for a Hybrid System Control2 Outlines Hybrid System Viability of a Control System Determining Viability Multi-Lyapunov ( Nonsmoooth Lyapunov) Function Population Control for a City University of Shanghai for Science and Technology

3. Viability for a Hybrid System Control3 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. University of Shanghai for Science and Technology

4. Viability for a Hybrid System Control4 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. University of Shanghai for Science and Technology

5. Viability for a Hybrid System Control5 A hybrid system t x University of Shanghai for Science and Technology

6. Viability for a Hybrid System Control6 Modelling of hybrid control As a complex system, hybrid systems has various forms. A representative model for hybrid control system is as the following. University of Shanghai for Science and Technology

7. Viability for a Hybrid System Control7 continuous part is discrete part is state is control University of Shanghai for Science and Technology

8. Viability for a Hybrid System Control8 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. University of Shanghai for Science and Technology

9. Viability for a Hybrid System Control9 A switched system University of Shanghai for Science and Technology

10. Viability for a Hybrid System Control10 Trajectory of switched control system is nonsmooth. Nonsmoothness appears at switching point. University of Shanghai for Science and Technology

11. Viability for a Hybrid System Control11 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) University of Shanghai for Science and Technology

12. Viability for a Hybrid System Control12 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 ． University of Shanghai for Science and Technology

13. Viability for a Hybrid System Control13 Some periodicals have some special issue on hybrid system IEEE Transactions on Automatic Control Automatica Lecture Notes in Computer Science University of Shanghai for Science and Technology

14. Viability for a Hybrid System Control14 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. University of Shanghai for Science and Technology

15. Viability for a Hybrid System Control15 x0x0 x(t ) Fig1 Viability for a continuous system University of Shanghai for Science and Technology

16. Viability for a Hybrid System Control16 ． Fig2 Viability for a hybrid system University of Shanghai for Science and Technology 0 x0x0 1 x1x1 2 x2x2 3 x3x3 K

17. Viability for a Hybrid System Control17 There are three topics in research on viability: 1.Determining viability; 2.Finding a variable solution; 3.Design of viable region. University of Shanghai for Science and Technology

18. Viability for a Hybrid System Control18 Applications of viability Safety region design Stability and stabilization for a control system Differential game ( dynamical game) Actually,viable region is a safety region. University of Shanghai for Science and Technology

19. Viability for a Hybrid System Control19 University of Shanghai for Science and Technology

20. Viability for a Hybrid System Control20

21. Viability for a Hybrid System Control21

22. Viability for a Hybrid System Control22 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. University of Shanghai for Science and Technology

23. Viability for a Hybrid System Control23 Fig3 The bigest varible region University of Shanghai for Science and Technology is viable region in

24. Viability for a Hybrid System Control24 Consider a differential inclusion (2) Where is a mapping from to its subset. 3.Viabity Condition University of Shanghai for Science and Technology

25. Viability for a Hybrid System Control25 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

26. Viability for a Hybrid System Control26 Differential inclusion (2) reduces to a nonlinear control system if that is University of Shanghai for Science and Technology

27. Viability for a Hybrid System Control27 Definition Let be nonempty. The tangent cone of at is defined by where University of Shanghai for Science and Technology

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

29. Viability for a Hybrid System Control29 Viability condition:The closed set is viable for the differential inclusion (2) if and only if University of Shanghai for Science and Technology

30. Viability for a Hybrid System Control30 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. University of Shanghai for Science and Technology

31. Viability for a Hybrid System Control31 In most applications, there are no common Lyapunov function. Multi- Lyapunov function, one is a nonsmooth, Lyapunov function, for instance, a piecewise smooth function. University of Shanghai for Science and Technology

32. Viability for a Hybrid System Control32 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. University of Shanghai for Science and Technology

33. Viability for a Hybrid System Control33 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 ． University of Shanghai for Science and Technology

34. Viability for a Hybrid System Control34 We get imigration to Shanghai: University of Shanghai for Science and Technology

35. Viability for a Hybrid System Control35 Thank You Very Much!