1. MESA Lab Two Interesting Papers Introduction at ICFDA 2014 Xiaobao Jia MESA (Mechatronics, Embedded Systems and Automation) Lab School of Engineering, University of California, Merced E: xjia3@ucmerced.edu Phone: 13925273161 Lab: CAS Eng 820 (T: 228-4398) Jun. 30, 2014. Monday 4:00-6:00 PM Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced

2. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-2/1024 1. Hybrid Systems and Control With Fractional Dynamics (I): Modeling and Analysis 2. Hybrid Systems and Control With Fractional Dynamics (II): Control Two papers

3. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-3/1024 1. Hybrid Systems Conception 2. Hybrid Systems layers 3. Fractional-order Switching System 4. Fractional-order reset control systems 5. Stability of Fractional-Order Hybrid Systems 6. Robust Fractional-Order Control of Switching System 7. Summa Outline

4. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-4/1024 1. Hybrid Systems Conception A hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior – a system that can both flow and jump. Often, the term "hybrid dynamic system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines.

5. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-5/1024 2. Hybrid Systems layers The control of a continuous state dynamical plant is often divided into two layers. At the lower or servo layer is a controller that regulates the plant in the traditional manner. At the upper layer the supervisor issues symbolic commands. Some time after a command is issued the supervisor receives a symbolic response indicating either that the command has been executed or that an error condition occurred.

6. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-6/1024 2. Hybrid Systems layers Mathematically, the servo layer and plant are modeled together as a differential equation evolving in real time. The supervisor, however, views the lower layer as a discrete event system whose behavior is described by sequences of events that unfold in logical time.

7. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-7/1024 3. Fractional-order Switching System Switching system is a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching among them. A general formulation of the switching systems with fractional-order is:

8. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-8/1024 3. Fractional-order Switching System a first-order system with two different dynamics Function fractional-order PI controllers the closed-loop transfer function of the system

9. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-9/1024 4. Fractional-order Reset Control Systems Reset controllers are linear controllers that reset some of their states to zero when their input is zero. A general Block diagram of a reset control is:

10. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-10/1024 4. Fractional-order Reset Control Systems The dynamics of the reset controller FDI equation The linear controller C(s) state space representations

11. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-11/1024 4. Fractional-order Reset Control Systems The plant P(s) state space representations The closed-loop reset control system can be then described by the following FDI as follows:

12. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-12/1024 5. Stability of Fractional-Order Hybrid Systems the stability of fractional-order switching systems by common Lyapunov functions and its equivalence in frequency domain.

13. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-13/1024 5. Stability of Fractional-Order Hybrid Systems Example:

14. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-14/1024 5. Stability of Fractional-Order Hybrid Systems Example:

15. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-15/1024 6. Robust Fractional-Order Control of Switching System A scheme of the strategy is shown in Fig. 1 for a general system with subsystems Gi(s), with i = 1;2; :::;L subsystems.

16. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-16/1024 Controller Type: 6. Robust Fractional-Order Control of Switching System

17. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-17/1024 Example: 6. Robust Fractional-Order Control of Switching System

18. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-18/1024 Result: 6. Robust Fractional-Order Control of Switching System

19. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-19/1024 Result: 6. Robust Fractional-Order Control of Switching System

20. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Slide-20/1024 7 Summa Additional flexibility to the Hybrid system More stability to the Hybrid system Strong robustness to the Hybrid system

21. MESA Lab 04/21/2014 AFC Workshop Series @ MESALAB @ UCMerced Thank You! Slide-21/1024