S. Hassan HosseinNia Blas M. Vinagre Universidad de Extremadura

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1 S. Hassan HosseinNia Blas M. Vinagre Universidad de Extremadura

2 Main Topics Introduction to switching systems Introduction to fractional order calculus Sliding Mode Control DC-DC buck converter Review of previous works Sliding Mode Control Integer order SMC Fractional order SMC 2

3 Introduction to switching systems Hybrid system: interaction between time driven and event driven system Many hybrid system can be modeled by continuous system with switching continuous system are easier to be analyzed and to be controlled 3

4 Introduction to switching systems The main issues then become stability analysis and control synthesis A switching system can be a differential equation whose right-hand side can be chosen from a family of functions based on a switching signal. 4

5 Introduction to switching systems Switching system 1. State dependent switching The family of switching surfaces and the resulting operating regions The family of continuous-time subsystems, one for each operating region; The reset map. Switching surface Continuous portion of trajectory jump 5

6 Introduction to switching systems 2. Switching in time : f q is continuous for each q. 6

7 Introduction to switching systems 3. Autonomous and controlled switching Embedded system Computer controlled system 7

8 Introduction to fractional order calculus 8

9 Introduction to fractional order calculus 9

10 Introduction to fractional order calculus 10

11 Introduction to fractional order calculus Caputo definition of Fractional derivative Laplace transform R-L Caputo 11

12 Sliding Mode Control 12

13 Sliding Mode Control 13

14 DC-DC buck Converter 14

15 DC-DC buck Converter This converter has two states u=1 and u=0 which preform a switched system. 15

16 Previous works Linear: Phase lag compensation Smith predictor structure Discrete bode ideal function Nonlinear: SMC (PI, PID) SMC (FPI, FPID) 16

17 Linear Controller 17

18 Nonlinear Controller PID: Fractional PID: 18

19 SMC using augmented system Augmented system 19

20 SMC using augmented system Surface: Sliding Condition: In absence of uncertainty, and: 20

21 SMC using augmented system Lyapunov function: Reachability condition and 21

22 SMC using augmented system Applying in buck converter 22

23 SMC using augmented system 23

24 24

25 Problem? Replacing fictitious control law to Boolean input, 1. The control law don t have the information of SMC or other controller and the output is not what is would be desired. 2. The steady state error may happen To overcome this problem following theories is proposed. 25

26 Sliding Mode Control: Theory I Consider the dynamics of buck converter. The hitting condition of the sliding mode is satisfied, the control u(t) is given by, Sliding surface: 26

27 Proof. (Theory I) Lyapunov function: Reachability condition Assuming and as a constant value, The sliding condition is satisfied if. 27

28 Sliding Mode Control: Theory II Consider the dynamics of buck converter. The hitting condition of the sliding mode is satisfied, the control u(t) is given by, Sliding surface: 28

29 Proof. (Theory II) Lyapunov function: Reachability condition Assuming and v d as a constant value, The sliding condition is satisfied if. 29

30 Sliding Mode Control: Theory III Consider the dynamics of buck converter. The hitting condition of the sliding mode is satisfied, the control u(t) is given by, Sliding surface: 30

31 Proof. (Theory III) Lyapunov function: Reachability condition Assuming and v d as a constant value, 31

32 Proof. (Theory III) Define, Fractional differentiation of f(s), 32

33 Proof. (Theory III) 33

34 Simulation Results =0.5 SMC (PID) =0.7 SMC (PID) SMC (FPID) 14 SMC (FPID) 14 v c v c time(s) time(s) 34

35 Simulation Results =0.8 SMC (PID) SMC (FPID) =0.9 SMC (PID) SMC (FPID) v c v c time(s) time(s) 35

36 Simulation Results SMC (PD) SMC (PID) SMC (FPID) v c time(s) 36

37 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

38 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

39 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

40 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

41 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

42 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

43 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

44 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as control input and the is no steady state error. 37

45 Conclusion In previous methods: Fractional SMC has almost better results than the other controller. The switching is not directly obtained from the control input and some information will be lost in modulation. Pulse width modulation in some cases are caused steady state error. In the proposed method: SMC and fractional SMC is proposed to obtain switching in buck converter. The Boolean input is directly designed using SMC which make the pulse width modulation useless. The switching has the same information as the control input and there is no steady state error. 37

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1.2 Software tools for analysis and design of control systems Terminology. Formulation of the control problem... 11

Contents 1 Introduction 1 1.1 Theimportanceofcontrol... 1 1.2 Software tools for analysis and design of control systems... 5 1.3 Ashorthistoryofcontrol... 6 2 Introduction to feedback control 11 2.1 Introduction...