1. 12 th international conference on Sciences and Techniques of Automatic control and computer engineering 1 Sousse, Tunisia, December 18-20, 2011 Flatness Robust Control of a Hydraulic Process Nizar CHELLY Hassen MEKKI

2. 2  Introduction ◦ Differential Flatness ◦ Flatness control ◦ Robust Flatness control  Robust Flatness Control Design ◦ Control Law Design ◦ Simulation Results  Conclusion d computer engine 12 th international conference on Sciences and Techniques of Automatic control and computer engineering

3. 3  Origin ◦ Differential flatness has been first introduced by M. Fliess, J. Lévine, P. Martin, P. Rouchon. 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Definition A system is said to be differentially flat if and only if the state and input can be determined by a set of variables, called flat output, which may possess physical meanings.  Application The control input of a flat system can be determined from a given trajectory of the flat output.

4. 4  A nonlinear system given as: 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  is differentially flat if there exists a vector  The dynamics behaviour of the system is resumed by the behaviour of its flat output z. such that :  The z is called a flat output or linearizing output.

5. 5 12 th international conference on Sciences and Techniques of Automatic control and computer engineering Flat output state t=0 t=T  Our goal : to take a system from initial situation (X 0,U 0 ) to a final situation (X f,U f )  Any fonction between [0,T] ; t ⟼ z(t) satisfed the limites conditions gives us a trajectory for the state and its associed control.

6. 6 12 th international conference on Sciences and Techniques of Automatic control and computer engineering Z ref (t), U ref (t) Non Linear System Trajectory Generation U(t) +- δz(t) δu(t) Z ref (t) + Z(t) Robust controller Flat output Perturbations PID controller Backstepping Controller LPV Controller Sliding Mode Controller

7. 7 12 th international conference on Sciences and Techniques of Automatic control and computer engineering The states :h 1,h 2,h 3 The inputs (commands) :Q 1,Q 2 The outputs : h 1,h 3

8. 8 12 th international conference on Sciences and Techniques of Automatic control and computer engineering Our goal is to control the water levels in tank1 from : 0.4 m to 0.45m and tank 3 from : 0.3 m to 0.35m In the presence of perturbation on the initial conditions 0.02 m in tank 1 0.01 m in tank 3 We want to reject this perturbation on 14 second in tank 1and on 10 second in tank 3. We want to assure a good robustness margin and a small tracking error

9. 9 12 th international conference on Sciences and Techniques of Automatic control and computer engineering Step 1 : determination of the flat outputs

10. 10 12 th international conference on Sciences and Techniques of Automatic control and computer engineering the command associated for [x1ref, x3ref] : We choose [z1ref, z2ref]= [x1ref, x3ref] as follow:  Step 2: Trajectories Generation

11. 11 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Step 4: Construction of the error model Linearization around the references trajectories Linear Parameter Variant LPV  Step 5 :  Step 3: coordinates transformation of the flat output

12. 12 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Step 6: Transformation LPV/LFT 4 variant parameters: P(s) Δ(t) δuδuδyδy wΔwΔ zΔzΔ

13. 13 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Step 7 : Setting specifications by weighting functions M 1, M 2 : to assure robustness margin 1, 2 : to assure a small error We(s) w 1, w 2 : to set the time rise

14. 14 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Step 8 : Construction of the augmented model

15. 15 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Step 9 : Using the Matlab LMI control toolbox :  Step 10 : We implement on Matlab this LPV Controller in the following control scheme :

16. 16 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Case 1: Perturbation on the Initial conditions 0.02 m in Tank 1 0.01 m in Tank 3

17. 17 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Case 2: Exogenous Perturbation 1/2

18. 18 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Case 2: Exogenous Perturbation 2/2

19. 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Case 3: Parametric Perturbation 1/2

20. 12 th international conference on Sciences and Techniques of Automatic control and computer engineering  Case 3: Parametric Perturbation 2/2

21. 12 th international conference on Sciences and Techniques of Automatic control and computer engineering The Flatness can resolve the trajectories generation for a non linear system without solving the differentiel system To eliminate uncertainties or perturbation on the system we have to design robust regulator The LPV controller is a robust controller

22. 12 th international conference on Sciences and Techniques of Automatic control and computer engineering Thank you for your attention