0
Outline   Introduction     Chapter 2   Chapter 3   Chapter 4    Chapter 5      Chapter 6   Conclusions




          Stabi...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Outline
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Introduction
Outline       Introduction   Chapter 2    Chapter 3   Chapter 4        Chapter 5     Chapter 6   Conclusions



Practical ...
Outline   Introduction      Chapter 2   Chapter 3     Chapter 4    Chapter 5     Chapter 6       Conclusions



Theoretica...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Special Structure

   ...
Outline      Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Objective



      ...
Outline      Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Objective



      ...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Literature

          ...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Major Contribution...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Lyapunov Stability for...
Outline         Introduction     Chapter 2       Chapter 3     Chapter 4          Chapter 5   Chapter 6   Conclusions



L...
Outline      Introduction   Chapter 2   Chapter 3      Chapter 4   Chapter 5   Chapter 6   Conclusions



Lyapunov Stabili...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions


Extension of local line...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Extension of linea...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



PWA Differential Inclus...
Outline   Introduction   Chapter 2           Chapter 3       Chapter 4   Chapter 5       Chapter 6   Conclusions



PWA Di...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Extension of linear co...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Extension of linea...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Extension of linea...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Extension of linea...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions


Controller synthesis fo...
Outline      Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions


Controller synthesis...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



L2 Gain Analysis for P...
Outline      Introduction   Chapter 2   Chapter 3       Chapter 4   Chapter 5   Chapter 6   Conclusions



A duality-based...
Outline         Introduction       Chapter 2           Chapter 3       Chapter 4        Chapter 5   Chapter 6    Conclusio...
Outline         Introduction       Chapter 2       Chapter 3      Chapter 4       Chapter 5      Chapter 6        Conclusi...
Outline   Introduction   Chapter 2   Chapter 3    Chapter 4    Chapter 5   Chapter 6   Conclusions



PWA L2 gain controll...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



A duality-based co...
Outline       Introduction   Chapter 2    Chapter 3   Chapter 4    Chapter 5   Chapter 6   Conclusions



A duality-based ...
Outline      Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



A duality-based con...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions


Backstepping Controller...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions


Backstepping Contro...
Outline       Introduction   Chapter 2   Chapter 3     Chapter 4     Chapter 5   Chapter 6   Conclusions


Backstepping Co...
Outline   Introduction   Chapter 2     Chapter 3       Chapter 4   Chapter 5   Chapter 6     Conclusions


Backstepping Co...
Outline      Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Sampled-Data PWA Sy...
Outline      Introduction      Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Sampled-Data PWA...
Outline   Introduction             Chapter 2       Chapter 3      Chapter 4     Chapter 5   Chapter 6   Conclusions



Sam...
Outline   Introduction              Chapter 2        Chapter 3   Chapter 4       Chapter 5   Chapter 6   Conclusions



Sa...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Sampled-Data PWA Syste...
Outline   Introduction   Chapter 2     Chapter 3      Chapter 4   Chapter 5      Chapter 6   Conclusions



Sampled-Data P...
Outline   Introduction     Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Sampled-Data PWA Sys...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Sampled-Data PWA S...
Outline      Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Sampled-Data PWA Sy...
Outline   Introduction              Chapter 2        Chapter 3   Chapter 4       Chapter 5   Chapter 6   Conclusions



Sa...
Outline   Introduction              Chapter 2        Chapter 3   Chapter 4       Chapter 5   Chapter 6   Conclusions



Sa...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Conclusions
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Summary of Major C...
Outline       Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Publications

    ...
Outline       Introduction   Chapter 2   Chapter 3    Chapter 4   Chapter 5    Chapter 6    Conclusions



Publications
  ...
Outline   Introduction   Chapter 2   Chapter 3   Chapter 4   Chapter 5   Chapter 6   Conclusions



Questions
Upcoming SlideShare
Loading in...5
×

Stability Analysis and Controller Synthesis for a Class of Piecewise Smooth Systems

1,405

Published on

Dr. Behzad Samadi Presentation at Dept of EE at Sharif University of Technology

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
1,405
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
31
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Transcript of "Stability Analysis and Controller Synthesis for a Class of Piecewise Smooth Systems"

  1. 1. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Stability Analysis and Controller Synthesis for a Class of Piecewise Smooth Systems The Oral Examination for the Degree of Doctor of Philosophy Behzad Samadi Department of Mechanical and Industrial Engineering Concordia University 18 April 2008 Montreal, Quebec Canada
  2. 2. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Outline
  3. 3. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Introduction
  4. 4. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Practical Motivation c Quanser Memoryless Nonlinearities Saturation Dead Zone Coulomb & Viscous Friction
  5. 5. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Theoretical Motivation Richard Murray, California Institute of Technology: “ Rank Top Ten Research Problems in Nonlinear Control 10 Building representative experiments for evaluating controllers 9 Convincing industry to invest in new nonlinear methodologies 8 Recognizing the difference between regulation and tracking 7 Exploiting special structure to analyze and design controllers 6 Integrating good linear techniques into nonlinear methodologies 5 Recognizing the difference between performance and operability 4 Finding nonlinear normal systems for control 3 Global robust stabilization and local robust performance 2 Magnitude and rate saturation 1 Writing numerical software for implementing nonlinear theory ...This is more or less a way for me to think online, so I wouldn’t take any of this too seriously.”
  6. 6. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Special Structure Piecewise smooth system: x = f (x) + g (x)u ˙ where f (x) and g (x) are piecewise continuous and bounded functions of x.
  7. 7. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Objective To develop a computational tool to design controllers for piecewise smooth systems using convex optimization techniques.
  8. 8. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Objective To develop a computational tool to design controllers for piecewise smooth systems using convex optimization techniques. Why convex optimization? There are numerically efficient tools to solve convex optimization problems. Linear Matrix Inequalities (LMI) Sum of Squares (SOS) programming
  9. 9. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Literature Hassibi and Boyd (1998) - Quadratic stabilization and control of piecewise linear systems - Limited to piecewise linear controllers for PWA slab systems Johansson and Rantzer (2000) - Piecewise linear quadratic optimal control - No guarantee for stability Feng (2002) - Controller design and analysis of uncertain piecewise linear systems - All local subsystems should be stable Rodrigues and How (2003) - Observer-based control of piecewise affine systems - Bilinear matrix inequality Rodrigues and Boyd (2005) - Piecewise affine state feedback for piecewise affine slab systems using convex optimization - Stability analysis and synthesis using parametrized linear matrix inequalities
  10. 10. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Major Contributions To propose a two-step controller synthesis method for a class 1 of uncertain nonlinear systems described by PWA differential inclusions. To introduce for the first time a duality-based interpretation of 2 PWA systems. This enables controller synthesis for PWA slab systems to be formulated as a convex optimization problem. To propose a nonsmooth backstepping controller synthesis for 3 PWP systems. To propose a time-delay approach to stability analysis of 4 sampled-data PWA systems.
  11. 11. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Lyapunov Stability for Piecewise Smooth Systems
  12. 12. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Lyapunov Stability for Piecewise Smooth Systems Theorem (2.1) For nonlinear system x(t) = f (x(t)), if there exists a continuous ˙ function V (x) such that V (x ⋆ ) = 0 V (x) > 0 for all x = x ⋆ in X t1 ≤ t2 ⇒ V (x(t1 )) ≥ V (x(t2 )) then x = x ⋆ is a stable equilibrium point. Moreover if there exists a continuous function W (x) such that W (x ⋆ ) = 0 W (x) > 0 for all x = x ⋆ in X t2 t1 ≤ t2 ⇒ V (x(t1 )) ≥ V (x(t2 )) + W (x(τ ))dτ t1 and x → ∞ ⇒ V (x) → ∞ then all trajectories in X asymptotically converge to x = x ⋆ .
  13. 13. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Lyapunov Stability for Piecewise Smooth Systems Why nonsmooth analysis? Discontinuous vector fields Piecewise smooth Lyapunov functions Vector Field Lyapunov Function Continuous Discontinuous Smooth Piecewise Smooth
  14. 14. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of local linear controllers to global PWA controllers for uncertain nonlinear systems
  15. 15. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Objectives: Global robust stabilization and local robust performance Integrating good linear techniques into nonlinear methodologies
  16. 16. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions PWA Differential Inclusions Consider the following uncertain nonlinear system x = f (x) + g (x)u ˙ Let x ∈ Conv{σ1 (x, u), . . . , σK (x, u)} ˙ where σκ (x, u) = Ai κx + ai κ + Bi κ u, x ∈ Ri , with Ri = {x|Ei x + ei ≻ 0}, for i = 1, . . . , M ≻ represents an elementwise inequality.
  17. 17. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions PWA Differential Inclusions PWA bounding envelope for a nonlinear function 10 0 -10 -20 -30 -40 f (x) -50 δ1 (x) δ2 (x) -60 -4 -3 4 -2 -1 0 1 2 3
  18. 18. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers
  19. 19. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Theorem (3.2) ¯ ¯ ¯ ¯ ¯ Let there exist matrices Pi = PiT , Ki , Zi , Zi , Λi κ and Λi κ that verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a given decay rate α > 0, desired equilibrium point x ⋆ , linear ¯ controller gain Ki ⋆ and ǫ > 0, then all trajectories of the nonlinear system in X asymptotically converge to x = x ⋆ .
  20. 20. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Theorem (3.2) ¯ ¯ ¯ ¯ ¯ Let there exist matrices Pi = PiT , Ki , Zi , Zi , Λi κ and Λi κ that verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a given decay rate α > 0, desired equilibrium point x ⋆ , linear ¯ controller gain Ki ⋆ and ǫ > 0, then all trajectories of the nonlinear system in X asymptotically converge to x = x ⋆ . If the conditions are feasible, the resulting PWA controller provides global robust stability and local robust performance.
  21. 21. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Theorem (3.2) ¯ ¯ ¯ ¯ ¯ Let there exist matrices Pi = PiT , Ki , Zi , Zi , Λi κ and Λi κ that verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a given decay rate α > 0, desired equilibrium point x ⋆ , linear ¯ controller gain Ki ⋆ and ǫ > 0, then all trajectories of the nonlinear system in X asymptotically converge to x = x ⋆ . If the conditions are feasible, the resulting PWA controller provides global robust stability and local robust performance. The synthesis problem includes a set of Bilinear Matrix Inequalities (BMI). In general, it is not convex.
  22. 22. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Controller synthesis for PWA slab differential inclusions: a duality-based convex optimization approach
  23. 23. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Controller synthesis for PWA slab differential inclusions: a duality-based convex optimization approach Objective: To formulate controller synthesis for stability and L2 gain performance of piecewise affine slab differential inclusions as a set of LMIs.
  24. 24. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions L2 Gain Analysis for PWA Slab Differential Inclusions PWA slab differential inclusion: x ∈Conv{Ai κ x + ai κ + Bwi κ w , κ = 1, 2}, (x, w ) ∈ RX ×W ˙ i y ∈Conv{Ci κ x + ci κ + Dwi κ w , κ = 1, 2} RX ×W ={(x, w )| Li x + li + Mi w < 1} i Parameter set:     Ai κ1 ai κ1 Bwi κ1  Φ =  Li Mi  i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2 li Ci κ2 ci κ2 Dwi κ2  
  25. 25. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Dual Parameter Set  T LT CiT 2   Ai κ1  i κ  ΦT =  aiT 1 ciT 2 li  i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2 κ κ T MiT T Bwi κ Dwi κ   1 2
  26. 26. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions LMI Conditions for L2 Gain Analysis Parameter Set P > 0, AT 1 P + PAi κ1 + CiT 2 Ci κ2 ∗ iκ κ <0 T T −γ 2 I + Dwi κ Dwi κ2 T Bwi κ P + Dwi κ Ci κ2 1 2 2 for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and λi κ1 κ2 < 0 AT 1 P + PAi κ1     iκ +CiT 2 Ci κ2 ∗ ∗     κ   +λi κ1 κ2 LT Li   i   aiT 1 P + ciT 2 Ci κ2 + λi κ1 κ2 li Li λi κ1 κ2 (li2 − 1) + ciT 2 ci κ2 ∗   <0 κ κ κ    T   −γ 2 I Bwi κ P      1 T T Dwi κ ci κ2 + λi κ1 κ2 li MiT T  +Dwi κ Dwi κ   +Dwi κ Ci κ2      2 2  2 2 TM +λi κ1 κ2 MiT Li +λi κ1 κ2 Mi i for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 /
  27. 27. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions LMI Conditions for L2 Gain Analysis Dual Parameter Set Q > 0, Ai κ1 Q + QAT 1 + Bwi κ1 Bwi κ T ∗ iκ <0 1 T −γ 2 I + Dwi κ2 Dwi κ T Ci κ2 Q + Dwi κ2 Bwi κ 1 2 for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and µi κ1κ2 < 0 Ai κ1 Q + QAT 1     iκ T  +B wi κ1 Bwi κ ∗ ∗    1     +µi κ1 κ2 ai κ1 aiT 1   κ   Li Q + Mi Bwi κ + µi κ1 κ2 li aiT 1 T µi κ1 κ2 (li2 − 1) + Mi MiT   ∗ <0  κ 1    −γ 2 I   T Ci κ2 Q + Dwi κ2 Bwi κ    +Dw D T Dwi κ2 MiT + µi κ1 κ2 li ci κ2   1 wi κ2 i κ2 T   +µi κ1 κ2 ci κ2 ai κ1  +µi κ1 κ2 ci κ2 ciT 2 κ for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 /
  28. 28. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions PWA L2 gain controller synthesis The goal is to limit the L2 gain from w to y using the following PWA controller: u = Ki x + ki , x ∈ Ri New variables: = Ki Q Yi = µi ki Zi = µi ki kiT Wi Problem: Wi is not a linear function of the unknown parameters µi , Yi and Zi .
  29. 29. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Proposed solutions: Convex relaxation: Since Wi = µi ki kiT ≤ 0, if the synthesis inequalities are satisfied with Wi = 0, they are satisfied with any Wi ≤ 0. Therefore, the synthesis problem can be made convex by omitting Wi .
  30. 30. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Proposed solutions: Convex relaxation: Since Wi = µi ki kiT ≤ 0, if the synthesis inequalities are satisfied with Wi = 0, they are satisfied with any Wi ≤ 0. Therefore, the synthesis problem can be made convex by omitting Wi . Rank minimization: Note that Wi = µi ki kiT ≤ 0 is the solution of the following rank minimization problem: min Rank Xi Wi Z i s.t. Xi = ≤0 ZiT µi Rank minimization is also not a convex problem. However, trace minimization (maximization for negative semidefinite matrices) works practically well as a heuristic solution Wi Z i max Trace Xi , s.t. Xi = ≤0 ZiT µi
  31. 31. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach The following problems for PWA slab differential inclusions with PWA outputs were formulated as a set of LMIs: Stability analysis (Propositions 4.1 and 4.2) L2 gain analysis (Propositions 4.3 and 4.4) Stabilization using PWA controllers (Propositions 4.5 and 4.6) PWA L2 gain controller synthesis (Propositions 4.7 and 4.8)
  32. 32. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach
  33. 33. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Objective: To formulate controller synthesis for a class of piecewise polynomial systems as a Sum of Squares (SOS) programming.
  34. 34. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Objective: To formulate controller synthesis for a class of piecewise polynomial systems as a Sum of Squares (SOS) programming. SOSTOOLS, a MATLAB toolbox that handles the general SOS programming, was developed by S. Prajna, A. Papachristodoulou and P. Parrilo. m fi 2 (x) ≥ 0 p(x) = i =1
  35. 35. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach PWP system in strict feedback from   x1 = f1i1 (x1 ) + g1i1 (x1 )x2 , ˙ for x1 ∈ P1i1 x1   ˙  x2 = f2i (x1 , x2 ) + g2i (x1 , x2 )x3 , for ∈ P2i2  2 2 x2   .  . .    x1   x2      ˙  xk = fki (x1 , x2 , . . . , xk ) + gki (x1 , x2 , . . . , xk )u, for   ∈ Pkik  .   . k k .       xk  where    x1   .  E (x , . . . , x ) ≻ 0   .  jij 1 Pjij =   . j   xj  
  36. 36. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Motivation: Toycopter, a 2 DOF helicopter model
  37. 37. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Example: Pitch model of the experimental helicopter: x1 =x2 ˙ 1 x2 = (−mheli lcgx g cos(x1 ) − mheli lcgz g sin(x1 ) − FkM sgn(x2 ) ˙ Iyy − FvM x2 + u) where x1 is the pitch angle and x2 is the pitch rate. Nonlinear part: f (x1 ) = −mheli lcgx g cos(x1 ) − mheli lcgz g sin(x1 ) PWA part: f (x2 ) = −FkM sgn(x2 )
  38. 38. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach 0.4 f (x1 ) ˆ 0.3 f (x1 ) 0.2 0.1 f (x1 ) 0 -0.1 -0.2 -0.3 -0.4 3.1416 -3.1416 -1.885 -0.6283 0.6283 1.885 x1 PWA approximation - Helicopter model
  39. 39. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach 2 1.5 1 0.5 0 x2 -0.5 -1 -1.5 -2 2 3 -3 -2 -1 0 1 x1 Continuous time PWA controller
  40. 40. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach
  41. 41. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Sampled-data PWA controller u(t) = Ki x(tk ) + ki , x(tk ) ∈ Ri Continuous−time Hold PWA systems PWA controller
  42. 42. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach The closed-loop system can be rewritten as x(t) = Ai x(t) + ai + Bi (Ki x(tk ) + ki ) + Bi w , ˙ for x(t) ∈ Ri and x(tk ) ∈ Rj where w (t) = (Kj − Ki )x(tk ) + (kj − ki ), x(t) ∈ Ri , x(tk ) ∈ Rj The input w (t) is a result of the fact that x(t) and x(tk ) are not necessarily in the same region.
  43. 43. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Theorem (6.1) For the sampled-data PWA system, assume there exist symmetric positive matrices P, R, X and matrices Ni for i = 1, . . . , M such that the conditions are satisfied and let there be constants ∆K and ∆k such that w ≤ ∆K x(tk ) + ∆k Then, all the trajectories of the sampled-data PWA system in X converge to the following invariant set Ω = {xs | V (xs , ρ) ≤ σa µ2 + σb } θ
  44. 44. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Solving an optimization problem to maximize τM subject to the constraints of the main theorem and η > γ > 1 leads to ⋆ τM = 0.2193
  45. 45. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach 2.5 2 1.5 1 0.5 0 x2 -0.5 -1 -1.5 -2 -2.5 3 -3 -2 -1 0 1 2 x1 Sampled data PWA controller for Ts = 0.2193
  46. 46. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach 2 1.5 1 0.5 0 x2 -0.5 -1 -1.5 -2 2 3 -3 -2 -1 0 1 x1 Continuous time PWA controller
  47. 47. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Conclusions
  48. 48. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Summary of Major Contributions To propose a two-step controller synthesis method for a class 1 of uncertain nonlinear systems described by PWA differential inclusions. To introduce for the first time a duality-based interpretation of 2 PWA systems. This enables controller synthesis for PWA slab systems to be formulated as a convex optimization problem. To propose a nonsmooth backstepping controller synthesis for 3 PWP systems. To propose a time-delay approach to stability analysis of 4 sampled-data PWA systems.
  49. 49. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Publications B. Samadi and L. Rodrigues, “Extension of local linear 1 controllers to global piecewise affine controllers for uncertain nonlinear systems,” accepted for publication in the International Journal of Systems Science. B. Samadi and L. Rodrigues, “Controller synthesis for 2 piecewise affine slab differential inclusions: a duality-based convex optimization approach,” under second revision for publication in Automatica. B. Samadi and L. Rodrigues, “Backstepping Controller 3 Synthesis for Piecewise Polynomial Systems: A Sum of Squares Approach,” in preparation B. Samadi and L. Rodrigues, “Sampled-Data Piecewise Affine 4 Systems: A Time-Delay Approach,” to be submitted.
  50. 50. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Publications 1 B. Samadi and L. Rodrigues, “Backstepping Controller Synthesis for Piecewise Polynomial Systems: A Sum of Squares Approach,” submitted to the 46th Conference on Decision and Control, cancun, Mexico, Dec. 2008. 2 B. Samadi and L. Rodrigues, “Sampled-Data Piecewise Affine Slab Systems: A Time-Delay Approach,” in Proc. of the American Control Conference, Seattle, WA, Jun. 2008. 3 B. Samadi and L. Rodrigues, “Controller synthesis for piecewise affine slab differential inclusions: a duality-based convex optimization approach,” in Proc. of the 46th Conference on Decision and Control, New Orleans, LA, Dec. 2007. 4 B. Samadi and L. Rodrigues, “Backstepping Controller Synthesis for Piecewise Affine Systems: A Sum of Squares Approach,” in Proc. of the IEEE International Conference on Systems, Man, and Cybernetics (SMC 2007), Montreal, Oct. 2007. 5 B. Samadi and L. Rodrigues, “Extension of a local linear controller to a stabilizing semi-global piecewise-affine controller,” 7th Portuguese Conference on Automatic Control, Lisbon, Portugal, Sep. 2006.
  51. 51. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Questions
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×