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- 1. References A MATLAB Toolbox for Piecewise-Aﬃne Controller Synthesis Mohsen Zamani Behzad Samadi Luis Rodrigues HYCONS Lab, Concordia University American Control Conference Montreal, Canada June 2012
- 2. References Outline Motivation PWA systems PWA approximation PWA controller synthesis Stability analysis of the closed loop system Example Conclusion
- 3. References Motivation Gain Scheduling Controller: Design an autopilot to: minimize steady state tracking error maximize robustness to wind gust subject to varying ﬂight conditions For controller design, consider the following issues: Theory of Linear Systems is very rich in terms of analysis and synthesis methods and computational tools. Real world systems, however, are usually nonlinear. What can be done to extend the good properties of linear systems theory to nonlinear systems?
- 4. References Motivation Gain Scheduling Controller: Gain scheduling is an attempt to address this issue. Divide and conquer Approximating nonlinear systems by a combination of local linear systems Designing local linear controllers and combining them Started in 1960s, very popular in a variety of ﬁelds from aerospace to process control Problem: proof of stability! Problem: By switching between two stable linear system, you can create an unstable system.
- 5. References PWA Systems Piecewise Smooth Systems The dynamics of a piecewise smooth (PWS) is deﬁned as: ˙x = fi (x), x ∈ Ri where x ∈ X is the state vector. A subset of the state space X is partitioned into M regions, Ri , i = 1, . . . , M such that: ∪M i=1 ¯Ri = X, Ri ∩ Rj = ∅, i = j where ¯Ri denotes the closure of Ri .
- 6. References PWA Systems Piecewise Aﬃne Systems The dynamics of a piecewise aﬃne (PWA) is deﬁned as: ˙x = Ai x + ai + Bi u, x ∈ Ri where Ri = {x|Ei x + ei ≥ 0}, i = 1, 2, . . . , M PWA systems are: nonlinear locally linear
- 7. References PWA Systems PWA Slab Systems The regions of PWA slab system are intervals on a scalar variable: Ri = {x|βi < cT x < βi+1} The regions of a PWA slab system can be written as degenerate ellipsoids: Ri = {x| |Ei x + ei | < 1}
- 8. References PWA Systems PWA Diﬀerential Inclusions In this toolbox, we have considered PWA diﬀerential inclusions (PWADI) described by: ˙x ∈ conv{Ai1x + ai1 + Bi1u, Ai2x + ai2 + Bi2u}, x ∈ Ri PWADIs are a form of uncertain PWA systems. Nonlinear systems can be included by PWADIs.
- 9. References PWA Approximation Given a nonlinear system: ˙x = Ax + a + f (x) + B(x)u PWATOOLS can create a PWA approximation using the following approaches: Uniform: connecting the dots of a uniform grid on the nonlinear surface Optimal Uniform: minimizing the error for a uniform grid Multi-resolution: recursively building a nonuniform grid based on error minimization (divide and conquer)
- 10. References PWA Controller Synthesis Global Lyapunov function Find a PWA controller u = Ki x + ki such that: V (x) = xTPx > 0 dV dt < −αV for α > 0 the control signal is continuous (optional) This problem is formulated as a set of Bilinear Matrix Inequalities (BMI). For PWA slab systems, the problem can formulated as a set of Linear Matrix Inequalities (LMI).
- 11. References PWA Controller Synthesis Piecewise quadratic Lyapunov function Find a PWA controller u = Ki x + ki such that: V (x) = xTPi x + 2qT i x + ri > 0 dV dt < −αV for α > 0 the Lyapunov function is continuous the dynamics of the closed loop system is continuous
- 12. References Stability of the Closed Loop System The dynamics of the closed loop system is described by: ˙x = (Ai + Bi Ki )x + ai + Bi ki The same inequalities are used assuming that the controller known. The BMIs become LMIs.
- 13. References Example Flutter Phenomenon Mechanism of Flutter Inertial Forces Aerodynamic Forces (∝ V 2 ) (exciting the motion) Elastic Forces (damping the motion) Flutter Facts Flutter is self-excited Two or more modes of motion (e.g. ﬂexural and torsional) exist simultaneously Critical Flutter Speed, largely depends on torsional and ﬂexural stiﬀnesses of the structure [1]
- 14. References Example State Space Equations: M ¨h ¨α + (C0 + Cµ) ˙h ˙α + (K0 + Kµ) h α + 0 αKα(α) = Bβo Kα(α) = 2.82α − 62.322α2 + 3709.71α3 − 24195.6α4 + 48756.954α5 State variables: plunge deﬂection (h), pitch angle (α), and their derivatives (˙h and ˙α) Inputs: angular deﬂection of the ﬂaps (βo ∈ R2) Constraints: on states and actuators [2; 3]
- 15. References PWATOOLS Create the system description (4 state variables, 2 inputs): pwacreate(4,2,ActiveFlutter_non.m); Edit the system description: ActiveFlutter non.m model.A = [0 0 1 0; 0 0 0 1; -M(Ko+Ku) -M(Co+Cu)]; model.Bx = [0 0; 0 0;muu*inv(M)*B]; model.aff = [0;0;0;0]; model.fx = @Flutter_Nonlinearity; model.xcl = [0;0;0;0]; model.Domain ={[-1 1],[-1 1],[-1 1],[-5 5]} model.NonlinearDomain = [0;1;0;0]; model.mtd = Uniform; model.M =6; % Number of regions
- 16. References PWATOOLS Create the nonlinear function: function F = Flutter_Nonlinearity(x) qx2 = 2.82*x(2)-62.322*x(2)^2+3709.71 *x(2)^3-24195.6*x(2)^4 +48756.954*x(2)^5;| M = [12.387 0.418; 0.418 0.065]; F = [0;0;-inv(M)*[0;qx2]]; Run the system description: ActiveFlutter non.m. It created two objects: nlsys: the nonlinear system pwainc: the PWADI
- 17. References PWATOOLS Create the nonlinear function: x0=[.4 .63 .5 1.2]’; setting.alpha=.1; setting.SynthMeth=’bmi’; setting.RandomQ= 0; setting.RandomR= 0; setting.QLin= 1.0e+003 * [2.7378 1.9536 2.1565 1.5109 1.9536 1.8326 1.5729 0.8522 2.1565 1.5729 1.9228 1.2733 1.5109 0.8522 1.2733 1.1032]; setting.RLin= 1.0e+003* [2.7512 3.0551 3.0551 3.3928];
- 18. References PWATOOLS Run the controller synthesis command: ctrl=pwasynth(pwainc, x0, setting); PWA controller: K1 = −2.14 64.26 −5.54 −18.41 17.92 2.06 −61.22 2.90 6.43 −7.79 , K2 = −2.14 −15.79 −5.54 −18.41 0 2.06 5.39 2.90 6.43 0 , K3 = −2.14 −2.59 −5.54 −18.41 0 2.06 −28.19 2.90 6.43 0 , K4 = −2.14 77.35 −5.54 −18.41 26.64 2.06 −70.89 2.90 6.43 −14.24 , K5 = −2.14 85.10 −5.54 −18.41 −48.41 2.06 −44.49 2.90 6.43 20.10 , K6 = −2.14 40.75 −5.54 −18.41 −18.85 2.06 −34.06 2.90 6.43 13.15
- 19. References PWATOOLS Simulation results
- 20. References Conclusions PWATOOLS is an easy to use toolbox for PWA systems. The plan is to extend PWATOOLS algorithms to piecewise polynomial (PWP) systems. Try PWATOOLS for your problem and let us know how it works.
- 21. References References I [1] M. R. Waszak. Modeling the benchmark active control technology wind tunnel model for application to ﬂutter suppression. AIAA, 96 - 3437, http: //www.mathworks.com/matlabcentral/fileexchange/3938. [2] J. Ko and T. W. Strganacy,. Stability and control of a structurally nonlinear aeroelastic system. Journal of Guidance, Control, and Dynamics, 21, 718-725. [3] S. Afkhami and H. Alighanbari. Nonlinear control design of an airfoil with active ﬂutter suppression in the presence of disturbance. Control Theory and Applications, 1(6):1638–1649, 2007. [4] A. Hassibi and S. Boyd. Quadratic stabilization and control of piecewise-linear systems. Proceedings of the American Control Conference, 6:3659 – 3664, 1998. [5] J. L¨ofberg. YALMIP : A Toolbox for Modeling and Optimization in MATLAB. Proceedings of the CACSD Conference, Taipei, Taiwan, 2004.
- 22. References References II [6] Manual. PWATOOLS. http://hycons.encs.concordia.ca/Projects/PWAtoolbox, May2011. [7] L. Rodrigues and S. Boyd. Piecewise-aﬃne state feedback for piecewise-aﬃne slab systems using convex optimization. Systems and Control Letters, 54:835–853, 2005. [8] B. Samadi and L. Rodrigues. Extension of local linear controllers to global piecewise aﬃne controllers for uncertain non-linear systems. International Journal of Systems Science, 39(9):867–879, 2008. [9] B. Samadi and L. Rodrigues. A duality-based convex optimization approach to L2-gain control of piecewise aﬃne slab diﬀerential inclusions. Automatica, vol. 45, no. 3, pp. 812 - 816, Mar. 2009. [10] B. Samadi and L. Rodrigues. A uniﬁed dissipativity approach for stability analysis of piecewise smooth systems. Automatica, vol. 47, no. 12, pp. 2735 - 2742, Dec. 2011.

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