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- 1. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Modeling, Control, and Optimization for Aerospace Systems HYCONS Lab, Concordia University Behzad Samadi HYCONS Lab, Concordia University American Control Conference Montreal, Canada June 2012
- 2. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Outline Motivation Aircraft design Parameter estimation Model order reduction Model based control design Convex Optimization Sum of Squares Lyapunov Analysis Controller Synthesis Safety Verification Polynomial Controller Synthesis Gain Scheduling Piecewise Smooth Systems References
- 3. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Motivation There are many problems that can be formulated as optimization problems: Aircraft design Modeling: Parameter estimation Modeling: Model order reduction Model based control design (Landing gear semi active suspension)
- 4. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Aircraft Design The aircraft designer wants to: maximize range minimize weight maximize lift to drag ratio minimize cost minimize noise subject to physical, geometrical, environmental, budget and safety constraints Multidisciplinary Optimization (MDO) problem: aerodynamics, structure, aeroelasticity, propulsion, noise and vibration, dynamics, stability and control, manufacturing
- 5. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Parameter Estimation Unmanned Rotorcraft Technology Demonstrator ARTIS at DLR (German Aerospace Center)
- 6. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Parameter Estimation The dynamic equations are given by: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ˙u ˙v ˙p ˙q ˙a ˙b ˙w ˙r ˙𝜑 ˙𝜃 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Xu 0 0 0 Xa 0 0 0 0 −g 0 Yv 0 0 0 Yb 0 0 g 0 Lu Lv 0 0 0 Lb Lw 0 0 0 Mu Mv 0 0 Ma 0 Mw 0 0 0 0 0 0 −1 −1 𝜏f Ab 𝜏f 0 0 0 0 0 0 −1 0 Ba 𝜏f −1 𝜏f 0 0 0 0 0 0 0 0 Za Zb Zw Zr 0 0 0 Nv Np 0 0 Nb Nw Nr 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u v p q a b w r 𝜑 𝜃 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 Yped 0 0 0 0 0 0 0 0 Mcol Alat 𝜏f Alon 𝜏f 0 0 Blat 𝜏f Blon 𝜏f 0 0 0 0 0 Zcol 0 0 Nped Ncol 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ 𝛿lat 𝛿lon 𝛿ped 𝛿col ⎤ ⎥ ⎥ ⎦ y = [︀ u v w p q r 𝜑 𝜃 ]︀T
- 7. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Parameter Estimation Discrete time linear model: x(k + 1) = A(𝜃)x(k) + B(𝜃)u(k) y(k) = C(𝜃)x(k) + D(𝜃)u(k) where x is the state vector, u denotes the input vector and y is the measurement vector. This is a parametric model, based on physical principles. In order to have a virtual model of the UAV, we need to find the best parameter vector using input-output data of a few flight tests.
- 8. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Parameter Estimation Assume that we are given a set of flight test data: 𝒟N = {(uft(k), yft(k)) |k = 0, . . . , N} The parameter estimation problem can be formulated as: minimize 𝜃,x(0) ΣN i=1 ‖y(tk) − yft(tk)‖2 2 subject to x(k + 1) = A(𝜃)x(k) + B(𝜃)uft(k) for k = 0, . . . , N − 1 y(k) = C(𝜃)x(k) + D(𝜃)uft(k) for k = 1, . . . , N
- 9. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Model Order Reduction After estimating the parameter vector, we have a high-order linear model. To design a controller for the pitch dynamics, we don’t need all the degrees of freedom. If G(s) is the transfer function of the original model, we need to compute ˆG(s) such that it captures the main characteristics of the pitch dynamics. Model order reductoion, in this case, can be formulated as the following optimization problem: minimize ^G(s) ‖G(s) − ˆG(s)‖∞
- 10. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Model Based Control Design Design a semi-active landing gear to: maximize stability on the ground maximize stability during taxi minimize noise minimize cost minimize weight subject to physical, geometrical, environmental, budget and safety constraints
- 11. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Convex optimization problems have extensive, useful theory have a unique optimal answer occur often in engineering problems often go unrecognized [cvxbook]
- 12. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Convex optimization problem minimize f (x) subject to x ∈ C where f is a convex function and C is a convex set.
- 13. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Convex set C ⊆ Rn is convex if x, y ∈ C, 𝜃 ∈ [0, 1] =⇒ 𝜃x + (1 − 𝜃)y ∈ C [cvxbook]
- 14. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Convex function f : Rn −→ R is convex if x, y ∈ Rn, 𝜃 ∈ [0, 1] ⇓ f (𝜃x + (1 − 𝜃)y) ≤ 𝜃f (x) + (1 − 𝜃)f (y) [cvxbook]
- 15. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Linear programming minimize aT 0 x subject to aT i x ≤ bi, i = 1, . . . , m [cvxbook]
- 16. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Semidefinite programming minimize cTx subject to x1F1 + · · · + xnFn + G ⪯ 0 Ax = b, where P ⪯ 0 for a matrix P ∈ Rn×n means that for any vector v ∈ Rn, we have: vT Pv ≤ 0 This is equivalent to all the eigenvalues of P being nonpositive. P is called negative semidefinite.
- 17. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Why Convex Optimization? In fact the great watershed in optimization isn’t between linearity and nonlinearity, but convexity and nonconvexity (R. Tyrrell Rockafellar, in SIAM Review, 1993). Convex optimization problems can be solved almost as quickly and reliably as linear programming problems.
- 18. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Nonnegativity of polynomials Polynomials of degree d in n variables: p(x) p(x1, x2, . . . , xn) = ∑︁ k1+k2+···+kn≤d ak1k2...kn xk1 1 xk2 2 · · · xkn n How to check if a given p(x) (of even order) is globally nonnegative? p(x) ≥ 0, ∀x ∈ Rn For d = 2, easy (check eigenvalues). What happens in general? Decidable, but NP-hard when d ≥ 4. “Low complexity” is desired at the cost of possibly being conservative. [Parrilo]
- 19. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion A sufficient condition A “simple” sufficient condition: a sum of squares (SOS) decomposition: p(x) = m∑︁ i=1 f 2 i (x) If p(x) can be written as above, for some polynomials fi, then p(x) ≥ 0. p(x) is an SOS if and only if a positive semidefinite matrix Q exists such that p(x) = ZT(x)QZ(x) where Z(x) is the vector of monomials of degree less than or equal to deg(p)/2 [Parrilo]
- 20. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Example p(x, y) = 2x4 + 5y4 − x2 y2 + 2x3 y = ⎡ ⎣ cx2 y2 xy ⎤ ⎦ T ⎡ ⎣ q11 q12 q13 q21 q22 q23 q13 q23 q33 ⎤ ⎦ ⎡ ⎣ cx2 y2 xy ⎤ ⎦ = q11x4 + q22y4 + (q33 + 2q12)x2 y2 + 2q13x3 y + 2q23xy3 An SDP with equality constraints. Solving, we obtain: Q = ⎡ ⎣ 2 −3 1 −3 5 0 1 0 5 ⎤ ⎦ = LTL, L = 1 √ 2 [︂ 2 −3 1 0 1 3 ]︂ And therefore p(x, y) = 1 2 (2x2 − 3y2 + xy)2 + 1 2 (y2 + 3xy)2 . [Parrilo]
- 21. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Sum of squares programming A sum of squares program is a convex optimization program of the following form: Minimize J∑︁ j=1 wj 𝛼j subject to fi,0 + J∑︁ j=1 𝛼jfi,j(x) is SOS, for i = 1, . . . , I where the 𝛼j’s are the scalar real decision variables, the wj’s are some given real numbers, and the fi,j are some given multivariate polynomials. [Prajna]
- 22. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Numerical Solvers SOSTOOLS handles the general SOS programming. MATLAB toolbox, freely available. Requires SeDuMi (a freely available SDP solver). Natural syntax, efficient implementation Developed by S. Prajna, A. Papachristodoulou and P. Parrilo Includes customized functions for several problems [Parrilo]
- 23. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Global optimization Consider for example: min x,y F(x, y) with F(x, y) = 4x2 − 21 10 x4 + 1 3 x6 + xy − 4y2 + 4y4 [Parrilo]
- 24. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Global optimization Not convex, many local minima. NP-Hard in general. Find the largest 𝛾 s.t. F(x, y) − 𝛾 is SOS. A semidefinite program (convex!). If exact, can recover optimal solution. Surprisingly effective. Solving, the maximum value is −1.0316. Exact value. Many more details in Parrilio and Strumfels, 2001 [Parrilo]
- 25. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Lyapunov stability analysis To prove asymptotic stability of ˙x = f (x), V(x) > 0, x ̸= 0, ˙V(x) = (︂ 𝜕V 𝜕x )︂T f (x) < 0, x ̸= 0 For linear systems ˙x = Ax, quadratic Lyapunov functions V(x) = xTPx P > 0, ATP + PA < 0 With an affine family of candidate Lyapunov functions V, ˙V is also affine. Instead of checking nonnegativity, use an SOS condition [Parrilo]
- 26. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Lyapunov stability - Jet Engine Example A jet engine model (derived from Moore-Greitzer), with controller: ˙x = −y + 3 2 x2 − 1 2 x3 ˙y = 3x − y Try a generic 4th order polynomial Lyapunov function. Find a V(x, y) that satisfies the conditions: V(x, y) is SOS. − ˙V(x, y) is SOS. Can easily do this using SOS/SDP techniques... [Parrilo]
- 27. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Lyapunov stability - Jet Engine Example After solving the SDPs, we obtain a Lyapunov function. [Parrilo]
- 28. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Lyapunov stability - Jet Engine Example Consider the nonlinear system ˙x1 = −x3 1 − x1x2 3 ˙x2 = −x2 − x2 1 x2 ˙x3 = −x3 − 3x3 x2 3 + 1 + 3x2 1 x3 Looking for a quadratic Lyapunov function s.t. V − (x2 1 + x2 2 + x2 3 ) is SOS, (x2 3 + 1)(− 𝜕V 𝜕x1 ˙x1 − 𝜕V 𝜕x2 ˙x2 − 𝜕V 𝜕x3 ˙x3) is SOS, we have V(x) = 5.5489x2 1 + 4.1068x2 2 + 1.7945x2 3 . [sostools]
- 29. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Parametric robustness analysis - Example Consider the following linear system d dt ⎡ ⎣ cx1 x2 x3 ⎤ ⎦ = ⎡ ⎣ −p1 1 −1 2 − p2 2 −1 3 1 −p1p2 ⎤ ⎦ ⎡ ⎣ cx1 x2 x3 ⎤ ⎦ where p1 ∈ [p1, p1] and p2 ∈ [p2, p2] are parameters. Parameter set can be captured by a1(p) (p1 − p1)(p1 − p1) ≤ 0 a2(p) (p2 − p2)(p2 − p2) ≤ 0 [sostools]
- 30. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Parametric robustness analysis - Example Find V(x; p) and qi,j(x; p), such that V(x; p) − ‖x‖2 + ∑︀2 j=1 q1,j(x; p)ai(p) is SOS, − ˙V(x; p) − ‖x‖2 + ∑︀2 j=1 q2,j(x; p)ai(p) is SOS, qi,j(x; p) is SOS, for i, j = 1, 2. [sostools]
- 31. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Safety verification - Example Consider the following system ˙x1 =x2 ˙x2 = − x1 + 1 3 x3 1 − x2 Initial set: 𝒳0 = {x : g0(x) = (x1 − 1.5)2 + x2 2 − 0.25 ≤ 0} Unsafe set: 𝒳u = {x : gu(x) = (x1 + 1)2 + (x2 + 1)2 − 0.16 ≤ 0} [sostools]
- 32. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Safety verification - Example Barrier certificate B(x) B(x) < 0, ∀x ∈ 𝒳0 B(x) > 0, ∀x ∈ 𝒳u 𝜕B 𝜕x1 ˙x1 + 𝜕B 𝜕x2 ˙x2 ≤ 0 SOS program: Find B(x) and 𝜎i(x) −B(x) − 0.1 + 𝜎1(x)g0(x) is SOS, B(x) − 0.1 + 𝜎2(x)gu(x) is SOS, − 𝜕B 𝜕x1 ˙x1 − 𝜕B 𝜕x2 ˙x2 is SOS 𝜎1(x) and 𝜎2(x) are SOS [sostools]
- 33. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Safety verification - Example [sostools]
- 34. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Nonlinear control synthesis Consider the system ˙x = f (x) + g(x)u State dependent linear-like representation ˙x = A(x)Z(x) + B(x)u where Z(x) = 0 ⇔ x = 0 Consider the following Lyapunov function and control input V(x) = ZT(x)P−1 Z(x) u(x) = K(x)P−1 Z(x) [Prajna]
- 35. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Nonlinear control synthesis For the system ˙x = A(x)Z(x) + B(x)u, suppose there exist a constant matrix P, a polynomial matrix K(x), a constant 𝜖1 and a sum of squares 𝜖2(x), such that: vT(P − 𝜖1I)v is SOS, −vT(PAT(x)MT(x) + M(x)A(x)P + KT(x)BT(x)MT(x) + M(x)B(x)K(x) + 𝜖2(x)I) is SOS, where v ∈ RN and Mij(x) = 𝜕Zi 𝜕xj (x). Then a controller that stabilizes the system is given by: u(x) = K(x)P−1 Z(x) Furthermore, if 𝜖2(x) > 0 for x ̸= 0, then the zero equilibrium is globally asymptotically stable. [Prajna]
- 36. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Nonlinear control synthesis - Example Consider a tunnel diode circuit: ˙x1 = 0.5(−h(x1) + x2) ˙x2 = 0.2(−x1 − 1.5x2 + u) where the diode characteristic: h(x1) = 17.76x1 − 103.79x2 1 + 229.62x3 1 − 226.31x4 1 + 83.72x5 1 [Prajna]
- 37. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Nonlinear control synthesis - Example [Prajna]
- 38. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion How conservative is SOS? It is proven by Hilbert that “nonnegativity” and “sum of squares” are equivalent in the following cases. Univariate polynomials, any (even) degree Quadratic polynomials, in any number of variables Quartic polynomials in two variables When the degree is larger than two it follows that There are signitcantly more nonnegative polynomials than sums of squares. There are signitcantly more sums of squares than sums of even powers of linear forms. [soscvx]
- 39. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Flutter Phenomenon Mechanism of Flutter Inertial Forces Aerodynamic Forces (∝ V2) (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 [flutter96]
- 40. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Flutter Phenomenon [flutter96]
- 41. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Flutter Phenomenon State Space Equations: M [︂ ¨h ¨𝛼 ]︂ + (C0 + C 𝜇) [︂ ˙h ˙𝛼 ]︂ + (K0 + K 𝜇) [︂ h 𝛼 ]︂ + [︂ 0 𝛼K 𝛼(𝛼) ]︂ = B 𝛽o 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 [flutter07] [flutter98]
- 42. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Active Flutter Suppression Bombardier Q400 HYCONS Lab, Concordia University
- 43. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion LQR Controller Very large control inputs R = 10I, Q = 10 4 I
- 44. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion LQR Controller Divergence: the effect of actuator saturation maximum admissible flap angles: 15 deg
- 45. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion LQR Controller Region of attraction: plung deflection - pitch angle plane
- 46. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion LQR Controller Region of attraction: plung deflection - plung deflection rate plane
- 47. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion LQR Controller Region of attraction: pitch angle - pitch rate plane
- 48. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Nonlinear Model Open loop:
- 49. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion PD Controller Open loop:
- 50. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Polynomial Controller Consider x3 and x4 as inputs of the following system: ˙x1 = x3 ˙x2 = x4 Consider the controller [︂ x3 x4 ]︂ = −10 [︂ x1 x2 ]︂ for the above system. Similar to backstepping approach, we construct the following Lyapunov function: V(x) = 1 2 {︀ x2 1 + x2 2 + (x3 + 10x1)2 + (x4 + 10x2)2 }︀ Find a polynomial u(x) such that −∇V.f (x) − V(x) is SOS where f (x) is the vector field of the closed loop system.
- 51. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Polynomial Controller smaller control inputs
- 52. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Polynomial Controller Divergence: the effect of actuator saturation maximum admissible flap angles: 15 deg
- 53. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Polynomial Controller Future work: To construct a nonlinear model of Q400 To design a nonlinear controller in order to enlarge the region of convergence in the presence of input saturation
- 54. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Gain Scheduling 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?
- 55. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Gain Scheduling 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.
- 56. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Piecewise Smooth Systems The dynamics of a piecewise smooth 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.
- 57. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion Conclusion Sum of squares, conservative but much more tractable than nonnegativity Many applications in control theory Try your problem!
- 58. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion References I [cvxbook] Convex optimmization, Stephen Boyd and Lieven Vandenberghe, http://www.stanford.edu/~boyd/cvxbook [Parrilo] Certificates, convex optimization, and their applications, Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich, http: //www.mat.univie.ac.at/~neum/glopt/mss/Par04.pdf [Prajna] Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach, Stephen Prajna et al, the 5th Asian Control Conference, 2004 [sostools] SOSTOOLS: control applications and new developments, Stephen Prajna et al, IEEE Conference on Computer Aided Control Systems Design, 2004
- 59. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion References II [soscvx] A convex polynomial that is not sos-convex, Amir Ali Ahmadi and Pablo A. Parrilo, http://arxiv.org/pdf/0903.1287.pdf [yalmip] YALMIP, A Toolbox for Modeling and Optimization in MATLAB, J. Löfberg. In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004, http://users.isy.se/johanl/yalmip [sos] Pre- and post-processing sum-of-squares programs in practice. J. Löfberg. IEEE Transactions on Automatic Control, 54(5):1007-1011, 2009. [dual] Dualize it: software for automatic primal and dual conversions of conic programs. J. Löfberg. Optimization Methods and Software, 24:313 - 325, 2009.
- 60. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion References III [sedumi] SeDuMi, a MATLAB toolbox for optimization over symmetric cones, http://sedumi.ie.lehigh.edu [flutter96] Modeling the benchmark active control technology windtunnel model for application to flutter suppression, M. R. Waszak, AIAA 96 - 3437, http://www.mathworks.com/ matlabcentral/fileexchange/3938 [flutter98] Stability and control of a structurally nonlinear aeroelastic system, Jeonghwan Ko and Thomas W. Strganacy, Journal of Guidance, Control, and Dynamics, 21 , 718-725. [flutter07] Nonlinear control design of an airfoil with active flutter suppression in the presence of disturbance, S. Afkhami and H. Alighanbari, IET Control Theory Appl., vol. 1 , 1638-1649.

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