This document discusses various applications of optimization techniques in aerospace systems modeling, control, and design. It covers topics such as aircraft design formulation as a multidisciplinary optimization problem, parameter estimation for unmanned aerial vehicles using input-output flight test data, model order reduction via optimization, and control design problems cast as convex and sum-of-squares optimizations. Specific examples discussed include stability analysis of a jet engine model using polynomial Lyapunov functions.
Measures of Dispersion and Variability: Range, QD, AD and SD
Aerospace Systems Modeling and Control Optimization Techniques
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)
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]
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]
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]
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
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.