Modeling, Control and Optimization for Aerospace Systems

Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Modeling, Control, and Optimization
for Aerospace Systems
HYCONS Lab, Concordia University
Behzad Samadi
American Control Conference
Montreal, Canada
June 2012

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

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)

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

Parameter Estimation
Unmanned Rotorcraft Technology Demonstrator ARTIS at DLR
(German Aerospace Center)

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

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.

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

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)‖∞

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

Convex optimization problems
have extensive, useful theory
have a unique optimal answer
occur often in engineering problems
often go unrecognized
[cvxbook]

Convex optimization problem
minimize f (x)
subject to x ∈ C
where f is a convex function and C is a convex set.

Convex set
C ⊆ Rn is convex if
x, y ∈ C, 𝜃 ∈ [0, 1] =⇒ 𝜃x + (1 − 𝜃)y ∈ C
[cvxbook]

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]

Linear programming
minimize aT
0
x
subject to aT
i x ≤ bi, i = 1, . . . , m
[cvxbook]

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.

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.

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]

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]

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]

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]

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]

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]

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]

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]

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]

After solving the SDPs, we obtain a Lyapunov function.
[Parrilo]

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]

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]

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]

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]

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]

[sostools]

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]

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]

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]

Nonlinear control synthesis - Example
[Prajna]

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]

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]

Flutter Phenomenon
[flutter96]

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]

Active Flutter Suppression
Bombardier Q400

LQR Controller
Very large control inputs
R = 10I, Q = 10
4
I

LQR Controller
Divergence: the effect of actuator saturation
maximum admissible flap angles: 15 deg

LQR Controller
Region of attraction: plung deflection - pitch angle plane

LQR Controller
Region of attraction: plung deflection - plung deflection rate plane

LQR Controller
Region of attraction: pitch angle - pitch rate plane

Nonlinear Model
Open loop:

PD Controller
Open loop:

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.

smaller control inputs

Divergence: the effect of actuator saturation
maximum admissible flap angles: 15 deg

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

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?

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.

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.

Conclusion
Sum of squares, conservative but much more tractable than
nonnegativity
Many applications in control theory
Try your problem!

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

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.

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.

Modeling, Control and Optimization for Aerospace Systems

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