Modeling, Control and Optimization for Aerospace Systems

Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion

Modeling, Control, and Optimization

for Aerospace SystemsHYCONS Lab, Concordia University

Behzad Samadi

HYCONS Lab, Concordia University

American Control ConferenceMontreal, Canada

June 2012


Outline

Motivation

Aircraft designParameter estimationModel order reductionModel 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 optimizationproblems:

Aircraft design

Modeling: Parameter estimation

Modeling: Model order reduction

Model based control design (Landing gear semi activesuspension)


Aircraft Design

The aircraft designer wants to:

maximize rangeminimize weightmaximize lift to drag ratiominimize costminimize noisesubject to physical, geometrical, environmental, budget and safetyconstraintsMultidisciplinary 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

��

𝜃

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Xu 0 0 0 Xa 0 0 0 0 −g0 Yv 0 0 0 Yb 0 0 g 0Lu Lv 0 0 0 Lb Lw 0 0 0Mu Mv 0 0 Ma 0 Mw 0 0 0

0 0 0 −1 −1𝜏f

Ab

𝜏f0 0 0 0

0 0 −1 0 Ba

𝜏f−1𝜏f

0 0 0 0

0 0 0 0 Za Zb Zw Zr 0 00 Nv Np 0 0 Nb Nw Nr 0 00 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

u

v

p

q

a

b

w

r

𝜑𝜃

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 00 0 Yped 00 0 0 00 0 0 Mcol

Alat

𝜏f

Alon

𝜏f0 0

Blat

𝜏f

Blon

𝜏f0 0

0 0 0 Zcol0 0 Nped Ncol

0 0 0 00 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎣𝛿lat𝛿lon𝛿ped𝛿col

⎤⎥⎥⎦

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 themeasurement vector.This is a parametric model, based on physical principles. In order tohave a virtual model of the UAV, we need to find the bestparameter 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:

minimize𝜃,x(0)

ΣNi=1‖y(tk) − yft(tk)‖22

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-orderlinear model.

To design a controller for the pitch dynamics, we don’t needall the degrees of freedom.

If G (s) is the transfer function of the original model, we needto compute G (s) such that it captures the main characteristicsof the pitch dynamics.

Model order reductoion, in this case, can be formulated as thefollowing optimization problem:

minimizeG(s)

‖G (s) − G (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 andsafety 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 aT0 x

subject to aTi x ≤ bi , i = 1, . . . ,m

[cvxbook]


Semidefinite programming

minimize cT x

subject to x1F1 + · · · + xnFn + G ⪯ 0

Ax = b,

where P ⪯ 0 for a matrix P ∈ Rn×n means that for any vectorv ∈ Rn, we have:

vTPv ≤ 0

This is equivalent to all the eigenvalues of P being nonpositive. Pis called negative semidefinite.


Why Convex Optimization?

In fact the great watershed in optimization isn’t betweenlinearity and nonlinearity, but convexity and nonconvexity (R.Tyrrell Rockafellar, in SIAM Review, 1993).

Convex optimization problems can be solved almost as quicklyand 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...knxk11 xk22 · · · xknn

How to check if a given p(x) (of even order) is globallynonnegative?

p(x) ≥ 0,∀x ∈ Rn

For d = 2, easy (check eigenvalues). What happens ingeneral?

Decidable, but NP-hard when d ≥ 4.

“Low complexity” is desired at the cost of possibly beingconservative.

[Parrilo]


A sufficient condition

A “simple” sufficient condition: a sum of squares (SOS)decomposition:

p(x) =m∑i=1

f 2i (x)

If p(x) can be written as above, for some polynomials fi , thenp(x) ≥ 0.

p(x) is an SOS if and only if a positive semidefinite matrix Qexists such that

p(x) = ZT (x)QZ (x)

where Z (x) is the vector of monomials of degree less than orequal to deg(p)/2

[Parrilo]


Example

p(x , y) = 2x4 + 5y4 − x2y2 + 2x3y

=

⎡⎣cx2y2

xy

⎤⎦T ⎡⎣q11 q12 q13q21 q22 q23q13 q23 q33

⎤⎦⎡⎣cx2y2

xy

⎤⎦= q11x

4 + q22y4 + (q33 + 2q12)x2y2 + 2q13x

3y + 2q23xy3

An SDP with equality constraints. Solving, we obtain:

Q =

⎡⎣ 2 −3 1−3 5 01 0 5

⎤⎦ = LTL, L =1√2

[2 −3 10 1 3

]

And therefore p(x , y) = 12

(2x2 − 3y2 + xy)2 + 12

(y2 + 3xy)2.[Parrilo]


Sum of squares programming

A sum of squares program is a convex optimization program of thefollowing form:

MinimizeJ∑

j=1

wj𝛼j

subject to fi ,0 +J∑

j=1

𝛼j fi ,j(x) is SOS, for i = 1, . . . , I

where the 𝛼j ’s are the scalar real decision variables, the wj ’s aresome given real numbers, and the fi ,j are some given multivariatepolynomials.[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:minx ,y

F (x , y)

with F (x , y) = 4x2 − 2110x4 + 1

3x6 + 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 functionsV (x) = xTPx

P > 0, ATP + PA < 0

With an affine family of candidate Lyapunov functions V , V isalso 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

2x2 − 1

2x3

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 = −x31 − x1x23

x2 = −x2 − x21 x2

x3 = −x3 −3x3

x23 + 1+ 3x21 x3

Looking for a quadratic Lyapunov function s.t.

V − (x21 + x22 + x23 ) is SOS,

(x23 + 1)(−𝜕V

𝜕x1x1 −

𝜕V

𝜕x2x2 −

𝜕V

𝜕x3x3) is SOS,

we have V (x) = 5.5489x21 + 4.1068x22 + 1.7945x23 .

[sostools]


Parametric robustness analysis - Example

Consider the following linear system

d

dt

⎡⎣cx1x2x3

⎤⎦ =

⎡⎣ −p1 1 −12− p2 2 −1

3 1 −p1p2

⎤⎦⎡⎣cx1x2x3

⎤⎦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) isSOS, for i , j = 1, 2.

[sostools]


Safety verification - Example

Consider the following system

x1 =x2

x2 = − x1 +1

3x31 − x2

Initial set:

𝒳0 = {x : g0(x) = (x1 − 1.5)2 + x22 − 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−1Z (x)

u(x) = K (x)P−1Z (x)

[Prajna]


Nonlinear control synthesis

For the system x = A(x)Z (x) + B(x)u, suppose there exist aconstant matrix P , a polynomial matrix K (x), a constant 𝜖1 and asum 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−1Z (x)

Furthermore, if 𝜖2(x) > 0 for x = 0, then the zero equilibrium isglobally 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 − 103.79x21 + 229.62x31 − 226.31x41 + 83.72x51

[Prajna]


Nonlinear control synthesis - Example

[Prajna]


How conservative is SOS?

It is proven by Hilbert that “nonnegativity” and “sum ofsquares” 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 thansums of squares.

There are signitcantly more sums of squares than sums of evenpowers 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-excitedTwo or more modes of motion (e.g. flexural and torsional)exist simultaneouslyCritical Flutter Speed, largely depends on torsional and flexuralstiffnesses of the structure

[flutter96]


Flutter Phenomenon

[flutter96]


Flutter Phenomenon

State Space Equations:

M

[h

��

]+ (C0 + C𝜇)

[h

��

]+ (K0 + K𝜇)

[h

𝛼

]+

[0

𝛼K𝛼(𝛼)

]= B𝛽o

State variables: plunge deflection (h), pitch angle (𝛼), andtheir derivatives (h and ��)

Inputs: angular deflection of the flaps (𝛽o ∈ R2)

Constraints: on states and actuators

[flutter07] [flutter98]


Active Flutter Suppression

Bombardier Q400

HYCONS Lab, Concordia University


LQR Controller

Very large control inputs

R = 10I ,Q = 104I


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

[x3x4

]= −10

[x1x2

]for the above system.

Similar to backstepping approach, we construct the followingLyapunov function:

V (x) =1

2

{x21 + x22 + (x3 + 10x1)2 + (x4 + 10x2)2

}Find a polynomial u(x) such that −∇V .f (x) − V (x) is SOSwhere 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 regionof 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 andsynthesis methods and computational tools.

Real world systems, however, are usually nonlinear.

What can be done to extend the good properties of linearsystems 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 locallinear systems

Designing local linear controllers and combining them

Started in 1960s, very popular in a variety of fields fromaerospace to process control

Problem: proof of stability!

Problem: By switching between two stable linear system, youcan 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 ispartitioned into M regions, ℛi , i = 1, . . . ,M such that:

∪Mi=1ℛi = 𝒳 , ℛi ∩ℛj = ∅, i = j

where ℛi denotes the closure of ℛi .


Conclusion

Sum of squares, conservative but much more tractable thannonnegativity

Many applications in control theory

Try your problem!


References I

[cvxbook] Convex optimmization, Stephen Boyd and LievenVandenberghe, http://www.stanford.edu/~boyd/cvxbook

[Parrilo] Certificates, convex optimization, and theirapplications, Pablo A. Parrilo, Swiss Federal Institute ofTechnology Zurich, http://www.mat.univie.ac.at/~neum/glopt/mss/Par04.pdf

[Prajna] Nonlinear control synthesis by sum of squaresoptimization: a Lyapunov-based approach, Stephen Prajna etal, the 5th Asian Control Conference, 2004

[sostools] SOSTOOLS: control applications and newdevelopments, Stephen Prajna et al, IEEE Conference onComputer Aided Control Systems Design, 2004

http://www.stanford.edu/~boyd/cvxbook

http://www.mat.univie.ac.at/~neum/glopt/mss/Par04.pdf

http://www.mat.univie.ac.at/~neum/glopt/mss/Par04.pdf


References II

[soscvx] A convex polynomial that is not sos-convex, Amir AliAhmadi and Pablo A. Parrilo,http://arxiv.org/pdf/0903.1287.pdf

[yalmip] YALMIP, A Toolbox for Modeling and Optimization inMATLAB, J. Löfberg. In Proceedings of the CACSDConference, Taipei, Taiwan, 2004,http://users.isy.se/johanl/yalmip

[sos] Pre- and post-processing sum-of-squares programs inpractice. J. Löfberg. IEEE Transactions on Automatic Control,54(5):1007-1011, 2009.

[dual] Dualize it: software for automatic primal and dualconversions of conic programs. J. Löfberg. OptimizationMethods and Software, 24:313 - 325, 2009.

http://arxiv.org/pdf/0903.1287.pdf

http://users.isy.se/johanl/yalmip


References III

[sedumi] SeDuMi, a MATLAB toolbox for optimization oversymmetric cones, http://sedumi.ie.lehigh.edu

[flutter96] Modeling the benchmark active control technologywindtunnel 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 nonlinearaeroelastic system, Jeonghwan Ko and Thomas W. Strganacy,Journal of Guidance, Control, and Dynamics, 21 , 718-725.

[flutter07] Nonlinear control design of an airfoil with activeflutter suppression in the presence of disturbance, S. Afkhamiand H. Alighanbari, IET Control Theory Appl., vol. 1 ,1638-1649.

http://sedumi.ie.lehigh.edu

http://www.mathworks.com/matlabcentral/fileexchange/3938

http://www.mathworks.com/matlabcentral/fileexchange/3938

Modeling, Control and Optimization for Aerospace Systems

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