Control Synthesis by Sum of Squares Optimization

Control Synthesis by Sum ofSquares Optimization

An Introduction

Behzad Samadi

[email protected]

Concordia University

Montreal, Canada

Outline

◮ Convex Optimization

▽Control Synthesis by Sum of Squares Optimization – p.1/30

Outline


◮ Sum of Squares


Outline


◮ Sum of Squares

◮ Control Applications


Outline


◮ Sum of Squares

◮ Control Applications

◮ Conclusion

Control Synthesis by Sum of Squares Optimization – p.1/30

Convex set

C ⊆ Rn is convex if

x, y ∈ C, θ ∈ [0, 1] =⇒ θx + (1 − θ)y ∈ C

"Convex Optimization with Engineering Applications", Professor Stephan Boyd, Stanford University


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)



Convex optimization problem

minimize f(x)

subject to x ∈ C

f convex, C convex



Convex optimization problem

minimize f(x)

subject to x ∈ C

f convex, C convex

Convex optimization problems

◮ can be solved numerically with great efficiency

◮ have extensive useful theory

◮ occur often in engineering problems

◮ often go unrecognized



Linear programming

minimize aT0 x

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



Semidefinite programming

minimize cT x

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

Ax = b,



In fact the great watershed inoptimization isn’t betweenlinearity and nonlinearity ,

butconvexityand nonconvexity.

Rockafellar 1993


Nonnegativity of polynomialsPolynomials of degree d in n variables:

p(x) , p(x1, x2, . . . , xn) =∑

k1+k2+···+kn≤d

ak1k2...knxk1

1xk2

2· · ·xkn

n



p(x) , p(x1, x2, . . . , xn) =∑

k1+k2+···+kn≤d

ak1k2...knxk1

1xk2

2· · ·xkn

n

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

p(x) ≥ 0,∀x ∈ Rn



p(x) , p(x1, x2, . . . , xn) =∑

k1+k2+···+kn≤d

ak1k2...knxk1

1xk2

2· · ·xkn

n

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 happensin generel?

◮ Decidable, but NP-hard when d ≥ 4.

◮ "Low complexity" is desired at the cost of possiblybeing conservative.

"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich


A sufficient conditionA "simple" sufficient condition: a sum of squares (SOS)decomposition:

p(x) =m

∑

i=1

f2i (x)

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


A sufficient conditionA "simple" sufficient condition: a sum of squares (SOS)decomposition:

p(x) =m

∑

i=1

f2i (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 semidefinitematrix Q exists such that

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

where Z(x) is the vector of monomials of degree lessthan or equal to deg(p)/2



Examplep(x, y) = 2x4 + 5y4 − x2y2 + 2x3y

=

x2

y2

xy

T

q11 q12 q13

q21 q22 q23

q13 q23 q33

x2

y2

xy

= q11x4 + q22y

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

3y + 2q23xy3

An SDP with equality constraints.


Examplep(x, y) = 2x4 + 5y4 − x2y2 + 2x3y

=

x2

y2

xy

T

q11 q12 q13

q21 q22 q23

q13 q23 q33

x2

y2

xy

= q11x4 + q22y

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

3y + 2q23xy3

An SDP with equality constraints. Solving, we obtain:

Q =

2 −3 1

−3 5 0

1 0 5

= LT L, 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.



Sum of squares programming

A sum of squares program is a convex optimizationprogram of the following form:

MinimizeJ

∑

j=1

wjαj

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, thewj ’s are some given real numbers, and the fi,j are somegiven multivariate polynomials.

"Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach", Stephen Prajna et al, ASCC 2004


SOSTOOLS: Sum of squares toolbox

◮ 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.Parrilio

◮ Includes customized functions for several problems

Get it from:http://www.aut.ee.ethz.ch/~parrilo/sostoolshttp://www.cds.caltech.edu/sostools



http://www.aut.ee.ethz.ch/~parrilo/sostools

http://www.cds.caltech.edu/sostools

Global optimizationConsider for example:

minx,y

F (x, y)

with F (x, y) = 4x2 − 21

10x4 + 1

3x6 + xy − 4y2 + 4y4



Global optimization

◮ Not convex, many local minima. NP-Hard in general.


Global optimization


◮ Find the largest γ s.t.

F (x, y) − γ is SOS.


Global optimization




◮ A semidefinite program (convex!).


Global optimization





◮ If exact, can recover optimal solution.


Global optimization






◮ Surprisingly effective.


Global optimization






◮ Surprisingly effective.

Solving, the maximum value is −1.0316. Exact value.Many more details in Parrilio & Strumfels, 2001



Lyapunov stability analysis◮ To prove asymptotic stability of x = f(x),

V (x) > 0, x 6= 0, V (x) =

(

∂V

∂x

)T

f(x) < 0, x 6= 0



V (x) > 0, x 6= 0, V (x) =

(

∂V

∂x

)T

f(x) < 0, x 6= 0

◮ For linear systems x = Ax, quadratic Lyapunovfunctions V (x) = xT Px

P > 0, AT P + PA < 0



V (x) > 0, x 6= 0, V (x) =

(

∂V

∂x

)T

f(x) < 0, x 6= 0


P > 0, AT P + PA < 0

◮ With an affine family of candidate Lyapunov functionsV , V is also affine.



V (x) > 0, x 6= 0, V (x) =

(

∂V

∂x

)T

f(x) < 0, x 6= 0


P > 0, AT P + PA < 0

◮ With an affine family of candidate Lyapunov functionsV , V is also affine.

◮ Instead of checking nonnegativity, use an SOScondition



Lyapunov stability - Example

A jet engine model (derived from Moore-Greitzer), withcontroller:

x = −y +3

2x2 − 1

2x3

y = 3x − y

Try a generic 4th order polynomial Lyapunov function.



A jet engine model (derived from Moore-Greitzer), withcontroller:

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...




After solving the SDPs, we obtain a Lyapunov function.



Lyapunov stability - Example◮ Consider the nonlinear system

x1 = −x31 − x1x

23

x2 = −x2 − x21x2

x3 = −x3 −3x3

x23+ 1

+ 3x21x3


Lyapunov stability - Example◮ Consider the nonlinear system

x1 = −x31 − x1x

23

x2 = −x2 − x21x2

x3 = −x3 −3x3

x23+ 1

+ 3x21x3

◮ Looking for a quadratic Lyapunov function s.t.

V − (x21 + x2

2 + x23) is SOS,

(x23 + 1)(− ∂V

∂x1x1 − ∂V

∂x2x2 − ∂V

∂x3x3) is SOS,

we have V (x) = 5.5489x21 + 4.1068x2

2 + 1.7945x23.

"SOSTOOLS: control applications and new developments", Stephen Prajna et al, CACSD 2004


Parametric robustness analysis - Example◮ Consider the following linear system

d

dt

x1

x2

x3

=

−p1 1 −1

2 − p2 2 −1

3 1 −p1p2

x1

x2

x3

where p1 ∈ [p1, p1] and p2 ∈ [p2, p2] are parameters.


Parametric robustness analysis - Example◮ Consider the following linear system

d

dt

x1

x2

x3

=

−p1 1 −1

2 − p2 2 −1

3 1 −p1p2

x1

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



Parametric robustness analysis - Example

Find V (x; p) and qi,j(x; p), such that

◮ V (x; p) − ‖x‖2 +∑2

j=1q1,j(x; p)ai(p) is SOS,

◮ −V (x; p) − ‖x‖2 +∑2

j=1q2,j(x; p)ai(p) is SOS,

qi,j(x; p) is SOS, for i, j = 1, 2.



Safety verification - Example

◮ Consider the following system

x1 = x2

x2 = −x1 +1

3x3

1 − 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}



Safety verification - ExampleBarrier certificate B(x)

◮ B(x) < 0,∀x ∈ X0

◮ B(x) > 0,∀x ∈ Xu

◮ ∂B∂x1

x1 + ∂B∂x2

x2 ≤ 0


Safety verification - ExampleBarrier 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



Safety verification - Example



Nonlinear control synthesis◮ Consider the system

x = f(x) + g(x)u



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



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 controlinput

V (x) = ZT (x)P−1Z(x)

u(x) = K(x)P−1Z(x)



Nonlinear control synthesisFor the system x = A(x)Z(x) + B(x)u, suppose thereexist a constant matrix P , a polynomial matrix K(x), aconstant ǫ1 and a sum of squares ǫ2(x), such that

◮ vT (P − ǫ1I)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) + ǫ2(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 ǫ2(x) > 0 for x 6= 0, then the zeroequilibrium is globally asymptotically stable.



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 + 229.62x3

1 − 226.31x41 + 83.72x5

1



Nonlinear control synthesis - Example



How conservative is SOS?

◮ It is proven by Hilbert that "nonnegativity" and "sumof squares" are equivalent in the following cases.⊲ Univariate polynomials, any (even) degree⊲ Quadratic polynomials, in any number of variables⊲ Quartic polynomials in two variables


How conservative is SOS?

◮ It is proven by Hilbert that "nonnegativity" and "sumof 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.[G. Blekherman, University of Michigan, submitted forpublication]


Conclusion

◮ Sum of squares, conservative but much moretractable than nonnegativity


Conclusion


◮ Many applications in control theory


Conclusion


◮ Many applications in control theory

◮ Try your problem!


References

◮ Convex optimmization by Stephen Boyd and LievenVandenberghe, available onlinehttp://www.stanford.edu/~boyd/cvxbook

◮ SOSTOOLS, MATLAB toolbox, freely availablehttp://www.cds.caltech.edu/sostools


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

http://www.cds.caltech.edu/sostools

Control Synthesis by Sum of Squares Optimization

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