The document outlines a presentation on control synthesis using sum of squares optimization. It begins with an introduction to convex optimization and sum of squares analysis. It then discusses applications of these techniques to control systems and stability analysis. The document provides examples of using sum of squares to solve global optimization problems and verify stability of nonlinear systems.
Objectives: This course will provide a comprehensive overview of power system stability and control problems. This includes the basic concepts, physical aspects of the phenomena, methods of analysis, the integration of MATLAB and SINULINK in the analysis of power system .
Course Content: 1. Power System Stability: Introduction
2. Stability Analysis: Swing Equation
3. Models for Stability Studies
4. Steady State Stability
5. Transient Stability
6. Multimachine Transient Stability
7. Power System Control: Introduction
8. Load Frequency Control
9. Automatic generation Control
10. Reactive Power Control
Generation shift factor and line outage factorViren Pandya
This is animated presentation to let students have an idea about use of generation shift factor and line outage distribution factor to assess power system security by contingency analysis. Entire presentation is prepared from a very nice book authored by Wood.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
Objectives: This course will provide a comprehensive overview of power system stability and control problems. This includes the basic concepts, physical aspects of the phenomena, methods of analysis, the integration of MATLAB and SINULINK in the analysis of power system .
Course Content: 1. Power System Stability: Introduction
2. Stability Analysis: Swing Equation
3. Models for Stability Studies
4. Steady State Stability
5. Transient Stability
6. Multimachine Transient Stability
7. Power System Control: Introduction
8. Load Frequency Control
9. Automatic generation Control
10. Reactive Power Control
Generation shift factor and line outage factorViren Pandya
This is animated presentation to let students have an idea about use of generation shift factor and line outage distribution factor to assess power system security by contingency analysis. Entire presentation is prepared from a very nice book authored by Wood.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
state space representation,State Space Model Controllability and Observabilit...Waqas Afzal
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
Introduction, Types of Stable System, Routh-Hurwitz Stability Criterion, Disadvantages of Hurwitz Criterion, Techniques of Routh-Hurwitz criterion, Examples, Special Cases of Routh Array, Advantages and Disadvantages of Routh-Hurwitz Stability Criterion, and examples.
INTRODUCTION BASIC TECHNIQUES TYPE OF BUSES
Y BUS MATRIX POWER SYSTEM COMPONENTS BUS ADMITTANCE MATRIX
Power (Load) flow study is the analysis of a power system in normal steady-state operation
This study will determine:
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
state space representation,State Space Model Controllability and Observabilit...Waqas Afzal
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
Introduction, Types of Stable System, Routh-Hurwitz Stability Criterion, Disadvantages of Hurwitz Criterion, Techniques of Routh-Hurwitz criterion, Examples, Special Cases of Routh Array, Advantages and Disadvantages of Routh-Hurwitz Stability Criterion, and examples.
INTRODUCTION BASIC TECHNIQUES TYPE OF BUSES
Y BUS MATRIX POWER SYSTEM COMPONENTS BUS ADMITTANCE MATRIX
Power (Load) flow study is the analysis of a power system in normal steady-state operation
This study will determine:
A crash coarse in stochastic Lyapunov theory for Markov processes (emphasis is on continuous time)
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https://netfiles.uiuc.edu/meyn/www/spm_files/MarkovTutorial/MarkovTutorialUCSB2010.html
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Inferring cause-effect relationships between variables is of primary importance in many sciences. In this talk, I will discuss two approaches for making valid inference on treatment effects when a large number of covariates are present. The first approach is to perform model selection and then to deliver inference based on the selected model. If the inference is made ignoring the randomness of the model selection process, then there could be severe biases in estimating the parameters of interest. While the estimation bias in an under-fitted model is well understood, I will address a lesser known bias that arises from an over-fitted model. The over-fitting bias can be eliminated through data splitting at the cost of statistical efficiency, and I will propose a repeated data splitting approach to mitigate the efficiency loss. The second approach concerns the existing methods for debiased inference. I will show that the debiasing approach is an extension of OLS to high dimensions, and that a careful bias analysis leads to an improvement to further control the bias. The comparison between these two approaches provides insights into their intrinsic bias-variance trade-off, and I will show that the debiasing approach may lose efficiency in observational studies.
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availability of algorithms that can solve them efficiently and within
the sampling time of the controlled system.
In this paper we propose a dual accelerated proximal
gradient algorithm which is amenable to parallelization and
demonstrate that its GPU implementation affords high speed-up
values (with respect to a CPU implementation) and greatly outperforms
well-established commercial optimizers such as Gurobi.
Non-linear optimization applications in finance including volatility estimation with ARCH and GARCH models, line search methods, Newton's method, steepest descent method, golden section search method, and conjugate gradient method.
The main machine learning algorithms are built upon various mathematical foundations such as statistics, optimization, and probability. Will this also hold true for Artificial Intelligence? In this presentation, I will showcase some recent examples of interactions between machine learning and mathematics.
Colloquium @ CEREMADE (October 3, 2023)
We present an overview of mathematical tools for building model emulators. We will primarily discuss forward emulation, where one seeks to predict the output of a model given an input. We will emphasize methods that boast stability, accuracy, and computational efficiency, and will discuss emulators built from non-adapted polynomials, and from adapted function spaces. The talk will highlight some notable advances made in the field of building emulators and will identify frontiers where mathematical or computational advances are needed.
We study an elliptic eigenvalue problem, with a random coefficient that can be parametrised by infinitely-many stochastic parameters. The physical motivation is the criticality problem for a nuclear reactor: in steady state the fission reaction can be modeled by an elliptic eigenvalue
problem, and the smallest eigenvalue provides a measure of how close the reaction is to equilibrium -- in terms of production/absorption of neutrons. The coefficients are allowed to be random to model the uncertainty of the composition of materials inside the reactor, e.g., the
control rods, reactor structure, fuel rods etc.
The randomness in the coefficient also results in randomness in the eigenvalues and corresponding eigenfunctions. As such, our quantity of interest is the expected value, with
respect to the stochastic parameters, of the smallest eigenvalue, which we formulate as an integral over the infinite-dimensional parameter domain. Our approximation involves three steps: truncating the stochastic dimension, discretizing the spatial domain using finite elements and approximating the now finite but still high-dimensional integral.
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does not hold in our case.
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This talk will report briey on some findings from the problem of picking the weights for a weighted function space in QMC. Then it will be mostly about importance sampling. We want to estimate the probability _ of a union of J rare events. The method uses n samples, each of which picks one of the rare events at random, samples conditionally on that rare event happening and counts the total number of rare events that happen. It was used by Naiman and Priebe for scan
statistics, Shi, Siegmund and Yakir for genomic scans and Adler, Blanchet and Liu for extrema of Gaussian processes. We call it ALOE, for `at least one event'. The ALOE estimate is unbiased and we find that it has a coefficient of variation no larger than p (J + J�1 � 2)=(4n). The coefficient of variation is also no larger than p (__=_ � 1)=n where __ is the union bound. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the J rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by 5772
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Cuckoo Search Algorithm: An IntroductionXin-She Yang
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4. Outline
◮ Convex Optimization
◮ Sum of Squares
◮ Control Applications
▽Control Synthesis by Sum of Squares Optimization – p.1/30
5. Outline
◮ Convex Optimization
◮ Sum of Squares
◮ Control Applications
◮ Conclusion
Control Synthesis by Sum of Squares Optimization – p.1/30
6. 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
Control Synthesis by Sum of Squares Optimization – p.2/30
7. 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 with Engineering Applications", Professor Stephan Boyd, Stanford University
Control Synthesis by Sum of Squares Optimization – p.3/30
8. Convex optimization problem
minimize f(x)
subject to x ∈ C
f convex, C convex
"Convex Optimization with Engineering Applications", Professor Stephan Boyd, Stanford University
▽Control Synthesis by Sum of Squares Optimization – p.4/30
9. 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
"Convex Optimization with Engineering Applications", Professor Stephan Boyd, Stanford University
Control Synthesis by Sum of Squares Optimization – p.4/30
10. Linear programming
minimize aT
0 x
subject to aT
i x ≤ bi, i = 1, . . . , m.
"Convex Optimization with Engineering Applications", Professor Stephan Boyd, Stanford University
Control Synthesis by Sum of Squares Optimization – p.5/30
11. Semidefinite programming
minimize cT
x
subject to x1F1 + · · · + xnFn + G ≤ 0
Ax = b,
"Convex Optimization with Engineering Applications", Professor Stephan Boyd, Stanford University
Control Synthesis by Sum of Squares Optimization – p.6/30
12. In fact the great watershed in
optimization isn’t between
linearity and nonlinearity,
but
convexity and nonconvexity.
Rockafellar 1993
Control Synthesis by Sum of Squares Optimization – p.7/30
13. 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
▽Control Synthesis by Sum of Squares Optimization – p.8/30
14. 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
▽Control Synthesis by Sum of Squares Optimization – p.8/30
15. 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 generel?
◮ Decidable, but NP-hard when d ≥ 4.
◮ "Low complexity" is desired at the cost of possibly
being conservative.
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.8/30
16. A sufficient condition
A "simple" sufficient condition: a sum of squares (SOS)
decomposition:
p(x) =
m
i=1
f2
i (x)
If p(x) can be written as above, for some polynomials fi,
then p(x) ≥ 0.
▽Control Synthesis by Sum of Squares Optimization – p.9/30
17. A sufficient condition
A "simple" sufficient condition: a sum of squares (SOS)
decomposition:
p(x) =
m
i=1
f2
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
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.9/30
18. Example
p(x, y) = 2x4
+ 5y4
− x2
y2
+ 2x3
y
=
x2
y2
xy
T
q11 q12 q13
q21 q22 q23
q13 q23 q33
x2
y2
xy
= q11x4
+ q22y4
+ (q33 + 2q12)x2
y2
+ 2q13x3
y + 2q23xy3
An SDP with equality constraints.
▽Control Synthesis by Sum of Squares Optimization – p.10/30
19. Example
p(x, y) = 2x4
+ 5y4
− x2
y2
+ 2x3
y
=
x2
y2
xy
T
q11 q12 q13
q21 q22 q23
q13 q23 q33
x2
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
= 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.
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.10/30
20. Sum of squares programming
A sum of squares program is a convex optimization
program of the following form:
Minimize
J
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, the
wj’s are some given real numbers, and the fi,j are some
given multivariate polynomials.
"Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach", Stephen Prajna et al, ASCC 2004
Control Synthesis by Sum of Squares Optimization – p.11/30
21. 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/sostools
http://www.cds.caltech.edu/sostools
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.12/30
22. Global optimization
Consider for example:
min
x,y
F(x, y)
with F(x, y) = 4x2 − 21
10x4 + 1
3x6 + xy − 4y2 + 4y4
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.13/30
23. Global optimization
◮ Not convex, many local minima. NP-Hard in general.
▽Control Synthesis by Sum of Squares Optimization – p.14/30
24. Global optimization
◮ Not convex, many local minima. NP-Hard in general.
◮ Find the largest γ s.t.
F(x, y) − γ is SOS.
▽Control Synthesis by Sum of Squares Optimization – p.14/30
25. 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!).
▽Control Synthesis by Sum of Squares Optimization – p.14/30
26. 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.
▽Control Synthesis by Sum of Squares Optimization – p.14/30
27. 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.
▽Control Synthesis by Sum of Squares Optimization – p.14/30
28. 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 & Strumfels, 2001
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.14/30
29. 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
▽Control Synthesis by Sum of Squares Optimization – p.15/30
30. 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) = xT Px
P > 0, AT
P + PA < 0
▽Control Synthesis by Sum of Squares Optimization – p.15/30
31. 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) = xT Px
P > 0, AT
P + PA < 0
◮ With an affine family of candidate Lyapunov functions
V , ˙V is also affine.
▽Control Synthesis by Sum of Squares Optimization – p.15/30
32. 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) = xT Px
P > 0, AT
P + PA < 0
◮ With an affine family of candidate Lyapunov functions
V , ˙V is also affine.
◮ Instead of checking nonnegativity, use an SOS
condition
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.15/30
33. Lyapunov stability - 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.
▽Control Synthesis by Sum of Squares Optimization – p.16/30
34. Lyapunov stability - 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...
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.16/30
35. Lyapunov stability - Example
After solving the SDPs, we obtain a Lyapunov function.
"Certificates, convex Optimization, and their applications", Pablo A. Parrilo, Swiss Federal Institute of Technology Zurich
Control Synthesis by Sum of Squares Optimization – p.17/30
36. Lyapunov stability - Example
◮ Consider the nonlinear system
˙x1 = −x3
1 − x1x2
3
˙x2 = −x2 − x2
1x2
˙x3 = −x3 −
3x3
x2
3 + 1
+ 3x2
1x3
▽Control Synthesis by Sum of Squares Optimization – p.18/30
37. Lyapunov stability - Example
◮ Consider the nonlinear system
˙x1 = −x3
1 − x1x2
3
˙x2 = −x2 − x2
1x2
˙x3 = −x3 −
3x3
x2
3 + 1
+ 3x2
1x3
◮ 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: control applications and new developments", Stephen Prajna et al, CACSD 2004
Control Synthesis by Sum of Squares Optimization – p.18/30
38. 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.
▽Control Synthesis by Sum of Squares Optimization – p.19/30
39. 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
"SOSTOOLS: control applications and new developments", Stephen Prajna et al, CACSD 2004
Control Synthesis by Sum of Squares Optimization – p.19/30
40. 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: control applications and new developments", Stephen Prajna et al, CACSD 2004
Control Synthesis by Sum of Squares Optimization – p.20/30
41. 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: control applications and new developments", Stephen Prajna et al, CACSD 2004
Control Synthesis by Sum of Squares Optimization – p.21/30
43. 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: control applications and new developments", Stephen Prajna et al, CACSD 2004
Control Synthesis by Sum of Squares Optimization – p.22/30
44. Safety verification - Example
"SOSTOOLS: control applications and new developments", Stephen Prajna et al, CACSD 2004
Control Synthesis by Sum of Squares Optimization – p.23/30
45. Nonlinear control synthesis
◮ Consider the system
˙x = f(x) + g(x)u
▽Control Synthesis by Sum of Squares Optimization – p.24/30
46. 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
▽Control Synthesis by Sum of Squares Optimization – p.24/30
47. 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)
"Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach", Stephen Prajna et al, ASCC 2004
Control Synthesis by Sum of Squares Optimization – p.24/30
48. 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 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−1
Z(x)
Furthermore, if ǫ2(x) > 0 for x = 0, then the zero
equilibrium is globally asymptotically stable.
"Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach", Stephen Prajna et al, ASCC 2004
Control Synthesis by Sum of Squares Optimization – p.25/30
49. 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
"Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach", Stephen Prajna et al, ASCC 2004
Control Synthesis by Sum of Squares Optimization – p.26/30
50. Nonlinear control synthesis - Example
"Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach", Stephen Prajna et al, ASCC 2004
Control Synthesis by Sum of Squares Optimization – p.27/30
51. 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
▽Control Synthesis by Sum of Squares Optimization – p.28/30
52. 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.
[G. Blekherman, University of Michigan, submitted for
publication]
Control Synthesis by Sum of Squares Optimization – p.28/30
53. Conclusion
◮ Sum of squares, conservative but much more
tractable than nonnegativity
▽Control Synthesis by Sum of Squares Optimization – p.29/30
54. Conclusion
◮ Sum of squares, conservative but much more
tractable than nonnegativity
◮ Many applications in control theory
▽Control Synthesis by Sum of Squares Optimization – p.29/30
55. Conclusion
◮ Sum of squares, conservative but much more
tractable than nonnegativity
◮ Many applications in control theory
◮ Try your problem!
Control Synthesis by Sum of Squares Optimization – p.29/30
56. References
◮ Convex optimmization by Stephen Boyd and Lieven
Vandenberghe, available online
http://www.stanford.edu/~boyd/cvxbook
◮ SOSTOOLS, MATLAB toolbox, freely available
http://www.cds.caltech.edu/sostools
Control Synthesis by Sum of Squares Optimization – p.30/30