Sum-of-Squares Applications in Nonlinear Controller Synthesis

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2.2 Sum-of-Squares Programming Control Lyapunov Functions and Sum-of-Squareswhich satisfies the condition of theorem 1 asminu ˙V =⎧⎨−x T Qx if x T P B = 0⎩−∞ if x T P B ≠ 0 .(13)We introduce this concept not only for its own interest but also to use a similar approach inchapter 3 to parametrize a controller for nonlinear systems. Furthermore, the parametrizationpresented here is directly applicable to nonlinear systems that are input-to-state feedbacklinearizable. As for arbitrary polynomial systems no systematic method of finding CLFsis yet discovered, we use the relatively recent approach of sum-of-squares programming [22].2.2 Sum-of-Squares ProgrammingWe start with a short review on polynomials and semidefinite programming to familarizethe reader with the basic ideas. The application of sum-of-squares programming to findcontrol Lyapunov functions is covered in section 2.3.First note that a monomial of n variables is a function of the form{}z(x 1 , x 2 , . . . , x n ) ∈ M (x 1 , x 2 , . . . , x n ) = x α 11 xα 22 · · · x αnn ∣ α i ∈ N 0 . (14)Equivalently, the short notation z(x) will be used to indicate a monomial in the componentsof a vector x. Furthermore, a vector representation will be convenient, ordering the entrieslexically:z(x) =[1 x 1 x 2 · · · x n x 2 1 x 1 x 2 · · · x 1 x n · · · x 2 1 x 2 · · ·] T. (15)A polynomial is then a finite linear combination of monomials with real coefficientsp(x 1 , x 2 , . . . , x n ) = ∑ ic i z i , c i ∈ R , z i ∈ M(x 1 , x 2 , . . . , x n ) , (16)which can also be written in vector notation asp(x 1 , x 2 , . . . , x n ) = c z , c ∈ R 1×m , z ∈ M m .The space of polynomials will be denoted by P and the space of sum-of-squares polynomialswill be defined as{S defn }= p ∈ P∣ p = ∑qi 2 , q i ∈ P, i = 1, . . . , n . (17)i=16
2.2 Sum-of-Squares Programming Control Lyapunov Functions and Sum-of-SquaresIt is obvious that if p ∈ S, then p is positive semidefinite everywhere. As the converseis not true, replacing the requirement of a polynomial to be positive semidefinite with asum-of-squares constraint is a relaxation to sufficient conditions.In general, sum-of-squares problems, i.e., the question whether a given polynomialis a sum-of-squares polynomial, can be solved computational tractably via semidefiniteprogramming. This is due toTheorem 2: Sum-of-Squares Decomposition.Given a polynomial p of degree 2d in the variables x, p is a sum-of-squares ifand only if there exists a symmetric positive semidefinite matrix Q such thatp = z T Q zwith z being the vector of all monomials of x up to degree d. Q is then calledthe Gram matrix of the sum-of-squares polynomial p.This result was first obtained by Choi et. al. [5] and shown by Parillo [15] to be a semidefiniteprogramming problem. Note that Q is not unique since the monomials are not algebraicallyindependent.Semidefinite programs can be seen as a generalization of linear programs. The goalis to minimize a linear function, subject to the constraint that an affine combination ofsymmetric matrices is positive semidefinite.min c T yr∑subject to: Q 0 + Q k y k ≽ 0k=1(18a)(18b)These constraints are equivalently known as linear matrix inequalities (LMI) in the controlcommunity, although they are usually nonlinear (see Boyd et. al. [4] for an in-depth survey).They are however always convex and efficient solvers such as SeDuMi are freely available.The computational method to check whether a given polynomial is a sum-of-squares isto use a parametric sum-of-squares polynomial in the monomials of the function in question.With theorem 2, this means to introduce a parametric Gram matrix. The entries are thenconstrained to match the coefficients of the given polynomial. A sum-of-squares constraintthus is equivalent to a set of equality constraints and a linear matrix inequality constraint.If the problem is feasible, the solution is a sum of squares decomposition, if it is infeasible,the polynomial in question is not a sum-of-squares.7
Page 4 and 5: NomenclatureGreek Small Lettersδγ
Page 6 and 7: Introductionmance. Arguments agains
Page 8 and 9: 2.1 Artstein-Sontag Theorem Control
Page 12 and 13: 2.3 Sum-of-Squares Relaxation for C
Page 14 and 15: 2.3 Sum-of-Squares Relaxation for C
Page 16 and 17: Sontag Formula Feedback3 Sontag For
Page 18 and 19: 3.1 Modifications for Performance S
Page 20 and 21: 3.2 Modifications for Systems with
Page 22 and 23: 3.3 Modifications for Disturbance A
Page 24 and 25: 3.3 Modifications for Disturbance A
Page 26 and 27: Illustrative Examples4 Illustrative
Page 28 and 29: 4.1 Quadratic Cost Performance Illu
Page 30 and 31: 4.1 Quadratic Cost Performance Illu
Page 32 and 33: 4.2 Controller Tuning Illustrative
Page 34 and 35: 4.2 Controller Tuning Illustrative
Page 36 and 37: REFERENCESREFERENCESReferences[1] A

Sum-of-Squares Applications in Nonlinear Controller Synthesis

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