Sum-of-Squares Applications in Nonlinear Controller Synthesis

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2.3 Sum-of-Squares Relaxation for CLFs Control Lyapunov Functions and Sum-of-Squaresand by fixing k 2 = 1 and s 2 = ŝ 2 l 2 , s 3 = ŝ 3 l 2 , p = ˆp l 2 the second equation is reduced to())( ))l 2 ŝ 2 + ŝ 3(L f V + l 2 + ˆp L g V = 0 ⇒ −(ŝ 3 L f V + l 2 + ˆp L g V = ŝ 2 .A sufficient condition for V to be a control Lyapunov function then is∃ s ∈ S, p, l 1 , l 2 ∈ P, l 1 , l 2 ≻0 such thatV − l 1 ∈ S( ( ))− s L f V + l 2 + p L g V ∈ S .(22)Using semidefinite programming, it is possible to evaluate condition (22) and to verifywhether a given candidate function V is a control Lyapunov function. 2 This is stated asLemma 2: Sum-of-Squares Certificate for Control Lyapunov Functions.V is a control Lyapunov function for the polynomial system ẋ = f(x) + g(x) ifV − l( ( )) 1− s L f V + l 2 + p L g Vs∈ S∈ S∈ Sp, l 1 , l 2 ∈ Pl 1 , l 2 ≻ 0is a feasible semidefinite programming problem.Note, that this can only be used to assess a candidate function V that has to be provided,e.g., by guessing.A way to actually synthesize a control Lyapunov function is to allow the coeffcients ofa parametric polynomial V , i.e., V = c z, c ∈ R 1×m , z ∈ M m (x), to be decision variables.The problem stated in lemma 2 can then be solved asProgram 1.Find V such that:V − l( ( )) 1− s L f V + l 2 + p L g Vs∈ S∈ S∈ Sp, l 1 , l 2 ∈ Pl 1 , l 2 ≻ 0As both the polynomial multiplier’s and the CLF’s coefficients are now decision variables,2 For actual implementation, the positive definite polynomials l 1 and l 2 need to be specified. A commonapproach is to use l i = ɛ i x T x with a decision variable ɛ.10
2.3 Sum-of-Squares Relaxation for CLFs Control Lyapunov Functions and Sum-of-Squaresthe associated semidefinite program will include bilinear constraints of the formn∑n∑ n∑Q 0 + Q k y k + R kl y k y l ≽ 0. (23)k=1k=1 l=1The numerical solution of these bilinear matrix inequalities (BMI) is still subject to ongoingresearch, since these problems are no longer convex. Current approaches include iterativeschemes which fix one bilinear variable and solve the resulting linear matrix inequality inthe other variables. In a next step, a different bilinear variable is fixed and the procedureis repeated. This is computanially expensive and convergence to a global extremum cannotbe guaranteed. The only available direct solver for bilinear problems currently is PENBMI,a commercial solver that uses a local penalty method. YALMIP [13] and its sum-of-squaresmodule [14] can be used as a parser.We consider this software to be still in an early stage and in our opinion it lacksboth performance and reliability compared to the dedicated polynomial manipulation toolboxMULTIPOLY [2]. For linear sum-of-squares programs, i.e., to verify that a candidatefunction is a CLF, we therefore recommend the use of the toolbox SOSTOOLS [16], thatprovides a number of advantages including the integration of MULTIPOLY. Furthermore,we recommend to verify results obtained from bilinear sum-of-squares problems throughYALMIP/PENBMI with SOSTOOLS. We nevertheless conclude that the available toolsallow for our method to attack the problem and if they are successful produce a meaningfulcertificate. If unsuccessful, however, no conclusions about possible intractability of theproblem can be drawn. We expect further development in this field.11
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 10 and 11: 2.2 Sum-of-Squares Programming Cont
Page 12 and 13: 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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