Sum-of-Squares Applications in Nonlinear Controller Synthesis

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Introductionmance. Arguments against the use of available nonlinear design techniques include thatthey usually require full state access. However, with the ongoing development in microelectromechanicalsystems, more sensors become cheaply available and consequently enablemeasurement of all states. The importance of state feedback for application is thereforelikely to further increase in the near future.In this research project, the application of sum-of-squares programming to the taskof regulator design for polynomial systems is investigated. A static state feedback law basedon the Sontag formula is introduced and shown to resemble LQR control for linear systems.The method provides easy means of tuning and still produces controllers that comeinherently with a Lyapunov certificate, anticipating a separate model based validation. Furtherextensions to ensure robustness for bounded parameter uncertainty and to guaranteedisturbance attenuation properties in terms of the L 2 norm are outlined.2
Control Lyapunov Functions and Sum-of-Squares2 Control Lyapunov Functions and Sum-of-SquaresThroughout this text, polynomial control affine dynamic systemsẋ(t) = f(x(t)) + g(x(t)) u(t)(1)with x(t) ∈ R n , f(t) ∈ R n , g(t) ∈ R n × R nu , u(t) ∈ R nuwith possibly multiple inputs are considered. From this point on, the time dependence isimplicitly assumed and therefore omitted. For historical reasons, the fundamental resultspresented in this chapter are given in a formulation for single input systems, i.e., n u = 1.They extend naturally to the multi input case that is considered from chapter 3 on. LiederivativesL f V def = ∇ V · f (2)are used for notational convenience whenever derivatives along trajectories of a system aredenoted.2.1 Artstein-Sontag TheoremThe conceptual important foundation for our approach to nonlinear controller synthesis isTheorem 1: Artstein-Sontag Theorem.A system ẋ = f(x) + g(x) u is globally asymptotically stabilizable by a statefeedback law u(x) if and only if there exists a positive definite, radially unboundedscalar function V (x) with()minu L f V + L g V u < 0 ∀x ≠ 0 .V is then called a control Lyapunov function (CLF).The theorem is an extension of classical Lyapunov theory to systems with inputs. It wasstated by Artstein [1] and was proved in a constructive way by Sontag [20] using the feedbacklaw⎧⎪⎨ 0 if L g V = 0( √ )u =( ) 2 ( ) 2⎪⎩ − 1 L f V + L f V + L g V if L g V ≠ 0 ,L g V(3)which is now known as the Sontag formula. This control law is continuous everywhere,except possibly at the origin. It requires full state access and results in global stability as3
Page 4 and 5: NomenclatureGreek Small Lettersδγ
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 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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