Sum-of-Squares Applications in Nonlinear Controller Synthesis

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NomenclatureGreek Small Lettersδγuncertainty boundL 2 gain boundIndicesdfeasILQRudisturbancebased on the feasibility problembased on the identitiy matrixbased on the linear quadratic regulatorinputMiscellaneous CharactersA ≻ (≽)0A ≺ (≼)0minA is positive (semi)definiteA is negative (semi)definiteb a minimize the objective a over the parameter b∇A gradient of A∅∀∃the empty setfor allthere existsNotationxxXẋx TX −1‖x‖‖x‖ 2x ∗scalarvectormatrixtime derivativetransposeinverseEuclidian norm of xL 2 norm of xoptimizeriii
Introduction1 IntroductionControl Lyapunov functions introduced by Artstein and Sontag [1, 20] in the 80’s offera valuable source for systematic design of controllers for nonlinear systems. They extendthe classical Lyapunov theory to systems with inputs, making it directly applicable in thedesign process. Moreover, by use of differential geometric concepts, it is possible to derivean explicit universal feedback law, known as the Sontag formula, once a control Lyapunovfunction for a system is found.Interestingly, a slightly modified version of this feedback law was shown to be asolution to the infinite time optimal control problem by Freeman and Kokotovic [6]. Theirnotion of inverse optimality can consequently be seen as a natural extension of the linearquadratic regulator theory to nonlinear systems. The extension of optimal control designtechniques to nonlinear systems is tempting, recalling that since Kalman [11] solved theLQR problem in the 60’s, optimal pole-placement has proved useful for actual application.Up to today, it is a useful starting point for the design of regulatory controllers. Reasons forthis are the systematic way of tuning, the guaranteed stability and good classical robustnessmargins that are associated with the method.One of the problems with the Sontag formula so far was to find suitable controlLyapunov functions, as no general analytical method is developed yet. The sum-of-squaresmethodolgy yields an approach to this issue for polynomial systems and provides a practicalway of finding control Lyapunov functions. Polynomial systems are an important class ofnonlinear systems and can be seen as a logical next step to extend theoretical results to. Theyarise wherever empirically obtained data is fit by polynomials, e.g., to obtain aerodynamiccoefficients for aircraft. Furthermore, it is often possible to turn a general nonlinear systeminto polynomial form with a change of variables. Last but not least, Taylor’s theorem orleast square regression allows to approximate every function by polynomials.Restricting polynomials to be a sum-of-squares implies their positive semidefinitenessand is a condition that can be computationally verified with semidefinite programming,a still growing discipline of convex optimization that was started by the introduction ofefficient interior point methods in the 90’s. Several software tools [13, 14, 16] are availableand can be employed in the process, providing a promising framework for the computeraided search for control Lyapunov functions.A reason for our interest in the topic is that nonlinear control is still widely disregardedfor actual application. Despite the existence of theoretically well established techniques, theoverwhelming majority of controllers used in industry is still designed using linear methodology.Regarding that most real world systems are nonlinear in nature, validity of linearapproximations in the design is often questionable and leads to a loss of potential perfor-1
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 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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