Sum-of-Squares Applications in Nonlinear Controller Synthesis

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Sontag Formula Feedback3 Sontag Formula FeedbackIt might seem that having established a method to find a control Lyapunov function, thetask of global regulatory control with the use of the Sontag formula is already solved. Butas this feedback law is in fact a function of the control Lyapunov function, it yields differentresults for different CLFs. This makes the intentional design of the controller particularlydifficult, especially considering the fact that the original Sontag feedback formula (3) doesnot include any design parameters.3.1 Modifications for PerformanceFreeman and Primbs [7] therefore use a modified Sontag formula feedback law that, besidesincorporating multi-input systems, introduces a tuning parameter to exchange state decayspeed for control action.⎧⎪⎨ 0 if L g V = 0( √ )u =(⎪⎩ − (L g V ) 2 ( ) ( ) T )TL(L g V ) (L g V ) T f V + L f V + q(x) L g V L g V if L g V ≠ 0(24)Primbs and Doyle [17] showed that the tuning parameter q(x) can be interpreted in thecontext of the Hamilton-Jacobi-Bellman equation. This interesting result will be brieflyoutlined here, as it motivates our own approach.Consider the infinite horizon nonlinear optimal control problemminu∫ ∞0()q(x) + u T u dtsubject to: ẋ = f + g uwith q(x) continuously differentiable, positive definite. Applying the Bellman principle ofoptimality yields the steady-state Hamilton-Jacobi equation(25)− ∂V ∗∂t= 0 = q + u ∗T u ∗ + L f V ∗ + L g V ∗ u ∗ . (26)The unknown V ∗ is called the value function and V ∗ (x 0 ) represents the optimal cost to gofrom an initial condition x 0 to the origin. The optimal input u ∗ can be calculated from thefirst order optimality condition0 = ∂∂u ∗ (q + u ∗T u ∗ + L f V ∗ + L g V ∗ u ∗) = 2 u ∗T + L g V ∗ (27)12
3.1 Modifications for Performance Sontag Formula Feedbackwhich yields the state feedback lawu ∗ = − 1 2(L g V ∗) T. (28)This is substituted into equation (27) to obtain the Hamilton-Jacobi-Bellman equation0 = L f V ∗ − 1 4(L g V ∗) ( L g V ∗) T+ q . (29)Assume now that the respective level sets of the value function V ∗ and the control Lyapunovfunction V posess the same shape. This implies that their gradients are co-linear at everypoint, i.e., ∇V ∗ = λ ∇V . Using this together with equation (29) allows to solve for theparameter λ which turns out to beλ =(L g V2) (L g VConsequently, equation (28) can be written as√ )( ) 2 ( ) ( ) T) T(L f V + L f V + q L g V L g V. (30)u ∗ = − λ 2= −( ) TL g V( ) TL g V) (L g V(L g V√ )( ) 2 ( ) ( ) T) T(L f V + L f V + q L g V L g V.(31)The optimizer for problem (25) thus coincides with the modified Sontag formula (24).This of course is only the case if V and V ∗ have level sets with the same shape, anassumption that is neither assessable nor likely to be fulfilled by chance. The control law(24) therefore cannot be expected to be optimal when using an arbitrary CLF. AlthoughFreeman and Kokotovic [6] emphasize inverse optimality of the control law i.e., the existenceof a functional that is minimized by it, actual performance decreases dramatically whenvalue function and control Lyapunov function differ significantly in the shape of their levelsets. The fact that the feedback law (24) uses the directional information provided bythe gradient of the CLF explains this dependence on the shape of the underlying controlLyapunov function.This is, however, widely ignored by researchers following the sum-of-squares approachto Lyapunov based control. The naïve use of program 1 to find a CLF with semidefiniteprograming will usually yield the first feasible point that the solver can find. The resultthus depends on the algorithm and starting point that the solver uses.We therefore see the need to introduce design objectives to the search for a CLF. Ourapproach is to impose a certain shape on the control Lyapunov function, that consequently13
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 14 and 15: 2.3 Sum-of-Squares Relaxation for C
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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