Sum-of-Squares Applications in Nonlinear Controller Synthesis

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3.2 Modifications for Systems with Bounded Uncertainties Sontag Formula Feedbacknear optimal performance for small deviations from the origin is expected. Focusing on theperformance around the origin is reasonable, as any trajectory will eventually enter thisregion due to global stability. Even more important from a practical point of view, we haveintroduced convenient tuning parameters for both state decay rates and control action.3.2 Modifications for Systems with Bounded UncertaintiesAn important question for actual application is how to deal with uncertainties in a system.Often uncertainty can be reasonably bounded, as parameters are usually known to varywithin a certain range or because estimates for external disturbances are available. This typeof uncertainty for nonlinear systems is very successfully addressed by sliding mode control(see Slotine and Li [18]). These (usually discontinuous) controllers seek to overpower an apriori bounded uncertainty term with control action. A crucial assumption for this is thematching condition, i.e., that the uncertainty enters the system through the same channelas the input. Recently developed dynamic surface control [19] generalizes these ideas alsoto mismatched systems. Although sliding mode controllers are usually used for trackingcontrol, we regard them as a valuable inspiration for the regulator task.We consider a polynomial system, affine in both control and disturbance, given bythe equationẋ = f(x) + g u (x) u + g d (x) dwith x(t) ∈ R n , f(t) ∈ R n , u(t) ∈ R nu , d(t) ∈ R n d,(42)g d (t) ∈ R n × R n d, g u (t) ∈ R n × R nu ,and restrict the unknown disturbance d to satisfy an a priori bound‖d‖ < δ ∀d with δ ∈ R n d(43)This can be used, e.g., to model parameter uncertainty for a family of systems. Recall thatwe seek to decrease the control Lyapunov function V to eventually drive the system to theorigin. For the disturbed system, the time derivative of the CLF is˙V = L f V + L guV u + L gdV d . (44)The disturbance d is unknown and thus sign indefinite, but with the uncertainty bound(43) the worst case is governed by the inequality˙V ≤ L f V + L guV u + ‖ L gdV δ‖ . (45)16
3.2 Modifications for Systems with Bounded Uncertainties Sontag Formula FeedbackThus, using the reasoning from the Artstein-Sontag theorem, the conditionL f V + ‖ L gdV δ‖ < 0 ∀x ≠ 0 such that L guV = 0 . (46)guarantees the existence of a globally stabilizing controller for the uncertain system. Sucha controller is easily constructed by replacing L f V in the Sontag feedback formula withα = L f V + ‖ L gdV δ‖. For the closed loop system, this yields˙V = L f V + L gdV d < L f V + ‖ L gdV δ‖ < 0˙V = −√α 2 + x T Q xif L guV = 0 , x ≠ 0( )L guV R −1( ) TL guV + Lgd V d − ‖ L gdV δ‖ < 0(47)if L guV ≠ 0 , x ≠ 0which proves stability even in the presence of bounded uncertainty. Note, that the controllaw still is continuous and that the formulation requires no matching condition. In fact, wecan conclude that for matched systems 3 L gdV and L guV have the same roots and thus anycontrol Lyapunov function will satisfy condition (46) for arbitrary δ. This is expected andreflects the conceptual similarity to sliding mode controllers: it is possible to overpower anymatched uncertainty with sufficient control action.Condition (46) is equivalent to the two conditionsL f V + L gdV δ < 0 ∀x ≠ 0 such that L guV = 0L f V − L gdV δ < 0 ∀x ≠ 0 such that L guV = 0(48)for whom a sum-of-squares relaxation can be obtained using theorem 3 and the proceduredescribed in section 2.3. A control Lyapunov function can then be found withProgram 3.Find V such that:(−(s 1−V − l)) 1L f V + L gdV δ + l 2 + p 1 L guV))+ l 3 + p 2 L guV(s 2(L f V − L gdV δ∈ S∈ S∈ Ss 1 , s 2 ∈ Sp 1 , p 2 , l 1 , l 2 , l 3 ∈ Pl 1 , l 2 , l 3 ≻ 03 Recall that the precise definition according to Khalil [12] is that the disturbance term belongs to therange space of the input matrix.17
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 16 and 17: Sontag Formula Feedback3 Sontag For
Page 18 and 19: 3.1 Modifications for Performance S
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

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