Sum-of-Squares Applications in Nonlinear Controller Synthesis

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4.1 Quadratic Cost Performance Illustrative ExamplesClosed loop simulations are used to illustrate the benefits that can be gained by usingthe proposed method. Direct application of the linear quadratic regulator, which does notprovide global stability, is used as a performance benchmark. As a metric, the quadraticcost determined by the functional ∫ τ0 xT Q x + u T R u dt is used, with τ being a sufficientlylarge time to allow the system to settle at the equilibrium. A disk of radius 2 around theorigin is uniformly sampled in state-space, which can be seen as a circular approximationof the LQR’s region of stability. Using polar coordinates with steps of 0.05 in radial and 5 ◦in angular direction, the sample consists of 2993 points. Each of these is used as an initialcondition and the cost along all trajectories is summed up. Table 1 provides the results fordifferent weighting matrices Q and R.Table 1: Quadratic cost of the different feedback laws. Uniform sample within a disk of radius 2around the origin with 2993 sample points. Cost of all trajectories added up to τ = 5.Weighting factors LQR Sontag (V LQR ) Sontag (V I ) Sontag (V feas )Q = [ 1.0 0 1.0 0 R = [ 1.0 0 1.0 0 ] 2117 2068 2331 3240Q = [ 1.0 0 1.0 0 R = [ 1.0 0 4.0 0 ] 2026 2003 2132 3115Q = [ 1.0 0 1.0 0 R = [ 30 0 20 0 ] 1253 1100 1494 2420Q = [ 5.0 0 0.5 0 R = [ 1.0 0 1.0 0 ] 3688 3579 3723 4977Q = [ 5.0 0 0.5 0 R = [ 1.0 0 4.0 0 ] 3689 3205 3420 4487Q = [ 5.0 0 0.5 0 R = [ 30 0 20 0 ] 3776 3348 3274 4404Q = [ 0.1 0 0.3 0 R = [ 1.0 0 1.0 0 ] 1446 1565 1901 2684Q = [ 0.1 0 0.3 0 R = [ 1.0 0 4.0 0 ] 1334 1399 1332 2159Q = [ 0.1 0 0.3 0 R = [ 30 0 20 0 ] 290 273 410 717Overall 19619 18267 20017 28203Summarized, the LQR based Sontag formula provides good results throughout all tuningsconsidered. In some cases the guessed control Lyapunov function and the linear quadraticregulator provide slightly better results. The control law based upon the feasibility problemyields consistently the worst performance. Our results thus suggest that improved performancecan be achieved with the systematic way of constructing the Lyapunov functioncompared to using some arbitrary feasible point.26
4.2 Controller Tuning Illustrative Examples4.2 Controller TuningAs a second system, we consider the inverted pendulum which is a well studied example inboth nonlinear dynamics and control. It has some interesting properties that make it prototypicalfor various real world systems. Most obviously it is unstable at its upper equilibriumpoint and therefore requires stabilizing control. It is also an underactuated non-minimumphase system and therefore inherently difficult to control. Examples for this class of systemsinclude rockets and aircraft.m 2m 1(a) Modelgπ2lx 2π4x40− π 4ux 1− π 2− π 2− π 40π4x 2(b) Phase portrait of the angular dynamicsπ2Figure 6: Second example: the inverted pendulum.The model is illustrated in figure 6, together with the phase portrait of the angular dynamics.The model equations are derived from physical interpretation of the system. The simplestmodel that captures the nonlinear dynamics of the inverted pendulum is a point mass m 2 ,attached to a cart of mass m 1 through a rigid rod of length l without mass. Effects of frictionand damping are neglected. A force u is applied to the cart and used as the control input.⎡ ⎤ ⎡ẋ 1ẋ 2=⎢ẋ ⎣ 3 ⎥⎦ ⎢⎣ẋ 4x 3x 4m 2 g cos(x 2 ) sin(x 2 )−m 2 l x 2 4 sin(x 2)m 1 +m 2 −m 2 cos 2 (x 2 )(m 1 +m 2 ) g sin(x 2 )−m 2 l x 2 4 sin(x 2) cos(x 2 )m 1 l+m 2 l−m 2 l cos 2 (x 2 )⎤⎡+⎥ ⎢⎦ ⎣001m 1 +m 2 −m 2 cos 2 (x 2 )cos(x 2 )m 1 l+m 2 l−m 2 l cos 2 (x 2 )⎤u . (71)⎥⎦As this model is rational with trigonometric terms and it is not possible to turn it into apolynomial form with a change of variables, an approximation for a certain region around27
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 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 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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