A control strategy for the cart-pendulum system Abstract 1 ...

As ϕ is not definite, H d is not a Lyapunov functionin the whole state space. Consider now the followingLyapunov function candidate:V = ϕ(x 1 ,x 2 )(H d − 1/3)(= ϕ(x 1 ,x 2 ) − 1 3 cos3x 1 + 1 2 x2 2 − 1 ). (10)3Notice that V ≥ −2/3 and continuous. Nevertheless,V is not differentiable at the switching curve of (8).Outside this curve,This implies that:˙V = −ϕ 2 x 2 2 cos 2 x 1 ≤ 0• V is a Lyapunov function in the region Ω inside thecentral closed curve of Fig. 2. This closed curve isthe level curve V = 0.• The switching curve in (8) is attractive.In fact, a sliding motion is produced along the switchingcurve. The motion inside this curve can be deducedfrom the first equation of (9). Thus, only four equilibriain the sliding motion are possible (with x 1 ∈ [−π,π]):E 1,2 : (±π/3,0) and E 3,4 (±π,0). Besides, outside thesliding curve there are more limit sets: the maximuminvariant set in Ω for which ˙V = 0 is (±2π/3,0) ⋃ (0,0),being all equilibrium points. The latter is the desiredequilibrium while it is clear that the former ones areunstable. Therefore, the only problematic points areE 1,2 and E 3,4 .curves. It can be seen that the undesirable equilibriumpoints E 1,2 and E 3,4 still exist but they are outsidethe sliding manifold and, since they are unstable,they do not preclude the almost-global stability property.Nevertheless, this property is not proved yet. Noticethat with the choice (12), the Lyapunov functioncandidate is continuous along the dotted curve of Fig.3. But now, function ϕ ε introduces new switching linesat x 1 = ±π/3 (segments AB and CD in Fig. 3). Function(12) is not continuous at these segments and theanalysis must be performed carefully.In the following, it will be proved that, for smallenough ε, when the system reaches these points of discontinuityfor (12), the trajectory evolves towards theorigin. Due to the symmetry of the problem only thecase of x 2 > 0 will be considered. When the systemstarts on the upper half of segment AB as x 2 > 0 itwill evolve to the right. In this region Ḣd > 0 and thesystem will evolve towards the sliding curve (the dottedcurve of Fig. 3). After reaching x 1 = π it will continuefrom x 1 = −π towards x 1 = −π/3. Therefore, segmentCD (with x 2 > 0) will eventually be reached. When thesystem starts on this segment as x 2 > 0 it will evolvetowards the right but, in this region Ḣd < 0. If ε issmall enough the level curve H d = 1/3 (the solid curve)will be reached before reaching segment AB. Once thislevel curve is reached and crossed the system can not goout of the compact set {x : H d (x) < 1/3} where ˙V ≤ 0.1.512.2 Achieving almost-global stabilityIn order to avoid the undesirable equilibria E 1,2 andE 3,4 , the switching curve is changed from H d = 1/3 toH d = 1/3+ε with 0 < ε ≪ 1. This change is performedsubstituting the switching function ϕ byx 20.50−0.5−1CDAB{△ −k1 if Hϕ ε = d ≤ (1/3 + ε) and cosx 1 < 0.5k 1 elsewhereThe control law is now:u = 2sin 2x} {{ } 1 +ϕ ε (x 1 ,x 2 )x 2 cos x 1 (11)} {{ }u c u dThe same analysis can be performed with the Lyapunovfunction candidateV = ϕ ε (x 1 ,x 2 )(H d − 1/3 − ε) (12)The behaviour of the system can now be explained withthe help of Fig. 3. In this figure the level curves correspondingto H d = 1/3 and H d = 1/3 + ε are plotted.The sliding motion is now along the second of these−1.5−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1x 1/πFigure 3: Level curves H d = 1/3 (solid) and H d =1/3 + ε (dotted)Since no other points of discontinuity exist, the followingproposition can be stated:Proposition 1. The origin of system (1) with controllaw (11) is almost globally asymptotically stable.2.3 Avoiding the sliding modeIn many control applications sliding modes are not admissibledue to the chattering phenomenon. The controllaw proposed in the previous section presents slidingmotions due to the discontinuity along the level curve3