Abstract
This chapter starts with stability of various systems with state jumps: Lyapunov stability of Measure Differential Equations, vibro-impact systems, and impact oscillators. Then the so-called grazing bifurcations are introduced. The Lyapunov stability of complementarity Lagrangian mechanical systems is analyzed in detail, and it is shown how the Zhuravlev-Ivanov nonsmooth transformation introduced in Chap. 1 may be used for finite-time stabilization with a sliding-mode controller. The chapter ends with the analysis of Lyapunov stability of a simple system hitting a unilateral spring-like environment, and the use of copositive matrices for studying the stability of linear complementarity systems.
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Notes
- 1.
Notice that in general we do not have \(\dot{V}=\frac{\partial V}{\partial x}\left( \{\dot{x}\}+\sigma _{x}\delta _{t_{k}}\right) \), but we do have \(\dot{V}=\{\dot{V}\}+\sigma _{V}(t_{k})\delta _{t_{k}}\). If \(V(\cdot )\) is quadratic in x and \(x \in \textit{RCLBV}\), then Moreau’s rule can be applied to get \(dV=d(x^{T}Px)=(x^{+}+x^{-})^{T}Pdx\) [867]. Outside jumps \(dV=2x^{T}Pdx\) and at discontinuity instants \(dV=(x^{+}+x^{-})^{T}P(x^{+}-x^{-})\delta _{t}=[(x^{+})^{T}Px^{+}- (x^{-})^{T}Px^{-}]\delta _{t}=\sigma _{V}\delta _{t}\). dV and dx are called differential measures or Stieltjes measures of \(V(\cdot )\) and \(x \in \textit{RCLBV}\).
- 2.
Nonmonotonic Lyapunov functions are often met in systems with state jumps, see Chap. 8. This is because possible strict decrease at state-jump times may compensate for (not too big) increase between the jumps.
- 3.
Such maps are also called first-return mapping, monodromy operator, successor mapping.
- 4.
Due to its simplicity, the bouncing-ball dynamics velocity is of special bounded variation, which allows one to get rid of the third term of its derivative as a measure, denoted \(\mu _{na}\) in Remark A.4.
- 5.
To simplify the presentation, we denote \(V(x(t),\dot{x}(t))\) as V(t), and its derivative along the system’s trajectories as \(\dot{V}(t)\).
- 6.
A similar study was also published at the same time by Feigin [390].
- 7.
Notice that it is not possible in this case to have a periodic trajectory with collisions occuring repeatedly on the same constraint, since the mass m is horizontally free between impacts. This however is possible with other systems, like the inverted pendulum in a box.
- 8.
In relationship with the fact that the calculated jump in the velocity, is independent of the fact that the used frame is Galilean or not (see Chap. 4, Sect. 4.1.5), it is clear here that the frame fixed with respect to the block is not Galilean, and does not satisfy the smoothness requirements discussed in Sect. 4.1.5, Eq. (4.25), since \(\dot{x}\) is discontinuous at impacts. In other words, the jump of the absolute velocity \(\dot{y}_{a}(\cdot )\) is clearly different from that of the relative velocity \(\dot{y}_{r}(\cdot )\).
- 9.
The reason for this apparently complicated notation is that we shall need several steps to go from u to \(\bar{u}_{\tiny {\varSigma }}\).
- 10.
In most cases it is clear that this will imply a reordering of the generalized coordinates. For instance if \(f(q)=q_{2}\), then evidently one will not define \(G(\cdot )\) as above, but rather first exchange \(q_{1}\) and \(q_{2}\) in q, or simply define \(G(\cdot )\) with \(\bar{q}_{1}\) as the second component of \(\bar{u}\).
- 11.
Apart from the examples presented here, the interested reader may have a look at [911] Eq. (1.3), [1252] Eq. (1.3), [1252] Eqs. (24)–(29), [998] Eq. (4), [1100] Eq. (2), [1095] equations (4) (5) for examples of implicit Poincaré maps as in (7.27).
- 12.
If \(g=f\circ h\) with \(h: \mathbb {R}^{n}\rightarrow \mathbb {R}^{p}\), \(f: \mathbb {R}^{p} \rightarrow \mathbb {R}^{k}\), then \(\nabla g(x_{0})=\nabla h(x_{0}) \nabla f(y_{0})\) and \(Dg(x_{0})=Df(y_{0}) Dh(x_{0})\), where \(y_{0}=h(x_{0})\).
- 13.
In the case of Hamiltonian (i.e. conservative) systems with an analytic Hamiltonian function, Lyapunov’s holomorphic integral theorem states that for every pair of pure imaginary roots \(\pm j\lambda \) of the system’s characteristic equation, and when there are no other roots, a family of periodic solutions exists whose period tend to \(\frac{2\pi }{\lambda }\) as their amplitude tends to zero.
- 14.
This, in case of feedback control of a juggling robot, should be guaranteed by the controller, but not a priori supposed.
- 15.
In other words, the trajectories of the impact Poincaré map increase in velocity to infinity.
- 16.
Note anyway that as pointed out in [92], if the case \(e_{\mathrm{n}}=1\) did not exist in nature, then all molecular motion would long since have ceased. But we leave here engineering. Such problems were discussed by Huygens and Leibniz at a time when scientists were trying to discover whether springiness or hardness (to be understood here as non-penetrability) is the real physical phenomenon that produces rebound [1050].
- 17.
The study of rattling-noise in gearboxes is fundamental to reduce the noise level and the vibrations in engines.
- 18.
Dissipativity holds between the harmonic force input and the velocity, i.e. with supply rate \(w(u,y)=F(t)\dot{q}(t)\). Dissipative systems with no input define Lyapunov stable systems [218]. Hence the eigenvalues of the system’s matrix A must have magnitude \(\le 1\).
- 19.
i.e. all phenomena involving an infinity of events in a finite time interval.
- 20.
The asymptotic Lyapunov stability is not shown.
- 21.
Basically, this is a sufficient condition for the Krasovskii-LaSalle invariance principle to hold, because it implies that positive limit sets of solutions are positively invariant [730, Proposition 6.12]. The autonomy property is central in the proof of [730, Proposition 6.12], and recall from Sect. 1.3.2 that it is implied by the uniqueness of solutions which is itself assured by the continuous dependence on initial data.
- 22.
- 23.
This type of set-valued controller is quite popular in the Sliding Mode Control scientific community, was introduced in [734] and its finite-time stability studied in [948].
- 24.
At this step, one may also invoke LaSalle’s invariance principle, though the differential inclusion is time-dependent, transforming the perturbed time-varying differential inclusion in an equivalent autonomous differential equation with rectangular uncertainties [948, §2].
- 25.
This last point will be important to assure via a suitable controller when one wants to stabilize a system on a surface.
- 26.
Notice that we have not proved the asymptotic stability of \(P_{\tiny {\varSigma }}\).
- 27.
We do not use the notation \(t_{k}\) because the \(t_{i}\)’s may correspond to detachment. In fact if contact is made at \(t_{2i}\) and lost at \(t_{2i+1}\), then \(t_{k}=t_{2i}\).
- 28.
This is not the case for the compliant model since \(\dot{V}_{nc}(t)=-\lambda _{2}\dot{x}(t)^{2}-k\dot{x}(t)x(t)\) during contact phases and \(\dot{V}_{c}(t)=-\lambda _{2}\dot{x}(t)^{2}+k\dot{x}(t)x(t)\) during free-motion phases.
- 29.
The tangent cone to K at x.
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Brogliato, B. (2016). Stability of Nonsmooth Dynamical Systems. In: Nonsmooth Mechanics. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-28664-8_7
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DOI: https://doi.org/10.1007/978-3-319-28664-8_7
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