5.5 Sliding Motion
5.5.1 Sliding Motion for Ordinary Differential Equations
a) b)
c) d)
Γ Γ
Γ Γ
Γ+ Γ+
Γ+ Γ+
Γ− Γ−
Γ− Γ−
fI fI
fI fI
fII
fII fII
fII
Figure 5.6: Phase space behavior at a switching surface
and we can consider the following system of differential equations
˙ x=
(fI(x) for x∈Γ+,
fII(x) for x∈Γ−, (5.24)
where fI =f|Γ+ and fII =f|Γ−. The system (5.23) is completely described by (5.24) in the domains Γ+ and Γ−, but on the switching surface Γ the standard definition of solution for ODEs may not be applicable, as the behavior of the solution of (5.23) on the switching surface is not defined. To cope with this, the discontinuous right-hand side of (5.23) can be replaced by a differential inclusion, see e.g. [40], i.e.,
˙
x(t)∈η(t, x). (5.25)
If at a point (t, x) the function f is continuous, then the set η(t, x) consist only of one point which is the value of the functionf at this point. If (t, x) is a point of discontinuity of f, then the set η(t, x) is to be defined in some other way.
In general, there are four types of solution behavior in the neighborhood of a switching surface characterized by the directions of the vector fieldsfI andfII as depicted in Figure 5.6. If the vector fields point towards the surface from one side and away from the surface from the other side as in cases a) and b) in Figure 5.6, the solution trajectory crosses the discontinuity and the system has a classical solution. On the other hand, if both vector fields point towards the switching surface Γ as ind) in Figure 5.6, then the solution cannot leave this manifold, but sticks to the manifold, and the solution can be defined via the differential inclusion (5.25). In this case chattering behavior during the numerical integration can occur as depicted in Figure 5.7. In the last case, where the vector fields on both sides point away from the surface as inc) in Figure 5.6, the switching surface cannot be crossed and there exists a point beyond which no classical solution exists.
In reality, small parameters in the system prevent the system from chattering and induce a smooth motion along the surface. In sliding motion the system dynamics are approximated
Γ Γ+
Γ−
Figure 5.7: Chattering behavior along a switching surface
in such a way that the state trajectory moves along the switching surface. There are two main approaches to describe the dynamics of the system on a switching surface. The first approach is calledequivalence in dynamicsorFilippov regularization, see e.g. [39, 40, 145].
Here, approximations of the solution trajectories on both sides in a small neighborhood around the surface are used to determine the average velocity on the surface. Another approach calledequivalence in controlis presented in [145]. Here, free solution components, i.e., controls, in the system are chosen such that the solution trajectory moves along the switching surface. It has been stated in [145] that if the true system behavior near the switching surface can be attributed to hysteresis phenomena, then the method of equivalent dynamics derives sliding behavior closer to the true system behavior than the method of equivalent control. On the other hand, if there are no hysteresis effects the equivalent control method may generate better approximation. In the following, we will describe the approach of equivalence in dynamics in detail and afterwards shortly present the main ideas of the method of equivalence in control.
In the Filippov regularization, for each point x ∈ Dx, the differential inclusion η(t, x) is defined to be the smallest closed convex set containing all the limit values of f(x⋆) for x⋆ ∈/ Γ, x⋆ → x. Then, a function x(t) is said to be a solution of (5.23) if it is absolutely continuous and satisfies (5.25) almost everywhere. For x⋆ approaching the point x ∈ Γ from Γ− and Γ+, let the functionf(x⋆) have the limit values
xlim⋆∈Γ− x⋆→x
fII(x⋆) =fΓII(x) and lim
x⋆∈Γ+ x⋆→x
fI(x⋆) = fΓI(x).
Then, the set η(t, x) is the line segment joining the end points of the vectors fΓII(x) and fΓI(x) for x ∈ Γ. If this line is on one side of the tangent plane to the switching surface Γ, then the solution passes from one side of the surface to the other side, see Figure 5.8.
On the other hand, if the line segment intersects the tangent plane, then the solutions approach Γ from both sides, see Figure 5.9. In this case, the standard notion of solution is not suitable as there is no indication of how a solution can be continued. Nevertheless, if the line segment intersects the tangent plane, the intersection point is the endpoint of the vectorfΓ(x) which determines the velocity of the motion ˙x=fΓ(x) along the surface Γ at x. From (5.25) the solution x(t) of the differential equation satisfies
˙
x=fΓ(x), (5.26)
Γ
Γ−
Γ+ fΓI
fΓII (t, x)
Figure 5.8: Regular switching at a switching surface
Γ
Γ−
Γ+ fΓI
fΓII (t, x) fΓ
Figure 5.9: Filippov’s construction of equivalent dynamics
wherefΓ is a linear combination of fΓI andfΓII and therefore it is also a solution of (5.23).
Ifx∈Γ+, then fΓ equals fI and ifx∈Γ−, thenfΓ equals fII. Note thatfΓ is a particular selection from the setη(t, x). It is also possible to define other differential inclusions as we will see below. The velocity vectorfΓ of sliding motion in (5.26) lies on a plane tangential to the surface, and therefore its end point is the intersection point of the tangential plane and the straight line connecting the end points of fΓI and fΓII. This line segment can be written as a convex combination of fΓI and fΓII, such that the equation for sliding motion is given by
˙
x=fΓ(x) =αfΓI(x) + (1−α)fΓII(x), (0≤α≤1). (5.27) In the following, we assume that g;x(x) 6= 0 in a neighborhood of the switching surface Γ. The parameter α should be selected such that the velocity vector is tangential to the switching surface , i.e.,g;x(x)fΓ(x) = 0 and therefore α is given by
α = [g;x(fΓII −fΓI)]−1g;xfΓII.
The Filippov construction of equivalent dynamics is depicted in Figure 5.9.
∆x
fI Γ fII
fTI fTII
fNI fNII
ε ε
Figure 5.10: Equivalent dynamics via hysteresis effects
In real systems delays, hysteresis and other nonidealities result in real sliding. The sliding equations (5.27) derived by equivalent dynamics on the surface can be considered as the motion of a limiting process. If we consider an infinitesimal hysteresis band of width ε around the switching surface, then the dynamics on the surface are defined as the behavior in the limit as ε → 0. Once the system hits the surface, oscillation in a neighborhood of width 2ε occur. If ε is small, then the velocity vectors fΓII and fΓI in the neighborhood of the discontinuity surface can be represented by their normal components fNII, fNI and tangential components fTII, fTI, i.e., we neglect curvature and the gradient of the surface is assumed to be constant. To determine the direction of the motion along the surface, we calculate the average velocity on the surface. The time to cross theε band is ∆t1 = f2εI
N for fΓI and ∆t2 =−f2εII
N for fΓII, where fNI =g;xfΓI and fNII =g;xfΓII are the normal projections of fΓI and fΓII onto the switching surface, see also Figure 5.10. The time to move back and forth over the band is therefore given by
∆t= ∆t1+ ∆t2
and the tangential distance the system travels over the time interval ∆t is
∆x=fΓI∆t1+fΓII∆t2.
Then the average state velocity of the motion on the surface is given by
˙
xav = ∆x
∆t = ∆t1
∆t fΓI+
1− ∆t1
∆t
fΓII, (5.28)
with
∆t1
∆t = fNII fNII −fNI .
Thus, the average velocity (5.28) equals fΓ in (5.27) and the equivalent dynamics on a sliding surface corresponds to the limiting behavior when switching tends to be infinitely fast.
Γ
Γ−
Γ+ fΓI
fΓII
(t, x) feq
Figure 5.11: Equivalence in control vs. equivalence in dynamics
Another way to construct the set η(t, x) in (5.25) is given by the equivalence in control method [145, 147]. In this case, we consider a system
˙
x=f(x, u(x)), (5.29)
where f : Dx ×Du → Rn is a continuous function and the function u : Dx → R is discontinuous on a smooth switching surface Γ ={x∈Dx|g(x) = 0}. At points belonging to the surface Γ we assume that the equation of sliding motion is given by
˙
x=f(x, ueq(x)), (5.30)
where the equivalent control ueq is defined such that the vector f lies tangentially to the surface Γ and the valueueqis contained in an interval [u−, u+], whereu±are limiting values of u on both sides of the surface Γ. In contrast to the Filippov construction, the endpoint of the vectorf(x, ueq(x)) lies on the intersection of the tangential plane to Γ at the point xwith the arc that is spanned by the endpoint of the vectorf(x, u) whenu varies fromu− tou+. Thus, in this case the setη(t, x) is an arc while in the Filippov construction η(t, x) is the straight line connecting f(x, u+) and f(x, u−), see Figure 5.11. If the function f is linear in u, then near the switching surface Γ the equation (5.29) can be written in the form
˙
x=f0(x) +B(x)u(x). (5.31)
To obtain the motion along the surface Γ the equivalent control ueq must be chosen such that ˙x is tangential to the surface Γ, i.e.,
g;x(x) ˙x=g;x(x)f0(x) +g;x(x)B(x)ueq= 0, and thus
ueq =−[g;x(x)B(x)]−1g;x(x)f0(x), (5.32)
ifg;x(x)B(x) is nonsingular. The regularity ofg;x(x)B(x) is also known as thetransversality conditionand establishes that the control vector fieldB(x) is not tangential to the switching surface Γ at any point x∈Dx. If ueq from (5.32) satisfies
u− ≤ueq ≤u+ or u+ ≤ueq ≤u−,
then by substituting the vector ueq into (5.31), we obtain the velocity vector of sliding motion along Γ as
˙
x=f0(x)−B(x)[g;x(x)B(x)]−1g;x(x)f0(x).
The equivalence in control method is also applicable if discontinuities occur along the intersection of several switching surfaces Γi and the controlu is a vector with components ui that are discontinuous on Γi, see [40, 145]. For systems that are linear with respect to the control the equivalence in control approach coincides with the Filippov construction.
In the case of sliding motion we pursue the solution along the switching manifold Γ. The Filippov construction (5.27) and also the equivalent control method approximate this mo-tion as a momo-tion tangential to the switching surface to construct an ordinary differential equation for sliding motion. From a DAE point of view a better way to define the system behavior during sliding is to append the condition that the solution should stay on the manifold Γ as an algebraic constraint and define thedifferential-algebraic system in sliding motion by
˙
x=αfI(x) + (1−α)fII(x),
0 = g(x), (5.33)
where the algebraic variable α is chosen such that the solution remains in Γ.
Theorem 5.28. Consider an ordinary differential system (5.23)where the right-hand side f(x) is discontinuous on a smooth switching surface Γ = {x ∈ Dx|g(x) = 0} such that (5.23) can be separated into fI(x) and fII(x) as in (5.24). If
g;x(x)(fI(x)−fII(x))
is nonsingular for all x ∈ Dx, then the equivalent dynamics of the system (5.23) during sliding motion are described by the DAE in sliding motion (5.33), and the DAE (5.33) is of strangeness index µ= 1.
Proof. Every solution of (5.23) is also a solution of (5.33) forα= 0 andα= 1, respectively.
Differentiation of the algebraic constraints yields 0 = d
dtg(x) =g;x(x) ˙x
=g;x(x)(αfI(x) + (1−α)fII(x))
=αg;x(x)(fI(x)−fII(x)) +g;x(x)fII(x),
which can be solved for α, if g;x(x)(fI(x) − fII(x)) is nonsingular. Thus, we get a strangeness-free system after one differentiation and therefore µ= 1.