Hybrid dynamical systems vs. ordinary differential equations: Examples of ”pathological” behavior

Elena Litsyn ^{∗} Yurii V. Nepomnyashchikh ^{†} Arcady Ponosov ^{‡}

Keywords. Stabilization, hybrid feedback control, functional differential equations.

AMS (MOS) subject classification. 93D15

1 Introduction

We study the following scalar equation

ξ̈+ ξ = u.  (1)

Here, the control u depends on the variable ξ. This is a controlled harmonic oscillator in which the external force u is allowed to depend only on the displacement ξ, but not on the velocity ξ̇ of the pendulum.

Equation (1) with the constraint u(ξ) is equivalent to the following controlled linear system

\begin{matrix}ξ̇ = η, \\ η̇ = −ξ + u, \\ y = ξ \end{matrix}  (2)

with a control u = u(y). Here u again depends only on the output y = ξ.

It can be shown (see e.g. [1]) that there is no output feedback control of the form u = f(ξ) = f(ξ(t)) that makes the system (2) asymptotically stable. Therefore, it was suggested in [1] to use hybrid feedback controls (abbr. HFC), which indeed can stabilize the system (2).

The idea used in [1] can be roughly described as follows. We incorporate a discrete device (an automaton) into the considered system ( a plant). The device is able to switch on and off certain control functions at certain instances. The time interval between two consecutive switchings depends on the last observation of ξ. As it was demonstrated in [1], careful choice of design procedure and switching instances provides asymptotic stability of the system (2). The discrete nature of hybrid outputs makes their practical implementation simpler.

More results on stabilization of linear and nonlinear systems via HFC with a finite number of automata’s locations are available (see e.g. [2], [3], [4], [5], [6], [8], [10]). In [7] it was proved that it is possible to stabilize an arbitrary linear system by using HFC with infinitely many locations.

∗Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, ISRAEL; Partially supported by the Ministry of Science and the Ministry of Absorption, Center for Absorption in Science, Israel. Email:

elenal@wisdom.weizmann.ac.il

†Department of Mechanics & Mathematics, Perm State University, Bukirev str. 15, 614600 Perm, RUSSIA; Supported by the Norwegian Research Council. Email: yuvn@psu.ru

‡Institutt for matematiske fag, NLH, Postboks 5035, N-1432, Ås, NORWAY; Email: matap@imf.nlh.no

In this paper we show that the dynamics of solutions x(t) of the system (2) which is controlled by the hybrid output designed in [1, Example 5.2], is quite erratic (see Figure 1). Trajectories’ behavior indicates that the observed dynamics cannot be described by ”classical” dynamical systems defined by ordinary differential equations. We suspect that this dynamics stems from differential equations with time lags, where the delay functions depend on solutions. We are planning to study this problem in the future.

The whole dynamics of hybrid dynamical systems is given by the triplet (x(t), q(t), τ(t)), where q(t) is the present location of the automaton, and τ(t) is the time remaining untill the next transition instance. We are interested here in dynamic properties of the first, most important, component, x(t), which describes the plant. To be able to ”track down” x(t) we need however to study the dynamics of the whole triplet.

Figure 1

2 Main results

We consider the controlled harmonic oscillator (2) assuming that u is a specific HFC designed in [1, Example 5.2]. This control procedure provides asymptotic stability of the zero solution of the system. For the sake of brevity we, as in [8], denote this HFC by u = A(δ), where δ is to be specified.

The HFC A(δ) is given by the following diagram

Figure 2

The automaton has 3 locations called q_{+}, q_{−} and q_{d}, and the values of T indicate the time of staying in the respective locations.

Remark 1 We have slightly modified the definition of A(δ) suggested in [1], where T (q_{d}) = π/4− 2δ. Our alteration is technical and does not influence the main results.

As was already mentioned the dynamics of the system (1) governed by the HFC, u = A(δ) is a triplet (x(t), q(t), τ(t)). However it is clear that the value τ(t) is uniquely determined by the value q(s), where s ≤ t is the moment of the last observation. In particular, τ(0) is a function of q(0). In what follows we fix an arbitrary initial location q(0) (as we will show, all the results below are independent of the choice of q(0)). Then, given an initial value x(0), the trajectory x(t) is uniquely defined, so that we, at least formally, can set up a single functional-differential equation

ẋ = F x = F (q(0))x  (3)

for all x(t) (see details in [7] and [9]). We are interested in the dynamics of this equation.

We start with some technical remarks. Consider a solution x(t) = (ξ(t), η(t)) of the equation (3), i.e. of the system (2) governed by the HFC u = A(δ) with q(0) fixed. The trajectory x(t) is assumed to start at x(0) ≠ 0.

We will use polar coordinates in the plane, so that any solution x(t) = (ξ(t), η(t)) of (3) is described by the (uniquely defined) pair of functions r : [0, ∞) → [0, ∞), φ : [0, ∞) → R/(2πZ), where ξ(t) = r(t) cos φ(t), η(t) = r(t) sin φ(t).

In what follows we assume that the function φ takes on values from the interval (−π, π].

Within any interval S = (s_{1}, s_{2}) ⊂ [0, ∞), where no change of locations occurs, the solution x(t) satisfies one of the following systems of differential equations:

( q(t) = q_{+}, t ∈ S ) ∨ ( q(t) = q_{−}, t ∈ S ) ⟹ \{ \begin{matrix}\frac{ṙ}{r} = 0 \\ φ̇ = −1, \end{matrix}  (4)

( q(t) = q_{d}, t ∈ S ) ⟹ \{ \begin{matrix}\frac{ṙ}{r} = −\frac{3}{2} sin 2φ \\ φ̇ = −1 − 3 cos^{2} φ. \end{matrix}  (5)

Theorem 1 There exist δ > 0, t^{∗} > 0 and two distinct initial states x_{1}(0), x_{2}(0), for which the corresponding solutions x_{1}(t) and x_{2}(t) to (2) governed by the HFC u = A(δ) coincide for t ≥ t^{∗}, i.e. x_{1}(t) = x_{2}(t), t ≥ t^{∗}. Moreover, in this case the ”true” hybrid trajectories H_{1}(t) = (x_{1}(t), q_{1}(t), τ_{1}(t)) and H_{2}(t) = (x_{2}(t), q_{2}(t), τ_{2}(t)) coincide for t ≥ t^{∗}, too.

Proof. We use equations (4) and (5) to derive estimates for solutions of (3) (i. e. of the system (2) with u = A(δ)). Assume that s_{1}, s_{2} ∈ [0, ∞), s_{1} ≤ s_{2}.

1) If either q(·) ≡ q_{+}, or q(·) ≡ q_{+} on [s_{1}, s_{2}] , then

φ(s_{1}) − φ(s_{2}) = s_{2} − s_{1},  (6)

r(s_{1}) = r(s_{2}).  (7)

2) If q(·) ≡ q_{d} on [s_{1}, s_{2}] and φ([s_{1}, s_{2}]) ⊂ (−π/2, π/2), then

s_{2} − s_{1} = \frac{1}{2} (arctan \frac{tan φ(s_{1})}{2} − arctan \frac{tan φ(s_{2})}{2})  (8)

\frac{r(s_{2})}{r(s_{1})} = \sqrt{\frac{1 + 3 cos^{2} φ(s_{1})}{1 + 3 cos^{2} φ(s_{2})}}.  (9)

Put T_{d} = π/4 − δ as the time of stay in the location q_{d}. Let t̄ be the moment of switching to the location q_{d}. Then t̄+ T_{d} is the moment of switching from q_{d} to another location. We define a function θ : [π/2 − δ, π/2] → R by θ(φ) = φ(t̄+ T_{d}), if φ = φ(t̄). Due to (8),

θ(φ) = − arctan (2 · cot (arctan \frac{tan φ}{2} + 2δ)) , φ ∈ (\frac{π}{2} − δ, \frac{π}{2} ) ,  (10)

so that θ(φ) is well-defined for sufficiently small δ, namely for those satisfying

θ (\frac{π}{2} ) − θ (\frac{π}{2} − δ) > δ.  (11)

We also define β : [π/2 − δ, π/2] → R by

β(φ) = \sqrt{\frac{1 + 3 cos^{2} φ}{1 + 3 cos^{2} θ(φ)}}.  (12)

From (11) and continuity of the function θ, it is easy to derive the existence of ψ_{1}, ψ_{2} ∈ R, for which

\frac{π}{2} − δ < ψ_{1} < ψ_{2} < \frac{π}{2} ,  θ(ψ_{2}) − θ(ψ_{1}) = δ.  (13)

From now on we fix a positive and sufficiently small δ as well as two constants ψ_{i} satisfying (13).

We pick two different trajectories x_{1}(t), x_{2}(t) being the ”shadows” of the ”true” hybrid trajectories

H_{i}(t) = (x_{i}(t), q_{i}(t), τ_{i}(t)), i = 1, 2,

We assume that at t = t_{0} ≥ 0 the automaton either switches from q_{−} to q_{d}, or keeps staying in q_{−}. In polar coordinates one has

φ_{1}(t_{0}) = ψ_{1} + δ, r_{1}(t_{0}) = r_{0} > 0, φ_{2}(t_{0}) = ψ_{2}, r_{2}(t_{0}) = r_{0} \frac{β(ψ_{1})}{β(ψ_{2})}.  (14)

An example of such a situation is given by q(0) = q_{−} and t_{0} = nδ, where n is a nonnegative integer satisfying nδ < \frac{π}{2} − δ.

Clearly, r_{2}(t_{0}) < r_{1}(t_{0}) (the function β is strictly increasing).

The two observations below can easily be derived from (13). See also Figure 3.

1) In the case of the trajectory x_{1}(t), the automaton keeps staying in the location q_{−} near t = t_{0}; i.e., H_{1}(t) = (x_{1}(t), q_{−}, δ) for t_{0} < t < t_{0} + δ. The first transition to the location q_{d} occurs at t = t_{0} + δ; i.e., H_{1}(t_{0} + δ) = (x_{1}(t_{0} + δ), q_{d}, T_{d}).

2) In the case of the trajectory x_{2}(t), the automaton switches from q_{−} to q_{d} at t = t_{0}; i.e., H_{2}(t_{0}) = (x_{2}(t_{0}), q_{d}, T_{d}). At the moment t = t_{0} + T_{d} the automaton switches from q_{d} to q_{+}; i.e., H_{2}(t_{0} + T_{d}) = (x_{1}(t_{0} + T_{d}), q_{+}, δ).

These observations imply that

q_{i}(t^{∗}) = q_{+}, τ_{i}(t^{∗}) = δ, i = 1, 2,  (15)

where t^{∗} = t_{0} + \frac{π}{4} = t_{0} + T_{d} + δ.

At the same time, from (6), (14) and the observations above it follows that

\begin{matrix}φ_{1}(t_{0} + δ) = (ψ_{1} + δ) − δ = ψ_{1}, & φ_{1}(t^{∗}) = θ(ψ_{1}), \\ φ_{2}(t_{0} + T_{d}) = θ(φ_{2}(t_{0})) = θ(ψ_{2}), & φ_{2}(t^{∗}) = θ(ψ_{2}) − δ = θ(ψ_{1}). \end{matrix}  (16)

Thus, it is shown that

φ_{1}(t^{∗}) = φ_{2}(t^{∗}).  (17)

Since φ_{i} is strictly monotone on [t_{0}, t^{∗}], there exist functions ρ_{i} (i = 1, 2), defined on the set D_{i} = φ_{i}^{−1}([t_{0}, t^{∗}]) and satisfying

r_{i}(t) = ρ_{i}(φ_{i}(t)).  (18)

Figure 3

Then (7), (9), (12), (16) imply

r_{1}(t^{∗}) = ρ_{1}(φ_{1}(t^{∗})) = ρ_{1}(θ(ψ_{1})) = β(ψ_{1})ρ_{1}(ψ_{1}) = β(ψ_{1})r_{0},
r_{2}(t^{∗}) = r_{2}(t_{0} + T_{d}) = ρ_{2}(θ(ψ_{2})) = β(ψ_{2})ρ_{2}(ψ_{2}) = β(ψ_{2})r_{2}(t_{0}).

From this and (14) one gets

r_{1}(t^{∗}) = r_{2}(t^{∗}).  (19)

Now, (17), (19) imply x_{1}(t^{∗}) = x_{2}(t^{∗}). Taking this and (15) into account one observes that H_{1}(t) = H_{2}(t) for t ≥ t^{∗} and, in particular, x_{1}(t) = x_{2}(t) for t ≥ t^{∗}.

Theorem 2 There exist positive δ, ε, t^{∗}, t^{∗∗} (t^{∗} < t^{∗∗}) and distinct initial states x_{1}(0), x_{2}(0), for which the corresponding solutions x_{1}(t) and x_{2}(t) to (2) governed by the HFC u = A(δ) satisfy the following properties

1) x_{1}(t) = x_{2}(t) for t^{∗} ≤ t ≤ t^{∗∗},

2) x_{1}(t) ≠ x_{2}(t) for t^{∗} − ε < t < t^{∗} and t^{∗∗} < t < t^{∗∗} + ε, while for the ”true” hybrid trajectories H_{1}(t) = (x_{1}(t), q_{1}(t), τ_{1}(t)) and H_{2}(t) = (x_{2}(t), q_{2}(t), τ_{2}(t)) one has

3) H_{1}(t) ≠ H_{2}(t) for t^{∗} ≤ t ≤ t^{∗∗}.

Proof. Putting again T_{d} = \frac{π}{4} we will use the same functions θ and β as in the course of the proof of Theorem 1. Similar to (13) we fix sufficiently small δ > 0, μ > 0 and two constants ψ_{1} and ψ_{2}, giving an increasing function β and relations

\frac{π}{2} − δ < ψ_{1} < ψ_{2} < \frac{π}{2} ,  θ(ψ_{2}) − θ(ψ_{1}) = δ + μ,  0 < μ < ψ_{1} + δ − \frac{π}{2} .  (20)

Figure 4

Consider two different trajectories x_{1}(t), x_{2}(t), t ≥ t_{0} > 0 of the system (2) governed by the HFC u = A(δ), for which (14) hold true. Assume that

q_{1}(t_{0}) = q_{−},  τ_{1}(t_{0}) = μ,  q_{2}(t_{0}) = q_{d},  τ_{2}(t_{0}) = T_{d}.  (21)

Clearly, (20) and (21) imply

q_{1}(t_{0} + μ + δ) = q_{d},  τ_{1}(t_{0} + μ + δ) = T_{d},  q_{2}(t_{0} + T_{d}) = q_{+},  τ_{2}(t_{0} + T_{d}) = δ, (22)

and hence

q_{1}(t^{∗}) = q_{+},  τ_{1}(t^{∗}) = δ,  q_{2}(t^{∗}) = q_{+},  τ(t^{∗}) = δ − μ,  (23)

where t^{∗} = t_{0} + T_{d} + δ + μ (see Figure 4).

From (8), (14), (20), (23) it follows that

\begin{matrix}φ_{1}(t_{0} + δ + μ) = ψ_{1} + δ − δ = ψ_{1},  φ_{1}(t^{∗}) = θ(ψ_{1}), \\ φ_{2}(t_{0} + T_{d}) = θ(φ_{2}(t_{0})) = θ(ψ_{2}),  φ_{2}(t^{∗}) = θ(ψ_{2}) − δ − μ = θ(ψ_{1}). \end{matrix}  (24)

Thus, the condition (17) is verified.

On the other hand, an argument similar to that used in the proof of Theorem 1 shows that

\begin{matrix}r_{1}(t^{∗}) = β(ψ_{1})r_{1}(t_{0}) \\ r_{2}(t^{∗}) = r_{2}(t_{0} + T_{d}) = β(ψ_{2})r_{2}(t_{0}) \end{matrix}

due to (7), (9), (12), (23), (24).

From this and (14) one easily derives (19).

Due to (22), q_{1}(t^{∗} −ε) = q_{d} and q_{2}(t^{∗} −ε) = q_{+} for sufficiently small ε. This and (7) together with (9) imply

r_{1}(t) ≠ r_{2}(t), t^{∗} − ε < t < t^{∗}.  (25)

By (6), (7), (17), we have that (19) and (23) imply the existence of t^{∗∗} > t^{∗}, for which

x_{1}(t) = x_{2}(t) for t^{∗} ≤ t ≤ t^{∗∗},  (26)

and

\begin{matrix}φ_{1}(t^{∗∗}) ∈ [−\frac{π}{2} − δ, −\frac{π}{2}),  τ_{1}(t) = τ_{2}(t) + μ for t^{∗} ≤ t ≤ t^{∗∗}, \\ q_{1}(t^{∗∗}) = q_{+},  τ_{1}(t^{∗∗}) = μ,  q_{2}(t^{∗∗}) = q_{d},  τ_{2}(t^{∗∗}) = T_{d}. \end{matrix}

The last four equalities say that in the case of the trajectory x_{2}(t) the automaton switches from q_{+} to q_{d} at time t = t^{∗∗}, while in the case of the trajectory x_{1}(t) switching occurs at time t = t^{∗∗} + μ.

From (9) and (26) one obtains

\begin{matrix}H_{1}(t) ≠ H_{2}(t) for t^{∗} ≤ t ≤ t^{∗∗} \\ r_{1}(t) ≠ r_{2}(t) for t^{∗∗} < t < t^{∗∗} + ε, \end{matrix}  (27)

for sufficiently small ε > 0.

The relations (25) - (27) prove the theorem.

Theorem 3 . There exist δ > 0 and two distinct initial states x_{1}(0), x_{2}(0), for which the corresponding solutions to (2) governed by the HFC u = A(δ) meet transversely at some time t^{∗} > 0. In other words, x_{1}(t^{∗}) = x_{2}(t^{∗}), x_{1}(t) ≠ x_{1}(t) for small |t − t^{∗}| ≠ 0, and the vectors ẋ_{1}(t^{∗}), ẋ_{2}(t^{∗}) are linearly independent.

Proof. As in the proof of theorem 2 let us fix sufficiently small δ > 0, μ > 0 and some constants ψ_{1}, ψ_{2}, so that the function β defined by (12) is increasing and (20) holds.

Consider two solutions x_{1}(t), x_{2}(t) of the system (2) governed by the HFC u = A(δ). The solutions are assumed to satisfy

φ_{1}(t_{0}) = ψ_{1} + δ, r_{1}(t_{0}) = r_{01}, φ_{2}(t_{0}) = ψ_{2}, r_{2}(t_{0}) = r_{02} < r_{01}  (28)

at some time t_{0} ≥ 0. We also assume that there occurs switching to a different location at t = t_{0}. According to (6) and (20) switching from q_{−} to q_{d} occurs at t = t_{0} + δ in the case of the trajectory x_{1}(t), and at t = t_{0} in the case of the trajectory x_{2}(t) (see Figure 5).

Figure 5

As in the proof of Theorem 1, the relations (28) imply that

φ_{1}(t_{0} + δ + T_{d}) = θ(ψ_{1}), φ_{2}(t_{0} + δ + T_{d}) = θ(ψ_{2}) − δ = θ(ψ_{1}) + μ.

Moreover, using the second inequality in (5), the mean value theorem and (20), (28) one can easily show that

φ_{1}(t_{0} + T_{d}) > φ_{2}(t_{0} + T_{d}), φ_{1}(t_{0} + T_{d} + δ) < φ_{2}(t_{0} + T_{d} + δ)

for sufficiently small μ > 0.

Due to the continuity of φ_{i}(t) there exists t^{∗} ∈ (t_{0} + T_{d}, t_{0} + T_{d} + δ), for which (17) holds true. We also put φ^{∗} = φ_{i}(t^{∗}).

Let ω : [\frac{π}{2} − δ, \frac{π}{2}] → R be a function defined by

ω(ψ_{1}, ψ_{2}) = \sqrt{\frac{1 + 3 cos^{2} ψ_{1}}{1 + 3 cos^{2} ψ_{2}}} .

Putting ψ_{i} = φ(t_{i}) and comparing the definition of ω with (9) we, as in Theorem 1, obtain

\frac{ρ_{i}(s_{2})}{ρ_{i}(s_{1})} = ω(s_{1}, s_{2}),  s_{1}, s_{2} ∈ φ_{i}(I)  (29)

being valid for any time interval I ⊂ [t_{0}, t_{1}], during which the automaton keeps staying in the location q_{d}. This applies to both of solutions x_{1}(t) and x_{2}(t), so that we may put ρ_{1} = ρ_{2} = ρ.

According to our calculations, neither t^{∗}, nor φ^{∗} depends on r_{0i}. This means that we can always find a pair r_{01}, r_{02}, for which the following additional assumption holds:

ω(ψ_{2}, θ(ψ_{2})) r_{02} = ω(ψ_{1}, φ^{∗}) r_{01}.  (30)

According to (17) and (29),

r_{1}(t^{∗}) = ρ_{1}(φ^{∗}) = ω(ψ_{1}, φ^{∗}) r_{01},
r_{2}(t^{∗}) = r_{2}(t_{0} + T_{d}) = ρ_{2}(θ(ψ_{2})) = ω(ψ_{2}, θ(ψ_{2})) r_{02},

so that (30) implies (19). From (17) and (19) it immediately follows that x_{1}(t^{∗}) = x_{2}(t^{∗}). Since r_{1}(t) is strictly monotone and ρ_{2}(t) is a constant in some neighbourhood O_{t^{∗}} of the point t^{∗}, we see that x_{1}(t) ≠ x_{2}(t) (∀t ∈ O_{t^{∗}} \ {t^{∗}}).

Finally, we observe that

q_{1}(t^{∗}) = q_{d}, q_{2}(t^{∗}) = q_{+}.  (31)

Put (ξ, η)^{T} = x_{1}(t^{∗}) = x_{2}(t^{∗}). Evidently, ξη ≠ 0. Hence,

αẋ_{1} + βẋ_{2} = ( \begin{matrix}0 & α + β \\ −α − 4β & 0 \end{matrix} ) ( \begin{matrix}ξ \\ η \end{matrix} ) ≠ 0

for |α| + |β| ≠ 0. Thus, the velocity vectors are linearly independent, and the theorem is proved.

Acknowledgments. We would like to thank the anonymous referee for many valuable remarks which helped us to improve the paper.

References

[1] Z. Artstein, Example of stabilization with hybrid feedback, in Hybrid Systems III. Verification and Control. Springer-Verlag, Berlin, Lecture Notes in Computer Science., 1066 (1996), pp. 173–185.

[2] C. Benassi, A. Gavioli, Hybrid stabilization of planar linear systems with one-dimensional outputs, preprint, Dipartamenta di Matematica Pura e Applicata, Universita di Modena, Italy, August 1999.

[3] M. S. Branicky, Stability of Switched and Hybrid Systems, in Proc. 33rd IEEE Conf. Decision Control, Lake Buena Vista, FL (1994), pp. 3498-3503.

[4] M. S. Branicky, Studies in hybrid systems: modeling, analysis and control, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, 1995.

[5] D. Liberzon, Stabilizing a linear system with finite-state hybrid output feedback, The 7th IEEE Mediterranean Conference on Control and Automation (1999).

[6] D. Liberzon, A.S.Morse, Benchmark problems in stability and design of switched systems, preprint, March 1999.

[7] E. Litsyn, Yu.V. Nepomnyaschchikh, and A. Ponosov, Stabilization of linear differential systems via hybrid feedback controls, accepted for publication in SIAM J. Control and Optimization (1999).

[8] E. Litsyn, Yu.V. Nepomnyaschchikh, and A. Ponosov, Stabilization of Quasilinear Systems and Hybrid Feedback Controls, The 7-th Mediterranean Conference on Control and Automation, Haifa (1999), pp. 37 - 52.

[9] E. Litsyn, Yu.V. Nepomnyaschchikh, and A. Ponosov, Uniform exponential stability of controlled quasilinear systems and functional differential equations, ”Topics on Functional Differential and Difference Equations” T. Faria and P. Freitas Eds, Fields Institute Communications (submitted for publication in 2000).

[10] A. Nerode and W. Kohn, Models for hybrid systems: automata, topologies, controllability, observability, in Hybrid Systems, R. L. Grossman et al eds., Springer-Verlag, Berlin, Lecture Notes in Computer Scient., 736 (1993), pp. 317–356.