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

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.


Introduction
We study the following scalar equation ξ + 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 ξ of the pendulum.
Equation (1) with the constraint u(ξ) is equivalent to the following controlled linear system ξ = η, η = −ξ + u, y = ξ 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.
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.

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 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 − EJQTDE, 2000 No. 9, p. 2 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 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 : In what follows we assume that the function ϕ takes on values from the interval (−π, π]. , where no change of locations occurs, the solution x(t) satisfies one of the following systems of differential equations: (
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 r(s 1 ) = r(s 2 ).( 7) EJQTDE, 2000 No. 9, p. 3 Put T d = π/4 − δ as the time of stay in the location q d .Let t be the moment of switching to the location q d .Then t + T d is the moment of switching from q d to another location.We define a function so that θ(ϕ) is well-defined for sufficiently small δ, namely for those satisfying We also define From ( 11) and continuity of the function θ, it is easy to derive the existence of 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 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 An example of such a situation is given by q(0) = q − and t 0 = nδ, where n is a nonnegative integer satisfying nδ < 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 ).
These observations imply that where t * = t 0 + π 4 = t 0 + T d + δ.At the same time, from ( 6), ( 14) and the observations above it follows that EJQTDE, 2000 No. 9, p. 4 Thus, it is shown that Since ϕ i is strictly monotone on [t 0 , t * ], there exist functions ρ i (i = 1, 2), defined on the set Transition point for the i-th trajectory: x .
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 By ( 6), ( 7), (17), we have that ( 19) and ( 23) imply the existence of t * * > t * , for which and 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 for sufficiently small ε > 0.
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 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).Transition point for the i-th trajectory: x r(t*) As in the proof of Theorem 1, the relations (28) imply that Moreover, using the second inequality in (5), the mean value theorem and (20), (28) one can easily show that 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 Putting ψ i = ϕ(t i ) and comparing the definition of ω with (9) we, as in Theorem 1, obtain 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 .