CORNER-SLIDING BIFURCATIONS FOR CONTROL DESIGN

CORNER-SLIDING BIFURCATIONS FOR CONTROL DESIGN Gerard Olivar ∗,1 Fabiola Angulo ∗ Mario di Bernardo ∗∗

∗

Department of Electrical and Electronical Engineering, and Computer Science, Campus La Nubia, Universidad Nacional de Colombia sede Manizales, Manizales, Columbia ∗∗ Dipartimento di Informatica e Sistemistica, Universita degli Studi di Napoli Federico II, Naples, Italy

Abstract: This paper shows a recent bifurcation found in discontinuous dynamical systems. The vector field which defines the system has discontinuous right-hand side, and thus it belongs to the class of Filippov systems. Sometimes, in a smooth dynamical system, such a type of bifurcation can be introduced for control purposes. In this communication this bifurcation is briefly described, analysed and a control application is shown through a representative example. Keywords: Differential equations, Discontinuous control, Dynamics, Nonlinearity, Numerical simulation, Phase plane, Simulation, Sliding, State-space models, Switching, Two-dimensional systems, Variable structure systems.

1. INTRODUCTION In recent years, an explosion of knowledge regarding non-smooth bifurcations in dynamical systems has spread out in almost every scientific and technical area. This has specially happened in DC-DC converters, since this is a source of non-smooth systems due to the switching action of transistors an diodes which are present in the circuits. Non-linear phenomena in power electronics have been shown in (Banerjee and Verghese, 2001), although nonlinear dynamics in a DCDC converters was reported initially in (Deane and Hamill, 1990),(Fossas and Olivar, 1996). From then on several researchers have reported non-smooth bifurcations in applications, and there 1

Some of the authors aknowledge Grants from AECI, and projects CORMORAN and SICONOS

are also quite important results in the theory. Mario di Bernardo and collaborators have contributed very much in the developing theory in the earlier 99’s regarding the sliding bifurcations. In (di Bernardo M. and Vasca, 1999b)(di Bernardo M. and Vasca, 1999a) they studied bifurcations of sliding type in relay feedback systems and in piecewise smooth dynamical systems. Later, grazing and border-collision in piecewise-smooth systems were studied in a unified analytical framework (di Bernardo M. and Champneys, 2001a) through appropiate normal forms making use of the concept of discontinuity maps, which were introduced previously by Nordmark. These bifurcations were specified with more detail through examples in (di Bernardo M. and Champneys, 2001b). Derivation of normal-form mappings for bifurcations of dynamical systems with sliding was reported in (di Bernardo M.

and Nordmark, 2002). In this paper, piecewisesmooth systems and sliding motion is studied and four possible bifurcation scenarios corresponding to sliding trajectories are described. Namely, sliding type I, grazing-sliding, switching-sliding (or sliding type II), and multisliding. More recently, in (Kowalczyk, 2005), chaos and border-collision bifurcations in non-invertible piecewise-linear maps were reported and partially classified through the method developed by Feigin (di Bernardo M. and Homer, 1999). Two-parameter non-smooth grazing bifurcations, including degenerate sliding bifurcations are still being studied by the Applied Nonlinear Mathematics research team in Bristol. Once the theory of nonsmooth bifurcations has been partially developed, one of the most challenging problems is how to use this knowledge for control purposes. Thus smooth systems displaying for example a non-desired limit cycle are controlled through non-smooth bifurcations in order to change the stability or even the trajectories. Once a discontinuous control is introduced into the system, it belongs to a class known as Filippov system (Filippov, 1984). Several papers and communications have been produced with this aim in mind, putting together results from control theory and from nonsmooth bifurcation theory (Angulo F. and Olivar, 2004b), (Angulo F. and Olivar, 2004a), (Angulo F. and Olivar, 2005), (di Bernardo M. and Angulo, 2005), (Angulo F. and Olivar, 2006). The features of a convenient Poincar´e map can be controlled through nonsmooth bifurcations and even the resulting dynamics can be predicted and classified through Feigin’s method (di Bernardo M. and Homer, 1999). The structure of the remaining of this paper is the following: in Section 2 we briefly describe the corner-collision bifurcation. Then in Section 3 a Poincar´e map for a planar version of this bifurcation is derived, Finally, the theory is applied to a representative example in Section 4, and some conclusions are stated.

2. CORNER-SLIDING BIFURCATION Let us consider a general system of the form: x˙ = F(x) + u

(1)

where F := (F1 , F2 , . . . , Fn ) : Rn 7→ Rn is a sufficiently smooth and differentiable vector field over the region of interest, say D ⊆ Rn . We suppose that, at some parameter value, the system exhibits a stable limit cycle of period T , i.e. x(t) = x(t + T ) ∀t ≥ 0.

In many control systems and electronic switching devices, switching conditions may be governed by several overlapping inequalities. A generic feature of such examples is that the discontinuity boundary has a corner-type singularity formed by the intersection between smooth codimension one surfaces Σ1 := {x ∈ Rn : H1 (x) = 0} and Σ2 := {x ∈ Rn : H2 (x) = 0} at a non-zero angle. The passage (or collision) of a trajectory through a point in a certain subset Γ ⊂ Σ1 ∪ Σ2 is a non-smooth bifurcation event because, in a neighborhood of the collision, there are distinct trajectories that do not behave similarly with respect to regions S1 and S2 on either side of Σ0 (the border of regions S1 and S2 ), which is a subset of Σ1 ∪ Σ2 . In nonsmooth Filippov systems sliding motions are possible due to vector fields on adjacent regions of the state space can be oriented towards the switching surface, one opposite to the other. Thus limit cycles can have part of the orbit in the sliding region of the switching surface. As some parameter is varied, a corner-collision can occur where a limit cycle hits the tip of the corner region (see Fig.1). Further parameter variations can lead to several different scenarios. To classify the possible scenarios following a corner collision the key issue is to be able to construct the Poincar´e normal form map of the cycle undergoing the bifurcation. Recently it was shown that a local map describing the dynamics of the system close to a corner-collision point can be derived by using the concept of discontinuity map (di Bernardo M. and Champneys, 2001a). If, after the corner collision, the orbit slides on the border of the corner region, we will refer to that as a corner-sliding bifurcation. Thus, this is a special case of a corner-collision bifurcation. This bifurcation arises when one consider twoparameter nonsmooth bifurcations of limit cycles. It is a type of degenerate grazing point, that is, there is a degeneracy of one of the analytical conditions determining the properties of the vector fields local to the grazing event. And this is likely to influence the leading order term of the normal form map derived via the discontinuity mapping (Kowalczyk P. and Piiroinen, 2005). The degenerate corner-collision bifurcation occurs when there is a tangency between the vector field inside the corner region S1 and the border Σ1 . One of the unfoldings of this scenario leads to the corner-sliding bifurcation (named corner-collision with a sliding motion, in (Kowalczyk P. and Piiroinen, 2005)). As written in (Kowalczyk P.

Σ2

S2

S

S1 Σ1

Σ2

2

S 1

Σ

Σ

1

(a) Before the bifurcation

Σ2

S2

S1 Σ1

(x 10,0) 1 (x 13,0) (x 1,0)

(xcenter,0)

Fig. 2. A Poincar´e map is computed. (b) Corner-Sliding bifurcation

S2

x 1 = b− m Σ 2

S 1 Σ

1

(c) After the bifurcation Fig. 1. A scheme of a corner-sliding bifurcation in a planar system. and Piiroinen, 2005), this scenario has not been analysed as yet in the literature, and thus it represents a novelty in the nonsmooth bifurcation theory. The authors plan to develope this theory in n-dimensional systems in the next future. In the planar case, the Poincar´e map is described in the next section.

2.1 Control through corner-sliding bifurcation We want to design a feedback controller to suppress such periodic oscillations or, alternatively, to select its characteristics (periodicity, amplitude etc.). Note that while the control action will be applied on the continuous-time system, the aim is to change the properties of its Poincar´e map. For this purpose our aim is to synthesise a controller based on the theory of nonsmooth bifurcations. Namely, we will select a switching feedback controller u in order to vary the main features of the local Poincar´e map associated to the limit cycle of interest. This in turn will allow the variation of the properties of such local map and hence the local control of the cycle. As it will be seen, locally to a corner-sliding bifurcation point the Poincar´e map can be estimated analytically as a non-invertible piecewise-linear

one. Thus, the main idea is for the controller to put the system close to a corner-sliding bifurcation event of the cycle of interest with an appropriately defined switching strategy in state space. In so doing, the controller will switch from one configuration to the other whenever the system trajectories cross the boundaries defining a cornerlike switching manifold in phase space. By varying the functional form of the control signal, we will change the properties of the local map and hence the main features of the fixed point corresponding to the cycle of interest.

´ MAP FOR THE PLANAR 3. A POINCARE STATE-SPACE According to the theory of corner-sliding, the Poincar´e map of a limit cycle undergoing such a bifurcation is non-invertible and piecewise-linear. The corner is also supposed such that sliding on one of its boundaries is forced to occur. Without loss of generality, we select u as the switching controller defined by: ( 0 if x ∈ S2 u= (2) φ(x, t) if x ∈ S1 with S1 ⊆ R2 being the region (corner) limited by the manifolds defined as Σ1 := {x ∈ Rn : H1 (x) = 0} and Σ2 := {x ∈ Rn : H2 (x) = 0} as depicted in Fig. 1. With this choice of u the controlled system becomes: ( F(x) if x ∈ S2 x˙ = (3) F(x) + φ(x, t) := G(x) otherwise In order for the control to be effective we need to select the boundaries of regions S1 and S2 , i.e. define the corner in phase space. According to the theory of corner-sliding, the corner must be such that

• sliding or Filippov solutions are forced to occur on one of its boundaries; • it penetrates the cycle to be controlled as one of its defining parameters is changed so that at some critical parameter value the target limit cycle undergoes a corner-collision bifurcation. In order to force sliding mode we choose H1 (x) and H2 (x) so that h∇H1 (x), F(x)i < 0, h∇H1 (x), G(x)i > 0. The point at which corner collision occurs will be identified as x0 and corresponds to the intersection of Σ1 and Σ2 . For simplicity, we suppose a counterclockwise direction of the vector field in a neighborhood of the corner collision point. We select H1 (x) := −x2 , and H2 (x) := x2 − mx1 + b, with m and b being real constants. As before we rescale the system coordinates so that a corner-collision occurs when b = 0 at the point x0 = (0, 0). Therefore varying b we can move the tip of the corner and hence yield a corner-collision bifurcation. We then have that the local interaction of the cycle with the corner can be described by using the local mapping Π defined in a convenient Poincar´e section Σ (for x1 in the neighborhood b ) (see Fig.2): of m x ( = (x1 , x2 ) 7→ Π(x) := (P x1 + xc (1 − P ), 0) Π(x) := (x31 , 0)

if x1 ∈ (x11 , xc ) if x1 < x11 (4) where P corresponds to the attraction of the original stable limit cycle and can be approximately computed, as ximage − xc = P (xinicial − xc ) and xc stands for xcenter . Note that we only need to know c1 ≥ 0 and c2 ≥ 0 such that |Fi (x)| ≤ ci , i = 1, 2 for x ∈ [xmin , xmax ], being x ∈ [xmin , xmax ] a neighborhood of the intersection of the cycle with the corner. Then we choose φ(x, t) = (−c1 , −c2 ) in the corner region to make sure that we have sliding, and that the sliding dynamics points towards the center of the cycle. Even, one can obtain an estimate of the amplitude of the reduced cycle to be δ =P ·(

b − xcenter ) m

b Thus one can choose xcorner := m in order to modify this amplitude δ arbitrarily. Some simulations in a representative example will be shown later.

4. A REPRESENTATIVE EXAMPLE Next, we show a simple example to illustrate the control design presented above. We choose the planar normal form of a Hopf bifurcation, described by:   p x˙1 = ε(x1 + 1) a − (x1 + 1)2 + x22 − x2 := F1   p x˙2 = εx2 a − (x1 + 1)2 + x22 + x1 + 1 := F2 (5) This system exhibits a limit cycle, which is a perfect circle of radius a centered in (x∗1 , x∗2 ) = (−1, 0). The system flow moves in anti-clockwise direction as time increases and hence crosses the line {x2 = 0} upwards. We choose the region S1 as the phase space set (corner) bounded by H1 (x) := −x2 = 0 and H2 (x) := x2 − mx1 = 0. We fix a = 1 and  = 0.1 in the example. We also fix m = 0.001. Note that when b = −0.9 a corner-sliding bifurcation occurs, as the limit cycle of radius 1 hits the tip of the corner defined above at the point (0, 0). Varying the control parameter b will cause the corner to penetrate the limit cycle and hence change its properties. By varying m and b, we expect the properties of the Poincar´e map to change and hence those of the limit cycle. For example, Fig.3 shows how the amplitude of the original limit cycle is reduced. Figure 4 shows the control signal, which is only non-vanishing for small intervals. Figure 5 shows the waveform of the first variable. The ripple is considerably reduced, as desired. Fig. 6 shows the piecewise-linear map for b = −0.9, m = 0.001, both the approximate computed P , and the numerically computed. As it can be seen, the PWL map is non-invertible on one part of it, as it is predicted by the theory. The noninvertible part agrees completely both with the approximated and numerically computed maps. Note that m can be reduced and thus the corner region can be designed as thin as desired without interfering with the target dynamics. Here we can observe that the controller is indeed effective in varying the map and therefore reduce the amplitude of the limit cycle or even make it disappear.

5. CONCLUSIONS In this paper we have shown that it is possible to synthesize a switching control law to suppress or change the main features of a target limit cycle in planar smooth dynamical system. In so doing, the theory of non-smooth bifurcations was explicitly used in the design process. Namely, by

−1.04

2

−1.06

without control

approximated Poincaré map

−1.08

1.5

−1.1 x1(k+1)

1

−1.12

x2

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numerical Poincaré map

−1.18

−0.5

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−1

−1.22

−1.5 −2 −3

controlled −2.5

−2

−1.5

−1 x1 −0.5

0

0.5

−1.25

1

Fig. 3. The original limit cycle is plotted together with the actual limit cycle of the controlled system. The amplitude is reduced.

2.5

2

signal control

1.5

1

0.5

0 0

20

40

60

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x (k)=x (k+1) 1

1

−1.24

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time

Fig. 4. Control signal.

2

−1.2

−1.15

x (k) 1

−1.1

−1.05

Fig. 6. Piecewise-linear non-invertible Poincar´e map. appropriately selecting the control constants and the switching manifolds, it is possible to change the properties of the Poincar´e map associated to the cycle of interest. The resulting control action is acting on the system in a relatively small neighborhood of the nonsmooth bifurcation point and hence guarantees the achievement of the control goal with a minimal control expenditure. We wish to emphasize that rather than being a technique for the control of bifurcations in nonlinear systems, the strategy presented here aims at exploiting the theory of non-smooth bifurcations for control system design. Ongoing research is aimed at further exploring the ideas presented in this paper and establish formal links between the controller gains and the properties of Ω-limit set of the closed-loop system. Also, the extra degrees of freedom corresponding to the control law parameters c1 and c2 can be further exploited to have solutions satisfying certain performance criteria. Future work will investigate this further and will also be concerned with the experimental validation of this control strategy.

1.5 1

REFERENCES

0.5 0 x1

−0.5 −1 −1.5 −2 −2.5 −3 0

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Fig. 5. Dynamics of x1 .

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