Csernák, Gábor

Stépán, Gábor

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ENOC 2014, July 6-11, 2014, Vienna, AustriaLocal and global bifurcations in a simple model of digitally controlled systemsGábor Csernák∗, and Gábor Stépán∗∗∗MTA-BME Research Group on Dynamics of Machines and Vehicles∗∗Budapest University of Technology and Econoomics, Budapest, HungarySummary. Chaotic vibrations that are originated from the digital implementation of control are known for more than 20 years [1].The consideration of the effects of sampling and processing delay is a routine procedure among control engineers, but the quantizationin the analogue-digital converters is usually neglected. However, this latter effect frequently leads to deterministic but small scalechaotic behaviour – so-called micro-chaos [2, 3] – which is often considered simply as stochastic noise in the practice. In the presentcontribution a simple but quite general PD-controlled system is considered. It is shown that micro-chaotic oscillations may have aconsiderable influence on the accuracy and settling time of control systems. We focus on the numerical and analytical examination ofglobal bifurcations since these may lead to the abrupt change of the aforementioned properties of the control system.IntroductionConsider a linear oscillator with negative stiffnesswhose equation of motion is linearized about the upper equilibrium:Mx¨(t) + cx˙(t)− kx(t) = 0, (1)whereM denotes (generalized) mass, c is the damping coefficient, while k > 0 is the (generalized) stiffness. PD controlis applied on the system in order to stabilize the position [x, x˙] = [0, 0], according to the output of a digital control system.The control force is assumed to be kept constant between the jth and (j + 1)st samping instants. The samping time isdenoted by τ and we assume that the control force is sent out without delay. Due to the sampling of the state and thequantisation of the control force with resolution h, the equation of motion of the controlled system assumes the formx¨(t) +cMx˙(t)−kMx(t) = −hMInt(P˜ xj + D˜x˙jh), t ∈ [jτ, (j + 1)τ), (2)where function Int() rounds toward zero. Introducing dimensionless variables one arrives at the equationy′′(T ) + 2βy′(T )− α2y(T ) = −Int(Pyj +Dy′j), (3)where T ∈ [j, (j + 1)) and y′ ≡ dy/dT . Equation (3) can be solved between the successive sampling instants. Conse-quently, the solutions can be given by the following mapping – the micro-chaos map:yj+1 = Uyj + b Int(Pyj +Dy′j)≡ Uyj + bm, (4)wherem = Int(Pyj +Dy′j)and yj ≡ col[yj , y′j]. By introducing the notations s ≡ sinh(γ), c ≡ cosh(γ), e ≡ exp(β),and γ2 ≡ α2 + β2, one obtainsU =1eγ[γc+ βs sα2s γc− βs], b =[e−ceα2− βseγα2− seγ]. (5)Phase-space structureAccording to the value of the control force m, the phase-space can be divided into parallel bands that are separated byswitching lines:y′j =m− PyjD.The initial points of possible periodic orbits of length p – that visit the bandsm1,m2. . .mp – can be found by the followingformula:y1 = (I−Up)−1(m1Up−1 +m2Up−2 + · · ·+mpU0)b.Note, that yi must be in bandmi. For example, the fixed point rm = col[m/α2, 0]must be in bandm, thus|m|P≤|m|α2<(|m|+ 1)P.If the fixed point crosses the switching line, it becomes virtual via a border collision bifurcation. Since all the fixed pointsare saddles, strange attractors or repellers can be formed between neighbouring fixed points (and between a fixed pointand a virtual fixed point). These primary structures can merge with each other, forming new, secondary strange attractors.The micro-chaos map is similar to a series of Baker maps in a large portion of the domain of stability. In this caseENOC 2014, July 6-11, 2014, Vienna, Austria• The attractor farthest from the origin is close to the mmaxth switching line, at position ySWmax, where mmax is theindex of the virtual fixed point that is closest to the origin:mmax = Int(PP − α2), ySWmax =mmaxP.• If the size of the primary attractors is small (i.e., the trajectory cannot jump over a whole band and cannot escapefrom the structure at the virtual fixed point towards larger displacements), ySWmax provides an estimation for themaximal control error.Another estimation method is based on the fact that the micro-chaos map (4) can be rewritten to the following form:yj+1 = Sjy0 −j−1∑k=0Skcχk, χk ∈ (−1, 1), (6)where S would be the coeffcient matrix of the continuous, controlled system and χk corresponds to the fractional part ofthe control force. Because limj→∞ Sjy0 = 0, the maximal possible control error can be estimated by the proper choiceof the numbers χk. If the eigenvalues of S are positive real numbers, χk = 1, ∀k provides the maximal error. In this case,−∞∑k=0Skc = −T[ c11−µ1c21−µ2]=[1P−α20].µi, i = 1, 2 denote the eigenvalues of S, matrixT transforms S to diagonal form and ci are the elements of vector c from(6), transformed with T. Thus, the maximal distance from the origin is y∞ = 1/(P − α2).As Figure 1 shows, the two methods provide quite similar results. Certainly, the control error will be less than theseestimates, especially if the farthest Baker map-like structure turns to a repeller. This favourable situation can occur quitefrequently in the practice, as it is also illustrated in the figure. On the other hand, the choice χk = 1, ∀k does not providethe maximal control error if the eigenvalues of S are complex or negative real numbers. Our goal is to give a more generalestimate for the control error and to describe the bifurcations that lead to the merging of primary attractors.5 10x 10−5104105106AttractorpositionPy∞ySWmaxD5e−5 10e−5021PFigure 1: The estimated control error at realistic parameters α = 6.85e−3, β = 0 andD = 2.566e−3 (left panel) and the parameterswhere the farthest attractor disappears or merges with other attractors (red on the right panel).ConclusionsWe showed that several disconnected attractors may coexist in a simple model of digitally controlled systems. The positionof the attractor, farthest from the origin, was estimated by two different methods that led to similar results. These methodscan be used for the estimation of the largest possible position error of the control system in a broad parameter domain.AcknowledgmentsThis research was supported by the Hungarian National Science Foundation under grant no. OTKA K 83890.References[1] Delchamps F.D. (1990) Stabilizing a linear system with quantized state feedback, IEEE Trans. Autom. Contr., 35:916-924.[2] Haller G., Stépán G. (1996) Micro-Chaos in Digital Control, J. Nonlinear Sci., 6:415-448.[3] Csernák G., Stépán G. (2010) Digital Control as Source of Chaotic Behavior, International Journal of Bifurcation and Chaos, 20(5):1365-1378.

Local and global bifurcations in a simple model of digitally controlled systems

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