# Multistability and Its Control by the PWM Carrier Signal Modulation in a DC Electric Drive System

Authors: Oleksiy Kuznyetsov
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

Multistability is a universal nonlinear feature for many areas of physics and technology; for most applications, it is an undesired feature leading to uncertainty of the system’s functioning, and therefore, it should be controlled. We deal with the model of a multistable chopper-fed DC electric drive and study the control of multistability by the periodic modulation of a PWM (pulse-width modulation) carrier signal. The effect of utilizing sinusoidal modulation has been studied employing brute-force bifurcation diagrams as well as those obtained by numerical continuation, symbolic analysis of periodic modes, and normal form approximation of border-collision bifurcation. The result is formulated in terms of the acceptable amplitude of the modulation while the effect of phase shift is taken into account. As an alternative, a modified modulation in the form of the absolute value of the sine wave has been studied; the latter is characterized by a simpler bifurcation structure. Additionally, we propose an implementation of the adaptive control with the objective to drive the system away from a “nonnominal” mode. It has been demonstrated that such a transition should use smooth attenuation of the modulation signal to ensure the desired “nominal” operation of the system.


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## Body

## 1. Introduction

A power electronic converter is an inalienable part of a modern electric drive [1] impacting the behavior of the whole system. Nonlinear phenomena related to the presence of switching devices were known since the 1960s, but an extensive study of the topic has started since the last quarter of the XX century. As expected, nonlinearities regarding the switching operation (an inherent feature of a power converter) are hence derived by an electric drive. Other sources of nonlinearity in an electric drive are caused by slotting and commutation [2], the mutual effect of a field and armature winding if they are connected in series [3], mechanical subsystem [4], and the combination of the factors [5, 6].The first attempt to summarize the theory, numerical simulations, and experimental research of nonlinear phenomena in switching power converters [7] along with several others that appeared in the early 2000s [8, 9] has analyzed mainly basic converter topologies. From the practitioner’s point of view, such nonlinear behavior is an undesired feature; therefore, the mechanism of their onset has to be clear. Recent studies are also devoted to the analysis of more complex power converter systems, such as multicell flying capacitor converters [10], LLC resonant converters [11], cascaded DC/DC converters [12], just to name a few, and also those in connection with the nonlinear source or load, such as photovoltaic module [13]. Onset of instability in such complex systems is also caused by the reasons, which differ from those that were depicted in the classic literature, like an interaction between the load and source converter in a cascaded system [14].The switching operation of a power converter implies that mathematically, it is represented as a piecewise-smooth dynamical system. In particular, the discussed class of dynamical systems is prone to the special type of bifurcations, the border-collision bifurcations (BCB) that can cause a variety of possible phenomena, which include, but are not limited to chaos [15], quasiperiodicity [16], coexistence of multiple attractors [17], up to an infinite number of attractors [18], etc.In the current work, we study multistability–the coexistence of several stable modes for the same set of parameters and inputs. The phenomenon is observed experimentally and explained analytically in laser and semiconductor physics, ecology, chemistry, climate dynamics, neuroscience, etc. [19, 20].Particular cases of multistability are extreme multistability [21, 22] characterized by an infinite number of coexisting attractors and megastability [23, 24] producing an infinite number of nested attractors in a layered structure.Power electronic converters are also prone to multistable behavior [25]. From the practitioner’s point of view, multistability is an undesired behavior for most applications, in particular, for power electronic systems with the purpose of energy conversion aiming to provide parameters such as current and voltage ripples, total harmonic distortion below the predefined level. Moreover, multistability frequently causes a harmful effect on technical systems [26, 27]. Therefore, it is a natural objective to develop the control strategies to force the system’s operation in the desired mode among the coexisting [28].For the sake of completeness, we should note that despite the harmful unpredictability, multistability is also a ground for the development of new theories, materials, and technology. Thus, it has numerous practical applications [20]; these include signal processing [29], cybersecurity [30], microelectromechanical actuators [31], energy harvesting [32], composite structures, and metamaterials [33].The DC drive with a semiconductor chopper itself has been studied in terms of nonlinear phenomena since [34]. The “classical” example in [34] is thoroughly studied by means of analysis and control of chaotic behavior [35], and also, for multistability [36, 37]. Like many other applications, any mode other than a T-periodic mode (a limit cycle with the periodicity equal to clock period T) is characterized by the current and speed ripples increase, in turn, causing an increase in vibrations [38].In a generalized power electronic system, switching between two or more topologies occurs both periodically and between periodic instances. When the switching condition is altered, it is expected that the stability of the modes of operation is also affected. It can be useful from a practical point of view; for instance, the method of resonant parametric perturbation was implemented to control the chaos by injecting a sinusoidal perturbation into a DC/DC converter [39], studying the effect of amplitude and phase shift of such control signal. Later, the effect of different types of injected periodic signals that lead to various nonlinear phenomena was studied [40].The similar technique was implemented to extend the stability margins of the “nominal” (T-periodic) mode of operation of a switching power converter [41] with the analytic explanation based on monodromy and saltation matrices. Furthermore, the authors proposed a universal on-the-chip implementation of the developed controller [42].It is our idea that similarly to theT-periodic one, the coexisting modes of operation are also affected by such perturbation, and therefore, it can be employed to control multistability. To the best of our knowledge, this technique has never been analyzed by its feasibility to control multistable behavior. Previously, we provided the sketches of the technique in [43, 44]. The current study is an attempt to generalize and further analyze the proposed idea.Proceeding with the problem of controlling multistability, the diversity of techniques is classified into two main categories, depending on whether the control (i) drives the system to the desired mode by an external perturbation [45–47] or (ii) forces the collapse of the undesired modes [48–50]. The one studied in the work can belong to both categories within the above classification (depending on the particular implementation). As compared to the other mentioned techniques, the proposed one is a nonfeedback one, it requires no extra complicated computing, and thus, its implementation provides no overcomplication both to the schematic and control system. On the other hand, the technique under study requires an appropriate tuning; the effective suppression of the coexisting modes is possible with an understanding of the impact of the tuning parameter(s) on the multistable behavior.Therefore, the aim of this work is to analyze the feasibility of the multistability control by periodic perturbation in a PWM-driven system on the example of a chopper-fed DC drive. Toward this purpose, the remainder of this article is organized as follows. In Section2, a mathematical model of the system is presented. Section 3 gives an overview of methods that were used for the analysis: brute-force bifurcation diagrams, symbolic analysis of the periodic orbits, continuation-based bifurcation diagrams that were obtained using monodromy and saltation matrices, and conditions for BCB in the system. An organization of periodic orbits within bifurcation diagrams and basins of attraction in Section 4 provides the understanding of multistability in the considered system. The following sections propose a control driving the multistable system to monostability. Thus, Section 5 is devoted to the analysis of the system’s behavior under sinusoidal modulation of a carrier signal and the effect of the adjustment coefficients on its effectiveness, while Section 6 proposes an alternative approach–nonsinusoidal modulation of a carrier signal that can simplify the bifurcation structure, thus simplifying the adjustment of the controller. In Section 7, the possible practical implementation of the control is discussed. And finally, in Section 8, concluding remarks are given.


## 2. Mathematical Model of the System

Consider a permanent magnet DC drive fed by a chopper (Figure1) and governed by speed feedback. PWM (pulse-width modulation) signals determine the state of the switch VS. Taking into account the drive’s operation in continuous conduction mode, we distinguish two possible modes depending on the state of the switch VS. In turn, that depends on the relationship between the control voltage vcon and the carrier (ramp) signal vramp [34]. Thus, if vcon<vramp, the switch is on, and the model is(1a)ddtωtit=−BJKTJ−KEL−RLωtit+−TLJVinL,while if vcon≥vramp, the switch is off and(1b)ddtωtit=−BJKTJ−KEL−RLωtit+−TLJ0,where the state variables are the rotational speed ω and the armature current i; the mechanical variables are as follows: B is the viscous damping, J is the load inertia, and TL is the load torque; the electrical variables are as follows: R is the armature resistance, and L is the armature inductance, and the DC machine parameters are as follows: KT is the torque constant, and KE is the back-EMF constant. The values of the quantities used in the work are provided in Appendix A.Figure 1
Circuit diagram of a chopper-fed DC drive [34].Equations (1a) and (1b) are represented with the state-space notation as follows:(2)x.=Ax+Ek,k=on,off,where A=−B/JKT/J−KE/L−R/L, x=ωtit, Eoff=−TL/J0, Eon=−TL/JVin/L.The change in the switch state is related to the change in the relationship between controlvcon and carrier vramp signals (as captured by the comparator A2, Figure 1) after resetting the ramp signal, or between periodic instances, if the condition vcon=vramp is satisfied.The carrier signal (ramp) voltage in the clock periodT is expressed as follows:(3)vrampt=vlo+vup−vlotTmod1,where vlo is the lower level and vup is the upper level of the signal (Figure 1).For the proportional speed control law implemented on the comparator A1 (Figure1) with the gain g,(4)vcont=gωt−ωref,where ωt is the instantaneous speed, and ωref is the reference speed.To study multistability, the model is considered in discrete time in the form of a stroboscopic map (Poincaré map synchronized with the clock periodT). As a result, the “nominal” T-periodic mode corresponds to a fixed point of the map (period-1 orbit), while the nT-periodic mode corresponds to a period-n orbit.


## 3. Methodology for the Analysis of the System

### 3.1. Brute-Force Bifurcation Diagrams

With increasing the gaing, we observe a well-known (see [34, 36]) period-doubling route to chaos after losing the stability of the “nominal” operating mode of the drive (a stable period-1 orbit) with the further sudden expansion of a chaotic attractor via the interior crisis (the expansion bifurcation) and with periodicity windows inside chaotic area (Figure 2).Figure 2
Several coexisting stable modes at the brute-force bifurcation diagrams with the feedback gaing as a parameter.In addition, besides the main period-1 orbit undergoing a period-doubling cascade, there exist period-3, -4, and four-band chaotic orbits, coexisting with the stable period-1 one, period-7 mode coexisting with a period-2 one (Figure2), as well as the others after the bifurcations of the mentioned above. The search for coexisting orbits can be performed employing brute-force bifurcation diagrams combined with the scan of the state space at several values of the bifurcation parameters with further brute-force continuation of the orbits found, as implemented in the software AnT 4.662 [51]. The procedure can be accompanied by a numerical continuation, allowing for a detection of unstable periodic orbits.Each of the coexisting period-n orbits undergoes the period-doubling cascade, causing their transformation into period-2n, -4n, …, modes, followed by chaotic modes collapsing via the boundary crises (the final bifurcations) as shown in Figure 2; the mechanisms lying behind are explained in detail in [7].


### 3.2. Symbolic Analysis of Periodic Orbits

A periodic orbit is depicted by its symbolic representation introducing the following notation:1 denotes “off” and 2—“on” switch’s state. Thus, a “nominal” T-periodic mode (a period-1 orbit of the stroboscopic map) is represented by the sequence 12, i.e., in the clock cycle: first, the switch operates in the off state and then in the on state. Likewise, the sequence 121−1−1−12 denotes the 4T-periodic mode (a period-4 orbit) with two switching instances per first clock cycle, no–per second and third, and one–per the fourth cycle.


### 3.3. Stability Analysis of the Periodic Orbits in a DC Drive

The studied system is governed by theT-periodic carrier signal vramp; therefore, the nT-periodic modes are analyzed on the basis of Floquet theory. Let the system for each set of the state of the power switches be a linear time-invariant (LTI) system governed by an equation x.=Ax+Bu. Therefore, the transition from xt0 to xt is governed by the state transition matrix Φt,t0=eAt−t0, where(5)xt=Φt,t0xt0.The same matrix is also used to depict the evolution of the perturbed initial statext0+∆xt0, as well as for the evolution of the perturbation itself(6)∆xt=Φt,t0∆xt0.Eigenvalues of the state transition matrixλii=1,…,N, where N is the dimensionality of the state space, are utilized to analyze the stability of the periodic mode, so that all of them are situated inside a unit circle λi<1 for a stable periodic mode.In a generalized power converter, the state space is divided into at least two partitions, and a trajectory during a periodic mode of operation crosses the border at least twice. A generalizednT-periodic mode (n-periodic orbit of the corresponding stroboscopic map) corresponds to a transition from an initial state x0=x0 to the same state after n clock periods x0=xnT. Similarly to the classical Floquet theory, the stability of the periodic mode is analyzed by the evolution of the perturbation; however, here, we have to account the evolution of the perturbation crossing the border (see [52] for details).Thus, a relation of the perturbation just after the event∆xt+ to that just before ∆xt− is described by the following state transition matrix (in this case, it is called the saltation matrix):(7)∆xt+=S∆xt−.The latter is depicted by the equation [52]:(8)S=IN+f+−f−n⊤n⊤f−+∂h/∂tt=tsw,where we assume that the trajectory crosses the border between the two subsystems represented by the vector fields f− and f+ at t=tsw, IN is an identity matrix of order N, and the switching hypersurface in the state space Σ is represented by the switching condition hx,t=0, n is the vector normal to Σ, and ⋅⊤ is the transpose operator.Let the dynamical system moving around the periodic orbitm times cross the switching hypersurfaces Σii=1,…,m, and let the transitions between the subsystems be governed by the saltation matrices Si. The transition over a cycle nT is governed by the monodromy matrix:(9)M=Sm×Φm×Sm−1×Φm−1×⋯×S2×Φ2×S1×Φ1,so that the evolution of the state is(10)xt0+nT=Mxt0,and the evolution of the perturbation is as follows:(11)∆xt0+nT=M∆xt0.Therefore, its eigenvalues are used to study the stability of a generalizednT-periodic orbit in a switched system.


### 3.4. Saltation Matrices for the DC Drive System

The expressions for the switching hypersurfacesΣi and the normals to them ni are adapted from [36], where the DC drive system is examined.The first case of switching corresponds to the equality of (3) and (4); the switching hypersurface is defined by the following condition:(12)Σ1=x,th1x,t=0,where(13)h1x,t=vcont−vrampt=gx1−ωref−vlo+vup−vlotTmod1.Therefore, the partial derivative is(14)∂h1∂t=−vup−vloT,and the normal vector is(15)n=∂h1∂x1∂h1∂x2=g0.Hence, the expression ofS1 related to the crossing of switching hypersurface Σ1 is obtained by substituting (13) and (14) into (8).In the second case, when the switching is associated with the end of the clock periodt=nT, discontinuity of vrampt defines that ∂h2/∂t=∞. As a result, when infinity is substituted in the denominator of (8), we obtain the saltation matrix S2=IN.


### 3.5. Numerical Continuation Utilizing Monodromy Matrix Approach

Stable and unstable periodic orbits were obtained semianalytically based on the algorithms proposed in [53, 54]. In our study, we developed our own software with similar purposes, implementing the same algorithm, as explained in the following.The starting point for a numerical continuation is an initial guessx0, which can be obtained as a result of a straightforward integration of the ODE system. The next step x0′ is obtained as a result of a Newton–Raphson search routine:(16)x0′=x0−∂xm∂x0−IN−1xm−x0.Since(17)∂xm∂x0=∂xm∂xm−1×Sm−1×∂xm−1∂xm−2×Sm−2×⋯×∂x2∂x1×S1×∂x1∂x0=M,then (16) is rewritten as(18)x0′=x0−M−IN−1xm−x0.Afterwards, the obtained value is taken as an initial guess with the parameter slightly varied, and utilizing the search routine (18), a new periodic orbit is obtained. The stability of the orbit is concluded based on the eigenvalues of the monodromy matrix.The developed software utilizes a Matlab ODE solver with built-in event detection. The equation for calculating a saltation matrixSi is selected depending on the type of event.In the following figures, representing the results of a numerical continuation, we use the following notation: blue lines stand for the stable orbits, and yellow, for the unstable ones.


### 3.6. BCB in a DC Drive

In a considered system, BCB is related to the conditionvcon=vramp exactly at the periodic instance; vcon collides with the upper or lower corner of the sawtooth wave. Therefore, at BCB, one of two possible conditions can be fulfilled:(19a)ωup=ωref+vupg,(19b)ωlo=ωref+vlog,where the first one (19a) is related to the collision with the upper corner, while the second (19b), with the bottom one. It is worth noticing that both conditions (19a) and (19b) are independent on it; accordingly, the BCB condition can be analyzed utilizing the bifurcation diagrams of the rotating speed only, omitting the second state variable, the armature current.In the following bifurcation diagrams, the corresponding borders are marked with the dark red.In the current work, the exact type of border-collision bifurcations in the model of a DC drive is detected employing the border-collision normal form (Appendix B); the example of using the normal form to analyze the type of BCBs is given in Appendix C.


## 3.1. Brute-Force Bifurcation Diagrams

With increasing the gaing, we observe a well-known (see [34, 36]) period-doubling route to chaos after losing the stability of the “nominal” operating mode of the drive (a stable period-1 orbit) with the further sudden expansion of a chaotic attractor via the interior crisis (the expansion bifurcation) and with periodicity windows inside chaotic area (Figure 2).Figure 2
Several coexisting stable modes at the brute-force bifurcation diagrams with the feedback gaing as a parameter.In addition, besides the main period-1 orbit undergoing a period-doubling cascade, there exist period-3, -4, and four-band chaotic orbits, coexisting with the stable period-1 one, period-7 mode coexisting with a period-2 one (Figure2), as well as the others after the bifurcations of the mentioned above. The search for coexisting orbits can be performed employing brute-force bifurcation diagrams combined with the scan of the state space at several values of the bifurcation parameters with further brute-force continuation of the orbits found, as implemented in the software AnT 4.662 [51]. The procedure can be accompanied by a numerical continuation, allowing for a detection of unstable periodic orbits.Each of the coexisting period-n orbits undergoes the period-doubling cascade, causing their transformation into period-2n, -4n, …, modes, followed by chaotic modes collapsing via the boundary crises (the final bifurcations) as shown in Figure 2; the mechanisms lying behind are explained in detail in [7].


## 3.2. Symbolic Analysis of Periodic Orbits

A periodic orbit is depicted by its symbolic representation introducing the following notation:1 denotes “off” and 2—“on” switch’s state. Thus, a “nominal” T-periodic mode (a period-1 orbit of the stroboscopic map) is represented by the sequence 12, i.e., in the clock cycle: first, the switch operates in the off state and then in the on state. Likewise, the sequence 121−1−1−12 denotes the 4T-periodic mode (a period-4 orbit) with two switching instances per first clock cycle, no–per second and third, and one–per the fourth cycle.


## 3.3. Stability Analysis of the Periodic Orbits in a DC Drive

The studied system is governed by theT-periodic carrier signal vramp; therefore, the nT-periodic modes are analyzed on the basis of Floquet theory. Let the system for each set of the state of the power switches be a linear time-invariant (LTI) system governed by an equation x.=Ax+Bu. Therefore, the transition from xt0 to xt is governed by the state transition matrix Φt,t0=eAt−t0, where(5)xt=Φt,t0xt0.The same matrix is also used to depict the evolution of the perturbed initial statext0+∆xt0, as well as for the evolution of the perturbation itself(6)∆xt=Φt,t0∆xt0.Eigenvalues of the state transition matrixλii=1,…,N, where N is the dimensionality of the state space, are utilized to analyze the stability of the periodic mode, so that all of them are situated inside a unit circle λi<1 for a stable periodic mode.In a generalized power converter, the state space is divided into at least two partitions, and a trajectory during a periodic mode of operation crosses the border at least twice. A generalizednT-periodic mode (n-periodic orbit of the corresponding stroboscopic map) corresponds to a transition from an initial state x0=x0 to the same state after n clock periods x0=xnT. Similarly to the classical Floquet theory, the stability of the periodic mode is analyzed by the evolution of the perturbation; however, here, we have to account the evolution of the perturbation crossing the border (see [52] for details).Thus, a relation of the perturbation just after the event∆xt+ to that just before ∆xt− is described by the following state transition matrix (in this case, it is called the saltation matrix):(7)∆xt+=S∆xt−.The latter is depicted by the equation [52]:(8)S=IN+f+−f−n⊤n⊤f−+∂h/∂tt=tsw,where we assume that the trajectory crosses the border between the two subsystems represented by the vector fields f− and f+ at t=tsw, IN is an identity matrix of order N, and the switching hypersurface in the state space Σ is represented by the switching condition hx,t=0, n is the vector normal to Σ, and ⋅⊤ is the transpose operator.Let the dynamical system moving around the periodic orbitm times cross the switching hypersurfaces Σii=1,…,m, and let the transitions between the subsystems be governed by the saltation matrices Si. The transition over a cycle nT is governed by the monodromy matrix:(9)M=Sm×Φm×Sm−1×Φm−1×⋯×S2×Φ2×S1×Φ1,so that the evolution of the state is(10)xt0+nT=Mxt0,and the evolution of the perturbation is as follows:(11)∆xt0+nT=M∆xt0.Therefore, its eigenvalues are used to study the stability of a generalizednT-periodic orbit in a switched system.


## 3.4. Saltation Matrices for the DC Drive System

The expressions for the switching hypersurfacesΣi and the normals to them ni are adapted from [36], where the DC drive system is examined.The first case of switching corresponds to the equality of (3) and (4); the switching hypersurface is defined by the following condition:(12)Σ1=x,th1x,t=0,where(13)h1x,t=vcont−vrampt=gx1−ωref−vlo+vup−vlotTmod1.Therefore, the partial derivative is(14)∂h1∂t=−vup−vloT,and the normal vector is(15)n=∂h1∂x1∂h1∂x2=g0.Hence, the expression ofS1 related to the crossing of switching hypersurface Σ1 is obtained by substituting (13) and (14) into (8).In the second case, when the switching is associated with the end of the clock periodt=nT, discontinuity of vrampt defines that ∂h2/∂t=∞. As a result, when infinity is substituted in the denominator of (8), we obtain the saltation matrix S2=IN.


## 3.5. Numerical Continuation Utilizing Monodromy Matrix Approach

Stable and unstable periodic orbits were obtained semianalytically based on the algorithms proposed in [53, 54]. In our study, we developed our own software with similar purposes, implementing the same algorithm, as explained in the following.The starting point for a numerical continuation is an initial guessx0, which can be obtained as a result of a straightforward integration of the ODE system. The next step x0′ is obtained as a result of a Newton–Raphson search routine:(16)x0′=x0−∂xm∂x0−IN−1xm−x0.Since(17)∂xm∂x0=∂xm∂xm−1×Sm−1×∂xm−1∂xm−2×Sm−2×⋯×∂x2∂x1×S1×∂x1∂x0=M,then (16) is rewritten as(18)x0′=x0−M−IN−1xm−x0.Afterwards, the obtained value is taken as an initial guess with the parameter slightly varied, and utilizing the search routine (18), a new periodic orbit is obtained. The stability of the orbit is concluded based on the eigenvalues of the monodromy matrix.The developed software utilizes a Matlab ODE solver with built-in event detection. The equation for calculating a saltation matrixSi is selected depending on the type of event.In the following figures, representing the results of a numerical continuation, we use the following notation: blue lines stand for the stable orbits, and yellow, for the unstable ones.


## 3.6. BCB in a DC Drive

In a considered system, BCB is related to the conditionvcon=vramp exactly at the periodic instance; vcon collides with the upper or lower corner of the sawtooth wave. Therefore, at BCB, one of two possible conditions can be fulfilled:(19a)ωup=ωref+vupg,(19b)ωlo=ωref+vlog,where the first one (19a) is related to the collision with the upper corner, while the second (19b), with the bottom one. It is worth noticing that both conditions (19a) and (19b) are independent on it; accordingly, the BCB condition can be analyzed utilizing the bifurcation diagrams of the rotating speed only, omitting the second state variable, the armature current.In the following bifurcation diagrams, the corresponding borders are marked with the dark red.In the current work, the exact type of border-collision bifurcations in the model of a DC drive is detected employing the border-collision normal form (Appendix B); the example of using the normal form to analyze the type of BCBs is given in Appendix C.


## 4. Organization of Periodic Orbits in a DC Drive

### 4.1. Bifurcation Branch Arising from the “Nominal” Period-1 Orbit

The “main” bifurcation branch contains a period-1 orbitO1 and all periodic orbits arising from it, a cascade of period-doubling interrupted by border collisions (Figure 3). For instance, at gPD1, the flip (period-doubling) bifurcation occurs; period-1 orbit becomes unstable (changes from O1s to O1u in Figure 3(a)), and period-2 orbit O2 is born, in turn, undergoing the second period doubling at gPD2 etc.Figure 3
Period-2 and -4 orbits arising from the period-1 orbit obtained by the numerical continuation technique (a) and blow-up of the marked window (b).
(a)(b)The stability of the periodic orbits is analyzed employing the eigenvalues of the monodromy matrix as explained previously. Thus, atgBC1, one of the eigenvalues crosses the unit circle at −1 implying the flip (period-doubling) bifurcation (Figure 4).Figure 4
Eigenvaluesλ1, λ2 of period-1 orbit from Figure 3(a): their real and imaginary parts, and absolute values (a), and the corresponding eigenvalue loci (b) with respect to a unit circle.
(a)(b)The periodic orbitO2,1s arising from O1 through the flip bifurcation in gPD1 undergoes a period doubling in gPD2 and continues as unstable O2,1u. After the BCB at gBC2, it disappears, and a new unstable orbit O2,2u is born (Figure 3(b)). The eigenvalues of the corresponding orbits are demonstrated in Figure 5. Here and below, the type of the BCB is determined using the normal form (B.1); the details will be given in the following. In the subscript, we use the second number after the commas to distinguish between the orbits characterized by different symbolic sequences.Figure 5
Eigenvaluesλ1, λ2 of period-2 orbits from Figure 3(b): their real and imaginary parts, and absolute values (a), and the corresponding eigenvalue loci with respect to a unit circle before gBC (b) and after gBC2 (c).
(a)(b)(c)Another interaction with the border in Figure3(b) is related to the disappearance of two period-4 orbits: the stable O4,1s arising from O2,1s through the flip bifurcation at gPD2 and the unstable one O4,2u arising through the fold (saddle-node) bifurcation at gSN. Consequently, BCB disrupts a regular period doubling, causing the coexistence of orbits of period-2 O2,1s and period-4 O4,3s and then two period-4 orbits O4,3s and O4,2s (Figure 3(b)).


### 4.2. Branches of the Period-3 and Period-4 Orbits

The branches of period-3 orbits include two stable ones: the one that coexists with the period-1 orbit, and the other that forms a periodic window inside the chaotic area (Figure6). A group of period-4 orbits includes a stable orbit and a chaotic attractor that coexists with the main period-1 orbit (compare Figures 2 and 7(a)).Figure 6
Group of the period-3 orbits along with the corresponding brute-force bifurcation diagrams.Figure 7
Group of the period-4 orbits along with the corresponding brute-force bifurcation diagrams (a) and the birth of a four-band chaotic attractor at the BCB with embedded period-4 orbits (b).
(a)(b)The bifurcations of the period-3 and period-4 group of orbits on the third and fourth, correspondingly, iterates of the stroboscopic map are depicted in Figure8, where SN stands for fold (saddle-node) bifurcation, PDi, for flip (period-doubling), and BCi, for BCB.Figure 8
Third iterate of the stroboscopic map for the group of the period-3 orbits (a) and fourth iterate, for the group of the period-4 orbits (b).
(a)(b)One of BCBs, BC1 in Figure8(b), corresponds to the birth of a chaotic attractor.


### 4.3. Multistability in the DC Drive System

As it has been addressed previously, the authors of [36] had already demonstrated multistability in a DC drive system. Our studies by means of numerical continuation accompanied by the brute-force bifurcation diagrams demonstrated several ranges of g where the system is prone to multistability. The basins of attraction of the coexisting period-1 and four-band chaotic (Figure 9(a)), period-1, 3 and 4 (Figure 9(b)), and period-1 and 6 (Figure 9(c)) orbits depict an interwoven structure of the basins of attraction, where Bi stands for basins of period-i orbits and Bχ, for the basin of the chaotic orbit.Figure 9
Basins of attractionBi of coexisting stable orbits for several values of g.
(a)(b)(c)The shape of the basins in Figure9 is similar for all cases; in fact, the boundaries between the basins are formed by the same unstable manifolds of the saddle orbits and below, and only one case, g=2, will be examined in developing a control strategy. The blow-up of the marked region of the basins of attraction in Figure 9(b) is presented in Figure 10(a) with the corresponding periodic orbits. It is clear that all of them are situated in the same part of the state space. The continuous-time representation of the three coexisting modes (Figure 10(b), where circles stand for periodic instances, and crosses, for switching between them) also explains that the coexisting “nonnominal” modes are characterized by higher ripples of state variables and should be avoided in any practical application.Figure 10
Blow-up of the marked region of the basins of attraction in Figure9(b) with the corresponding sampled periodic modes (a), and continuous-time state-space representations of attractors Ai of the corresponding period-i orbits (b).
(a)(b)Another issue related to the practical exploitation of the system is that due to lowest distance to the border between the basins, period-4 orbit atg=2, as well as chaotic orbit at g=1.195, are the most sensitive to noise and external disturbances among the coexisting and are most probably driven to the coexisting modes under nonideal operating conditions.In the next sections, a multistability control is developed forg=2, keeping in mind that the coexistence scenario is mostly influenced by the period-3 orbit.


## 4.1. Bifurcation Branch Arising from the “Nominal” Period-1 Orbit

The “main” bifurcation branch contains a period-1 orbitO1 and all periodic orbits arising from it, a cascade of period-doubling interrupted by border collisions (Figure 3). For instance, at gPD1, the flip (period-doubling) bifurcation occurs; period-1 orbit becomes unstable (changes from O1s to O1u in Figure 3(a)), and period-2 orbit O2 is born, in turn, undergoing the second period doubling at gPD2 etc.Figure 3
Period-2 and -4 orbits arising from the period-1 orbit obtained by the numerical continuation technique (a) and blow-up of the marked window (b).
(a)(b)The stability of the periodic orbits is analyzed employing the eigenvalues of the monodromy matrix as explained previously. Thus, atgBC1, one of the eigenvalues crosses the unit circle at −1 implying the flip (period-doubling) bifurcation (Figure 4).Figure 4
Eigenvaluesλ1, λ2 of period-1 orbit from Figure 3(a): their real and imaginary parts, and absolute values (a), and the corresponding eigenvalue loci (b) with respect to a unit circle.
(a)(b)The periodic orbitO2,1s arising from O1 through the flip bifurcation in gPD1 undergoes a period doubling in gPD2 and continues as unstable O2,1u. After the BCB at gBC2, it disappears, and a new unstable orbit O2,2u is born (Figure 3(b)). The eigenvalues of the corresponding orbits are demonstrated in Figure 5. Here and below, the type of the BCB is determined using the normal form (B.1); the details will be given in the following. In the subscript, we use the second number after the commas to distinguish between the orbits characterized by different symbolic sequences.Figure 5
Eigenvaluesλ1, λ2 of period-2 orbits from Figure 3(b): their real and imaginary parts, and absolute values (a), and the corresponding eigenvalue loci with respect to a unit circle before gBC (b) and after gBC2 (c).
(a)(b)(c)Another interaction with the border in Figure3(b) is related to the disappearance of two period-4 orbits: the stable O4,1s arising from O2,1s through the flip bifurcation at gPD2 and the unstable one O4,2u arising through the fold (saddle-node) bifurcation at gSN. Consequently, BCB disrupts a regular period doubling, causing the coexistence of orbits of period-2 O2,1s and period-4 O4,3s and then two period-4 orbits O4,3s and O4,2s (Figure 3(b)).


## 4.2. Branches of the Period-3 and Period-4 Orbits

The branches of period-3 orbits include two stable ones: the one that coexists with the period-1 orbit, and the other that forms a periodic window inside the chaotic area (Figure6). A group of period-4 orbits includes a stable orbit and a chaotic attractor that coexists with the main period-1 orbit (compare Figures 2 and 7(a)).Figure 6
Group of the period-3 orbits along with the corresponding brute-force bifurcation diagrams.Figure 7
Group of the period-4 orbits along with the corresponding brute-force bifurcation diagrams (a) and the birth of a four-band chaotic attractor at the BCB with embedded period-4 orbits (b).
(a)(b)The bifurcations of the period-3 and period-4 group of orbits on the third and fourth, correspondingly, iterates of the stroboscopic map are depicted in Figure8, where SN stands for fold (saddle-node) bifurcation, PDi, for flip (period-doubling), and BCi, for BCB.Figure 8
Third iterate of the stroboscopic map for the group of the period-3 orbits (a) and fourth iterate, for the group of the period-4 orbits (b).
(a)(b)One of BCBs, BC1 in Figure8(b), corresponds to the birth of a chaotic attractor.


## 4.3. Multistability in the DC Drive System

As it has been addressed previously, the authors of [36] had already demonstrated multistability in a DC drive system. Our studies by means of numerical continuation accompanied by the brute-force bifurcation diagrams demonstrated several ranges of g where the system is prone to multistability. The basins of attraction of the coexisting period-1 and four-band chaotic (Figure 9(a)), period-1, 3 and 4 (Figure 9(b)), and period-1 and 6 (Figure 9(c)) orbits depict an interwoven structure of the basins of attraction, where Bi stands for basins of period-i orbits and Bχ, for the basin of the chaotic orbit.Figure 9
Basins of attractionBi of coexisting stable orbits for several values of g.
(a)(b)(c)The shape of the basins in Figure9 is similar for all cases; in fact, the boundaries between the basins are formed by the same unstable manifolds of the saddle orbits and below, and only one case, g=2, will be examined in developing a control strategy. The blow-up of the marked region of the basins of attraction in Figure 9(b) is presented in Figure 10(a) with the corresponding periodic orbits. It is clear that all of them are situated in the same part of the state space. The continuous-time representation of the three coexisting modes (Figure 10(b), where circles stand for periodic instances, and crosses, for switching between them) also explains that the coexisting “nonnominal” modes are characterized by higher ripples of state variables and should be avoided in any practical application.Figure 10
Blow-up of the marked region of the basins of attraction in Figure9(b) with the corresponding sampled periodic modes (a), and continuous-time state-space representations of attractors Ai of the corresponding period-i orbits (b).
(a)(b)Another issue related to the practical exploitation of the system is that due to lowest distance to the border between the basins, period-4 orbit atg=2, as well as chaotic orbit at g=1.195, are the most sensitive to noise and external disturbances among the coexisting and are most probably driven to the coexisting modes under nonideal operating conditions.In the next sections, a multistability control is developed forg=2, keeping in mind that the coexistence scenario is mostly influenced by the period-3 orbit.


## 5. Control in the Form of a Sinusoidal Modulation of a Carrier Signal

Slight changes in the switching function in a switched system can dramatically affect the stability of periodic modes. Thus, sinusoidal modulation of a reference signal for a buck converter was applied in [39] to implement a method of resonant parametric perturbation to control the chaotic behavior of the converter. The variables affecting the control efficiency are the amplitude, the phase shift, and the frequency mismatch of the perturbing signal. In [41], the monodromy and saltation matrices are utilized to explain the effect of different forms of perturbation affecting the ramp or the control signal. Thus, it was demonstrated that with the adequate adjustment, the perturbation extends the stability margins of the “nominal” T-periodic mode.Likewise, such slight changes in the switching decision also affect coexisting modes in the case of multistability. Thus, several multistability control strategies implementing the changes were proposed for a self-oscillating H-bridge parallel resonant DC-DC converter [49]. The selection of the desired periodic mode among the coexisting is utilized with the modulation of the desired period (see e.g., [48]).Since the desired mode isT-periodic, we apply the modulation of the same period as is described in the following. Thus, with the switching condition given as (14), the periodic modulation can be applied to the reference or to the carrier signal; however, it is simply demonstrated that both are just reformulations of each other, while for our application, it is better implemented with the periodic modulation of the carrier signal [43]. Such implementation is also preferable for the purpose of analysis, as well as control signal is affected by only a state variable (a rotating speed). Therefore, the switching decision (13) is changed to(20)h1x,t=gωt−ωref−vlo+vup−vlotTmod1−asin2πtT+θ,where the two adjustment coefficients for the control are a, a modulation strength, and θ, a phase shift.


### 5.1. Saltation Matrices under the Control Strategy (20)

As it has already been mentioned, the injection of external disturbances for the purposes of extending the stability margins of the “nominal” mode was first explained as the manipulations with saltation matrices in [41]. Later, the theory obtained a practical on-chip implementation [42]. Manipulation with saltation matrices was also applied to select optimal controller tuning for DC-DC converters [55].The saltation matrix (8) has two degrees of freedom for manipulation, the normal vector n (for the uncontrolled system, it is given by (15)) and the time derivative ∂h1/∂t (given by (14) for an uncontrolled system).With the control given as (20), the normal vector remains unchanged, while the time derivative of the switching function is(21)∂h1∂t=−vup−vloT−acos2πtT+θ,and the rest of the components of the monodromy matrix remains unchanged. The corresponding changes have been applied to the algorithm for stability analysis and numerical continuation of periodic orbits.


### 5.2. Effect of the Modulation Strengtha

With the applied control strategy (20) and the phase shift fixed to θ=0, only the modulation strength a is changed; depending on the coefficient, the shape of the carrier signal is more or less distorted by the sinusoid (Figure 11).Figure 11
Carrier signalvrampi for the control strategy (20) with different modulation strength coefficient a and θ=0.We should note that the case of DC-DC converter with a smooth sinusoidal carrier signal is depicted in the literature [56, 57]. The distinguishing feature of that application in comparison with the sawtooth carrier signal from the point of view of its bifurcation behavior is the condition for BCBs [56] related to the grazing of the control signal with the sine wave, unlike the “corner collision” type of BCB under sawtooth carrier signal. As a result, for the smooth sinusoidal carrier signal, the condition for BCB cannot be as simply calculated as (19a) and (19b) for the sawtooth carrier signal. In the case of the control (20), both types of BCBs are possible.In a bifurcation diagram Figure12, only the borders depicted by conditions (19a) and (19b) are marked, and the coexisting modes are denoted with different colors. The group of orbits arising from the period-4 orbit exists in a narrow range of the parameter a limited by the saddle-node bifurcation and a boundary crisis (final bifurcation)–a “classical” scenario [7]. The other, the group of orbits arising from the period-3 orbits, is limited by final bifurcations at aF3,1 and aF3,2 and disrupted by border-collision bifurcations at aBC3,1, aBC3,2, and aBC3,3.Figure 12
Coexisting modes, at the brute-force bifurcation diagram, under the control strategy (20) with different modulation strengths a.The structure related to the “nominal” period-1 mode is given in detail in Figure13. The three regions, where the period-1 orbits are stable, are clearly observed. The stable regions on the left and right are limited by the period-doubling bifurcations at aPD1,1 and aPD1,2, respectively, and the central region is limited by the two BCBs at aBC1,1 and aBC1,2.Figure 13
Period-1 orbits and their bifurcations under the control strategy (20).Two remarks should be made; both are also explained by utilizing the time dependencies as shown in Figure14. Note that the “nominal” mode of operation is characterized by the symbolic sequence 12, depicted in Figure 14(a) for a=0 (an uncontrolled system).Figure 14
Time dependencies corresponding to different switching sequences of period-1 orbits under the control strategy (20) for different modulation strength coefficients a.
(a)(b)(c)The first remark is the type of BCB that the orbit is limited by. Thus, ataBC1,2, a collision with the upper corner of the ramp signal occurs, and due to the transition, the symbolic sequence is changed to 121 (see Figure 14(b) for a=0.7). The type of border collision occurring at aBC1,1 is related to grazing of the control signal with the sinusoidally distorted carrier one; therefore, the symbolic sequence of the new orbit is 1212 (see Figure 14(c) at a=−0.65).The second remark is also related to the symbolic sequences characterizing the periodic orbits. For a practical application, only the period-1 orbit characterized by the symbolic sequence12 is acceptable. The resulting region of the “acceptable” monostability is for negative a limited by aF3,1, a final bifurcation of the branch arising from the period-3 orbit, and aSN2, the birth of the coexisting period-2 orbit via a saddle-node bifurcation (with the region of bistability between aBC1,1 and aSN2).


### 5.3. Effect of the Phase Shiftθ

The effect of the phase shift is in the form of a carrier signal which is depicted in Figure15, where the modulation strength is fixed at a=0.3. It can be observed that the values of the original sawtooth and a sinusoidally distorted signal are equal twice over the period. This means that if for a “nominal” mode the switching occurs at that time instant, the duty cycle remains unaffected, as well are all indices of the dynamics (ripples of the state variables, average values, etc.) [42].Figure 15
Carrier signalvrampi for the control strategy (20) with the modulation strength coefficient a=0.3 for different phase shifts θ.For our particular application, it is undesirable to further complicate the system to gain a fraction of a percent of the nominal speed value (Figure12) while the ripples remain unchanged, as well as for the armature current [43]. On the other hand, the phase shift can be an additional degree of freedom for a controller; therefore, we analyze the possible effect of utilizing the additional setting.Thus, the effect of the phase shift is studied by means of 1D bifurcation diagrams withθ as a parameter for different values of a (Figures 16 and 17). For a=0, the phase shift, as it could be expected, has no effect on the orbit; while for others, also unsurprisingly, it has a sinusoidal shape affected by the amplitude a (Figure 16). For low a, those are stable orbits, while for higher values, there appears an unstable orbit close to the uncontrolled system’s state, where the slope of the bifurcation curve is negative. The width of the “region of instability” increases as increases the modulation strength.Figure 16
Numerical-continuation-based bifurcation diagrams of period-1 orbits for the control strategy (20) with phase shift θ as a bifurcation parameter for different modulation strengths a.Figure 17
Coexisting period-1 and period-3 modes for the control strategy (20) with phase shift θ as a bifurcation parameter for different modulation strengths a.
(a)(b)(c)(d)(e)(f)The first thing that should be mentioned is the phase-shifted symmetry for positive and negativea.The second one is the way of losing stability. Thus, ata=−0.02, the “bubble” at the bifurcation curve of the period-3 orbit is observed, limited by the pair of direct and reverse period-doubling bifurcations at θD3,1 and θD3,2. At a=−0.03, the bubble extends, and finally, at a=−0.1, the period-doubling cascade ends with the final bifurcations at θF3,1 and θF3,2. A region of monostability between the period-doubling of the period-1 orbit and the final bifurcation of period-3 one in the vicinity of 0 is the admissible range for the phase shift for the control; the same is due to the π-shifted symmetry in the vicinity of π for a=0.1.With the change ofa, the same pattern is observed. Thereafter, on the one hand, the evolution of the orbits leads to the final bifurcations at θF1 and θF3, while on the other hand, both coexisting chaotic attractors overlap resulting in the only big one.We should also note that with a frequency mismatch between the one frequency dictated by the clock and the one, by the injected signal, an intermittency is observed in the form of “bubbles” on the time-bifurcation diagrams [39, 40]. Those have the same nature as the one in the bifurcation diagrams in Figure 17.


## 5.1. Saltation Matrices under the Control Strategy (20)

As it has already been mentioned, the injection of external disturbances for the purposes of extending the stability margins of the “nominal” mode was first explained as the manipulations with saltation matrices in [41]. Later, the theory obtained a practical on-chip implementation [42]. Manipulation with saltation matrices was also applied to select optimal controller tuning for DC-DC converters [55].The saltation matrix (8) has two degrees of freedom for manipulation, the normal vector n (for the uncontrolled system, it is given by (15)) and the time derivative ∂h1/∂t (given by (14) for an uncontrolled system).With the control given as (20), the normal vector remains unchanged, while the time derivative of the switching function is(21)∂h1∂t=−vup−vloT−acos2πtT+θ,and the rest of the components of the monodromy matrix remains unchanged. The corresponding changes have been applied to the algorithm for stability analysis and numerical continuation of periodic orbits.


## 5.2. Effect of the Modulation Strengtha

With the applied control strategy (20) and the phase shift fixed to θ=0, only the modulation strength a is changed; depending on the coefficient, the shape of the carrier signal is more or less distorted by the sinusoid (Figure 11).Figure 11
Carrier signalvrampi for the control strategy (20) with different modulation strength coefficient a and θ=0.We should note that the case of DC-DC converter with a smooth sinusoidal carrier signal is depicted in the literature [56, 57]. The distinguishing feature of that application in comparison with the sawtooth carrier signal from the point of view of its bifurcation behavior is the condition for BCBs [56] related to the grazing of the control signal with the sine wave, unlike the “corner collision” type of BCB under sawtooth carrier signal. As a result, for the smooth sinusoidal carrier signal, the condition for BCB cannot be as simply calculated as (19a) and (19b) for the sawtooth carrier signal. In the case of the control (20), both types of BCBs are possible.In a bifurcation diagram Figure12, only the borders depicted by conditions (19a) and (19b) are marked, and the coexisting modes are denoted with different colors. The group of orbits arising from the period-4 orbit exists in a narrow range of the parameter a limited by the saddle-node bifurcation and a boundary crisis (final bifurcation)–a “classical” scenario [7]. The other, the group of orbits arising from the period-3 orbits, is limited by final bifurcations at aF3,1 and aF3,2 and disrupted by border-collision bifurcations at aBC3,1, aBC3,2, and aBC3,3.Figure 12
Coexisting modes, at the brute-force bifurcation diagram, under the control strategy (20) with different modulation strengths a.The structure related to the “nominal” period-1 mode is given in detail in Figure13. The three regions, where the period-1 orbits are stable, are clearly observed. The stable regions on the left and right are limited by the period-doubling bifurcations at aPD1,1 and aPD1,2, respectively, and the central region is limited by the two BCBs at aBC1,1 and aBC1,2.Figure 13
Period-1 orbits and their bifurcations under the control strategy (20).Two remarks should be made; both are also explained by utilizing the time dependencies as shown in Figure14. Note that the “nominal” mode of operation is characterized by the symbolic sequence 12, depicted in Figure 14(a) for a=0 (an uncontrolled system).Figure 14
Time dependencies corresponding to different switching sequences of period-1 orbits under the control strategy (20) for different modulation strength coefficients a.
(a)(b)(c)The first remark is the type of BCB that the orbit is limited by. Thus, ataBC1,2, a collision with the upper corner of the ramp signal occurs, and due to the transition, the symbolic sequence is changed to 121 (see Figure 14(b) for a=0.7). The type of border collision occurring at aBC1,1 is related to grazing of the control signal with the sinusoidally distorted carrier one; therefore, the symbolic sequence of the new orbit is 1212 (see Figure 14(c) at a=−0.65).The second remark is also related to the symbolic sequences characterizing the periodic orbits. For a practical application, only the period-1 orbit characterized by the symbolic sequence12 is acceptable. The resulting region of the “acceptable” monostability is for negative a limited by aF3,1, a final bifurcation of the branch arising from the period-3 orbit, and aSN2, the birth of the coexisting period-2 orbit via a saddle-node bifurcation (with the region of bistability between aBC1,1 and aSN2).


## 5.3. Effect of the Phase Shiftθ

The effect of the phase shift is in the form of a carrier signal which is depicted in Figure15, where the modulation strength is fixed at a=0.3. It can be observed that the values of the original sawtooth and a sinusoidally distorted signal are equal twice over the period. This means that if for a “nominal” mode the switching occurs at that time instant, the duty cycle remains unaffected, as well are all indices of the dynamics (ripples of the state variables, average values, etc.) [42].Figure 15
Carrier signalvrampi for the control strategy (20) with the modulation strength coefficient a=0.3 for different phase shifts θ.For our particular application, it is undesirable to further complicate the system to gain a fraction of a percent of the nominal speed value (Figure12) while the ripples remain unchanged, as well as for the armature current [43]. On the other hand, the phase shift can be an additional degree of freedom for a controller; therefore, we analyze the possible effect of utilizing the additional setting.Thus, the effect of the phase shift is studied by means of 1D bifurcation diagrams withθ as a parameter for different values of a (Figures 16 and 17). For a=0, the phase shift, as it could be expected, has no effect on the orbit; while for others, also unsurprisingly, it has a sinusoidal shape affected by the amplitude a (Figure 16). For low a, those are stable orbits, while for higher values, there appears an unstable orbit close to the uncontrolled system’s state, where the slope of the bifurcation curve is negative. The width of the “region of instability” increases as increases the modulation strength.Figure 16
Numerical-continuation-based bifurcation diagrams of period-1 orbits for the control strategy (20) with phase shift θ as a bifurcation parameter for different modulation strengths a.Figure 17
Coexisting period-1 and period-3 modes for the control strategy (20) with phase shift θ as a bifurcation parameter for different modulation strengths a.
(a)(b)(c)(d)(e)(f)The first thing that should be mentioned is the phase-shifted symmetry for positive and negativea.The second one is the way of losing stability. Thus, ata=−0.02, the “bubble” at the bifurcation curve of the period-3 orbit is observed, limited by the pair of direct and reverse period-doubling bifurcations at θD3,1 and θD3,2. At a=−0.03, the bubble extends, and finally, at a=−0.1, the period-doubling cascade ends with the final bifurcations at θF3,1 and θF3,2. A region of monostability between the period-doubling of the period-1 orbit and the final bifurcation of period-3 one in the vicinity of 0 is the admissible range for the phase shift for the control; the same is due to the π-shifted symmetry in the vicinity of π for a=0.1.With the change ofa, the same pattern is observed. Thereafter, on the one hand, the evolution of the orbits leads to the final bifurcations at θF1 and θF3, while on the other hand, both coexisting chaotic attractors overlap resulting in the only big one.We should also note that with a frequency mismatch between the one frequency dictated by the clock and the one, by the injected signal, an intermittency is observed in the form of “bubbles” on the time-bifurcation diagrams [39, 40]. Those have the same nature as the one in the bifurcation diagrams in Figure 17.


## 6. Alternative Control Strategy–Nonsinusoidal Modulation of the Carrier Signal

From the point of view of theory related to the saltation matrices, not only a sine wave but any other periodic function can be used to stabilize aT-periodic mode. Thus, the cases of injecting a triangular or a sawtooth signal are already analyzed in the literature [40]. The reason for injecting a sine wave is related to the fact that with minimal distortion of the switching pattern for T-periodic mode, it produces a maximal effect on the derivative in the saltation matrix (8) as the derivative is a cosine function.In general, the objectives of the control, minimum impact on theT-periodic mode with the destruction of the coexisting non-T-periodic modes, can be satisfied with an injection of the T-periodic function ft so that ft=ft+T. Thus, an alternative control can be applied in the form of the switching function as follows [44]:(22)h1x,t=gωt−ωref−vlo+vup−vlotTmod1−asinπtT,with a as the adjustment coefficient. The carrier signal is, therefore, distorted as in Figure 18.Figure 18
Carrier signalvrampi for the control strategy (22) with different modulation strength coefficients a.In the discussed case, it is obvious that the values ofa to satisfy the objectives of the control should be associated with the positive a (Figure 19). The period-4 orbit exists in the narrow range of the studied parameter, being limited for positive a by the saddle-node bifurcation at aSN4, and period-3 orbit, by BCB at aBC3. The bifurcation structure is simpler than the one associated with the application of the control (20) (see Figures 12 and 17); therefore, the adjustment analysis is simpler.Figure 19
Coexisting modes, at the brute-force bifurcation diagram, under the control strategy (22) with different modulation strengths a.


## 7. Possible Practical Implementation of Multistability Control

The last feature discussed is related to the practical implementation of the proposed control strategies, in particular. Thus, the first one is the continuous operation of the system with the applied control, thus guaranteeing monostable behavior, while the second one is related to the case of control temporarily applied when the system is driven to the undesired coexisting mode due to noise or an external perturbation and disabling afterwards. It is worth noting that there could be reasons for not applying the control continuously, e.g., related to the dynamic indices of the controlled system (settling time or overshoot, etc.). Additionally, the frequency mismatch of the clock and the modulation signal causes an undesirable intermittency, as demonstrated in [39, 40].The latter case is demonstrated in Figure20 in the case of applying the control strategy (22).When the implementation is associated with the abrupt switching of the value ofa, there can be a case when, after the control is disabled, the system is driven to an undesired mode (Figure 20(a)). That means, after the control is disabled, the trajectory occurs within the basin of attraction of the undesired mode. The possible solution is the smooth attenuation of the control action [44] as demonstrated in Figure 20(b).Figure 20
Different implementations of the control strategy (22): abrupt switching to and from the control strategy at t1 and t2, respectively, (a) and with the further smooth attenuation of the control action from t2 to t3 (b).
(a)(b)


## 8. Conclusion

For a multistable chopper-fed DC drive system, we propose a control of multistability in the form of periodic modulation of the PWM carrier signal. If the modulation signal is sinusoid, it has two degrees of freedom, an amplitudea and a phase shift θ. We demonstrate that with positive a, the coexisting modes are prone to complex bifurcation behavior, even with the unacceptable symbolic sequence of the T-periodic mode. With negative a, the coexisting modes collapse, and the system is driven to monostability. However, the settings of the control should also account for the effect of phase shift, as far as it sometimes can become a disturbing factor, causing additional bifurcations.An alternative control strategy is proposed due to the fact that in general, any other periodic function can be used for the modulation. As an example, we also propose a modulation function as an absolute value of the sine wave with the advantages over the sine-wave modulation, i.e., less complicated bifurcation behavior.As for the practical implementation of the control strategy, we propose to switch shortly to the controlled state just to drive the system to monostability. Afterwards, the smooth attenuation of the modulation strength to zero ensures the avoidance of transitions to the undesired mode when returning to the uncontrolled state.As far as the proposed control strategy can be implemented for any PWM-driven system, the future research can be directed towards the search of a suitable application where the emergence of multistability is an undesired factor. Another direction of research can be the discovery of possible features and drawbacks if the control is implemented within digital PWM control.


## 9. Disclosure

It was performed as part of the author’s employment at the Lviv Polytechnic National University.



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*Source: 5559763-2023-02-16.xml*