Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system

: The fractional-order controllers (FOCs) have recently had a signi ﬁ cant impact on control applications. However, they still need further research for feedback systems with hard nonlinearities, such as dead zones. The above compelling evidence motivates the design of a new robust FOC to avoid limit cycles caused by dead zones in the control loops. The proposed FOC consists of the cascade of two shifted in frequency, fractional-order lead networks. They provide high-value and su ﬃ ciently ﬂ at phase leads in su ﬃ ciently large frequency intervals. In this way, the linear part of the control loop can be easily shaped to achieve avoidance of limit cycles. The article applies classical concepts, such as the Nyquist plot and describing function method, to derive guidelines for designing the free parameters of the FOC. Moreover, a realization algorithm and a parameter setting procedure make the new FOC easily implementable in engineering practice.


Introduction
Control systems are negatively affected by the dead zone nonsmooth nonlinearity, which produces poor steady-state accuracy, limit cycles, and system instability [1].Besides, the dead zone nonlinearity (DZNL) is difficult to remove because it is described by nondifferentiable models with poorly known parameters [2].For this reason, the pioneered adaptive inverse schemes used by [1,3,4] in combination with many control design approaches do not achieve the perfect cancellation of the DZNL but only obtain bounded output errors [5].Other attempts to compensate for the DZNL have also been made by neural networks [6] and fuzzy logic methods [7].However, on the one hand, the above techniques are usually not suited for industrial practice because of the heavy computational demand, but, on the other hand, the common proportional-integral-derivative (PID) controllers cannot face the challenges of nonlinear control problems.
Recently, the so-called fractional-order controllers (FOCs for brevity), which are based on noninteger orders of integration and differentiation, have been proposed and, in many cases, gradually replaced the standard PID controllers in solving complex problems.Since they use integrators and differentiators of noninteger orders, the FOCs introduce two more tuning parameters for improving the performance of the classical PID controllers.Moreover, it is remarked that fractional derivatives and fractional-order differential equations offer an enhanced and more flexible tool to mathematical modeling than integer-order differential equations [8,9].These fundamental reasons motivated the success of pioneering applications.The Commande Robuste d'Ordre Non Entièr [10,11] has made relevant contributions to the automotive industry, and the PI D λ μ -controller [12] has generalized and made more flexible the PID controller.Moreover, in the tilted integral derivative controller [13], the operator ∕ s n 1 (with n integer) has successfully controlled systems with uncertain parameters.Finally, the fractional lead-lag networks have shown better robustness and flexibility than their integer counterparts [14,15].Auto-tuning was also developed [15,16], and neural networks were employed for the design and implementation of noninteger-order integrators [17].However, the FOCs still need further research in nonlinear applications.In this context, starting from the combination of fractional calculus and sliding-mode control for integer-order systems [18] or fractional-order systems [19], some contributions exist for sliding-mode control of certain classes of uncertain linear [20] or nonlinear fractionalorder systems [21].It is also recalled that nonlinear chaotic systems can be of noninteger (fractional) order, and the topic of chaos synchronization is investigated very much because of the benefits in different areas of science and engineering [22,23].However, in some cases, chaos is not desirable, and control is designed to suppress the chaotic oscillations of the fractional-order uncertain model of real systems like a permanent magnet synchronous generator [24].
In particular, with a nonlinear element in the control loop, the FOCs require specific control algorithms for industrial practice [25].Hence this article introduces a new, two-stage controllerrepresented by the transfer function ( ) H s

12
to avoid the occurrence of limit cycles due to a DZNL.The controller is made by two fractional-order lead networks, which are shifted in frequency relatrive to each other.The uncompensated system including the nonlinear dead zone element and a linear plant represented by a transfer function ( ) G s p , namely a servo system, can show a limit cycle.(By the way, nonlinear systems are frequently represented by the series of a memoryless nonlinearity and a linear transfer function, and the derived Hammerstein model is important in control engineering, since it can effectively approximate many devices [26,27].)Otherwise, the scheme in Figure 1 describes the system compensated by the FOC.The controller ( ) H s 12 consists of a cascade of two shifted fractional-order lead networks (an SFLN for brevity) that are placed at a given distance, one after the other, on the frequency axis.By providing a nearly constant phase in a sufficiently large frequency interval, the SFLNs inhibit the intersections between the plots of the negative inverse of the dead zone describing function and the loop transfer function, given that such intersections are necessary conditions for a limit cycle to occur.Finally, the design and rational realization of the SFLNs is easy and guarantees a secure and robust avoidance of limit cycles in servo systems with a dead zone in the control loop.
The organization of this article is as follows: Section 2 describes the proposed FOC, which is an irrational, twostage fractional-order lead network made of two consecutive stages, and moreover, the section shows the realization of the controller by a rational transfer function; Section 3 introduces a design approach of the proposed SFLN and gives an illustrative example at the same time; and Section 4 gives the conclusions.is set to obtain equal but shifted stages.Here, τ and τΔ are the numerator and denominator time constants (in seconds) of the first stage, F represents the shift of the second stage with respect to the first, and the noninteger (fractional) exponent ν, with < < ν 0 1, is the so-called fractional order.Then, the frequency response of the two-stage SFLN is given as follows: where the variable = u ωτ is the dimensionless angular frequency and ω is the angular frequency (in radians/sec- onds).The minimum value of Δ in a classical lead network can be fixed to 0.05 because it is limited by the physical construction of the compensator and by the limitations of the maximum phase lead and high-frequency gain that are provided by the network [28].Namely, if a maximum phase lead greater than 60°is needed, two cascaded lead networks with moderate values of Δ are advised.Moreover, high-frequency noise signals are amplified by ∕Δ 1 , while a unitary amplification applies to low-frequency control signals.Then, > Δ 0.07 is often recommended [29].The usual choice is ( ) ∈ Δ 0.05, 0.25 [28,29].Then, this article makes the common choice = Δ 0.1, which is recom- mended for lead networks [29], for both stages of the proposed SFLN.
Finally, the parameter F , with < < F 0 1, shifts the phase diagram of ( ) H juF 2 with reference to the position of the phase diagram of ( ) H ju 1 on the u-axis [30][31][32].Clearly, the fractional-order lead networks, ( ) H ju , respectively.These values coincide with the frequencies at which the maximum phase lead occurs in a network with the same base but with an integer-order exponent = ν 1.They are also the geometric mean of the two corner frequencies, i.e., the middle point, in the logarithmic scale, between the zero and pole corner frequency.So, according to De Moivre's theorem, the phase of ( ) H ju 1 given as follows: Similarly, the second stage provides the maximum value Moreover, if two generic frequencies u 1 and u 2 , with 2 .Now the frequency u m12 , where ( ) H ju 12 provides the maximum phase lead, say Φ m12 , must be determined.Namely, the pair (u , Φ m m 12 12 ) plays a major role in an SFLN design finalized to both avoiding the limit cycle and maintaining the stabilization mode.To determine u m12 , the following equation must be solved: , we obtain: ) ( ) After simple but tedious calculations, Eq. ( 5) and the positions = α FΔ, , and So putting = u q 2 in Eq. ( 6), an obvious choice of symbols leads to The coefficients of Eq. ( 8) only depend on F and Δ and can be easily determined.Since c 1 , c 2 , and c 3 are real, Eq. ( 7) has at least one real root.Moreover, by Descartes' Rule of signs, the number of positive real roots of Eq. ( 7) is either equal to the sign changes in the sequence { } c c c 1, , , 1 2 3 or is less than this number by a positive even integer [33] , so that the number of sign variations is 1.Hence, Eq. ( 7) has exactly one real positive root (say q 1 ).The value of q 1 can be obtained as a numerical solution of Eq. ( 7) or as a result of Cardano's analytical procedure [34] and leads to the frequency of the maximum phase lead.In this article, q 1 is obtained as a numerical solution of Eq. (7).Hence, = u q m12 1 .Now comments on further characteristics of ( ) H ju 12 are in order.The following analysis tries to fix the fractional order ν on the basis of preliminary design decisions.Thus, to provide the phase lead the control loop requires for compensation, the value of ν is first selected.For example, in the case of the next section, a good first choice is = ν 0.7.Moreover, the parameters F and Δ influence u m12 and 12 , which are the main frequencydomain measures assessing performance.Hence, if Δ is fixed (e.g., = Δ 0.1), each value given to F will identify a member of a function family.As shown in Figure 2, each phase-frequency plot is marked by an F -value.Moreover, the lower values of F make the phase curves flatter around their maximum, Φ m12 , whereas the more the value of F rises, the more Φ m12 increases and u m12 decreases (Figure 2 and Table 1).
In the second relation, each value assigned to Δ leads to a one-to-one function and to a curve of Figure 3. Lower values of Δ lead to higher values of u m12 corresponding to assigned F .
Figures 2 and 3 with formulas ( 6), (7), and ( 8) are the basis of an efficient grapho-analytical design procedure of ( ) H ju 12 .Of course, different pairs of the above figures can be obtained for different values of the fractional order ν.
Fractional-order lead networks to avoid limit cycle  3

Realization of the controller
The model of the SFLN is irrational and infinite-dimensional.However, a finite-dimensional, rational transfer function realization can be obtained in three steps.The first one introduces the rational transfer function approximating the irrational derivative operator x ν : where ∈ x (complex plane) and < < ν 0 1.Note that the coefficients − a N i and − b N i depend on ν and can be obtained by one of the existing methods (see [35] and references therein).However, this article proposes the following simple, effective closed-formulas for − a N i and − b N i : define the Pochhammer functions, with ( ) ( ) (see [36,37] for more details).In the second step, the transformation is introduced in Eq. ( 9).Then, some elementary operations lead to: Furthermore, a routine-based algebra yields ) according to the following linear combination: and where In conclusion, . Figures 2 and 4 show the amplitude and phase plots of an SFLN with some values assigned to ν and Δ. Figure 4 shows a fourth-order ( ) = N 4 approximation by a dash- dotted line.Note that the discrepancy between the irrational network ( ) H jω 12 and its approximation in the Bode phase diagram is not important.Namely, it is negligible for < u u m12 and gives no problem for = u u m12 , where the maximum phase Φ m12 is achieved and used for design purpose (Section 3).The phase difference only consists in few degrees for > u u m12 , but it does not significantly affect the design that requires specifications at u m12 and a certain flatness in the immediate range around u m12 .
As for the phase plot concerns, curve C ( ) = F 0.7 in Figures 2 and 4 shows a sufficiently flat phase behavior in a sufficiently large frequency range both for the irrational controller ( ) H s 12 and for its rational transfer function realization.The phase diagram is not really "flat" (constant phase) but its variation is limited, especially with reference to the corresponding phase change provided by integer-order networks.
3 Robust controller design to avoid the limit cycle 3.

The system with limit cycle
This section proposes the positional control of a common type of plant, i.e., a servomotor, as case study (Figure 5).).Moreover, a DZNL in the loop is described as follows: where = μ 1 and = a 0.8 are here used, w is the input to the dead zone, z is the output from the dead zone, and Figure 5(a) explains the meaning of the symbols μ (slope) and a (half of dead zone).Now the describing function approach can be successfully applied because, by replacing → s jω in Eq. ( 19), ( ) G jω p shows the necessary higher harmonics filtering capacity.Hence, the harmonic balance equation yields where W is the amplitude of the sinusoidal input ( ) ( ) = w t W ωt sin applied to the nonlinear element and Approxim.for F = 0.7 Compensator for F = 0.9 Amp. at m for F = 0.9 Approxim.for F = 0.9 = 0.7 = 0.1 is the describing function of the DZNL element [38,39].Since the dead zone is a single-valued nonlinearity, ( ) N W and its negative inverse ( ) V W are real and their plots lie on the real axis (Figure 6).Moreover, the intersection point P 1 of the curve ( ) G jω p and the curve ( ) V W corresponds to a limit cycle that is defined by the pair The amplitude W c corresponds to P 1 on the ( ) V W curve, and the same point P 1 on the plot of ( ) G jω p identifies the limit cycle frequency, ω c (Figure 6).In this case, it holds ( . The curve ( ) V W plays the role of a "critical curve" that is compar- able to the critical point ( ) − + j 1 0 in the simplified Nyquist stability criterion [38], according to Eq. ( 21) and the simplified scheme in Figure 5(a).Then, if no control is applied, the timedomain control system response exhibits a limit cycle oscillation, as shown in Figure 7. Finally, since the plot of ( ) V W meets the curve of ( ) G jω p from left to right of the cross-point P 1 , then a practical rule states that the limit cycle is unstable [38,40].As remarked in [39], the importance of the Nyquist plot and Nyquist criterion in control theory made the graphical implementation of the describing function method a popular tool available to control engineers, when facing nonlinear control systems [39].

Controller design
The robust limit cycle avoidance is committed both to the Gain Margin (GM) and Phase Margin (PM) of . Namely, one or another between the PM and GM is a suitable design metric because ( ) G jω HP is minimum-phase with a unique relation between the amplitude and phase.Hence, the controller ( ) H jω Fractional-order lead networks to avoid limit cycle  7 of ( ) H jω 12 is chosen so that the maximum lead of the controller will contribute wholly to the phase margin.Hence, let = τω u gcHP m12 and use the parameter > h 1 to express ω gcHP in terms of ω gcP as follows: = ω hω gcHP gcP .Then, τ can be written as: To obtain out of the circular difficulty of determining the (unknown) time constant τ in terms of the (unknown) parameter h, a simple numerical code is used to find the element = h h giving both the smallest value establishes a useful robustness measure, which shows the efficiency of the design approach.It guarantees a suitable distance between the intersection of the Nyquist plot of the compensated linear part with the negative real axis (square point in Figure 8) and the plot of the negative inverse of the dead zone describing function.Namely, Figure 8 indicates the design robustness to plant parametric variations.Part (a) considers a ±20% change in the plant gain, and part (b) considers a ±20% change in the lower time constant T 1 .These changes determine an uncer- tainty region whose limits are specified by the dashed red curves around the Nyquist plot of ( ) G jω p .Then, the resulting region of the compensated system has limits given by the dashed black curves around the Nyquist plot of ( ) ( ) . The first square point on the nominal Nyquist plot (solid line) located between the limit curves intersects the unit circle, thus providing the gain crossover frequency and the phase margin.The second square point on the nominal plot corresponds to the phase crossover frequency and the gain margin.Then, as the figures show, the design is quite robust with respect to parametric variations.
Remark.The controller structure could be extended by introducing a pure integrator, for example, when it is required to reject disturbances on the plant input or to improve steady-state accuracy in reference tracking.Then, the proposed design methodology remains applicable if one treats this integrator as part of the plant transfer function during the control design procedure.Fractional-order lead networks to avoid limit cycle  9

Integer-order network
An integer-order lead network can be designed by a classical procedure [28,29].The proposed fractional-order control design draws inspiration from this classical method.
In the integer-order case, since the required phase lead (75°) in the example is higher than the maximum achievable by a lead compensator (about 65°), a double integer- order lead network, i.e., two equal lead networks with the same parameters are designed.Then, the controller is Although the frequency-domain specifications are met and the Nyquist plot does not intersect the plot of the negative inverse of the describing function, the time-domain control error is not satisfactory because the steady-state error is not zero, as shown in Figure 9, where a reference step input = r 4.5 is considered to generalize the case of = r 0. We may con- clude that the integer-order controller is not suitable, thus showing the superiority of the FOC.

Conclusion
This article deals with prediction and avoidance of limit cycles generated by a dead zone in SISO servo systems.The prediction is committed to the describing function approach.The limit cycle avoidance is a remarkable achievement of a new, two-stages, FOC that provides nearly constant, high phase lead values within an interval around the frequency of the provided maximum lead.The controller shapes the Nyquist plot of the compensated system so as to respect robustness constraints based on a given distance from the describing function plot.Although the design approach is based on the classical theory of the describing function and Nyquist plots, it is remarked that classical techniques are still useful in the engineering practice [41].The controller is implemented by approximating irrational transfer functions through a simple, efficient technique providing reduced order transfer functions.Such approximation is necessary for applying FOC to real systems but control performance and robustness is not weakened.The simplicity and efficiency make the controller acceptable to the engineering practice, for example, in many industrial applications where control loops are negatively affected by nonlinearities (dead zone, saturation, and backlash).Namely, in this article, the controller is designed for nonlinear systems including a dead zone element and a linear plant, which is a servo system for associated applications.However, the developed method can be extended to other systems including different nonlinear elements and/or different linear plants.This analysis will be the focus of future work.

Figure 1 :
Figure 1: The compensated control system: FOC is the fractional-order controller compensating the DZNL and P is the plant having transfer function ( ) G s p .

2 can
the third step, the rational transfer function of the second stage ( ) H s be immediately obtained by replacing s with Fs:

Figure 3 :
Figure 3: Value of the dimensionless angular frequency ( ) u F Δ , m12 for the maximum phase lead as function of F and with = ν 0.7.

Fractional-order lead networks to avoid limit cycle  5
), respectively, the values of the gain and phase margins of ( ) m for F = 0.3 Approxim.for F = 0.3 Compensator for F = 0.5 Amp. at m for F = 0.5 Approxim.for F = 0.5 Compensator for F = 0.7 Amp. at m for F = 0.7

Figure 5 :
Figure 5: Block schemes describing the FOC of a servomotor with a dead zone.(a) The uncompensated system including the DZNL with = μ 1 and = a 0.8 and the plant transfer function ( ) G s p and (b) the controlled system: ( ) H s

Figure 6 :Figure 7 :
Figure 6: Nyquist plot of ( ) G jω p (red solid line); negative inverse of the describing function of the DZNL (blue solid line); point P ( ) W ω , c c 1 : limit cycle position; Nyquist plot of ( ) ( ) H jω G jω p 12(black solid line): no limit cycle occurs.

Figure 8 :Figure 9 :
Figure 8: Evaluation of robustness to plant parametric uncertainties.(a) Sensitivity to ±20% variation in the plant gain and (b) sensitivity to ±20% variation in the plant lower time constant.

Table 1 :
Maximum phase lead and its dimensionless angular frequency