Root Locus Practical Sketching Rules for Fractional-Order Systems

Abstract

For integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.

1. Introduction

Root locus (RL) analysis is a graphical method that shows how the poles of a closed-loop transfer function change with relation to a given system parameter [1, 2]. Usually, the chosen parameter is a proportional gain, 𝐾≥0, included in a unity feedback closed-loop controlled system (Figure 1).

Figure 1 

Unity feedback closed-loop controlled system.

The open and closed-loop transfer functions are given by 𝐺_OL(𝑠)=𝐾⋅𝐺(𝑠) and 𝐺_CL(𝑠)=𝐾⋅𝐺(𝑠)/[1+𝐾⋅𝐺(𝑠)], respectively. The denominator of 𝐺_CL(𝑠) is the characteristic equation, and its roots are the system closed-loop poles. Every point of the RL simultaneously satisfies the well-known argument (angle) and magnitude conditions given by arg{𝐾⋅𝐺(𝑠)}=(2ℎ+1)⋅180^∘{(𝑠,ℎ=0,±1,±2,…,abs𝐾⋅𝐺)}=1. (1)

The RL is a classical and powerful tool for the dynamical analysis and design of integer-order linear time-invariant (LTI) systems [1–6]. Nowadays, there are efficient numerical algorithms, implemented in several software packages (e.g., MATLAB, Octave, Scilab, and FreeMat) [7–10] that take advantage of the powerful digital processors of modern computers to perform RL analysis. For fractional-order (FO) systems, while several studies addressing RL are available [11–17], the problem is more difficult and researchers have mainly preferred to adopt frequency-based methods.

On the other hand, the ability to quickly sketch RL by hand is invaluable in making fundamental decisions early in the design process. For integer-order systems, there are well-known practical rules for RL sketching, but those cannot be directly applied to FO systems. Moreover, the existing literature on this topic exclusively focuses on the particular case of commensurate FO systems that occur when truncating real valued integro-differential orders up to a finite precision [15, 16]. This allows the generalisation of some rules to FO systems, but limiting the precision and the type of symbolic expressions [17, 18]. The rules for commensurate FO systems do not apply to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers when dealing with RL analysis.

In this paper, we extend several practical rules, available to sketch the RL of integer-order systems, to the FO domain. The main contribution is that the practical sketching rules apply to open-loop transfer functions expressed as the ratio of FO zeros and poles, contributing to fill the gap in the existing literature about this topic. The subject is presented in a didactic perspective, being the rules applied to several examples that contribute to reduce the lack of intuition about the corresponding system dynamics.

Bearing these ideas in mind, the paper is organized as follows. Section 2 introduces fundamental concepts related to fractional calculus. Section 3 analyses several FO systems and generalises the RL rules to a class of FO systems. Finally, Section 4 draws the main conclusions.

2. Fractional Calculus

Fractional calculus (FC) denotes the branch of calculus that extends the concepts of integrals and derivatives to noninteger and complex orders [19–23]. During the last years, FC was found to play a fundamental role in the modelling of a considerable number of phenomena [24–29] and emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Nowadays, the application of FC concepts includes a wide spectrum of studies [30–33], going from the dynamics of financial markets [34, 35], biological systems [36, 37], earth sciences [38], and DNA sequencing [39] up to mechanical [40–43], electrical [44–46], and control systems [21, 24].

The generalisation of the concept of derivative and integral to noninteger orders, 𝛼, has been addressed by several mathematicians. The Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions of fractional derivative are the most used and are given, respectively, by [47] ^RL_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡_𝑎𝑓(𝜏)(𝑡−𝜏)^{𝛼−𝑛+1}𝑑𝜏,𝑛−1<𝛼<𝑛,_GL𝑎𝐷^𝛼_𝑡𝑓(𝑡)=lim_ℎ→01ℎ^{𝛼[(𝑡−𝑎)/ℎ]}_𝑘=0(−1)^𝑘𝛼𝑘𝑓(𝑡−𝑘ℎ),_𝐶𝑎𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑓^(𝑛)(𝜏)(𝑡−𝜏)^{𝛼−𝑛+1}𝑑𝜏,𝑛−1<𝛼<𝑛, (2)(3)(4) where Γ (⋅) represents Euler’s gamma function, [𝑥] is the integer part of 𝑥, and ℎ is a time step.

The Laplace transform applied to (2) yields 𝐿^RL_𝑎𝐷^𝛼_𝑡𝑓(𝑡)=𝑠^𝛼𝐿𝑓(𝑡)−^𝑛−1_𝑘=0𝑠^𝑘RL₀𝐷_𝑡^{𝛼−𝑘−1}𝑓0⁺, (5) where 𝐿 and 𝑠 denote the Laplace operator and variable, respectively, and t represents time.

The general LTI, single-input-single-output (SISO), and FO incommensurate system can be represented by [48] 𝑎_𝑛𝐷^𝛼𝑛𝑦(𝑡)+𝑎_𝑛−1𝐷^𝛼𝑛−1𝑦(𝑡)+⋯+𝑎₀𝐷^𝛼0𝑦(𝑡)=𝑏_𝑚𝐷^𝛽𝑚𝑥(𝑡)+𝑏_𝑚−1𝐷^𝛽𝑚−1𝑥(𝑡)+⋯+𝑏₀𝐷^𝛽0𝑥(𝑡), (6) where 𝑥(𝑡) and 𝑦(𝑡) represent the system input and output, respectively, 𝐷^(⋅)is the derivative operator, {𝑎_𝑝,𝑏_𝑞}∈𝐑,{𝛼_𝑝,𝛽_𝑞}∈𝐑⁺₀, 𝑝=0,…,𝑛, and 𝑞=0,…,𝑚. Besides, it is considered that 𝛼_𝑛>𝛼_𝑛−1>⋯>𝛼₀,𝛽_𝑚>𝛽_𝑚−1>⋯>𝛽₀,𝛼_𝑛>𝛽_𝑚, and 𝑎_𝑛≠0.

In the Laplace domain, (6) results in a transfer function given by the ratio of two non-integer polynomials: 𝐺(𝑠)=𝑏_𝑚𝑠^𝛽𝑚+𝑏_𝑛−1𝑠^𝛽𝑚−1+⋯+𝑏₀𝑠^𝛽0𝑎_𝑛𝑠^𝛼𝑛+𝑎_𝑛−1𝑠^𝛼𝑛−1+⋯+𝑎₀𝑠^𝛼0. (7)

If 𝛼_𝑝=𝑘_𝑝/𝑣, 𝛽_𝑞=𝑘_𝑞/𝑣, with 𝑣∈𝐑⁺and 𝑘_𝑝, 𝑘_𝑞∈𝐍₀, then (7) is a commensurate FO system and can be written as 𝐺(𝑠)=∑^𝑚_𝑞=0𝑏_𝑞𝑠^{1/𝜈𝑘𝑞}∑^𝑛_𝑝=0𝑎_𝑝𝑠^{1/𝜈𝑘𝑝}. (8)

The FO system is said to be rational if 𝑣∈𝐍.

In general, a polynomial 𝑃(𝑠^𝛼) is a multivalued function, the domain of which is a Riemann surface with an infinite number of sheets [48]. Only in the particular case of 𝛼 being rational, the number of sheets will be finite. Such type of function becomes single-valued when an appropriate cut of the complex plane is assumed. This branch cut is not unique, but the negative real axis is usually chosen. In this case, the origin of the complex plane is a branch point and the first Riemann sheet, ℘, is defined as ℘=𝑟𝑒^𝑗𝜙𝑟∈R⁺,−𝜋<𝜙<𝜋. (9)

For example, Figure 2 depicts two Riemann surfaces corresponding to the function 𝑃(𝑠^𝛼)=𝑠^𝛼+𝑏(𝛼>0,𝑏>0), the roots of which are 𝑠=𝑏^1/𝛼⋅𝑒^{𝑗(𝜋+2ℎ𝜋)/𝛼},ℎ=0,±1,±2,…;𝑗=√−1. (10)

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(b)

Figure 2 

Riemann surfaces: (a) 𝑃(𝑠𝛼)=𝑠1/2+1 has two sheets; (b) 𝑃(𝑠𝛼)=𝑠4/3+1 has three sheets.

For 𝛼=1/2 and 𝑏=1, the Riemann surface has two sheets (Figure 2(a)), and for 𝛼=4/3 and 𝑏=1, the Riemann surface presents three different sheets (Figure 2(b)). In the former case, there are no roots, and in the latter case, two roots appear on the first sheet. Riemann surfaces are important when dealing with RL of FO systems, as will be seen in Section 3.

3. Root Locus

In this section, we assume that the system open-loop transfer function is given by the following: 𝐺(𝑠)∏=𝐾^𝑚𝑞=1𝑠+𝑏_𝑞^𝛽𝑞∏^𝑛_𝑝=1𝑠+𝑎_𝑝^𝛼𝑝, (11) where 𝑎_𝑝, 𝑏_𝑞∈𝐂 and 𝛼_𝑝, 𝛽_𝑞∈𝐑⁺.

Equation (11) represents a direct extension to the FO domain of the classical integer-order problem usually addressed by control engineers when dealing with RL analysis. Rules for RL sketching applicable to this case are summarised in Table 1. Only the first Riemann sheet will be considered.

Table 1 

Practical rules for RL sketching of FO systems as defined in (11).

In the sequel, several examples are presented, namely, (i) one FO real pole; (ii) two FO real poles; (iii) one FO pole and one FO zero; (iv) a pair of FO complex conjugate poles. The RL plots are generated using the numeric algorithm presented in [17]. The application of the practical sketching rules is detailed for a few examples, and for all cases, the RL plots serve the purpose of elucidating system dynamics. This will help readers to gain intuition about system behaviour as a function of poles and zeros fractional orders.

3.1. One Fractional-Order Real Pole

In this case, the open-loop transfer function is given by 𝐺₁(𝑠)=𝐾𝑠+𝑎₁^𝛼1, (12) where the RL corresponds to the roots of the characteristic equation 𝑠+𝑎₁^𝛼1+𝐾=0,𝑠=−𝑎₁+𝐾^1/𝛼1⋅𝑒^{𝑗(2ℎ+1)⋅𝜋/𝛼1},ℎ=0,±1,±2,…. (13)(14)

In general, the RL spreads along several Riemann sheets, meaning that RL branches can begin in one sheet, cross the branch cut, and enter in another sheet. For example, considering 𝐺₁(𝑠) with 𝑎₁=2 and 𝛼₁=1.5, we verify that the characteristic (13) has roots in two Riemann sheets (Figure 3(a)). However, choosing 𝛼₁=1.6 results in roots in five different sheets (Figure 3(b)).

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(b)

Figure 3 

Root-Locus of 𝐺1(𝑠): (a) 𝑎1=2 and 𝛼1=1.5 result in RL branches in two sheets; (b) 𝑎1=2 and 𝛼1=1.6 result in RL branches in five sheets.

It is well-known that just the first Riemann sheet has physical significance [49]. As such, in the sequel, we consider only the RL branches corresponding to the first sheet.

Observing the RL of 𝐺₁(𝑠), we verify that for 0<𝛼₁<1, there are no closed-loop poles. However, for 1≤𝛼<4, several graphs are obtained, as shown in Figure 4. Starting from the integer case (𝛼₁=1) represented in Figure 4(a), as the FO pole increases, two branches emerge from the open-loop pole 𝑠=−2 and flow towards infinity (Figure 4(b)). For 𝛼₁=2, we get the classical plot with two vertical branches (Figure 4(c)). Increasing 𝛼₁ (2<𝛼₁<3), two RL branches are still observed (Figure 4(d)). When 𝛼₁=3, the well-known three branches RL occurs (Figure 4(e)), and finally, when the FO pole is in the interval 3<𝛼₁<4, four branches emerge. Larger values of the FO pole (i.e., 𝛼₁≥4) were also investigated. We concluded that the RL sketching rules also apply. The results are of the same type, and therefore, we decided not to include them.

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Figure 4 

Root locus of 𝐺1(𝑠) for 1≤𝛼1<4 and 𝑎1=2.

The practical rules apply to all FO cases. For example, for the RL shown in Figure 4(f), as 𝛼₁=3.5, the RL has four branches. The asymptotes centroid and angles are 𝜎=−2 and 𝜑=−154.3^∘, −51.4^∘, 51.4^∘, and 154.3^∘, respectively. Solving the characteristic equation for 𝑠=𝑗𝜔, the RL branches intersect the imaginary axis at 𝜔=±2.51, for 𝐾=59.2.

3.2. Two Fractional-Order Real Poles

In this subsection, we consider the open-loop transfer function given by 𝐺₂(𝑠)=𝐾𝑠+𝑎₁^𝛼1𝑠+𝑎₂^𝛼2. (15)

The RL was computed for various values of {𝛼₁,𝛼₂} (𝑎₁=2, 𝑎₂=1) and the graphs analysed. It was observed that no RL branches exist when 𝛼₁₂=𝛼₁+𝛼₂<1. Several RL examples are depicted in Figures 5 to 7 for 1≤𝛼₁₂<4. The results are presented in three groups: (i) 1≤𝛼₁₂<2; (ii) 2≤𝛼₁₂<3; (iii) 3≤𝛼₁₂<4. Similar results were observed for 𝛼₁₂≥4 and 𝑎₁<𝑎₂. For both cases, the practical sketching rules still apply.

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Figure 5 

Root locus of 𝐺2(𝑠). Cases from group (i), 1≤𝛼12<2 (𝑎1=2,𝑎2=1).

Figure 5 shows the plots from group (i). When 𝛼₁₂=1, the RL has a single branch in the real axis (Figures 5(a) and 5(b)). As 𝛼₁₂ increases (1<𝛼₁₂<2), two branches emerge from the poles 𝑠=−2 or 𝑠=−1, depending on the values of 𝛼₁ and 𝛼₂, and tend to infinity (Figures 5(c) to 5(f)).

As said in Section 3.1, all practical rules are valid for 𝐺₂(𝑠), (15). Using the case shown in Figure 5(f), for example, we have 𝛼₁+𝛼₂=1.9, meaning that the RL has two branches. As we have two open-loop poles, rule 4 must be used to determine the pole from which the branches are departing. Thus, applying the angle condition to the test points 𝑝₁ and 𝑝₂, we obtain 𝜙₁=−90^∘ and 𝜙₂=90^∘, respectively, indicating that no branches can depart from 𝑠=−2. Rule 4 can be used in all cases; nevertheless, an easier to use specific rule about RL starting and ending points still requires more research before a definitive statement.

The angle condition is also used to determine the departing angles from pole 𝑠=−1, resulting in 𝜙=±138.46^∘. The asymptotes centroid and angles are 𝜎=−1.3 and 𝜑=±94.7^∘, respectively.

Figure 6 depicts results from group (ii). When 𝛼₁₂=2, we get the plots represented in Figures 6(a) and 6(b). We observe two RL branches that, as before, depending on the values of 𝛼₁ and 𝛼₂, can depart from one or the other open-loop poles. In both cases, the branches tend to infinity with angles 𝜑=±90^∘. Increasing the value of 𝛼₁₂ (2<𝛼₁₂<3), two RL branches are still observed (Figures 6(c) to 6(h)).

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Figure 6 

Root locus of 𝐺2(𝑠). Cases from group (ii), 2≤𝛼12<3 (𝑎1=2,  𝑎2=1).

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Figure 7 

Root locus of 𝐺2(𝑠). Cases from group (iii), 3≤𝛼12<4 (𝑎1=2,  𝑎2=1).

The results from group (iii) are illustrated in Figure 7. For 𝛼₁₂=3, the RL of Figures 7(a) to 7(c) shows three branches that depart from the same or different open-loop poles and flow to infinity with angles 𝜑=180^∘ and ±60^∘. Increasing 𝛼₁₂ (3<𝛼₁₂<4), four RL branches arise (Figures 7(d) to 7(g)).

The results obtained for two FO real poles are similar to those of a single real pole. This means a similar behavior, both in terms of the number of branches and the type of RL charts, whenever 𝛼₁ and 𝛼₁₂ are close. It should be noted that the RL depends not only on the equivalent order 𝛼₁₂ (by means of rules 2, 5, or 6) but also on the FO of each pole. By other words, the same value of 𝛼₁₂ may lead to different RL.

3.3. One Fractional-Order Pole and One Fractional-Order Zero

In this case, the open-loop transfer function is given by 𝐺₃(𝑠)=𝐾𝑠+𝑏₁^𝛽1𝑠+𝑎₁^𝛼1. (16)

The RL was obtained for various values of {𝛼₁,𝛽₁} (𝑎₁=1, 𝑏₁=2) and the graphs analysed as previously. It was observed that no RL branches exist when 𝛼₁<1. Figures 8 to 10 depict several RL for 1≤𝛼₁<4. As before, for easing the comparison, the results are presented in three groups: (i) 1≤𝛼₁<2; (ii) 2≤𝛼₁<3; (iii) 3≤𝛼₁<4. Additional experiments were carried out, both for different values of the FO pole and FO zero and for 𝑎₁>𝑏₁. We concluded that the sketching rules are valid for all cases and the results are similar to those presented.

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(b)

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Figure 8 

Root locus of 𝐺3(𝑠). Cases from group (i), 1≤𝛼1<2 (𝑎1=1,  𝑏1=2).

Figure 8 shows plots from group (i). We see that each RL has two branches that depend on the difference between the orders of the denominator and numerator, 𝛿=𝛼₁−𝛽₁: when 𝛿<1, both branches converge to the open-loop zero (Figure 8(a)); if 𝛿=1, one branch converges to the open-loop zero and the other tends to infinity (on the real axis) (Figure 8(b)); for 1<𝛿<2, the two branches flow to infinity (Figure 8(c)).

Applying rule 6 to the case depicted in Figure 8(b), for all real axis points in the line ]−∞,−2], we have 𝛿=1, meaning that this line belongs to the RL. The break-in point is computed using rule 8, resulting in 𝑠=−2.3.

Figure 9 depicts several plots from group (ii), that is, 2≤𝛼₁<3. All RL still have two branches, the paths of which depend on the difference between the FO of the open-loop pole and zero (Figures 9(a) to 9(e)).

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Figure 9 

Root locus of 𝐺3(𝑠). Cases from group (ii), 2≤𝛼1<3 (𝑎1=1,  𝑏1=2).

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Figure 10 

Root locus of 𝐺3(𝑠). Cases from group (iii), 3≤𝛼1<4 (𝑎1=1,  𝑏1=2).

Several RL for group (iii), 3≤𝛼₁<4, are shown in Figure 10. It can be observed that all RL have four branches, and as before, the paths depend on the difference between the orders of the open-loop pole and zero.

3.4. One Pair of Fractional-Order Complex Conjugate Poles

The open-loop transfer function is given by 𝐺₄(𝑠)=𝐾𝑠²+2𝜉𝜔_𝑛𝑠+𝜔²_𝑛^𝛼1/2=𝐾𝑠+𝑎₁^𝛼1/2⋅𝑠+𝑎^∗₁^𝛼1/2, (17) where 𝑎₁∈𝐂 and 𝑎^∗₁ denotes the conjugate of 𝑎₁.

Plotting the RL, it can be seen that there are no branches unless 𝛼₁≥1. In Figure 11, several RL graphs are shown for 1<𝛼₁<8. Figure 11(a) depicts the RL for 𝛼₁=1.2, where we can see that there are gaps between the open-loop poles and the points were the branches initiate. Recalling that the RL can spread along several Riemann sheets, meaning that RL branches can begin in one sheet, cross the branch cut, and enter in another sheet, the gaps correspond to points not belonging to the first Riemann sheet. As in the previous examples, when 1<𝛼₁<3, the RL has two branches (Figures 11(a) to 11(c)). When 3<𝛼₁<5, the number of branches is four. Even though, for 3<𝛼₁<4, there are gaps in two branches (Figure 11(d)), and for 4<𝛼₁<5, two extra small branches depart from the open-loop poles and end close to those points, entering in another Riemann sheet (Figure 11(e)). The same qualitative behaviour is observed for 5<𝛼₁<7 (Figures 11(f) to 11(g)). Figure 11(h) depicts the RL for 7<𝛼₁<8, revealing eight branches departing from the open-loop poles.

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Figure 11 

Root locus of 𝐺4(𝑠) for 1<𝛼1<8 (𝑎1=1−𝑗2).

To conclude the analysis, we use the case shown in Figure 11(g) to underline that all RL practical rules are applicable, namely, the asymptotes centroid and angles, which are 𝜎=−1 and 𝜑=−158.8^∘, −52.9^∘, 52.9, and 158.8^∘, respectively. The angle condition is used to determine the departing angles from pole 𝑠=−1+𝑗2, resulting in the values 𝜙=−142.9, −37.1^∘, 68.8, and 174.7^∘.

4. Conclusion

The Root-Locus (RL) is a classical method for the analysis and synthesis of linear time-invariant (LTI) integer-order systems, consisting of the plot of the paths of all possible closed-loop poles as a design parameter varies in a given range. Nowadays, there are efficient numerical algorithms devoted to RL analysis, implemented by several packages. For integer-order systems, there are well-known practical rules for RL sketching, but those cannot be directly applied to FO systems, and the existing literature on this topic almost exclusively focuses on particular cases, namely, the commensurate FO systems.

This paper generalises RL practical rules to a class of FO systems, which are defined by an open-loop transfer function expressed as a ratio of FO zeros and poles. As usual, using practical rules, even though the RL sketch might result somewhat incomplete, the ability to quickly sketch RL by hand is invaluable, from the control designer viewpoint, in making fundamental decisions early in the design process.