PID controller parameters.

## Abstract

In this chapter, we propose several Nyquist-like stability criteria for linear dynamical systems that are described by fractional commensurate order linear time-invariant (FCO-LTI) state-space equations (thus endowed with fractional-order transfer functions) by means of the argument principle for complex analysis. Based on the standard Cauchy integral contour or its shifting ones, the stability conditions are necessary and sufficient, independent of any intermediate poles computation, domain transformation, and distribution investigation, which can be implemented graphically with locus plotting or numerically without any locus plotting. The proposed criteria apply to both single and multiple fractional cases as well and can be exploited in regular-order systems without any modification. Case study is included.

### Keywords

- fractional-order
- commensurate
- stability
- meromorphic/holomorphic
- argument principle
- Cauchy integral contour

## 1. Introduction

Fractional-order calculus possesses a long history in pure mathematics. In recent decades, its involvements in systems, control, and engineering have attracted great attention; in the latest years, its significant extensions in various aspects of systems and control are frequently encountered [1, 2, 3, 4, 5, 6, 7, 8]. It turns out that phenomena modeled with fractional-order calculus much more widely exist than those based on regular-order ones. It has been shown that fractional-order calculus describes real-world dynamics and behaviors more accurately than the regular-order counterparts and embraces many more analytical features and numerical properties of the observed things; indeed, many practical plants and objects are essentially fractional-order. Without exhausting the literature, typical examples include the so-called non-integer-order system of the voltage–current relation of semi-infinite lossy transmission line [9] and diffusion of the heat through a semi-infinite solid, where heat flow is equal to the half-derivative of the temperature [10].

One of the major difficulties for us to exploit the fractional-order models is the absence of solution formulas for fractional-order differential equations. Lately, lots of numerical methods for approximate solution of fractional-order derivative and integral are suggested such that fractional-order calculus can be solved numerically. As far as fractional-order systems and their control are concerned, there are mainly three schools related to fractional-order calculus in terms of system configuration: (i) integer-order plant with fractional-order controller, (ii) fractional-order plant with integer-order controller, and (iii) fractional-order plant with fractional-order controller. The principal reason for us to bother with fractional-order controllers is that fractional-order controllers can outperform the integer-order counterparts in many aspects. For example, it has been confirmed that fractional-order PID can provide better performances and equip designers with more parametrization freedoms (due to its distributed parameter features [4, 11, 12, 13]).

An important and unavoidable problem about fractional-order systems is stability [13, 14, 15]. As is well known, stability in integer-order LTI systems is determined by the eigenvalues distribution; namely, whether or not there are eigenvalues on the close right-half complex plane. The situation changes greatly in fractional-order LTI systems, due to its specific eigenvalue distribution patterns. More precisely, on the one hand, eigenvalues of fractional-order LTI systems cannot generally be computed in analytical and closed formulas; on the other hand, stability of the fractional-order LTI systems is reflected by the eigenvalue distribution in some case-sensitive complex sectors [13, 15], rather than simply the close right-half complex plane for regular-order LTI systems. In this paper, we revisit stability analysis in fractional commensurate order LTI (FCO-LTI) systems by exploiting the complex scaling methodology, together with the well-known argument principle for complex analysis [16]. This work is inspired by the study for structural and spectral characteristics of LTI systems that is also developed by means of the argument principle [17, 18, 19]. The complex scaling technique is a powerful tool in stability analysis and stabilization for classes of linear and/or nonlinear systems; the relevant results by the author and his colleagues can be found in [20, 21, 22, 23, 24, 25]. Also around fractional-order systems, the main results of this chapter are several Nyquist-like criteria for stability with necessary and sufficient conditions [26], which can be interpreted and implemented either graphically with loci plotting or numerically without loci plotting, independent of any prior pole distribution and complex/frequency-domain facts.

Outline of the paper. Section 2 reviews basic concepts and propositions about stability in FCO-LTI systems that are depicted by fractional commensurate order differential equations or state-space equations. The main results of the study are explicated in Section 3. Numerical examples are sketched in Section 4, whereas conclusions are given in Section 5.

*Notations and terminologies of the paper*.

## 2. Preliminaries and properties in FCO-LTI systems

### 2.1 Preliminaries to fractional-order calculus

Based on [13, 15], fractional-order calculus can be viewed as a generalization of the regular (integer-order) calculus, including integration and differentiation. The basic idea of fractional-order calculus is as old as the regular one and can be traced back to 1695 when Leibniz and L’Hôpital discussed what they termed the half-order derivative. The exact definition formula for the so-called

where

Basic facts about fractional-order calculus are given as follows [13]:

• If

• If

• If

• Fractional-order differentiation and integration are linear operations. Thus

• Under some additional assumptions about

• If

The fractional-order calculus (1) and its properties are essentially claimed in the time domain. Therefore, it is generally difficult to handle these relations directly and explicitly. To surmount such difficulties, the Laplace transform of (1) is frequently used, which is given by

where

### 2.2 Definition and features of FCO-LTI state-space equations

A scalar fractional-order linear time-invariant system can be described with a fractional-order state-space equation in the form of

where

For simplicity, we employ

which is the fractional-order transfer function defined from

In the following, the fractional-order polynomial

is called the characteristic polynomial of the state-space equation (3).

We note by complex analysis ([16], p. 100) that

where

Bearing (6) in mind, our questions are (i) under what conditions

To address (i), let us return to (6) and observe for any

where we have used

To see under what conditions

where

Obviously,

Under the assumption that (7) and suppose that

Based on (6) and (9), the

By (10), it is not hard to see that

In the sequel, when the assumption (7) is true and

### 2.3 Closed-loop configuration with FCO-LTI systems

Consider the feedback system illustrated in Figure 1, in which we denote by

where

Fractional-order transfer functions for

where

Now we construct the state-space equations for the open- and closed-loop systems of Figure 1. The open-loop system can be expressed by the fractional-order state-space equation:

In the closed-loop system, we can write the closed-loop state-space equation as

where

By definition, the characteristic polynomial for the closed-loop system

where

with

Let us return to (15) and continue to observe that

In deriving (17), the determinant equivalence

which is nothing but the return difference relationship for the fractional-order feedback system

**Remark 1.** Recalling our discussion in Section 2.2 and assuming that there exists a number

## 3. Main results

### 3.1 Nyquist contours in the z -/s -domains

As another preparation for stability analysis in fractional-order systems by means of the argument principle for meromorphic functions, we need to choose appropriate Nyquist contours.

Firstly, the simply closed curve defined on the

Secondly, the simply closed curve defined on the

where

Remarks about the contours

• In both cases, the origin of the complex plane is excluded from the contours themselves and their interiors. The reason for these specific contours is that

• One might suggest that in order to detour the origin, the small arc in

### 3.2 Stability conditions related to FCO-LTI systems

Stability conditions in terms of the zeros distribution of

**Proposition 1.** Consider the fractionally commensurate system with commensurate order

where

### 3.3 Stability criterion in FCO-LTI systems

In what follows, a fractional-order polynomial

where

**Theorem 1.** Consider the fractional-order system with commensurate order

vanishes nowhere over

In the above, the clockwise/counterclockwise orientation of

*Proof of Theorem 1*. By introducing any fractional-order commensurately Hurwitz polynomial

Under the given assumption about the concerned characteristic polynomial and the fact that

Bearing these facts in mind, let us apply the argument principle to (22) counterclockwisely with

More precisely, since

where

Note that all the roots of

The above equation says that

Several remarks about Theorem 1.

• Theorem 1 is independent of the contour and locus orientations; or the locus orientations can be alternatively defined after the locus is already drawn. The fractionally commensurate Hurwitz polynomial

• When the stability locus with respect to the infinite portion of

Since

• Each and all the conditions in Theorem 1 can be implemented only by numerically integrating

• The clockwise/counterclockwise orientation of

Next, a regular-order polynomial

where

**Theorem 2.** Consider the fractional-order system with commensurate order

vanishes nowhere over

*Proof of Theorem 2.* Repeating those for Theorem 1 but in terms of

**Remark 2.** The proof arguments can also be understood by using the transformation

### 3.4 Stability criteria for closed-loop FCO-LTI systems

Based on the return difference equation (31) claimed in the feedback configuration of Figure 1, together with the argument principle, the following

**Theorem 3.** Consider the feedback system as in Figure 1 with the fractional-order subsystems

satisfies: (i)

*Proof of Theorem 3*. Under the given assumptions, the return difference equation (19) is well-defined on

To complete the proof, it remains to only show why we must work with the contour

which says clearly that if there are imaginary zeros of

On the contrary, if there exist no imaginary open-loop poles, it is not hard to see that working with

As a

**Theorem 4.** Under the same assumptions of Theorem 3; the closed-loop system is stable if and only if for any

satisfies: (i)

In the above,

Several remarks about Theorems 3 and 4:

• The shifted contour

• Clearly, the detouring treatments in Theorems 3 and 4 do not exist in Theorems 1 and 2, since the stability conditions in the latter ones are claimed directly on the fractional-order characteristic polynomials, in which transfer functions are not involved.

## 4. Numerical illustrations

### 4.1 Example description for Theorems 1 and 2

Consider a single fractional-order commensurate system [15] with the characteristic polynomial

where the commensurate order

In what follows, the

### 4.2 Numerical results for Theorems 1 and 2

The following cases are considered in terms of

•

The same conclusions can be drawn by examining the

•

The same conclusions can be drawn by examining the

•

The instability conclusions in each

•

The instability conclusions in each

•

The instability conclusions in each

•

Stability in each case of

Based on the numerical results, the stability/instability conclusions based on the

### 4.3 Example description for Theorem 3

Consider the feedback configuration of Figure 1 used for automatic voltage regulator (AVR) in generators, which is formed by the subsystems [6]:

where ^{1}, an amplifier modeled as

In the following, we focus merely on verifying the closed-loop stability based on Theorem 3, based on the parametrization results therein. To this end, the

The so-called optimal controller parameters are listed in Table 1.

Case 1 | 1.2623 | 0.5531 | 0.2382 | 1.2555 | 1.1827 |

Case 2 | 1.2623 | 0.5526 | 0.2381 | 1.2559 | 1.1832 |

### 4.4 Numerical results for Theorem 3

Based on Table 1, the stability loci in the two cases are plotted in Figure 10. The stability loci for the two cases cannot be distinguished from each other graphically. By counting the outmost circle as one clockwise encirclement around the origin, then one can count another counterclockwise encirclement after zooming into the local region around the origin; it follows that the net encirclements number is zero. Indeed, our numerical phase increment computations in either case yields that

## 5. Conclusions

Stability is one of the imperative and thorny issues in analysis and synthesis of various types of fractional-order systems. By the literature [28, 29, 30], the frequently adopted approaches are through single/multiple complex transformation such that fractional-order characteristic polynomials are transformed into standard regular-order polynomials, and then stability testing of the concerned fractional-order systems is completed by the root distribution of the corresponding regular-order polynomials. In view of the root computation feature, such existing approaches are direct in testing methodology.

In this paper, we claimed and proved an indirect approach that is meant also in the

## Acknowledgments

The study is completed under the support of the National Natural Science Foundation of China under Grant No. 61573001.

## Notes

- The fractional-order integral portion in the PID is approximated by K I / s λ + 0.0001 in order to avoid definition problem at the origin when it is in the form of K I / s λ .