Fractional order linear time invariant system stabilization by brute-force search

Abstract

Fractional calculus increases their applications in system design and analysis problems because of providing more realistic modeling of real systems. Owing to computational complexity of fractional calculus, the computer-aided design and analysis methods are required for engineering applications of fractional order systems. This study presents a numerical method for parametric robust stabilization of fractional order systems by employing single-parameter perturbation. This method implements a fractional order perturbation strategy on the basis of brute-force search technique for system stabilization problems. In order to meet a predefined minimum argument root design specification, the proposed algorithm searches for a desired placement of the minimum argument characteristic root within the first Riemann sheet by performing iterative perturbations of the fractional order. This approach can provide a straightforward numerical solution for robust stabilization problems of fractional order systems by employing an order perturbation scheme. Moreover, a possible utilization of a fractional order derivative operator as a system stabilizer is theoretically discussed. Illustrative examples show the utilization of the proposed stabilization algorithms for computer-aided fractional order system design applications.

Keywords

Systems linear system theory robust control stability pole placement control system design

Introduction

Fractional order system modeling is increasing importance in the solution of science and engineering problems. It has been reported that fractional order system models can provide a more accurate representation of real systems (Das, 2011). The fields of system science and control engineering have particularly benefitted from trends in the fractional order system modeling over the last decade (Chen et al., 2009; Monje et al., 2010; Petras, 2009).

Stability analysis plays a substantial role in control engineering, and several studies have accordingly addressed robust stability analysis problems of fractional order systems: Stability analysis, according to pole placement in the complex plane, has been discussed for fractional order systems (Matignon, 1996). Petras et al. (2005) presented a stability check procedure for uncertain linear time invariant (LTI) systems with interval fractional orders and coefficients and minimum argument eigenvalue based stability testing of the fractional order LTI interval systems has been shown (Chen et al., 2006). The use of linear matrix inequality (LMI) to stabilization of fractional order LTI systems has also been addressed in numerous studies (Adelipour et al., 2015; Ahn et al., 2007; Lu and Chen, 2009; N’Doye et al., 2013). The fractional order extension of the Lyapunov direct method has also been used to demonstrate the stability of nonlinear and time varying fractional order systems (Aguila-Camachoa et al., 2014). Moreover, well-established parametric, robust stability analysis methods have been applied to fractional order systems in several studies: Robust stability checking based on four Kharitonov’s polynomials has been shown for commensurate order LTI systems (Petras et al., 2004). Utilization of stability boundary locus for stabilization of feedback systems has also been carried out (Hamamci, 2007). The zero exclusion principle has considered for fractional order interval systems (Tan et al., 2009). A numerical method based on edge polynomial sampling from interval box of uncertain parameters has been demonstrated for fractional order interval systems (Senol et al., 2014).

Although several heuristic optimization methods, such as evolutionary methods (Biswas et al., 2009) and particle swarm methods (Zamani et al., 2009), have been proposed for the tuning of fractional order controllers, studies assessing the application of heuristic search methods for robust stabilization of fractional order LTI systems with interval uncertainty are quite rare in the literature. In order to develop a fractional order system design tool for engineering problems, it is very useful to employ straightforward and effective search methods for robust stabilization issues because of the difficulties involved in the development of a common analytical method that can be applied to stability analysis of any fractional order system. In fact, there is a need to develop computer-aided design tools based on numerical analysis techniques to simplify the design and analysis of such order systems for engineers. One of the aims of the present study was to develop a computational framework that enables application of brute-force search methods in the problem of robust stabilization of fractional order LTI systems for control engineering. The brute-force method is a fundamental, reliable search algorithm that tests all possible solutions in a finite solution set in order to find the most appropriate one, and it is very practical method to find out an optimal solution in discrete and finite search spaces (Haynal and Haynal, 2013). Their consistency and effectiveness in finding the best solution from a finite set of options have enabled brute-force based algorithms to be employed for computer-aided design of advanced systems (Amy et al., 2013).

A probabilistic robust stability evaluation scheme based on fractional order perturbation has previously presented for the robust stabilization of fractional order systems (Alagoz et al., 2015), and an analytical solution for robust stabilization and control was recently proposed for first-order uncertain systems by using a fractional order integrator (Hmed et al., 2016). Theoretical demonstration of system stabilization by means of single parameter perturbation promotes the development of single-parameter real-time system stabilization methodologies. However, straightforward numerical schemes or algorithms are required to achieve the online stabilization of complex fractional order systems for practical applications. In this sense, the current study also examines the utilization of closed loop $s^{α}$ compensation method as an implication of the order perturbation approach in control system stabilization problems.

Consecutive order perturbation enables directional movement of characteristic polynomial roots in the first Riemann sheet, and therefore it takes effect on the root arguments (Alagoz et al., 2015; Petras, 2009), namely, the polar angles of characteristic roots. In the current study, the author proposes a practical fractional order system stabilization scheme based on tuning of the characteristic root arguments for a given minimum root angle specification. A numerical algorithm based on the brute-force search is employed to search for a relevant fractional order perturbation satisfying the design specification. Illustrative stabilization examples are also presented to show application of the proposed method for computer-aided fractional order system design problems.

LMI-based methods for the solution of fractional order system stabilization have been extensively studied. Those types of analytical approaches present an increasing computational complexity depending on the matrix operations when system complexity grows. The matrix structures may need to be reestablished for more complex fractional order systems. Model-depended analytical solutions for stabilization problems of some fundamental forms of fractional order system models have been also proposed, however, the solutions are only valid for these fundamental forms. The proposed stabilization method-based brute-force search provides a straightforward scheme, of which algorithmic complexity does not grows severely when complexity of fractional order system models increase. This can be a key advantage when developing computer-aided system design tools. In addition, the proposed algorithm can achieve a minimum root angle specification to ensure a desired degree of robustness in terms of characteristic root locus.

Methodology

Basic definitions and remarks

Definition 1 (Hurwitz stability): A real coefficient integer order polynomial is said to be Hurwitz stable if all roots lie in the left half plane (Husek, 2008), ${p (s) | p (s) = 0 : \forall s \in C \land Re {s} < 0}$ .

Definition 2 (Hurwitz stability region and Hurwitz stability line): The left half side of the complex root plane, expressed as $π / 2 < \arg (s) < 3 π / 2$ , is called the Hurwitz stability region (HSR) and the bounds of the HSR, written as a combination of line segments with the argument of $- π / 2$ and $π / 2$ , are called the Hurwitz stability line (HSL).

Definition 3 (Fractional order polynomials and expanded degree integer order polynomials): Let us consider fractional order polynomials with real coefficients expressed in the form of $p_{f} (s) = \sum_{i = 0}^{n} c_{i} s^{α_{i}}$ , where $α_{i} \in R^{+}$ for $i = 1, 2, 3, . ., n$ , are the fractional orders of the polynomials. Setting $α_{0} = 0$ yields the constant term of polynomials. For the root locus analysis of fractional order LTI system, $s = v^{m}$ mapping has been proposed to transform fractional order characteristic polynomials to the expanded degree integer order characteristic polynomials defined in the form of $p_{m} (v) = \sum_{i = 0}^{n} c_{i} v^{m α_{i}}$ . Here, $\forall m α_{i}$ for $i = 1, 2 . ., n$ are positive integer numbers. In this manner, $s = v^{m}$ mapping facilitates the stability analysis of a fractional order system model (Chen et al., 2009; Das, 2011; Monje et al., 2010; Petras, 2009). Some fundamental remarks for further discussions can be summarized as follows:

Remark 1: Let us consider a fraction order real polynomial $p_{f} (s)$ , where the input variable is defined as $s = M e^{j θ} \in C$ . The $s = v^{m}$ transformation maps the function $p_{f} (s)$ to a complex function $p_{m} (v)$ , where $v = \tilde{M} e^{j ϕ} \in C$ such that the magnitude is $\tilde{M} = M^{(1 / m)}$ and the argument is $ϕ = θ / m$ .

Proof: This property was discussed by Petras (2009) without a need for proof. In order to better understand the mapping properties of $s = v^{m}$ transformation, reverse transformation can be written as $v = s^{1 / m} = M^{(1 / m)} e^{j θ / m}$ in the case of a complex point $s = M e^{j θ}$ . Parameters $M$ and $θ$ stand for the magnitude and angle of the point in the $s$ domain. It is evident that $s = v^{m}$ transformation maps a complex point in the $s$ domain to the point $v = \tilde{M} e^{j ϕ}$ in the $v$ domain, where the argument is $ϕ = θ / m$ and the magnitude is $\tilde{M} = M^{(1 / m)}$ .

Remark 2: In the case of $s = v^{m}$ mapping, HSR is confined to $(π / 2 m, π / m]$ for positive root arguments and $[- π / m, - π / 2 m)$ for negative root arguments.

Proof: Property 1 states that $s = v^{m}$ transformation maps a point $s = M e^{j θ}$ to the point $v = \tilde{M} e^{j ϕ}$ , where the argument is $ϕ = θ / m$ and the magnitude is $\tilde{M} = M^{(1 / m)}$ . As a consequence of this mapping, HSL is mapped from $θ_{s} = \pm π / 2$ to $ϕ_{s} = \pm π / 2 m$ . Accordingly, the HSR is mapped to $(π / 2 m, π / m]$ for positive root arguments and $[- π / m, - π / 2 m)$ for negative root arguments. Figure 1 illustrates mapping of the HSR in the $v$ -plane under $s = v^{m}$ transformation.

Problem formulation and preliminaries

This section presents a robust stabilization scheme based on the minimum root argument calculation. Fractional order LTI systems are represented by fractional order differential equations in the form of (Caponetto et al., 2010; Xue and Chen, 2014):

\begin{matrix} a_{n} D^{α_{n}} y (t) + a_{n - 1} D^{α_{n - 1}} y (t) + \dots + a_{1} D^{α_{1}} y (t) \\ + a_{0} y (t) = b_{m} D^{φ_{m}} u (t) + b_{m - 1} D^{φ_{m - 1}} u (t) + \\ \dots + b_{1} D^{φ_{1}} u (t) + b_{0} u (t) \end{matrix}

(1)

By using the property $L {D^{α} f (t)} = s^{α} F (s)$ for $f (0^{+}) = 0$ (Caponetto et al., 2010), fractional order transfer functions are expressed as:

T (s) = \frac{Y (s)}{U (s)} = \frac{\sum_{i = 0}^{m} b_{i} s^{φ_{i}}}{\sum_{i = 0}^{n} a_{i} s^{α_{i}}}

(2)

where denominator polynomial coefficients, $a_{i}$ , and numerator polynomial coefficients, $b_{i}$ , are positive real numbers. The fractional orders are denoted by $α_{i} \in R$ , $α_{i} > α_{i - 1} \geq 0$ $(i = 0, 1, 2, 3 \dots, n)$ and $φ_{i} \in R$ , $φ_{i} > φ_{i - 1} \geq 0$ $(i = 0, 1, 2, 3 \dots, m)$ (Xue and Chen, 2014). For $α_{0} = 0$ and $φ_{0} = 0$ , the transfer function has constant terms $a_{0}$ and $b_{0}$ . Since exact modeling of real systems is not always possible by means of nominal system models, system modeling with parametric uncertainty is a common way of representing the variable and uncertain nature of real systems. By considering Equation (2), the characteristic polynomial of fractional order transfer functions with interval uncertainty can be expressed as:

Δ (s) = \sum_{i = 0}^{n} (a_{i} \pm Δ a_{i}) s^{α_{i} \pm Δ α_{i}}

(3)

The interval uncertainty bounds of coefficient $a_{i}$ can be expressed by considering coefficient deviation $Δ a_{i}$ around the nominal value $a_{i}$ , which can also be expressed as: $[{\underline{a}}_{i} {\bar{a}}_{i}] = [a_{i} - Δ a_{i}, a_{i} + Δ a_{i}]$ .

In the present study, the fractional order perturbation of the system model is represented by the order deviation $\pm Δ α_{i} \in R$ .

In order to facilitate root locus analysis of fractional order LTI systems, $s = v^{m}$ mapping can be used to transform a fractional order characteristic polynomial to an expanded degree integer order characteristic polynomial (Chen et al., 2006; Radwan et al., 2009; Petras, 2009; Xue and Chen, 2014). Thus, stability analysis of fractional order models is carried out by examining the root locus of expanded degree integer order characteristic polynomials within the first Riemann sheet. By applying the $s = v^{m}$ transformation to Equation (3), expanded degree integer order characteristic polynomials are written as follows:

Δ_{m} (v) = \sum_{i = 0}^{n} (a_{i} \pm Δ a_{i}) v^{β_{i}}

(4)

where, $β_{i} \in Z$ is the expanded degree integer order that is defined as $β_{i} = m (α_{i} \pm Δ α_{i})$ , $m = {0, 1, 2 \dots}$ .

The stability analyses based on characteristic root placement in the first Riemann sheet have previously been discussed in detail (Alagoz et al., 2015; Matignon, 1996; Petras, 2009; Radwan et al., 2009). These studies showed that, by applying $s = v^{m}$ transformation to fractional order characteristic polynomials, the stability analysis can be carried out in the first Riemann sheet, which is defined in the argument range of $(- π / m, π / m)$ (Petras, 2009; Radwan et al., 2009). The first Riemann sheet includes stable and unstable root regions sliced by the stability line, which is defined by the angle of line $π / 2 m$ . The first Riemann sheet is symmetrical with respect to real axis in the complex root plane. The positive angle side HSR is limited to the root angle range $π / 2 m < ϕ \leq π / m$ . A fractional order system model is said to be stable, when all characteristic roots lie in HSR. For $m = 1$ , the stability line becomes the complex axis, separating the left and right half planes, and it refers to the case of conventional Hurwitz stability analysis of integer order LTI systems.

The current study demonstrates an application of the brute-force search method to find out fractional order perturbation $(\pm Δ α_{i})$ that stabilizes LTI systems with interval uncertainty. It is proposed for computer-aided stabilization and the design of fractional order systems. In general, the parametric uncertainty of systems can be characterized by interval coefficients of characteristic polynomials. The interval coefficient defines a hyperrectangle $(A)$ set, also called n-orthotope, in a polynomial coefficient space as:

A = {a : {\underline{a}}_{i} \leq a_{i} \leq {\bar{a}}_{i}, i = 1, 2, 3, \dots, n}

(5)

where, $a = [a_{1} a_{2} a_{3} \dots a_{n}]$ is a coefficient vector and represents a fractional order polynomial within the interval polynomial family. Hyperrectangle $A$ was also referred to as the uncertainty box. All polynomials, included in the hyperrectangle $A$ , form a family of characteristic polynomials, denoted by the set $Ω$ . Let us assume that each characteristic polynomial of $Ω$ has $ξ$ number of complex roots, denoted by $v_{r}$ in the complex plane. The complex root set of the interval polynomial family $Ω$ forms a set of roots in the first Riemann sheet, which is referred to as a root set (Senol et al., 2014), defined as:

\begin{array}{l} R (Ω) = {v_{r} : Δ_{m} (a, v_{r}) = 0 \land | \arg (v_{r}) | \\ < \frac{π}{m}, \forall a \in A, r = 1, 2, 3,,, ξ} \end{array}

(6)

The expanded degree integer order characteristic polynomials $Δ_{m} (v)$ are said to be robust stable polynomials, if the minimum argument condition $min {Arg (R (Ω))} > π / 2 m$ is satisfied (Chen et al., 2006; Petras, 2009).

For a given unstable fractional order interval transfer function, the basic steps of the robust stabilization method based on minimum argument consideration can be summarized as follows:

Step 1: Compose hyperrectangle $A$ by sampling $a_{i}$ with unit length $(δ_{i})$ and construct the root set $R (Ω)$ from the first Riemann sheet.

Step 2: Find the minimum root angle $ϕ_{min} = min | \arg (v_{r}) |$ from the root set $R (Ω)$ . If the minimum root angle is greater than the target angle $(ϕ_{T} = π / 2 m)$ , the condition $ε = (min | \arg (v_{r}) | - ϕ_{T}) > 0$ is satisfied, the system is robust stabilized, and finish the process. Otherwise, go to Step 3.

Step 3: Reduce the fractional order $α_{i}$ by $Δ α$ unit perturbation and go back to Step 1.

The fractional order perturbation takes effect on the both root angles and magnitudes, as suggested by Remark 1. Alteration of the fractional order can change root argument of commensurate order models directionally. Alagoz et al. (2015) introduced a probabilistic robust stability index and performed robust stabilization according to the robust stability probability of the system. In the current study, minimum root argument in the first Riemann sheet is utilized for robust stabilization of fractional order interval systems.

Example 1: Let us find out robust stabilizing $α$ for the unstable fractional order interval system defined in the form of $T (s) = \frac{k}{a_{2} s^{2 + α} + a_{1} s^{1 + α} + a_{0}}$ . The interval uncertain parameters are $a_{2} \in [1.2, 1.7]$ , $a_{1} \in [1.0, 2.0]$ , $a_{0} \in [0.5, 1.5]$ and the fractional order parameter is $α = 1.9$ .

The characteristic polynomial of the $T (s)$ can be written as:

Δ (s) = [1.2, 1.7] s^{2 + α} + [1.0, 2.0] s^{1 + α} + [0.5, 1.5]

(7)

By applying $s = v^{10}$ , one obtains expanded degree integer order characteristic polynomials in the form of:

Δ_{10} (v) = [1.2, 1.7] v^{20 + 10 α} + [1.0, 2.0] v^{10 + 10 α} + [0.5, 1.5]

(8)

For the robust stabilization of $T (s)$ , we configured $α^{(0)} = 1.9$ , $Δ α = 0.1$ for $m = 10$ and unit sampling lengths of interval coefficients ( $δ_{0} = 0.1$ , $δ_{1} = 0.1$ and $δ_{2} = 0.05$ ). Then, we applied the three steps of the proposed robust stabilization method. At 17^th iterations, robust stabilizing $α$ was obtained as $α^{(17)} = 0.3$ . Figures 2(a) and (b) demonstrate distribution of root set $R (Ω)$ of the unstable system (initial system) at $α^{(0)} = 1.9$ and the stabilized system at $α^{(17)} = 0.3$ , respectively. The subset in red, which is below the line $ϕ = π / 20$ , shows the roots causing system instability in Figure 2(a). After all roots in the first Riemann sheet move above the stability boundary indicated by the line $ϕ = π / 20$ , the interval system is robust stabilized according to Figure 2(b). Figures 2(c) and (d) show the decrease of minimum root angle error $(ε = min | \arg (v_{r}) | - ϕ_{T})$ and the change of $α$ , respectively.

In order to validate stabilization of the system model for $α^{(17)} = 0.3$ , Figure 3 presents the root locus of vertex and edge polynomials obtained from hyperrectangle $A$ and stable step responses of eight vertex polynomials { $V_{1} (s)$ , $V_{2} (s)$ ,. ., $V_{8} (s)$ } at $α^{(17)} = 0.3$ . According to Edge theorem, when all exposed edge polynomials are stable, the interval system is robust stable (Tepljakov, 2011). Figure 3 confirms the robust stability of the interval system for $α^{(17)} = 0.3$ . Vertex polynomial $V_{2} (s)$ is the polynomial that is the closest to stability boundary $ϕ_{T} = π / 2 m$ , and this polynomial has a stable response, as illustrated in Figure 3(b).

A possible application: system stabilization by $s^{α}$ compensator

This section demonstrates a possible utilization of fractional order derivative in the application of parametric stabilization of nominal systems. In this application, $α$ -order perturbation is used for tuning the minimum root argument of the closed loop system structure depicted in Figure 4. For positive fractional orders $(α > 0)$ , $s^{α}$ operator performs fractional order derivative, and it can be useful for the improvement of transient responses owing to the lead compensation. For negative fractional orders $(α < 0)$ , $s^{α}$ operator performs fractional order integration, and it is effective for dealing with steady state errors similar to lag compensators. A description of fractional order forms of lead-lag compensators can be found in several studies (Chen et al., 2009; Hmed et al., 2016; Tepljakov, 2011). In addition to control applications, fractional derivative operators have been used effectively for modeling complex systems (San Jose Martinez et al., 2010).

In this section, our control objective is to find $s^{α}$ compensator to meet a minimum argument characteristic root specification for closed loop system stabilization. Here, brute-force search method is employed to find out an appropriate $α$ value to reach this objective. Figure 4 depicts a block diagram of the closed loop $s^{α}$ compensator that is used for the parametric stabilization of the plant $G (s)$ .

The transfer function of the stabilizing system can be written as:

G_{f} (s) = {\begin{matrix} \frac{G (s)}{s^{α} + G (s)} & α < 0 \\ \frac{G (s)}{1 + G (s)} & α = 0 \\ \frac{s^{α} G (s)}{1 + s^{α} G (s)} & α > 0 \end{matrix}

(9)

One can assume the plant transfer function in a rational form as $G (s) = N (s) / D (s)$ . In this case, the characteristic polynomial of the closed loop system can be expressed depending on $α$ as:

Δ (s) = {\begin{matrix} s^{α} D (s) + N (s) & α < 0 \\ D (s) + N (s) & α = 0 \\ D (s) + s^{α} N (s) & α > 0 \end{matrix}

(10)

Equation (10) reveals that compensation with fractional order integrator takes effect on the part of characteristic polynomials that is associated with denominator polynomial, and the compensation with fractional order derivative affects the part of characteristic polynomials that is associated with numerator polynomial of $G (s)$ .

Figure 5 shows an algorithm implementing the brute-force search of $α$ for the stabilization of the feedback system with $s^{α}$ compensator. The algorithm searches $α$ in a predefined range $(α_{min}, α_{max})$ with a unit increment of $Δ α = 10^{- m}$ , and calculates the minimum argument $ϕ_{min}^{(n)} = min | \arg (v_{r}) |$ and corresponding angle margin $ϕ_{m}^{(n)} = ϕ_{min}^{(n)} - ϕ_{T}$ at each iteration $(n)$ . The increase of angle margin results in more robust systems in terms of stability. To find out $α$ order, where the system becomes stable, target angle $ϕ_{T}$ can be set to $π / 2 m$ , which is the argument of the HSL for $s = v^{m}$ mapping.

The system model is stable for the positive angle margin, $ϕ_{m}^{(n)} > 0$ . However, the best robust stabilizing $α_{s}$ in $(α_{min}, α_{max})$ can be obtained by searching for the positive largest angle margins, defined as follows:

α_{s} = {α^{(n_{max})} : n_{max} = max_{(n)} {ϕ_{min}^{(n)}} \land (ϕ_{m}^{(n_{max})} > 0)}

(11)

If the algorithm does not find a positive angle margin, it fails to stabilize systems in the range $(α_{min}, α_{max})$ with the increment $Δ α$ .

In practice, the closed loop $s^{α}$ compensation turns an unstable plant transfer function $G (s)$ into a stable transfer functions $G_{f} (s)$ for a desired minimum argument specification, and this can improve controller performance of practical control systems.

Example 2: Let us stabilize unstable fractional order plant model $G (s) = \frac{1}{0.8 s^{2.7} + 0.5 s^{1.3} + 1}$ by applying closed loop $s^{α}$ compensation in the $α \in (- 3.0, 3.0)$ range.

Our objective is to robust stabilize the system by locating the minimum argument characteristic root of the system far from the stability boundary. The initiation parameters of the algorithm were configured to $α_{min} = - 3.0$ , $α_{max} = 3.0$ in order to consider fractional order integrator and derivative options. Setting $m = 100$ provides two-digit precision for $α^{(n)}$ increment, so $Δ α$ is set to $0.01$ . In this search range, the characteristic polynomial configurations can be expressed as:

Δ (s) = {\begin{matrix} 0.8 s^{2.7 + α} + 0.5 s^{1.3 + α} + s^{α} + 1 & α < 0 \\ 0.8 s^{2.7} + 0.5 s^{1.3} + 1 & α = 0 \\ 0.8 s^{2.7} + 0.5 s^{1.3} + s^{α} + 1 & α > 0 \end{matrix}

(13)

Figure 6(a) shows the trace of the minimum argument characteristic root during the brute-force searching. The maximum value of root angle $(ϕ_{max})$ , which was found by the search algorithm, is indicated by the green dots on green dash lines in Figures 6(a) and (b). Figures 6(b) and (c) show the change of the minimum argument $ϕ_{min}$ and the angle margin $ϕ_{m}$ depending on the order $α^{(n)}$ . Although $α \in (0.35, 2.41)$ interval satisfies $ϕ_{m} > 0$ and stabilizes the system, as indicated in Figure 6(c), the best robust stabilizing order $α_{s}$ was obtained at 1.39 for $ϕ_{max} = 0.0186$ and $ϕ_{m}^{n_{max}} = 0.0032$ as indicated by the green dots at peaks in Figure 6. The stabilized closed loop system transfer function for $s^{1.36}$ compensator was obtained as:

G_{f} (s) = \frac{s^{1.39}}{0.8 s^{2.7} + s^{1.39} + 0.5 s^{1.3} + 1}

(14)

Figures 7(a), (b) and (c) illustrate time responses for the unstable fractional order plant function $G (s)$ , $G_{f} (s)$ for $α ≅ 0$ (the feedback system without $s^{α}$ compensation) and the stabilized $G_{f} (s)$ for $α_{s} = 1.39$ (feedback system with $s^{α}$ compensation). Consequently, Figures 6(c) and 7(c) validate the robust stabilization of unstable plant function $G (s)$ by using closed loop $s^{1.36}$ compensation.

Example 3: Let us stabilize the fractional order model of the permanent magnet synchronous motor (PMSM) velocity servo system, given as $G (s) = \frac{6.28}{0.00078 s^{1.74} + 0.097 s^{0.87} + 1}$ (Yu et al., 2016). We apply closed loop $s^{α}$ compensation by using the fractional order integrator for $α \in (- 3.0, 0)$ .

Our objective is to improve stability and control of the fractional order PMSM velocity servo system model by using a fractional order integrator and place the minimum argument characteristic root at a relevant distance from the stability boundary to enhance robust stability. The initiation parameters of the algorithm were configured to $α_{min} = - 3.0$ , $α_{max} = 0$ for the fractional order integrator. For two-digit precision $(m = 100)$ , $Δ α$ was set to $0.01$ . Figure 8(a) shows the trace of the minimum argument characteristic root during the iterative brute-force searching. The maximum value of root angle $(ϕ_{max})$ , found by the algorithm during the search, is indicated by the green dots in Figures 8(a), (b) and (c). Figures 8(b) and (c) show the change of minimum argument $ϕ_{min}$ and the angle margin $ϕ_{m}$ versus the order $α^{(n)}$ . The $α$ intervals, satisfying $ϕ_{m} > 0$ , stabilize the system, however the best robust stabilizing order in the search range was obtained at $α_{s} = - 1.11$ for $ϕ_{max} = 0.0266$ and $ϕ_{m}^{n_{max}} = 0.0109$ . This is indicated in Figure 8 via green dots at peaks. For the best stabilizing fractional order integrator, $\frac{1}{s^{1.11}}$ , the closed loop system transfer function was obtained as:

G_{f} (s) = \frac{6.28}{0.00078 s^{2.85} + 0.097 s^{1.39} + s^{1.11} + 6.28}

(12)

Figures 9(a), (b) and (c) illustrate time responses for the unstable $G_{f} (s)$ cases for $α ≅ - 3$ and $α ≅ - 2.01$ , and the stabilized $G_{f} (s)$ case for $α = - 1.11$ . Figure 9(c) clearly shows improvement of the stability and control of the fractional order model of PMSM velocity servo system.

Conclusions

This paper presented a numerical system stabilization scheme based on relocation of minimum argument characteristic roots. Illustrative examples were given to demonstrate possible applications of the proposed method for the robust stabilization of interval uncertain fractional order system and the closed loop $s^{α}$ compensation. The numerical results revealed that brute-force search technique can be implemented for the stabilization of closed loop control systems by using the $s^{α}$ compensator according to a desired minimum argument root specification. Some concluding remarks can be summarized as follows:

An increase in the complexity of fractional order system changes the characteristic polynomial. However, this does not lead to a significant modification of the algorithmic structure of the proposed method. This is an advantageous property, particularly, for the development of computer-aided design tools.

Brute-force search provides uniform searching of order perturbation with a desired resolution in a predefined range. Thus, stabilizing range of the order perturbation and the best robust stabilizer can be found for closed loop $s^{α}$ compensator design.

Computational cost strongly depends on the search range of order, $(α_{min}, α_{max})$ , and the unit increment, $Δ α$ . The root calculation is the most time-consuming operation of the algorithm. The number of root calculation operation can be estimated by $(α_{max} - α_{min}) / Δ α$ .

This is an encouraging result for the development of real-time system stabilizers including fractional order $s^{α}$ compensator loops. For control engineering applications, a future study should address the self-tuning of the $s^{α}$ compensator.

The proposed methodology can achieve replacements of system poles by using a root trace technique to meet minimum argument constraints. This is a noteworthy contribution to computer-aided system design. The computational scheme based on brute-force search is easy to implement and it is yet effective to manage stabilization of complex fractional order systems.

Footnotes

Conflict of interest statement

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Fractional order linear time invariant system stabilization by brute-force search

Abstract

Keywords

Introduction

Methodology

Basic definitions and remarks

Problem formulation and preliminaries

A possible application: system stabilization by s α compensator

Conclusions

Footnotes

Conflict of interest statement

Funding

References

A possible application: system stabilization by $s^{α}$ compensator