Robust D-Stability Test of LTI General Fractional Order Control Systems

Reza Mohsenipour and Xinzhi Liu

Abstract—This work deals with the robust D-stability test of linear time-invariant (LTI) general fractional order control systems in a closed loop where the system and/or the controller may be of fractional order. The concept of general implies that the characteristic equation of the LTI closed loop control system may be of both commensurate and non-commensurate orders, both the coefficients and the orders of the characteristic equation may be nonlinear functions of uncertain parameters, and the coefficients may be complex numbers. Some new specific areas for the roots of the characteristic equation are found so that they reduce the computational burden of testing the robust D-stability. Based on the value set of the characteristic equation, a necessary and sufficient condition for testing the robust D-stability of these systems is derived. Moreover, in the case that the coefficients are linear functions of the uncertain parameters and the orders do not have any uncertainties, the condition is adjusted for further computational burden reduction. Various numerical examples are given to illustrate the merits of the achieved theorems.

I. Introduction

FRACTIONAL order calculus, is nowadays a well-known theory which deals with integrals and derivatives of arbitrary orders [1]. The superiority of fractional order calculus with respect to its integer counterpart is its ability to more accurate model the behaviour of many systems in the real world,such as viscoelastic materials [2], chaotic systems [3], waves propagation [4], biological systems [5], multi-agent systems[6], and human operator behaviors [7]. Further applications can be found in [8] and references therein. Moreover, because of the robustness and fast performance of fractional order controllers, their implementation are spreading widely [9]–[12].Furthermore, modeling real world systems often leads to uncertain mathematical models. Hence, the robust stability and performance analysis of fractional order control systems,where the controller and/or the system may be of fractional order, have attracted much interest from many researchers.

Over the last decade, many papers have been published to present criteria for the robust stability analysis of linear timeinvariant (LTI) fractional order systems by different methods.Some sufficient criteria using the linear matrix inequality(LMI) approach were introduced in [13]–[15]. For the robust stability of fractional order systems of commensurate orders between 0 and 2, some necessary and sufficient criteria were produced in [16], [17]. However, finding a solution for the LMI may lead to conservatism [18], [19]. Moreover, the LMI approach can be applied only to systems of commensurate order, and not to the ones of non-commensurate order. The robust stability of fractional order systems of commensurate order was also studied in [20] in the roots space of the characteristic equation. The aforementioned papers studied systems having uncertainties only in the coefficients of their characteristic equations. Some criteria for the robust stability of fractional order systems of both commensurate and noncommensurate orders with uncertainties in both the coefficients and the orders were reported in [21] using Young and Jensen inequalities, but these criteria were sufficient; not necessary and sufficient.

Another efficient tool to check the robust stability of LTI fractional order systems, which can be less conservative and also employed for systems of both commensurate and noncommensurate orders, is the value set. Initially, a sufficient criterion was introduced in [22] for the robust stability of fractional order systems with uncertainties only in the coefficients by using the value set tools and extending the zero exclusion condition to fractional order systems. Then, in works [23]–[25], some necessary and sufficient criteria were presented. Sufficient criteria were also presented in [26] for fractional order systems having multi-linear uncertainties in their coefficients. Moreover, the robust stability of fractional order systems having uncertainties in both their orders and coefficients was studied in [27]–[29].

All the above-surveyed works focused on stability of the system; not performance.D-stability analysis is a method by which one can analyze both the stability and the performance.TheD-stability of an LTI system implies that all roots of its characteristic equation lie in a desired area of the complex plane [30]. This desired area can be chosen from the open left half-plane; therefore,D-stability can embrace performance in addition to the stability because the poles of a stable factional order system lie in the open left half-plane [31, Theorem 5].By extending some results on theD-stability of integer order systems to fractional order ones, the robustD-stability of fractional order systems was first investigated in [32], and some sufficient criteria were produced. Then, necessary and sufficient criteria for the robustD-stability of fractional order systems with linear and real uncertainties only in the coefficients were reported in [33], [34]. Based on [35], if the uncertainties exist anywhere other than in the coefficients, the value set of the characteristic equation gets a nonconvex shape, and consequently the results presented in [32]–[34] are not applicable anymore.

References [32]–[34] considered only specific cases of LTI fractional order systems, i.e., systems with linear and real uncertainties only in the coefficients. Generally, in LTI fractional order systems, both the coefficients and the orders of the characteristic equation can include uncertainties [21],[28]. These uncertain coefficients and orders can also be nonlinear functions of the uncertainties [27], [36]. Moreover,the coefficients of an LTI system may be complex numbers[37], especially in aerospace applications where the dynamics of a system in different directions with real coefficients are compacted into one dynamical equation with complex coefficients (for more detail see, e.g., [38]). Furthermore, an LTI fractional order system can be of either commensurate or non-commensurate order [39]. LTI fractional order control systems which have all aforementioned features together will be referred to as a general fractional order control system throughout this paper. Regarding the aforementioned works,the existence of a necessary and sufficient condition for checking the robustD-stability of general fractional order control systems is an open problem which is tackled in this paper.

The main contribution of this paper is presenting a condition for checking the robustD-stability of LTI general fractional order control systems with the following merits. It is a sufficient and necessary condition, applicable to the systems of both commensurate and incommensurate orders with uncertainties in both the coefficients and the orders, and applicable to the systems whose coefficients may be complex numbers and also may be nonlinear functions of the uncertain parameters. Since the condition is based on the characteristic equation of the systems, it is applicable to systems described by any state-space and transfer function models. As further contributions, some new specific areas for the roots of the characteristic equation of general fractional order control systems are found so that they reduce the computational burden of testing the condition in some important cases,including for robust stability. Moreover, in the case that the coefficients are linear function of the uncertain parameters and the orders do not have any uncertainties, the condition is adjusted to further reduce computational burden.

The reminding sections of this work are organized as follows. In Section II, some definitions and preliminaries are given. The main results are provided in Section III. In Section IV, illustrative examples are presented. A conclusion is given in Section V.

II. Definitions and Preliminaries

Definition 2:Consider an LTI general fractional order system described by any model with a controller in a closed loop control system where any one of the system and the controller may be of fractional order. The characteristic equation of the corresponding general fractional order control system can be written as

whereu∈U,U?RMis a closed, non-null, and bounded set,andM,I∈N . Suppose the functions αi:U→C and βj:U→R+are continuous onUfor anyi∈W≤Iandj∈N≤I.Assume βI(u)>βi(u)>0, andfor anyu∈Uandi∈N≤I?1. The characteristic equation in (1) is of commensurate order if and only if δ(s,u) is of commensurate order for anyu∈U. Otherwise, δ (s,u) is of non-commensurate order.

Definition 3:The principal branch of the characteristic equation of a general fractional order control system such as δ(s,u) in (1) is defined as δpb(s,u)=δ(spb,u) wherespb=|s|ejarg(s). Moreover, the value set of a fractional order function such as δ(s,u) for az∈C is defined as

Remark 1 ([40]):Consider δ(s) in Definition 1 as a fractional order function of commensurate order β. It follows that the roots of δint(s) are the mapped roots of δpb(s) on the first Riemann sheet by the mapping

Definition 4:Assume thatD?C be an open set. Then, the characteristic equation stated in (1) and its corresponding general fractional order control system are said to beD-stable for au0∈Uif and only if δpb(s,u0) has no roots inDC.Moreover, they are said to be robustD-stable if and only if δpb(s,u) has no roots inDCfor allu∈U.

Definition 5:IfDmentioned in Definition 4 is defined as{s∈C|Im(s)≥0,Re(se?j?1)<0,?1∈[0,π/2)}∪{s∈C|Im(s)≤0, Re(se j?2)<0 , ?2∈[0,π/2)}, then robustD-stability and regionDare referred to as robust ?-stability and region Φ,respectively. Furthermore, ifDis defined as {s∈C|Im(s)≥0,Re(s)<σ1}∪{s∈C|Im(s)≤0,Re(s)<σ2}, then robustDstability and regionDare referred to as robust σ-stability and region Σ, respectively. Examples of the regions Φ and Σ are shown in Fig. 1 [41]. If the coefficients of the characteristic equation are real, since the roots are symmetric with respect to the real axis,Dcan be chosen symmetrically, and therefore we use ?=?1=?2and σ=σ1=σ2. In the case where ?1=?2=0 or σ1=σ2=0 , the robust ? -stability or σstability (D-stability) is equivalent to the robust stability [31,Theorem 5].

Fig. 1. Regions Φ and Σ defined in Definition 5.

Remark 2:Suppose that δ(s) stated in Definition 1 is a fractional order function of commensurate order β. According to Remark 1, it is deduced that δ(s) is ?-stable if and only if

δint(s) has no roots in {s∈C|Im(s)≥0, arg(s)≤β(π/2+?1)}a rg(s)≥?β(π/2+?2)}.

Lemma 1 (Lemma 4 of [23]):Given δ(s) as defined in Definition 1, all roots of δpb(s) lie inwhere

Lemma 2 (Theorem 1.2 of [42]):Letf(z) be analytic interior to a simple closed Jordan curve Γ and continuous and different from zero on Γ . LetKbe the curve described in thew-plane by the pointw=f(z) and let ΘΓarg(f(z)) denote the net change in arg(f(z)) as pointztraverses Γ once over in the counterclockwise direction. Then the numberpof zeros off(z) interior to Γ, counted with their multiplicities, is

that is,pis the net number of times thatKwinds about the pointw=0.

III. Main Results

On the issues related to the robust stability of fractional order systems using the value set concept, the zero exclusion condition plays a key role. In this section, some areas for the roots of the characteristic equation of a general fractional order control system are obtained, and then by using these areas, the zero exclusion condition is extended for checking the robustD-stability of general fractional order control systems.

A. Areas for the Roots

The following theorem presents areas for the roots of the characteristic equation of a general fractional order control system. These areas will be used to extend and check the zero exclusion condition of the robustD-stability of general fractional order control systems.

Theorem 1:Consider the characteristic equation of a general fractional order control system, δ(s,u), as described in that αiand βjdenote αi(u) and βj(u) for anyi∈W≤Iandj∈N≤I. Then, for anyu∈Uall roots of δpb(s,u) lie in the area {s∈C|Emin≤|s|≤Emax} where Definition 1. For simplicity, to the end of the theorem assume

Proof:The proof is presented in the Appendix.

The following theorem provides areas for the roots of the characteristic equation of a general fractional order control system on the half-line {s∈C|arg(s)=Λ,Λ ∈(?π,π]}. These areas reduce the computational burden of checking the robust ?-stability compared with those were introduced in existing works.

Theorem 2:Consider the characteristic equation of a general fractional order control system, δ(s,u), as described in Definition 1. For simplicity, to the end of the theorem assume that αiand βjdenote αi(u) and βj(u) for anyi∈W≤Iandj∈N≤I. Then, for anyu∈Uif δpb(s,u) has any roots on the half-line {s∈C|arg(s)=Λ,Λ ∈(?π,π]}, all these roots lie in the area {s∈C|arg(s)=Λ,Rmin≤|s|≤Rmax} where

Proof:The proof is presented in the Appendix.

B. Zero Exclusion Condition

The zero exclusion condition is extended for checking the robustD-stability of general fractional order control systems as a necessary and sufficient condition as follows.

Theorem 3:Consider the characteristic equation of a general fractional order control system, δ(s,u), as described in Definition 1. Suppose thatUis pathwise connected. LetD?C be an open set, given for checking the robustDstability of δ(s,u). DefineD+={s∈D|Im(s)≥0} andD?={s∈D|Im(s)≤0}. Also, define everys∈{s∈?D?|Im(s)=0, Re(s) < 0} as |s|e?jπ, rather than ?|s|. Then, δ(s,u) is robustD-stable if and only if there exists au0∈Usuch thatδ(s,u0)isD-stable, and 0δvs(z,U) for allz∈S D={s∈?D+∪?D?|Emin≤|s|≤Emax}, whereEminandEmaxare calculated through Theorem 1.

Proof: The proof is presented in the Appendix.

Remark 3: From Theorems 1 and 3, the following special cases can be concluded:

1) If {s∈C|Emin≤|s|≤Emax}?D, then δ(s,u) is robustDstable.

2) IfD∩{s∈C|Emin≤|s|≤Emax}=? then δ(s,u) is notDstable for anyu∈U.

Remark 4:Assume thatD, mentioned in Theorem 3, can be considered as a region Φ defined in Definition 5. From Theorem 2, it follows thatS Dmentioned in Theorem 3 can be replaced byS D={s∈?D+∪?D?|Rmin≤|s|≤Rmax} where?D+∪?D?=?Φ={0}∪{s∈C|arg(s)=π/2+?1,?π/2??2}.(If one wants to avoid Theorem 2 calculations, he/she can ignore this remark). It may be noted that Theorem 2 employs Λ ( Λ =π/2+?1or Λ=?π/2??2) for calculatingRminandRmaxwhile Theorem 1 does not do so for calculatingEminandEmax. Therefore,Rmax?Rmin≤Emax?Eminand accordinglyS Dintroduced here is smaller than the one introduced in Theorem 1 for checking the robust Φ-stability, and thereby causes a reduction in the computational burden of checking the robustD-stability.

Note that a region Φ, as described in Definition 5,corresponds with points of the complex plane whose overshoots are less than a specific value. Furthermore,considering Theorem 5 of [31], a fractional order system is stable if and only if its characteristic equation does not have any roots in the closed right half-plane. Hence, if ?1=?2=0,then the ?-stability is equivalent to the stability. Therefore, the robust ?-stability is an important case of the robustDstability. It is noteworthy that the works related to the robust stability analysis of fractional order systems used theorems like Theorem 1 to check the robust stability [23]–[25], [28].However, here as stated in Remark 4, using Theorem 2 causes a reduction of the computational burden (see Example 1).

In the case where the coefficients of the characteristic equation of a general fractional order control system are real numbers, the following remark reduces the computational burden of checking the robustD-stability.

Remark 5:Suppose that the functions αi(u) of the characteristic equation δ(s,u) mentioned in Theorem 3 are real-valued functions for anyi∈W≤I. Therefore, the roots of δ(s,u)are mirror symmetric with respect to the real axis, and consequentlyDcan be chosen symmetrically with respect to the real axis. Givenz∈C, it can be shown easily thatwhere “ ?” denotes the complex conjugate. Hence,S Dmentioned in Theorem 3 can be replaced byfor reducing the computational burthen of checking the robustD-stability.

Remark 6:IfD, mentioned in Theorem 3, can be considered as a region Φ defined in Definition 5, and the functionsαi(u)of the characteristic equation δ(s,u) are real-valued functions for anyi∈W≤I, thenS Dfrom Theorem 3 can be replaced by the intersection of two setsSDintroduced in Remarks 4 and 5,i.e.,

For t esting whether or not δ(s,u0) from Theorem 3 isDstable, the following remark is presented.

Remark 7:Take into account Theorem 3. Assume that none of the special cases stated in Remark 3 apply. Checking whether or not δ(s,u0) (the characteristic equation of the nominal control system) isD-stable can be performed as follows:

1) In the general case, define ?+={s∈DC|Im(s)≥0,Emin ≤|s|≤Emax}, ??={s∈DC|Im(s)≤0,Emin≤|s|≤Emax},Γ+=??+, and Γ?=???. Also, define everys∈{s∈Γ?|Im(s)=

3) In the case whereDcan be considered as a region Φ and δ(s,u0) is of commensurate order, theD-stability ofδ(s,u0)can be verified by employing Remark 2.

In Theorem 3, if δ(s,u) does not have any uncertainties in its orders, and the coefficients αi(u) are linear functions for anyi∈W≤I, the following theorem is very effective for decreasing the computational burden of the robustD-stability.

Theorem 4:Assume that the characteristic equation of a fractional order control system is

whereu∈U,U?RMis a closed, non-null, bounded, pathwise connected, and convex set, andM,I∈N. Suppose that the functions αi:U→C are linear and continuous onUfor anyi∈W≤I. Let βi∈R+for anyi∈N≤I. Then,for anyz∈C whereUEis a set includ-

ing the exposed edges ofU(for more information about the edges and the exposed edges of an uncertain set see, e.g.,[22]).

Proof: The proof is presented in the Appendix.

Remark 8:Suppose that the functions αi(u) of the characteristic equation δ(s,u) mentioned in Theorem 3 are linear functions for anyi∈W≤I. Moreover, assume that the orders of δ(s,u) do not have any uncertainties. According to Theorem 4, the expressioncan be replaced by the expressionwhich significantly reduces the computational burden of calculating the zero exclusion condition.

Note that Theorems 1–3 are applicable to: systems of both commensurate and non-commensurate orders; systems with complex coefficients; and systems with nonlinear uncertainties in both the coefficients and the orders.Moreover, Theorem 4 is applicable to: systems of both commensurate and non-commensurate orders; systems with complex coefficients; and systems with linear uncertainties only in the coefficients.

Overall, the steps of checking the robustD-stability of a general fractional order control system with the characteristic equation described in (1) can be outlined as follows:

1) CalculateEminandEmaxthrough Theorem 1.

2) If one of the special cases stated in Remark 3 is applicable, determine the robustD-stability of δ(s,u) through Remark 3. Otherwise, go to the next step.

3) For au0∈U, investigate whether or not δ(s,u0) isDstable using Remark 7. If δ(s,u0) isD-stable, go to the next step. Otherwise, δ (s,u) is not robustD-stable.

4) Based on Theorem 3, Remarks 4–6, determineS Dfor which the zero exclusion condition should be checked.

5) According to Remark 8, determine whether the zero exclusion condition should be checked forUorUE.

6) Plot δvs(z,U) or δvs(z,UE), depending on the previous step, for allz∈S D. In the case of plotting δvs(z,U), choose appropriate number of the vectorsufrom bothUand its edges such that ? δvs(z,U) is recognizable clearly.

7) Regarding Theorem 3, determine whether δ (s,u) is robustD-stable or not.

IV. Illustrative Examples

In this section, three numerical examples are given to verify the obtained results in this paper. Example 1 investigates the robust stability of a closed loop control system whose characteristic equation is a fractional order function of commensurate order with real coefficients and linear uncertainties only in the coefficients. The efficiency of Theorem 2 is shown in this example. In Example 2, a fractional order controller of incommensurate order is suggested to σ-stabilizing a space tether system whose characteristic equation coefficients are nonlinear and realvalued functions of uncertainties. Finally, Example 3 studies the robust ?-stability of motion control system of a satellite whose characteristic equation has complex coefficients, and uncertainties exist in both the orders and the coefficients. The criteria introduced in the literature are challenged in Examples 2 and 3, while the presented theorems in this paper overcome the challenges well.

Example 1:In [25], the robust stability of the fractional order system

with the fractional order PI controllerC(s)=5+0.5s?0.4in a closed loop control system has been studied. Let the aim be to check the robust ?-stability of the closed loop control system for ?=0, that it is equivalent to the robust stability, by using the results presented in this paper. The characteristic equation of the closed loop control system is obtained as

whereu∈Uand

Fig. 2. Drawing δ vs(z,UE) for all z ∈S D in Example 1.

Example 2:Consider the deployment of a tethered satellite from a space shuttle studied in [43]. Let λ is the normalized length of the tether such that λ=1 when the tether deploys completely. Also, allow θ andTbe the pitch angle and the tether tension as the input, respectively. The differential equations of this system have been introduced in [43].Supposing 10 percent uncertainties in the system parameters and defining the state vectorx=[x1,x2,x3,x4]T=[λ?1,λ˙,θ,θ˙]T, the state-space equations of the system are formed as

whereu∈Uand

The goal is to deploy the tether from the initial stateto the final state[1,0,0,0]Tin a fast and low overshoot manner. Let us achieve this goal by σ-stabilizing the system for σ=?0.3. Consider a f ractional order control law as

δ(s,u)is of non-commensurate order. In the following, it is demonstrated that the closed loop control system is robust σstable. For using Theorem 1, one can calculate max|αi(u)| and min|αi(u)|similar to the way described in Example 1.According to Theorem 1, all roots of δpb(s,u) lie in the area{s∈C|Emin≤|s|≤Emax} whereEmin=0.1052 andEmax=1.0878×106. Letu0=[3,2,?2,?3]T. According to Remark 7,consider ?+and Γ+as displayed in Fig. 3. When the pointstraverses Γ+once over in the direction indicated, the path due to δpb(s,u0) denoted byK+is obtained as drawn in Fig. 4. It is visible thatK+does not pass the origin, and also the net number of times thatK+winds about the origin in the clockwise direction is zero, i.e.,p+=0. Moreover, δ(s,u0) is analytical inside Γ+and continuous on it. Therefore, regarding Remark 7, δ(s,u0) is σ-stable. Regarding Theorem 3 and Remark 5, the value set should be plotted for the setSD={s∈C|Im(s)=0,?0.3≤Re(s)≤?0.1052}∪{s∈C|Re(s)=?0.3,0 ≤Im(s)≤1.0878×106}. Note that Remark 8 can not be applied here because the coefficients of δ(s,u) have a nonlinear structure. The value set δvs(z,U) for 400 vectorsu,chosen fromUand its edges, per element ofS Dwith the step 0.005 over Re(s) or Im(s), depending on which is varying, is plotted in Fig. 5. It is seen from Fig. 5 that the value set does not include the origin. Hence, according to Theorem 3, the closed loop control system is robust σ-stable.

Fig. 3. Curve Γ + for checking the σ -stability of δ (s,u0) in Example 2.

Fig. 4. The path obtained by δ pb(s,u0) when s traverses Γ+ in Fig. 3.

Fig. 5. Plotting δ vs(z,U) for all z ∈S D in Example 2.

Let us compare the results achieved in this paper with those were published in the literature. There is no condition by which the control system studied in this example can be analyzed for the robustD-stability. However, someone can use approximations to be able to employ the theorems introduced in [32] to test the robustσ-stability of the characteristic equation in (5). For this, imagine thatis approximated as 0.14. With this approximation, the characteristic equation δ(s,u) in (5) transforms to a fractional order function of commensurate order β =0.02. Consider

Regarding part 1 of Theorem 3.7 of [32], the closed loop control system is robust σ- stable if δint(s,u) is robust stable.Because δint(s,u) is an integer order polynomial with the order 157, investigating its stability is too insufferable, but using the condition presented here and a simple graphical approach it was illustrated that the control system is robust σ-stable, without approximating. Now, Assume thatcan be approximated as 0.1. Using the theorems presented in this paper,it can be illustrated that the closed loop control system is not robust σ-stable. In this case, using Theorem 3.7 of [32] does not provide any result, because the control system is not robust σstable, and Theorem 3.7 presents just sufficient, not necessary and sufficient, conditions for the robust σ-stability. Note that the results presented in [33], [34] are associated with systems whose coefficients are equal to the uncertain parameters and accordingly are not applicable here where the coefficients are nonlinear functions of the uncertain parameters.

For the numerical simulation of the closed loop control system, consider a 220 km orbit altitude, 1 .1804×10?3rad/s orbital rate, and 100 km tether length [44]. The tether length obtained from numerical simulations is plotted in Fig. 6 with red solid line for 10 vectorsu∈U. In [44], an integer order control law was suggested by which the results of the simulations are also plotted in Fig. 6 with blue dashed line.From Fig. 6, it follows that the responses of the closed loop control system by the fractional order control law are faster and have lower overshoot than the integer order one.

Example 3:The characteristic equation of motion of a satellite, obtained from nonlinear equations by linearizing as stated in [38], is

Fig. 6. The responses of system (3) by the fractional order control law (4)and the integer order one suggested in [44].

whereq1=0.1646,q2=0.1583, andq3=2.3292. Consider the state feedback control inputwhere

T0λ,xc, andrespectively are a parameter corresponding to the thrust force of motor, displacement, and fractional order derivative operator with the orderq4∈[0.3,0.5]. By applyingT0λ, and embedding some uncertainties in the parametersq1,q2, andq3, the characteristic equation of the closed loop control system is obtained as

whereu∈Uand

Due to the changes of the orderu4, δ(s,u) is a fractional order function of both commensurate and non-commensurate orders. Let the aim be to check the robust ?-stability of the closed loop control system for ?1=?2=0.05π. Using Theorem 2, all roots of δpb(s,u) lie in the area{s∈C|Emin≤|s|≤Emax} whereEmin=0 andEmax=4.7004.Supposeu0=[0.1,0.3,2.5,0.4]T. Regarding Remark 7,consider ?+, ??, Γ+, and Γ?as shown in Fig. 7. Whenstraverses Γ+and Γ?once over in the directions indicated, the paths due to δpb(s,u0) respectively denoted byK+andK?are obtained as plotted in Fig. 8. As it is specified in Fig. 8, a window zooming in around the origin of Fig. 8 is also displayed in Fig. 9. From Figs. 8 and 9, it is visible thatK+andK?do not pass the origin, but the net number of times thatK+andK?wind about the origin in the clockwise direction is 0 and 2, respectively. Therefore,p+=p?=0 does not hold. It follows that δ(s,u0) is not ?-stable, and accordingly, the closed loop control system is not robust ?-stable.

Although it was determined that δ(s,u) is not robust ?stable, let us plot the value set of δ(s,u). One can use Remark 4, but let us ignore that here. Regarding Theorem 3, the value set should be plotted forS D={s∈C|arg(s)=0.55π,0<|s|≤4.7004}∪{0}∪{s∈C|arg(s)=?0.55π,0<|s|≤4.7004}. Note that Remark 8 can not be used here. The value set δvs(z,U) for 800 vectorsu, chosen fromUand its edges, per element ofS Dwith the step 0 .05 over |s| is depicted in Fig. 10. It can be seen that the value set includes the origin. Therefore, in addition to that δ(s,u0) is not ?-stable, the zero exclusion condition is not also held.

Fig. 7. Curves Γ+ and Γ? for checking the ? -stability of δ (s,u0) in Example 3.

Fig. 8. The path obtained by δ pb(s,u0) when s traverses Γ + and Γ ? in Fig. 7.

Fig. 9. Zoomed in around the origin of Fig. 8.

It is notable that the general fractional order control system studied in this example is of both commensurate and incommensurate orders with complex coefficients, and uncertainties in both the coefficients and the orders. There is no condition to analyze theD-stability of such systems in the literature while the condition presented in this paper is applicable to these systems.

Fig. 10. Plotting δ vs(z,U) for all z ∈S D in Example 3.

V. Conclusion

This paper has investigated the robustD-stability test of LTI general fractional order control systems. The characteristic equation of these systems may be of both commensurate and non-commensurate orders, may have complex coefficients,and may have uncertainties in both its coefficients and its orders. Moreover, the uncertainties can have a nonlinear structure. For the roots of the characteristic equation, some new specific areas have been found. These areas reduce the computational burden of testing the robustD-stability in some important cases, specially in the robust stability case. The zero exclusion condition has been extended for the robustDstability of these systems, and a necessary and sufficient condition has been derived. Furthermore, in the case that the coefficients have a linear structure and the uncertainties do not exist in the orders, the condition has been adjusted for further computational burden reduction. Three numerical examples have been studied to verify the merits of the presented results.For future works, extending the results achieved here to systems with time delays and also deriving conditions for designing a robustlyD-stabilizing fractional order controller may be considered. It is notable that since theD-stability deals with the location of roots of the characteristic equation, it is not extendable to linear parameter-varying systems and nonlinear systems.The authors would like to thank Dr. Mohsen Fathi Jegarkandi, with the Department of Aerospace Engineering,Sharif University of Technology, Tehran, Iran, for his support.

Acknowledgment

Appendix

Proof of Theorem 1:BecauseU?RMis a bounded and closed set, according to Theorem 17.T of [45], it is a compact set. Moreover, since αiand βjare continuous for anyi∈W≤Iandj∈N≤I, regarding Theorem 17.T of [45], there are finite values for min|αi|, max|αi|, minβj, and maxβjoveru∈Ufor anyi∈W≤Iandj∈N≤I. Now, forandstated in Lemma 1, regarding that the orders βjare uncertain, one can write

It is notable thatso minimizingandinvolves maximizing their powers. Using Lemma 1 onδ(s,u)for anyu∈Uand regarding the relations provided forE1min,E2min, andEmin, it follows that for anyu∈Uthe function δpb(s,u) does not have any roots in the area {s∈C||s|Emax}.

Proof of Theorem 2:It is obvious that ifthen δ(0,u)=0, and consequentlyRmin=0. For proving the rest of the theorem, it is sufficient to prove that if|s|>R1max,R2max,R3max,R4max,R5max,R6maxholds and also if|s|0. For the sake of brevity in the proof, let us prove only two cases |s|>R1maxand |s|

Hence, for we have |δpb(s,u)|>0, we must have for anyu∈U

Furthermore, according to the triangle inequality, we can write

Case |s|>R1max: Iffc=1, for anyu∈Uone can write

It is notable that iffc=0, one cannot drive (8), and accordingly it cannot be obtained some result forR1max.Therefore,R1max=∞. Supposingfc=1 the proof can be pursued for anyu∈Uas follows:

Case |s|

By defining

one has

Proof of Theorem 3:For proving this theorem, the following lemmas are needed. Here, forS1,S2?C defineand

Lemma 3 (Theorem 4 of [46]):AssumeQis a bounded and open subset of CNwhereN∈N. Suppose thatis a continuous mapping in its domain and analytical for anys∈Q. Further, suppose thatholds for anys∈?Qandt∈[0,1]. Then,h(s,0) andh(s,1) have the same number of zeros inQ.

Lemma 4:Consider the characteristic equation of a general fractional order control system, δ(s,u), as described in Definition 1. Suppose thatUis pathwise connected. LetD?C be an open set. Iffor allz∈S D={s∈?D+∪?D?|Emin≤|s|≤Emax} , whereEminandEmaxare define as in Theorem 1, andD+andD?as in Theorem 3, then δpb(s,u1) and δpb(s,u2) have the same number of roots inDCfor anyu1,u2∈U.

Proof:SinceUis a pathwise connected set, there exists the continuous functionu:[0,1]→Usuch thatu(0)=u1andu(1)=u2for anyu1,u2∈U. LetUs={u(t)|t∈[0,1]}.Usis a δpb(s,u) does not have any roots inE={s∈DC||s|>Emax} for anyu∈Us, and also ifholds for allthen it holds for allz∈?D, too. ConsiderQ={s∈DC?D||s|

Now, the proof of Theorem 3 can be presented as follows:Necessary condition: regarding Lemma 4, δpb(s,u) for anyu∈Uhas no roots inDC. Therefore, δ (s,u) is robustD-stable.Sufficient condition: according to Definition 4, if δ(s,u) is robustD-stable, thenfor anys∈{?D+∪?D?}?DCandu∈U.

Proof of Theorem 4:By definingu=[u1,u2,...,uM]T, the functions αi(u) for anyi∈W≤Ican be rewritten as

where αim∈C for anyi∈W≤Iandm∈W≤M. For as∈C andwe have

Substituting (22) in (2) and employing (23, one gets

From (24) it is seen that the real part is related with the imaginary part linearly, and viceversa. It follows that (24),which is a linear mapping fromUto C, is a polygon whose exposed edges and vertices are the mapping of the exposed edges and the vertices ofU. Therefore,can be obtained fromUE. Hence,for any