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Exact stability test and stabilization for fractional systems
Systems & Control Letters 85 (2015) 95–99
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Exact stability test and stabilization for fractional systems J.Y. Kaminski a , R. Shorten b,∗ , E. Zeheb c,d a
Department of Applied Mathematics, Faculty of Sciences, Holon Institute of Technology, Holon, Israel
b
Optimization, Control and Decision Science, IBM Research, Ireland
c
Faculty of Sciences, Holon Institute of Technology, Holon, Israel
d
Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel
article
abstract
info
Article history: Received 10 September 2014 Received in revised form 24 July 2015 Accepted 14 August 2015
In this paper we point out a connection between regular chains and the stability of fractional order systems. This observation leads to an elementary test for the stability of commensurate fractional systems. © 2015 Published by Elsevier B.V.
Keywords: Fractional order systems Stability Regular chains algorithms
1. Introduction We consider a fractional order system of the following form: dα X
= AX + BU dt α Y = CX + DU
(1)
and transfer matrix
−1
H (s) = Y (s)U (s)−1 = C sα I − A
B + D,
(2)
where α is a real number and ‘‘s’’ is the Laplace transform complex variable. These systems are a generalization of the regular linear time-invariant (LTI) systems, where α = 1 is the only legitimate value. Such systems arise, and are important, in two situations. The first is when the physics of the system itself is fractional, e.g., electrical circuits using fractional capacitors [1,2] and inductors [3], infinite transmission lines [4], Brownian motion [5]. The second situation is when we wish to design a fractional controller [6–8], for an integer order system or for a fractional order system, in order to increase the versatility of the performance of the controller. The interested reader is referred to the recent book on this topic [9] for a more detailed description of such applications, and background material on fractional systems.
∗
Corresponding author. E-mail address: [email protected] (R. Shorten).
http://dx.doi.org/10.1016/j.sysconle.2015.08.005 0167-6911/© 2015 Published by Elsevier B.V.
Our prime interest here focuses on commensurate fractional systems; namely of the form of Eq. (2) as opposed to the case where the terms in the polynomial describing the dynamic system have exponents that are not scalar multiples of each other. Before proceeding with our study, we mention that significant results have been recently obtained for both commensurate and non-commensurate fractional systems. A description of the recent directions in non-commensurate systems can be found in [10,11] (and the references therein). Much of the recent work on commensurate systems involves the use of linear matrix inequalities (LMI); see [12,13] for an overview of some of this work. A nice feature of using LMI to study fractional systems is their ability to accommodate uncertainty as in [14–16] in a relatively straightforward manner. In the contact of this latter comment, it is worth asking the question as to why develop new techniques that go beyond the use of LMI? While LMI are flexible tools, it is worth to mention that in order to check the validity of a linear matrix inequality numerical computations are required. This always involves some degree of approximation, and interpreting results in some physical manner, is always an issue in using numerical techniques. In contrast our approach is rather classical. As explained below, the tools developed in this paper involve only symbolic computations, which by definition, do not generate any numerical error and which always return the right answer. Further, the use of semi-algebraic systems offers the possibility to develop stabilization methods for fractional systems that mirror some of the backstepping approaches used in the nonlinear systems literature. In the remainder of this paper we use the term ‘fractional system’ to mean a ‘commensurate fractional system’. In this
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J.Y. Kaminski et al. / Systems & Control Letters 85 (2015) 95–99
context our contribution in this note is to show that elementary conditions for the stability of such systems can be expressed using results on regular chains for semi-algebraic sets. These in turn lead to exact stability conditions that can be checked easily. 2. Discussion of results and preliminary material In this section we present some basic concepts that are needed for the general presentation in this paper. Bounded-input–bounded-output (BIBO) stability of (1) requires that all the poles of the system should lie in the open left half complex plane. Assuming no pole-zero cancellation in H (s), the condition for stability becomes
det sα I − A ̸= 0
in Re [s] ≥ 0.
(3) Fig. 1. Stability region in the p-plane.
Denote sα = p = ρ ejθ ;
θ ∈ [−π , π].
(4)
Then, condition (3) becomes the requirement that the sector
−απ 2
≤θ ≤
απ
(5)
2
in the p-plane is devoid of zeros of the polynomial det[pI − A] [17,18]. For α ≥ 2, this condition is obviously not satisfied and the system is un-stable. For α = 1, the system is not fractional and the testing of the stability condition is carried out by a simple Routh–Hurwitz, or equivalent, criteria. However, for the intervals 0 < α < 1 and 1 < α < 2, the testing of the stability condition is more complicated from the computational complexity point of view. As we have already mentioned, the existing methods are mainly using LMI methods [12,13]. In the next sections we describe a new method which is relatively simple from the computational complexity viewpoint, and is carried out with a finite number of algebraic steps. For the interval 1 < α < 2 the test is just a Routh–Hurwitz criterion for a complex coefficients polynomial. For the interval 0 < α < 1 the test is based on the Regular Chains algorithm using resultants. This latter condition exploits the fact that stability is a test for ‘emptiness’ of a solution set, and this notion can be easily investigated in the context of semi algebraic systems. Finally, we note that if the system is unstable, it might be possible to stabilize the system by static output feedback. A simple algebraic method to determine all the intervals of the feedback gain which stabilize a certain class of such systems is given in [19]. It is observed that this result can also be used to stabilize certain fractional systems. This is discussed in Section 5. 2.1. Stability of fractional systems The stability condition of the system (1) is equivalent to: P (s) = det(sα I − A) ̸= 0
for ℜ(s) ≥ 0.
When α ≥ 2, the system is known to be unstable. If α = 1, one would use the classical Routh theorem to design a stability test. It remains to investigate the cases: 1 < α < 2 and 0 < α < 1. Case 1 (1 < α < 2): Using (4), condition (3) for stability becomes
det pI − A ̸= 0
in
−απ 2
≤θ ≤
απ 2
.
(6)
All the zeros of the polynomial det[pI − A] must lie in the lined area of the p-plane as depicted in Fig. 1. Since the coefficients of det[pI − A] are real, so that if p = pi is a zero of the polynomial, then so is p = p∗i , condition (6) is equivalent to requiring that all
Fig. 2. Equivalent stability region in the p-plane.
the zeros of det[pI − A] must lie in the lined half plane depicted in Fig. 2. Now the transformation: p = ej
π (α−1) 2
z
(7)
will rotate the axes of the p-plane by an angle of π2 (α − 1). Then (6), or (3), become equivalent to the fact that the following complex polynomial in the variable z:
det ej
π (α−1) 2
zI − A
(8)
is Hurwitz or in other words has no root in the half-plane defined by Re [z ] ≥ 0. This condition can clearly be checked by a finite number of simple algebraic steps using for example the Routh–Hurwitz algorithm for polynomials with complex coefficients (see [20]). Comment: Eq. (8) appears as Eq. (35) in the related Ref. [18]. Note however, that there are differences in the manner in which this equation is being used in the cited reference. Our method to solve the problem is to find a manipulation to enable the use of the Routh–Hurwitz criteria. The observation that enables this is the fact that requiring all zeros to be inside the sector is equivalent to requiring all zeros to be in a half plane. In [16], the method to solve the problem is by LMI, which is facilitated by a rotation of the sector lines. Thus, the methods in our paper and in [16] are absolutely different, even though our Eq. (8) appears in both manuscripts. Comment: In the spirit of the previous comment, we note that similar directions of research to that presented here can be found (implicitly) in [21]. In this book, Jury briefly deals with conditions for the roots of system’s characteristic equation to lie in a certain sector in the left half of the s-plane. While the discussion deals with
J.Y. Kaminski et al. / Systems & Control Letters 85 (2015) 95–99
regular systems and is qualitatively different from our discussion, the method suggested can, almost certainly, be extended to the case of Fractional systems in the case where 1 < α < 2. Case 2 (0 < α < 1): Here the previous approach fails. However one can use concepts from semi-algebraic systems to obtain the following results. Indeed write p = x + iy where x, y ∈ R. Then define: f (x, y) = ℜ det pI − A
ℑ det pI − A
and g (x, y) =
. With these notations, the system is unstable if
and only if the following system has at least one solution:
f (x, y) = 0 g (x, y) = 0 y + β x ≥ 0 −y + β x ≥ 0 where β = tan( απ ). To understand this result it is necessary 2 to introduce the concept of regular chains for semi-algebraic systems [22]. Comment: Note that the approach in Case 2 can also be used to solve Case 1 as well. 2.2. Regular chains for semi-algebraic sets Consider first a field k of characteristic zero, that is a field that contains the set of rational numbers. Let K be the algebraic closure of k. Recall that a field is said to be algebraically closed if any non constant polynomial with coefficients in this field has a root in the field. The algebraic closure of a given field is by definition the smallest field that satisfies this property. Consider the ring of polynomials Pn = k[x1 , . . . , xn ] with ordered variables x1 < · · · < xn . For each non-constant polynomial p ∈ Pn , we shall denote by mvar (p), init (p) and mdeg (p) respectively the greatest variable appearing in p (called the main variable), the leading coefficient or the initial and the degree of p seen as a univariate polynomial in mvar (p). A finite subset T ⊂ Pn is called a triangular set if no element of T is constant and all elements have pair-wise distinct main variables. It is clear that triangular sets are polynomial counterpart of triangular linear systems. In that sense, they are concise representation of their own solutions. The idea behind triangular decomposition is that given a polynomial system defined by a set of polynomials F ⊆ Pn , the set of solutions can be represented by a finite of triangular sets T1 , . . . , Te , in the sense that a point a = (a1 , . . . , an ) ∈ Kn satisfies F (a) = 0 if and only if Ti (a) = 0 for some i ∈ {1, . . . , e}. However a complication raises here since not all triangular sets are easy to work with. For instance consider the case: T = {x21 − x1 , x1 x2 − 1, (x1 − 1)x3 + x2 }. The main variables are respectively x1 , x2 and x3 . This shows that it is indeed a triangular set. From the first equation, we deduce that either x1 = 0 or x1 = 1. Both cases lead to a contradiction. Thus the set of solution is empty. In order to avoid this kind of situation, one shall introduce the concept a regular chain. Before that we need some further background. First recall, given a set of polynomials in Pn , say S ⊂ Pn , the ideal generated by S is, by definition, the set of combinations of elements of S, with coefficients that are themselves polynomials. Usually this ideal is denoted in the literature by ⟨S ⟩. Concisely, this can be written as
⟨S ⟩ = {f ∈ Pn /∃k, ∃f1 , . . . , fk ∈ S , ∃g1 , . . . , gk ∈ Pk , f = g1 f1 + · · · gk fk }. Furthermore we shall need the notion of a saturated ideal which we will define for a triangular set. Let hT be the product of all initials of elements in T : hT =
t ∈T
init (t ).
97
Then we define the saturated ideal of T , denoted by sat(T ), as n ⟨T ⟩ : h∞ T = {f ∈ Pn /∃n, fhT ∈ ⟨T ⟩}. Here ⟨T ⟩ denotes the ideal generated by T . We can rephrase this definition, by saying that the saturated ideal generated by T is the set of polynomials f for which the product f times hnT belongs to the mere ideal ⟨T ⟩ generated by T for some power n. The notation ⟨T ⟩ : h∞ T which might seem cumbersome, but remains in fact that this notion is a particular case of a more general concept called quotient ideal, which is irrelevant here. Then the quasi-component, say W (T ), of T is defined as V (T ) \ V (hT ), where V (S ) denotes the zero set of S in Kn for a polynomial subset S ⊂ Pn . One can prove that the Zariski closure [23] of W (T ) satisfies W (T ) = V (sat(T )) (recall that the Zariski closure denotes [23] the smallest algebraic that contains W (T )). This algebraic set is precisely the zero set of the ideal sat(T ). In order to grasp the idea underpinning the definition of the quasicomponent, let us observe that a point a lies in W (T ) if it is a zero of the original system T , while not vanishing any initial of any element of T . Such a point is called a regular zero of T . Now we are in position to define what a regular chain is. Definition 1. A triangular set T is called a regular chain if either T is empty or T \ {t } is a regular chain, where t is the polynomial in T with the maximal main variable, and the initial of t is regular (that is neither zero nor a zero divisor) modulo sat(T \ {t }). In order to make the last condition clear, let us mention that t is a zero divisor modulo sat(T \ {t }) if there exists a polynomial g ̸∈ sat(T \ {t }) such that the product gt lies in sat(T \ {t }). Concretely this means that the zero set of t cuts V (sat(T \ {t })) in such a way that makes the dimension of V (sat(T )) strictly smaller than the dimension of V (sat(T \ {t })). Regular chains have several desirable properties. One of the most important one is that the locally closed set W (T ) is guaranteed to be non-empty if T is a regular chain. This is the reason why, we shall define a triangular decomposition of a system F of polynomial equations over k, as a set of regular chains T1 , . . . , Te , such that V (F ) = ∪ei=1 W (Ti ). The concept of triangular set, regular chains, and triangular decomposition can be generalized to the case of semi-algebraic systems. For this purpose, one has first to define the concept of a regular semi-algebraic set. The details here are more cumbersome and we refer the reader to [22] for an exhaustive presentation. For our purpose, we shall only state the following result extracted from [22], upon which we shall rely. Theorem 1. Given a semi-algebraic system S, one can decompose it into finitely many regular semi-algebraic systems S1 , . . . , Se such that a point in Rn is a solution of S if and only if it is solution of Si for some i. Using this decomposition provides us a way to formulate a general theorem for a stability test of fractional systems. 3. Main result Theorem 2. A fractional dynamical system with fractional index α > 0: dα X
= AX + BU dt α Y = CX + DU is stable if and only if the following semi-algebraic set
f (x, y) = 0 g (x, y) = 0 y + β x ≥ 0 −y + β x ≥ 0
(9)
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J.Y. Kaminski et al. / Systems & Control Letters 85 (2015) 95–99
where f (x, y) = ℜ det (x + iy)I − A and g (x, y) = ℑ det (x + iy)I − A and β = tan( απ ) has an empty triangular decomposition. 2 Proof. As mentioned above, the stability of the system is equivalent to the statement that no pole of the transfer function lies in the sector − απ ≤ θ ≤ απ . This former condition is easily seen 2 2 to be equivalent to the emptiness of the semi-algebraic set defined in the theorem. According to Theorem 1, this is equivalent to the emptiness of the triangular decomposition. In [22], the authors give an example that illustrates how regular chains can be used in order to study the stability of a classical dynamical system (not fractional) depending on a single positive parameter. Here we have shown how stability of fractional dynamical systems can be established using regular chains. Moreover, we will show in Section 5, how to get benefit of regular chains to perform the stabilization of the system. 4. Examples In order to illustrate the concepts presented in the text, we now give some examples. We begin with the relatively simple case, and then move on to the case dealt with using regular chains. Example 1 (1 < α < 2). Let α =
−1 A= −1
4 3
and
+1 . −1 π
π
Then p = s 3 = ej 2 (α−1) z = ej 6 z. The condition for stability (8) becomes π
det(ej 6 zI − A) ̸= 0
for ℜ(z ) ≥ 0.
(10)
This condition reduces to z 2 + 2e
−j π
6
z + 2e
−j π
̸= 0 for ℜ(z ) ≥ 0.
3
This condition is satisfied if and only if [20] Var(1, ∆1 , ∆2 ) = 0 where ∆1 = 2 cos( π6 ) and where
2 cos
∆2 = det
π
0
6
2 sin
π
1
2 cos
0
−2 sin
3π 3
π
3 π . 2 sin 6 π
2 cos
6
Here Var(σ1 , .., σn ) denotes the number of sign changes in the sequence of real numbers σ1 , .., σn . Since ∆1 > 0, and ∆2 = 3, it follows that Var(1, ∆1 , ∆2 ) = 0 and stability is ensured. However, if α = 5/3 instead of α = 4/3, we have: p = s5/3 = ejπ/3 z and condition (8) becomes
det ejπ /3 zI − A ̸= 0
in Re [z ] ≥ 0,
which reduces to z 2 + 2e−jπ/3 z + 2e−j2π/3 ̸= 0 Here [20],
∆1 = 2 cos
π 3
=1
in Re [z ] ≥ 0.
2 cos
∆2 = det
π
0
3
1
2 cos
2 sin 2π 3
0
−2 sin
2π
3
2π
3 π . 2 sin 3 π
2 cos
3
Thus, Var(1, ∆1 , ∆2 ) = Var(1, 1, −1) = 1 ̸= 0 and the system is unstable. Example 2 (0 < α < 1). We now give an example for the more complicated situation. The computations have been done with Maple using the tool boxes ‘‘RegularChains’’ and ‘‘SemiAlgebraicSetTools’’, that implement the formalism of regular chains for semi-algebraic systems defined over the field of rational numbers Q. Consider the matrix: 5 −4 −1
A=
−1 2
−3
−3 −1 . 8
Note that A is a non-singular matrix. Fix α = 2/3. Consider now the polynomial Q (p) = det(pI − A) = f (x, y) + ig (x, y), where p = x + iy, f (x, y) = x3 − 15x2 − 3xy2 + 56x + 15y2 + 10 and g (x, y) = 3x2 y − 30xy − y3 + 56y. For the actual computations with Maple, we need to consider a ), given by β = 70 226/40 545. rational approximation of tan( απ 2 Then we consider the semi-algebraic set:
4
and
f ( x, y ) = 0 g ( x, y ) = 0 y + βx ≥ 0 −y + β x ≥ 0.
(11)
We are now in a position to compute a triangular decomposition of this set, invoking the Maple command ‘‘RealTriangularize’’. In that case, the decomposition is non-empty showing that there is a real solution to the system (11), or in other words, that the system in this example is unstable. More precisely, we get the following triangular decomposition:
2 (4y + 19)x − 20y2 − 155 = 0 16y6 + 456y4 + 3249y2 − 3941 = 0 129 744y5 + 1 232 568y3 + 2 927 349y + 1 123 616y4 + 14 045 200y2 + 41 363 114 > 0 −129 744y5 − 1 232 568y3 − 2 927 349y + 1 123 616y4 + 14 045 200y2 + 41 363 114 > 0. Note that we did not solve the system (11); rather, we only tested the emptiness of the set of solutions. On the other hand, if the value of α would be 1/16 = 0.0625, then the triangular decomposition would be empty, indicating that there is no real solution to the system (11) and therefore the dynamical system under consideration is stable. 5. Stabilization of fractional systems We now conclude the paper by noting that standard and well known results can be used to stabilize fractional feedback systems. Consider the class of feedback systems depicted in Fig. 3 where G(p) describes a transfer function matrix of a multivariable fractional system with n inputs and n outputs according to (1) and (2). p is defined in (4) and 0 < α < 2. Each output of G(p) is
J.Y. Kaminski et al. / Systems & Control Letters 85 (2015) 95–99
99
SISO fractional systems. A method to do the same for a general multivariable fractional system can be deduced from the results in [24]. Acknowledgment
Fig. 3. Feedback stabilization.
fed back to some input (one output to one input), multiplied by a common finite gain (−k). Thus In is the unit matrix. The closed loop transfer function matrix is given by H (p) = k[In + kG(p)]−1 G(p) and the characteristic equation is F (p, k) = det[In + kG(p)] = 0. As is well known [18], the condition for bounded-input–boundedoutput stability of the closed loop system is that the polynomial F (p, k) does not vanish in
−
απ 2
≤θ ≤
απ 2
,
(12)
where θ is the argument of p. The following problem has been solved by a simple algebraic method [19]: find all and only intervals of k such that condition (12) is satisfied. In other words, find all the feasible values of the gain k which stabilize the fractional system (1). Obviously, if a specific performance criterion is given, then computations should be performed to determine an optimal gain from the set of all feasible gains which is described by the method above. Comment: Another formulation of the problem is by noting that condition (12) is equivalent to det[pI − Ac ] not zero in −(απ /2) ≤ θ ≤ (απ /2) where Ac is the A-matrix of the closed loop system, and is given by Ac = A − B(k−1 I + D)−1 C .
(13)
For example, let A be as in Example 1 with α = 5/3 which is the unstable case. Let also n = 2 and
1 B= 0
0 1
0 C = 1
1 0
D = 0.
Using the method in [19], it is easily obtained that all and only values of feedback gain in the interval 2/3 < |k| <
√ 2
will stabilize the system. 6. Conclusion In this paper we point out a connection between regular chains and the stability of fractional order systems. This observation leads to an elementary and exact test for the stability of such systems. We use this to develop a stabilization procedure for a class of
The second author was supported in part by Science Foundation Ireland grant 11/PI/1177. References [1] G.W. Bohannan, Analog fractional order controller in temperature control application, in: Proc. of the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, July 2006. [2] P. Varshney, P. Gupta, G.S. Visweswaran, New Switched capacitor fractional order integrator, J. Active Passive Electron. Dev. 2 (3) (2007) 187–197. [3] I. Schafer, K. Kruger, Modelling of coils using fractional derivatives, J. Magn. Magn. Mater. 307 (1) (2006) 91–98. [4] T. Clarke, B.N.N. Achar, J.W. Hanneken, Mittag-Leffler functions and transmission lines, J. Molecular Liquids 114 (2004) 159–163. [5] M.D. Ortigueira, A.G. Batista, On the relation between the fractional Brownian motion and the fractional derivatives, Phys. Lett. A 372 (7) (2008) 958–968. [6] C. Bonnet, J.R. Partington, Analysis of fractional delay systems of retarded and neutral type, Automatica 38 (2002) 1133–1138. [7] Shankar P. Bhattacharyya, Aniruddha Datta, Ming-Tzu Ho, Structure and Synthesis of PID Controllers, Springer-Verlag, London, 2000. [8] P. Lanusse, V. Pommier, A. Oustaloup, Fractional control system design for a hydraulic actuator, in: Proc. of the First IFAC Conference on Mechatronics Systems, Mechatronic 2000, Darmstadt, Germany, September 2000. [9] C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu-Batlle, Fractional-Order Systems and Controls, in: Springer Series: Advances in Industrial Control, 2010. [10] J. Sabatier, C. Farges, J.C. Trigeassou, A stability test for non-commensurate fractional order systems, Systems Control Lett. 62 (2013) 739–746. [11] J. Trigeassou, A. Benchellal, N. Maamri, T. Poinot, A frequency approach to the stability of fractional differential equations, Trans. Syst. Signals Dev. 4 (1) (2009) 1–25. [12] M. Rivero, S.V. Rogosin, J.A. Tenreiro Machado, J.J. Trujillo, Stability of fractional order systems, Math. Probl. Eng. 2013 (2013) 15. Article ID 356215. [13] C. Farges, M. Moze, J. Sabatier, Pseudo state feedback stabilization of commensurate fractional order systems, Automatica 46 (10) (2010) 1730–1734. [14] J.C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations, Signal Process. 91 (3) (2011) 437–445. [15] J.G. Lu, Y.Q. Chen, Robust stability and stabilization of fractional order interval systems with the fractional order α : The 0 < α < 1 case, IEEE Trans. Automat. Control 55 (1) (2010) 152–158. [16] C. Farges, J. Sabatier, M. Moze, Fractional order polytopic systems: robust stability and stabilization, Adv. Differential Equations (35) (2001). [17] D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Enginnering in Systems Applications, 1996, pp. 963–968. [18] J. Sabatier, M. Moze, C. Farges, LMI stability conditions for fractional order systems, Comput. Math. Appl. 59 (2010) 1594–1609. [19] E.I. Jury, E. Zeheb, An algebraic algorithm for determining the desired gain of multivariable feedback systems, Internat. J. Control 37 (5) (1983) 1081–1094. [20] M. Marden, Geometry of Polynomials, second ed., AMS, Providence, Rhode Island, 1966, p. 179. Th. (40.1). [21] E. Jury, Inners and Stability of Dynamic Systems, Krieger, 1982. [22] C. Chen, J.H. Davenport, J.P. May, M.M. Maza, B. Xia, R. Xiao, Triangular decomposition of semi-algebraic systems, J. Symbolic Comput. 49 (2013) 3–26. [23] B. Hassett, Introduction to Algebraic Geometry, Cambridge University Press, 2007. [24] E. Walach, E. Zeheb, Generalized zero sets of multiparameter polynomials and feedback stabilization, IEEE Trans. Circuits Syst. CAS 29 (1) (1982) 15–23.
Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Exact stability test and stabilization for fractional systems J.Y. Kaminski a , R. Shorten b,∗ , E. Zeheb c,d a
Department of Applied Mathematics, Faculty of Sciences, Holon Institute of Technology, Holon, Israel
b
Optimization, Control and Decision Science, IBM Research, Ireland
c
Faculty of Sciences, Holon Institute of Technology, Holon, Israel
d
Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa, Israel
article
abstract
info
Article history: Received 10 September 2014 Received in revised form 24 July 2015 Accepted 14 August 2015
In this paper we point out a connection between regular chains and the stability of fractional order systems. This observation leads to an elementary test for the stability of commensurate fractional systems. © 2015 Published by Elsevier B.V.
Keywords: Fractional order systems Stability Regular chains algorithms
1. Introduction We consider a fractional order system of the following form: dα X
= AX + BU dt α Y = CX + DU
(1)
and transfer matrix
−1
H (s) = Y (s)U (s)−1 = C sα I − A
B + D,
(2)
where α is a real number and ‘‘s’’ is the Laplace transform complex variable. These systems are a generalization of the regular linear time-invariant (LTI) systems, where α = 1 is the only legitimate value. Such systems arise, and are important, in two situations. The first is when the physics of the system itself is fractional, e.g., electrical circuits using fractional capacitors [1,2] and inductors [3], infinite transmission lines [4], Brownian motion [5]. The second situation is when we wish to design a fractional controller [6–8], for an integer order system or for a fractional order system, in order to increase the versatility of the performance of the controller. The interested reader is referred to the recent book on this topic [9] for a more detailed description of such applications, and background material on fractional systems.
∗
Corresponding author. E-mail address: [email protected] (R. Shorten).
http://dx.doi.org/10.1016/j.sysconle.2015.08.005 0167-6911/© 2015 Published by Elsevier B.V.
Our prime interest here focuses on commensurate fractional systems; namely of the form of Eq. (2) as opposed to the case where the terms in the polynomial describing the dynamic system have exponents that are not scalar multiples of each other. Before proceeding with our study, we mention that significant results have been recently obtained for both commensurate and non-commensurate fractional systems. A description of the recent directions in non-commensurate systems can be found in [10,11] (and the references therein). Much of the recent work on commensurate systems involves the use of linear matrix inequalities (LMI); see [12,13] for an overview of some of this work. A nice feature of using LMI to study fractional systems is their ability to accommodate uncertainty as in [14–16] in a relatively straightforward manner. In the contact of this latter comment, it is worth asking the question as to why develop new techniques that go beyond the use of LMI? While LMI are flexible tools, it is worth to mention that in order to check the validity of a linear matrix inequality numerical computations are required. This always involves some degree of approximation, and interpreting results in some physical manner, is always an issue in using numerical techniques. In contrast our approach is rather classical. As explained below, the tools developed in this paper involve only symbolic computations, which by definition, do not generate any numerical error and which always return the right answer. Further, the use of semi-algebraic systems offers the possibility to develop stabilization methods for fractional systems that mirror some of the backstepping approaches used in the nonlinear systems literature. In the remainder of this paper we use the term ‘fractional system’ to mean a ‘commensurate fractional system’. In this
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J.Y. Kaminski et al. / Systems & Control Letters 85 (2015) 95–99
context our contribution in this note is to show that elementary conditions for the stability of such systems can be expressed using results on regular chains for semi-algebraic sets. These in turn lead to exact stability conditions that can be checked easily. 2. Discussion of results and preliminary material In this section we present some basic concepts that are needed for the general presentation in this paper. Bounded-input–bounded-output (BIBO) stability of (1) requires that all the poles of the system should lie in the open left half complex plane. Assuming no pole-zero cancellation in H (s), the condition for stability becomes
det sα I − A ̸= 0
in Re [s] ≥ 0.
(3) Fig. 1. Stability region in the p-plane.
Denote sα = p = ρ ejθ ;
θ ∈ [−π , π].
(4)
Then, condition (3) becomes the requirement that the sector
−απ 2
≤θ ≤
απ
(5)
2
in the p-plane is devoid of zeros of the polynomial det[pI − A] [17,18]. For α ≥ 2, this condition is obviously not satisfied and the system is un-stable. For α = 1, the system is not fractional and the testing of the stability condition is carried out by a simple Routh–Hurwitz, or equivalent, criteria. However, for the intervals 0 < α < 1 and 1 < α < 2, the testing of the stability condition is more complicated from the computational complexity point of view. As we have already mentioned, the existing methods are mainly using LMI methods [12,13]. In the next sections we describe a new method which is relatively simple from the computational complexity viewpoint, and is carried out with a finite number of algebraic steps. For the interval 1 < α < 2 the test is just a Routh–Hurwitz criterion for a complex coefficients polynomial. For the interval 0 < α < 1 the test is based on the Regular Chains algorithm using resultants. This latter condition exploits the fact that stability is a test for ‘emptiness’ of a solution set, and this notion can be easily investigated in the context of semi algebraic systems. Finally, we note that if the system is unstable, it might be possible to stabilize the system by static output feedback. A simple algebraic method to determine all the intervals of the feedback gain which stabilize a certain class of such systems is given in [19]. It is observed that this result can also be used to stabilize certain fractional systems. This is discussed in Section 5. 2.1. Stability of fractional systems The stability condition of the system (1) is equivalent to: P (s) = det(sα I − A) ̸= 0
for ℜ(s) ≥ 0.
When α ≥ 2, the system is known to be unstable. If α = 1, one would use the classical Routh theorem to design a stability test. It remains to investigate the cases: 1 < α < 2 and 0 < α < 1. Case 1 (1 < α < 2): Using (4), condition (3) for stability becomes
det pI − A ̸= 0
in
−απ 2
≤θ ≤
απ 2
.
(6)
All the zeros of the polynomial det[pI − A] must lie in the lined area of the p-plane as depicted in Fig. 1. Since the coefficients of det[pI − A] are real, so that if p = pi is a zero of the polynomial, then so is p = p∗i , condition (6) is equivalent to requiring that all
Fig. 2. Equivalent stability region in the p-plane.
the zeros of det[pI − A] must lie in the lined half plane depicted in Fig. 2. Now the transformation: p = ej
π (α−1) 2
z
(7)
will rotate the axes of the p-plane by an angle of π2 (α − 1). Then (6), or (3), become equivalent to the fact that the following complex polynomial in the variable z:
det ej
π (α−1) 2
zI − A
(8)
is Hurwitz or in other words has no root in the half-plane defined by Re [z ] ≥ 0. This condition can clearly be checked by a finite number of simple algebraic steps using for example the Routh–Hurwitz algorithm for polynomials with complex coefficients (see [20]). Comment: Eq. (8) appears as Eq. (35) in the related Ref. [18]. Note however, that there are differences in the manner in which this equation is being used in the cited reference. Our method to solve the problem is to find a manipulation to enable the use of the Routh–Hurwitz criteria. The observation that enables this is the fact that requiring all zeros to be inside the sector is equivalent to requiring all zeros to be in a half plane. In [16], the method to solve the problem is by LMI, which is facilitated by a rotation of the sector lines. Thus, the methods in our paper and in [16] are absolutely different, even though our Eq. (8) appears in both manuscripts. Comment: In the spirit of the previous comment, we note that similar directions of research to that presented here can be found (implicitly) in [21]. In this book, Jury briefly deals with conditions for the roots of system’s characteristic equation to lie in a certain sector in the left half of the s-plane. While the discussion deals with
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regular systems and is qualitatively different from our discussion, the method suggested can, almost certainly, be extended to the case of Fractional systems in the case where 1 < α < 2. Case 2 (0 < α < 1): Here the previous approach fails. However one can use concepts from semi-algebraic systems to obtain the following results. Indeed write p = x + iy where x, y ∈ R. Then define: f (x, y) = ℜ det pI − A
ℑ det pI − A
and g (x, y) =
. With these notations, the system is unstable if
and only if the following system has at least one solution:
f (x, y) = 0 g (x, y) = 0 y + β x ≥ 0 −y + β x ≥ 0 where β = tan( απ ). To understand this result it is necessary 2 to introduce the concept of regular chains for semi-algebraic systems [22]. Comment: Note that the approach in Case 2 can also be used to solve Case 1 as well. 2.2. Regular chains for semi-algebraic sets Consider first a field k of characteristic zero, that is a field that contains the set of rational numbers. Let K be the algebraic closure of k. Recall that a field is said to be algebraically closed if any non constant polynomial with coefficients in this field has a root in the field. The algebraic closure of a given field is by definition the smallest field that satisfies this property. Consider the ring of polynomials Pn = k[x1 , . . . , xn ] with ordered variables x1 < · · · < xn . For each non-constant polynomial p ∈ Pn , we shall denote by mvar (p), init (p) and mdeg (p) respectively the greatest variable appearing in p (called the main variable), the leading coefficient or the initial and the degree of p seen as a univariate polynomial in mvar (p). A finite subset T ⊂ Pn is called a triangular set if no element of T is constant and all elements have pair-wise distinct main variables. It is clear that triangular sets are polynomial counterpart of triangular linear systems. In that sense, they are concise representation of their own solutions. The idea behind triangular decomposition is that given a polynomial system defined by a set of polynomials F ⊆ Pn , the set of solutions can be represented by a finite of triangular sets T1 , . . . , Te , in the sense that a point a = (a1 , . . . , an ) ∈ Kn satisfies F (a) = 0 if and only if Ti (a) = 0 for some i ∈ {1, . . . , e}. However a complication raises here since not all triangular sets are easy to work with. For instance consider the case: T = {x21 − x1 , x1 x2 − 1, (x1 − 1)x3 + x2 }. The main variables are respectively x1 , x2 and x3 . This shows that it is indeed a triangular set. From the first equation, we deduce that either x1 = 0 or x1 = 1. Both cases lead to a contradiction. Thus the set of solution is empty. In order to avoid this kind of situation, one shall introduce the concept a regular chain. Before that we need some further background. First recall, given a set of polynomials in Pn , say S ⊂ Pn , the ideal generated by S is, by definition, the set of combinations of elements of S, with coefficients that are themselves polynomials. Usually this ideal is denoted in the literature by ⟨S ⟩. Concisely, this can be written as
⟨S ⟩ = {f ∈ Pn /∃k, ∃f1 , . . . , fk ∈ S , ∃g1 , . . . , gk ∈ Pk , f = g1 f1 + · · · gk fk }. Furthermore we shall need the notion of a saturated ideal which we will define for a triangular set. Let hT be the product of all initials of elements in T : hT =
t ∈T
init (t ).
97
Then we define the saturated ideal of T , denoted by sat(T ), as n ⟨T ⟩ : h∞ T = {f ∈ Pn /∃n, fhT ∈ ⟨T ⟩}. Here ⟨T ⟩ denotes the ideal generated by T . We can rephrase this definition, by saying that the saturated ideal generated by T is the set of polynomials f for which the product f times hnT belongs to the mere ideal ⟨T ⟩ generated by T for some power n. The notation ⟨T ⟩ : h∞ T which might seem cumbersome, but remains in fact that this notion is a particular case of a more general concept called quotient ideal, which is irrelevant here. Then the quasi-component, say W (T ), of T is defined as V (T ) \ V (hT ), where V (S ) denotes the zero set of S in Kn for a polynomial subset S ⊂ Pn . One can prove that the Zariski closure [23] of W (T ) satisfies W (T ) = V (sat(T )) (recall that the Zariski closure denotes [23] the smallest algebraic that contains W (T )). This algebraic set is precisely the zero set of the ideal sat(T ). In order to grasp the idea underpinning the definition of the quasicomponent, let us observe that a point a lies in W (T ) if it is a zero of the original system T , while not vanishing any initial of any element of T . Such a point is called a regular zero of T . Now we are in position to define what a regular chain is. Definition 1. A triangular set T is called a regular chain if either T is empty or T \ {t } is a regular chain, where t is the polynomial in T with the maximal main variable, and the initial of t is regular (that is neither zero nor a zero divisor) modulo sat(T \ {t }). In order to make the last condition clear, let us mention that t is a zero divisor modulo sat(T \ {t }) if there exists a polynomial g ̸∈ sat(T \ {t }) such that the product gt lies in sat(T \ {t }). Concretely this means that the zero set of t cuts V (sat(T \ {t })) in such a way that makes the dimension of V (sat(T )) strictly smaller than the dimension of V (sat(T \ {t })). Regular chains have several desirable properties. One of the most important one is that the locally closed set W (T ) is guaranteed to be non-empty if T is a regular chain. This is the reason why, we shall define a triangular decomposition of a system F of polynomial equations over k, as a set of regular chains T1 , . . . , Te , such that V (F ) = ∪ei=1 W (Ti ). The concept of triangular set, regular chains, and triangular decomposition can be generalized to the case of semi-algebraic systems. For this purpose, one has first to define the concept of a regular semi-algebraic set. The details here are more cumbersome and we refer the reader to [22] for an exhaustive presentation. For our purpose, we shall only state the following result extracted from [22], upon which we shall rely. Theorem 1. Given a semi-algebraic system S, one can decompose it into finitely many regular semi-algebraic systems S1 , . . . , Se such that a point in Rn is a solution of S if and only if it is solution of Si for some i. Using this decomposition provides us a way to formulate a general theorem for a stability test of fractional systems. 3. Main result Theorem 2. A fractional dynamical system with fractional index α > 0: dα X
= AX + BU dt α Y = CX + DU is stable if and only if the following semi-algebraic set
f (x, y) = 0 g (x, y) = 0 y + β x ≥ 0 −y + β x ≥ 0
(9)
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where f (x, y) = ℜ det (x + iy)I − A and g (x, y) = ℑ det (x + iy)I − A and β = tan( απ ) has an empty triangular decomposition. 2 Proof. As mentioned above, the stability of the system is equivalent to the statement that no pole of the transfer function lies in the sector − απ ≤ θ ≤ απ . This former condition is easily seen 2 2 to be equivalent to the emptiness of the semi-algebraic set defined in the theorem. According to Theorem 1, this is equivalent to the emptiness of the triangular decomposition. In [22], the authors give an example that illustrates how regular chains can be used in order to study the stability of a classical dynamical system (not fractional) depending on a single positive parameter. Here we have shown how stability of fractional dynamical systems can be established using regular chains. Moreover, we will show in Section 5, how to get benefit of regular chains to perform the stabilization of the system. 4. Examples In order to illustrate the concepts presented in the text, we now give some examples. We begin with the relatively simple case, and then move on to the case dealt with using regular chains. Example 1 (1 < α < 2). Let α =
−1 A= −1
4 3
and
+1 . −1 π
π
Then p = s 3 = ej 2 (α−1) z = ej 6 z. The condition for stability (8) becomes π
det(ej 6 zI − A) ̸= 0
for ℜ(z ) ≥ 0.
(10)
This condition reduces to z 2 + 2e
−j π
6
z + 2e
−j π
̸= 0 for ℜ(z ) ≥ 0.
3
This condition is satisfied if and only if [20] Var(1, ∆1 , ∆2 ) = 0 where ∆1 = 2 cos( π6 ) and where
2 cos
∆2 = det
π
0
6
2 sin
π
1
2 cos
0
−2 sin
3π 3
π
3 π . 2 sin 6 π
2 cos
6
Here Var(σ1 , .., σn ) denotes the number of sign changes in the sequence of real numbers σ1 , .., σn . Since ∆1 > 0, and ∆2 = 3, it follows that Var(1, ∆1 , ∆2 ) = 0 and stability is ensured. However, if α = 5/3 instead of α = 4/3, we have: p = s5/3 = ejπ/3 z and condition (8) becomes
det ejπ /3 zI − A ̸= 0
in Re [z ] ≥ 0,
which reduces to z 2 + 2e−jπ/3 z + 2e−j2π/3 ̸= 0 Here [20],
∆1 = 2 cos
π 3
=1
in Re [z ] ≥ 0.
2 cos
∆2 = det
π
0
3
1
2 cos
2 sin 2π 3
0
−2 sin
2π
3
2π
3 π . 2 sin 3 π
2 cos
3
Thus, Var(1, ∆1 , ∆2 ) = Var(1, 1, −1) = 1 ̸= 0 and the system is unstable. Example 2 (0 < α < 1). We now give an example for the more complicated situation. The computations have been done with Maple using the tool boxes ‘‘RegularChains’’ and ‘‘SemiAlgebraicSetTools’’, that implement the formalism of regular chains for semi-algebraic systems defined over the field of rational numbers Q. Consider the matrix: 5 −4 −1
A=
−1 2
−3
−3 −1 . 8
Note that A is a non-singular matrix. Fix α = 2/3. Consider now the polynomial Q (p) = det(pI − A) = f (x, y) + ig (x, y), where p = x + iy, f (x, y) = x3 − 15x2 − 3xy2 + 56x + 15y2 + 10 and g (x, y) = 3x2 y − 30xy − y3 + 56y. For the actual computations with Maple, we need to consider a ), given by β = 70 226/40 545. rational approximation of tan( απ 2 Then we consider the semi-algebraic set:
4
and
f ( x, y ) = 0 g ( x, y ) = 0 y + βx ≥ 0 −y + β x ≥ 0.
(11)
We are now in a position to compute a triangular decomposition of this set, invoking the Maple command ‘‘RealTriangularize’’. In that case, the decomposition is non-empty showing that there is a real solution to the system (11), or in other words, that the system in this example is unstable. More precisely, we get the following triangular decomposition:
2 (4y + 19)x − 20y2 − 155 = 0 16y6 + 456y4 + 3249y2 − 3941 = 0 129 744y5 + 1 232 568y3 + 2 927 349y + 1 123 616y4 + 14 045 200y2 + 41 363 114 > 0 −129 744y5 − 1 232 568y3 − 2 927 349y + 1 123 616y4 + 14 045 200y2 + 41 363 114 > 0. Note that we did not solve the system (11); rather, we only tested the emptiness of the set of solutions. On the other hand, if the value of α would be 1/16 = 0.0625, then the triangular decomposition would be empty, indicating that there is no real solution to the system (11) and therefore the dynamical system under consideration is stable. 5. Stabilization of fractional systems We now conclude the paper by noting that standard and well known results can be used to stabilize fractional feedback systems. Consider the class of feedback systems depicted in Fig. 3 where G(p) describes a transfer function matrix of a multivariable fractional system with n inputs and n outputs according to (1) and (2). p is defined in (4) and 0 < α < 2. Each output of G(p) is
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99
SISO fractional systems. A method to do the same for a general multivariable fractional system can be deduced from the results in [24]. Acknowledgment
Fig. 3. Feedback stabilization.
fed back to some input (one output to one input), multiplied by a common finite gain (−k). Thus In is the unit matrix. The closed loop transfer function matrix is given by H (p) = k[In + kG(p)]−1 G(p) and the characteristic equation is F (p, k) = det[In + kG(p)] = 0. As is well known [18], the condition for bounded-input–boundedoutput stability of the closed loop system is that the polynomial F (p, k) does not vanish in
−
απ 2
≤θ ≤
απ 2
,
(12)
where θ is the argument of p. The following problem has been solved by a simple algebraic method [19]: find all and only intervals of k such that condition (12) is satisfied. In other words, find all the feasible values of the gain k which stabilize the fractional system (1). Obviously, if a specific performance criterion is given, then computations should be performed to determine an optimal gain from the set of all feasible gains which is described by the method above. Comment: Another formulation of the problem is by noting that condition (12) is equivalent to det[pI − Ac ] not zero in −(απ /2) ≤ θ ≤ (απ /2) where Ac is the A-matrix of the closed loop system, and is given by Ac = A − B(k−1 I + D)−1 C .
(13)
For example, let A be as in Example 1 with α = 5/3 which is the unstable case. Let also n = 2 and
1 B= 0
0 1
0 C = 1
1 0
D = 0.
Using the method in [19], it is easily obtained that all and only values of feedback gain in the interval 2/3 < |k| <
√ 2
will stabilize the system. 6. Conclusion In this paper we point out a connection between regular chains and the stability of fractional order systems. This observation leads to an elementary and exact test for the stability of such systems. We use this to develop a stabilization procedure for a class of
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