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Periodic polynomials in the analysis of periodic cyclic systems
IFAC Workshop on Adaptation and Learning in Control and Signal Processing, and IFAC Workshop on Periodic Control Systems, Yokohama, Japan, August 30 – September 1, 2004
PERIODIC POLYNOMIALS IN THE ANALYSIS OF PERIODIC CYCLIC SYSTEMS S. Bittanti, P. Colaneri
Politecnico di Milano Dipartimento di Elettronica e Informazione Piazza Leonardo da Vinci 32, 20133 Milano (Italy) [email protected] [email protected]
Abstract: In this paper, we consider polynomials with periodic coefficients in discrete time, and we define some basic elements of periodic polynomials algebra, in particular the periodic Bezout identity and the notion of (weak and strong) coprimeness. With these notions, we treat some basic problems of periodic systems theory. Specifically, by the notion of periodic cyclic matrix, we takle the question of the existence of companion and canonical forms. Keywords: Periodic Systems, Periodic Cyclic Matrix, Companion Form, Coprimeness of Periodic Polynomials, Reachability and Observability
1. INTRODUCTION
The research activity in periodic control over more than three decades is witnessed by the papers in the Preprints of this IFAC workshop, as well as in the Proceedings of the previous and first workshop on the same subject organized in 2001 in Cernobbio-Como (Italy) (Bittanti, Colaneri, 2002). For survey papers, the interested reader is referred to (Bittanti, 2000) and (Bittanti, Colaneri, 1999). However, the polynomials approach to periodic systems is still in its early days and, in the authors’ opinion, it’s a subject deserving more attention. For a pioneer paper the reader is referred to (Colaneri, Kucera, Longhi, 2003). Herein we focus on the use of polynomials in some basic problems of periodic system theory, so providing a contribution towards the extension of the polynomial approach to control system design (see e.g. (Kucera, 1991)) from time invariant systems to periodic systems.
The main subject dealt with in this paper is the relation connecting the companion form for periodic matrices and the canonical form of periodic systems in discrete-time to the properties of certain periodic polynomials. The keys towards such connection are i) the notion of coprimenss of periodic polynomials and ii) the concept of periodic cyclic matrix. First some basic concepts on periodic polynomials are introduced. Then, the basic definitions of strong and weak coprimeness of two periodic polynomials are presented. Finally, the connection with periodic system theory is outlined.
2. ELEMENTARY ALGEBRA OF PERIODIC POLYNOMIALS Denote by pi (t), i = 0, 1, · · · n, a set of real coefficients evolving periodically in time with period T , i.e. pi (t + T ) = pi (t), ∀t, and let
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p(σ, t) = p0 (t)σ n + p1 (t)σ n−1 + · · · pn (t) be the associated T -periodic polynomial in the symbol σ. Such a symbol denotes the one-step ahead operator. In other words, p(σ, t) is an operator transforming a time signals v(t) into a time signal w(t) as follows:
The above property can be written as: σ k p(σ, t) = p(σ, t + k)σ k
(2)
and is referred to as the pseudo-commutative property. Obviously, if k is a multiple of the period, then σ k and p(σ, t) commute.
w(t) = p(σ, t)v(t) = p0 (t)v(t + n) + + p1 (t)v(t + n − 1) + · · · + pn (t)v(t) (1) A more precise symbol could have been adopted in order to stress the operator character of p(σ, t) (for instance (p ∗ w)(t)). However, for the sake of simplicity, we will use the above notation throughout the paper. Furthermore, notice that p(σ, t) is anticausal in that σ is the one-step ahead shift operator. Remark 2.1. If w(t) is a white noise, the equation p(σ, t)v(t) = w(t) defines an Autoregressive Model. As is well known, periodic AR models or, more in general, cyclostationary models are of wide interest in seasonal time-series analysis, see e. g. (Luetkepohl, 1993), (Lund, Seymour, 2002). Regular polynomial The degree of the polynomial is defined as the T -periodic function ρ(t) given by the power associated with the maximum-power coefficient which is non zero at time t. If p0 (t) = 0, ∀t, then ρ(t) = n, ∀t, and the polynomial is called regular. A regular polynomial with p0 (t) = 1, ∀t, is said to be monic.
We are now in a position to introduce the polynomial product, as p(σ, t)q(σ, t) = r0 (t)σ 2n + r1 (t)σ 2n−1 + + r2 (t)σ 2n−1 + · · · + r2n−1 σ + r2n (t) where r0 (t) = p0 (t)q0 (t + n) r1 (t) = p0 (t)q1 (t + n) + p1 (t)q0 (t + n − 1) .. . r2n−1 = pn−1 qn (t + 1) + pn (t)qn−1 (t) r2n (t) = pn (t)qn (t) In this way the set of T -periodic polynomials forms a non commutative ring. However, the subset constituted by the polynomials in the symbol σ T is a commutative ring. Characteristic equation and zeros For time-invariant polynomials the concept of zero has been widely studied. For a T -periodic polynomial, we introduce herein the notion of zero as follows. Consider the following system of equations, for t = 0, 1, · · · , T − 1:
Sum The sum of two T -periodic polynomials, say
0 = p(λσ, t)¯ y (t) + p0 (t)λn y¯(t + n) +
p(σ, t) and q(σ, t) = q0 (t)σ n + q1 (t)σ n−1 + · · · qn (t) is still a T -periodic polynomial, given by p(σ, t) + q(σ, t) = (p0 (t) + q0 (t))σ n + +(p1 (t) + q1 (t))σ n−1 + · · · (pn (t) + qn (t)) Product First observe that, for any integer k and signal v(t), the following identity holds true: [σ k p(σ, t)]v(t) = [p(σ, t + k)σ k ]v(t) Indeed, in view of (1), [σ k p(σ, t)]v(t) = σ k w(t) = w(t + k) = p(σ, t + k)v(t + k) = [p(σ, t + k)σ k ]v(t)
y (t) + p1 (t)λn−1 y¯(t + n − 1) + · · · + pn (t)¯ For a given (real or complex) λ, this is a system of homogeneous linear equations in the unknown function y¯(·). Restrict now the attention to T periodic functions y¯(·). Suppose that a non trivial (periodic) solution of the above system exists for a given λ. Then another T -periodic non trivial solution y¯(·) exists associated with any T -roots of λT . This leads to the definition of λT as a zero of the periodic polynomial p(σ, t). Therefore a regular n-degree T -periodic polynomial admits n zeros, as in the time-invariant case. For easy reference we will refer to the equation p(λσ, t)¯ y (t) = 0 as the characteristic periodic difference equation. In the trivial case T = 1 (time-invariant polynomial) the definition of zero and characteristic equation now given reduce to the well known standard definitions.
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Unimodular polynomial Following standard algebraic terminology, a T -periodic polynomial p(σ, t) is said to be unimodular if there exists a T -periodic polynomial q(σ, t) so that for each t,
two T -periodic polynomials are said to be weakly left coprime if any left divisor is unimodular. Analogously, they are said to be weakly right coprime if any right divisor is unimodular. Strong coprimeness
p(σ, t)q(σ, t) = q(σ, t)p(σ, t) = 1 where 1 denotes the identity operator. Contrary to the case of constant polynomials, there might exist unimodular T -periodic polynomials with degree different from zero at some time point. For instance, the 2-periodic polynomial p(σ, t) = p0 (t)σ + p1 (t)
A stronger concept of coprimeness can be directly formulated in terms of the celebrated Bezout identity. Precisely, we say that two T -periodic polynomials p(σ, t) and q(σ, t) are strongly left coprime if there exists two T -periodic polynomials x(σ, t) and y(σ, t) such that the identity p(σ, t)x(σ, t) + q(σ, t)y(σ, t) = 1
with p0 (0) = 0, p0 (1) = 1, p1 (0) = 1, p1 (1) = −1, is unimodular in that
holds for each t. Analogously, they are said to be strongly right coprime if the identity
(p0 (t)σ − p1 (t + 1))(p0 (t)σ + p1 (t)) = 1 Notice that the no regular T -periodic polynomial can be unimodular.
3. THE PERIODIC BEZOUT IDENTITY AND THE NOTION OF COPRIMENESS Divisor We say that a T -periodic polynomial q(σ, t) is a left divisor of a T -periodic polynomial p(σ, t) if there exists a T -periodic polynomial r(σ, t) such that p(σ, t) = q(σ, t)r(σ, t) Analogously, q(σ, t) is said to be a right divisor of p(σ, t) if there exists a T -periodic polynomial s(σ, t) such that p(σ, t) = s(σ, t)q(σ, t) Along this route we define the notion of common left (right) divisor. To this purpose, consider two T-periodic polynomials p(σ, t) and q(σ, t). If there exists a T -periodic polynomial r(σ, t) such that p(σ, t) = r(σ, t)s1 (σ, t),
(5)
q(σ, t) = r(σ, t)s2 (σ, t)
for some T -periodic polynomials s1 (σ, t) and s2 (σ, t), then r(σ, t) is a common left divisor of p(σ, t) and q(σ, t).
x(σ, t)p(σ, t) + y(σ, t)q(σ, t) = 1
(6)
holds for some T -periodic polynomials x(σ, t) and y(σ, t). Weak coprimeness vs Strong coprimeness It is important to stress that, differently from the time-invariant case, the notions of weak and strong coprimeness are not equivalent. Strong coprimeness implies weak coprimeness, but the converse is not true in general. To see this, assume that (6) holds for some T -periodic polynomials x(σ, t) and y(σ, t). Now, let r(σ, t) be any common right divisor according to (3). Therefore it is possible to write 1 = (x(σ, t)s1 (σ, t) + y(σ, t)s2 (σ, t))r(σ, t) which shows that the generic right divisor r(σ, t) is unimodular. This means that strong right copriminess implies weak right copriminess. Analogously, strong left copriminess implies weak left copriminess. In order to see that weak coprimeness does not imply strong coprimeness, it suffices to consider the simple 2 − periodic counterexample: p(σ, t) = σ − a(t),
Analogously if
a(0) = b(0) = 0,
q(σ, t) = σ − b(t) a(1) = 1, b(1) = 2
Weak coprimeness By means of the concept of
It is apparent that the polynomials have only unimodular (zero degree) divisors, and as such they are weakly right and left coprime. However, the Bezout identity is not satisfied by any pair x(σ, t), y(σ, t), a(t) and b(t) being both zero at the same time instant t = 0.
left (right) divisor, it is possible to define the notions of weak left (right) coprimeness. Precisely,
Another difference with respect to the case of time-invariant polynomials is that it may happen
p(σ, t) = s1 (σ, t)r(σ, t)
(3)
q(σ, t) = s2 (σ, t)r(σ, t)
(4)
then r(σ, t) is a common right divisor of p(σ, t) and q(σ, t).
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that two polynomials with the same singularities are strongly coprime. An elementary example illustrating this fact can be built by considering the two 2-periodic polynomials p(σ, t) = σ + a(t) q(σ, t) = σ + a(t + 1),
a(0) = 1, a(1) = 5
It is apparent that both polynomials share the same singularity, namely λT = 5. On the other hand, the Bezout equation x(σ, t)p(σ, t) + y(σ, t)q(σ, t) = 1 with x(σ, t) = k1 (t) − σ,
y(σ, t) = k2 (t) + σ
4. CONNECTION WITH PERIODIC MATRICES AND PERIODIC SYSTEMS The use of periodic polynomials in periodic systems is dealt with in a few papers only, such as (Mrabet, Bourles, 1998) and (Colaneri, Kucera, Longhi, 2003). In the following, we will use the notion of periodic cyclic matrix to point out the connection between the coprimeness of certain periodic polynomials and the canonical forms of periodic systems. In this way, some long standing issues concerning the realization of periodic systems, partially addressed in (Bittanti, Bolzern, Guardabassi, 1985) and in (Luetkepohl, 1993), are eventually clarified.
where k1 (0) = 4.75, k1 (1) = 1.25, k2 (0) = −0.75, k2 (1) = −5.25, is satisfied. An effective way to characterize strong coprimeness is to resort to the associated characteristic periodic difference equations. Precisely, two polynomials p(σ, t) and q(σ, t) are not strongly right coprime if and only if their characteristic periodic difference equations have a common solution. i.e. there exists a complex number λ and a non identically zero T -periodic function x(·) such that p(λσ, t)x(t) = 0
(7)
q(λσ, t)x(t) = 0
(8)
Interestingly, if (7), (8) hold for some λ = 0, then the two polynomials have one common right divisor, i.e. the two polynomials are not weak right coprime. Example 3.1. Take again the 2 − periodic polynomials p(σ, t) = σ − a(t), a(0) = b(0) = 0,
q(σ, t) = σ − b(t)
Now, take any two periodic polynomials of degree one, i.e. q(σ, t) = σ − b(t)
and assume that (7) and (8) hold for some λ = 0. Then λx(t + 1) = a(t)x(t),
In this section we introduce some canonical forms for the dynamical matrix A(·). This leads to the definition of the so-called companion forms. Precisely we will focus on the following two forms. 1) The n × n matrix ⎡ 0 Ahc (t) =
λx(t + 1) = b(t)x(t)
imply that a(t) = b(t) for each t, so that the two polynomials do coincide.
1 0 .. . 0
0 1 .. . 0
⎤
··· 0 ··· 0 ⎢ 0 ⎥ ⎢ .. ⎥ .. .. ⎢ . ⎥ . . ⎣ ⎦ 0 ··· 1 −αn (t) −αn−1 (t) −αn−2 (t) · · · −α1 (t)
where αi (t), i = 1, 2, · · · , n, are T -periodic coefficients, is said to be in h-companion form 2) The n × n matrix ⎡ 0 ⎢1 ⎢ ⎢ Avc (t) = ⎢ 0 ⎢ .. ⎣.
⎤ 0 −βn (t) 0 −βn−1 (t) ⎥ ⎥ 0 −βn−2 (t) ⎥ ⎥ ⎥ .. .. ⎦ . . 0 0 · · · 1 −β1 (t)
a(1) = 1, b(1) = 2
These polynomials are not strong right coprime, since (7) and (8) hold for λ = 0 and x(1) = 0. However, they are right weak coprime, as already discussed.
p(σ, t) = σ − a(t),
4.1 Companion forms of periodic matrices
0 0 1 .. .
··· ··· ··· .. .
where βi (t), i = 1, 2, · · · , n, are T -periodic coefficients, is said to be in v-companion form We now characterize the conditions under which A(·) is algebraically equivalent to a matrix in h-companion or v-companion form. To this end, introduce the symbol ΨA (t, τ ) to denote the transition matrix of A(·) over the interval from τ to t. Then, the following definition, first introduced in (Bittanti, Colaneri, 2002), is in order. Definition 4.1. A n × n T -periodic matrix A(·) is said to be T -cyclic if there exists a T -periodic vector x(·), x(t) = 0, ∀t, such that the following n × n matrix
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R(t) = x1 (t) x2 (t) x3 (t) · · · xn (t)
(9)
where xi (t) = ΨA (t, t − i + 1)x(t),
first class we consider is the class of periodic systems in thereachable canonical form, i.e. systems described by
i = 1, 2, · · · n
⎡
0 1 0 0 1 ⎢ 0 ⎢ .. . . . . ⎢ . . . ⎣ 0 0 0 −αn (t) −αn−1 (t) −αn−2 (t)
is invertible, for all t. If such a vector x(·) exists, it is said to be a periodic cyclic generator.
A(t) =
It is not difficult to show that for a T -cyclic matrix A(·) it is possible to find a set of T -periodic coefficients αi (·), i = 1, 2, · · · n such that
B(t) = 0 0 · · · 0 1 C(t) = γ1 (t) γ2 (t) · · · γn−1 (t) γn (t)
αn (t − 1)x(t) + αn−1 (t − 2)ΨA (t, t − 1)x(t − 1) + αn−2 (t − 3)ΨA (t, t − 2)x(t − 2) + · · · + α1 (t − n)ΨA (t, t − n + 1)x(t − n + 1) + ΨA (t, t − n)x(t − n) = 0
(10)
Indeed, being matrix R(t) invertible, any vector of Rn , and, in particular ΨA (t, t − n)x(t − n), can be seen as a linear combination of the columns of R(t). The periodicity of the coefficients is a consequence of the periodicity of R(t) and ΨA (t, t − n)x(t − n). It is also important to stress that a T -periodic matrix in h-companion form or in v-companion form is indeed T -cyclic. Precisely, the cyclic generator associated with a h-companion form is xhc = 0 0 · · · 1 , whereas the one associated with a v-companion form is xvc = 1 0 · · · 0 .
where αi (·) and γi (·) are T -periodic functions, ∀i. The dynamic matrix of this form is a matrix in h-canonical form. The system in the canonical reachable form can be equivalently rewritten by resorting to the two T -periodic polynomials d(σ, t) = σ n + α1 (t)σ n−1 + α2 (t)σ n−2 + · · · + αn (t) n(σ, t) = γ1 (t)σ n−1 + γ2 (t)σ n−2 + · · · + γn (t) which completely define the system. By means of the above defined polynomials, one can represent the system given in the canonical reachable form by the input-output polynomial model y(t) = n(σ, t)z(t)
As a matter of fact, the following result can be proven, connecting cyclicity to companion forms. Theorem 4.1. With reference to an n × n T periodic matrix A(·), the following statements are equivalent each other (i) A(·) is T -cyclic. (ii) A(·) is algebraically equivalent to a T periodic matrix in v-companion form. (iii) A(·) is algebraically equivalent to a T periodic matrix in h-companion form.
4.2 Canonical forms of periodic systems Consider now a SISO periodic system described by x(t + 1) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) where B(·) and C(·) are T -periodic as well. We say that the system is cyclic whenever A(·) is a T -cyclic matrix. Among cyclic periodic systems, a
⎤
··· 0 ··· 0 ⎥ ⎥ . .. . ⎥ . . ⎦ ··· 1 · · · −α1 (t)
u(t) = d(σ, t)z(t) where, as easily checkable, the variable z(t) coincides with the first state variable x1 (t). Hence, y(t) = n(σ, t)d(σ, t)−1 u(t) is the right fractional representation of the system. The system in the canonical reachable form defined above is, by construction, a fully reachable periodic system. In addition, thanks to the structure of matrices (A(·), B(·)), the reachability interval is no longer than the system order n. Notice that this property does not hold in general since, as well known, the interval of time required to reach the states of a reachable periodic system may be as long as nT steps. Now, we want to assess the observability properties of the system, which, as already said, depends on the properties of the two polynomials above. Theorem 4.2. The system in the reachable canonical form is observable for each t if and only if the two polynomials (d(σ, t), n(σ, t)) are strongly right coprime.
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It is easy to show that a dual result can be given if the system in the canonical observable form is considered, i.e. ⎡
⎤ 0 −βn (t) 0 −βn−1 (t) ⎥ ⎥ 0 −βn−2 (t) ⎥ ⎥ ⎥ .. .. ⎦ . . 0 0 · · · 1 −β1 (t)
0 ⎢1 ⎢ ⎢ A(t) = ⎢ 0 ⎢ .. ⎣.
0 0 1 .. .
··· ··· ··· .. .
B(t) = δ1 (t) δ2 (t) · · · δn−1 (t) δn (t) C(t) = 0 0 · · · 0 1 where βi (·) and δi (·) are T -periodic functions, ∀i. Now, the fractional polynomial representation of the system in canonical observable form is d(σ, t)y(t) = z(t) n(σ, t)u(t) = z(t) where d(σ, t) = σ n + σ n−1 β1 (t) + σ n−2 β2 (t) + · · · βn (t) n(σ, t) = σ n−1 δ1 (t) + σ n−2 δ2 (t) + · · · δn (t) Hence,
y(t) = d(σ, t)−1 n(σ, t)u(t)
is the left fractional representation of the system. Notice in passing that this model corresponds to the so-called PARMA representation, widely used for prediction and identification purposes. The system in the observable canonical form is, by construction, observable for each t. Moreover, observability can be performed in n steps at most. As for the reachability properties of the system, the following result can be proven in a totally analogous way of Theorem 4.2. Theorem 4.3. The system in observable canonical form is reachable for each t if and only if the two polynomials (d(σ, t), n(σ, t)) are strongly left coprime.
5. CONCLUSIONS This paper contributes to periodic control theory with a view towards the polynomial approach (in discrete time). Some elements of periodic polynomials algebra are introduced, in particular the periodic Bezout identity and the notion of (weak and strong) coprimeness. Then, we treat two basic problems of periodic systems theory, namely the notion of periodic cyclic matrix, and the question of the existence of companion and canonical forms.
ACKNOWLEDGEMENT Research supported by the Italian National Research Project “New Techniques of Identification and Adaptive Control for Industrial Systems” and partially by CNR - IEIIT.
REFERENCES S. Bittanti, Periodic Control - Theory and Application, The Sunahara Lecture, Conference on Stochastic Systems Theory and Applications, Tottori, Japan, 2000. S. Bittanti, P. Bolzern, G. Guardabassi (1985): Some critical issues concerning the state- representation of time-varying ARMA models. 7th IFAC Symposium on Identification and System Parameter Estimation, York (England), p. 1479-1484. S. Bittanti, P. Colaneri (1999) Periodic Control. John Wiley Encyclopedia of Electrical and Electronic Engineering, Vol. 16, p. 240 - 253, John Wiley and sons, J. Webster ed. S. Bittanti, P. Colaneri (Eds) (2002) Periodic Control Systems. Proc. of the IFAC Workshop PSYCO, Cernobbio-Como, Italy, August 27-28, 2001, Pergamon. S. Bittanti, P. Colaneri (2002) ”Companion forms and cyclic matrices for discrete-time periodic systems”, Proc. of the IFAC World Congress, Barcelona, Spain. P. Colaneri, V. Kucera, S. Longhi (2003) ”Polynomial approach to the control of SISO periodic systems subject to input constraint”, Automatica, 39, 8, pp. 1417-1424. Y. El Mrabet, H. Bourles (1998) ”Periodic polynomial interpretation for structural properties of linear periodic discrete-time systems”, Systems and Control Letters, 33, pp. 241-251. V. Kucera (1991) Analysis and Design of Discrete Time Control Systems. Prentice Hall. Kuijper M. (1999) ”A periodically time-varying minimal partial realization algorithm based on twisting”, Automatica, Vol. 35, pp. 15431548. Luetkepohl H. (1993) Introduction to Multiple Time-Series Analysis, Springer-Verlag. Lund R.B., L. Seymour, (2002) ”Periodic TimeSeries”, in Encyclopedia of Environmetics, Wiley.
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PERIODIC POLYNOMIALS IN THE ANALYSIS OF PERIODIC CYCLIC SYSTEMS S. Bittanti, P. Colaneri
Politecnico di Milano Dipartimento di Elettronica e Informazione Piazza Leonardo da Vinci 32, 20133 Milano (Italy) [email protected] [email protected]
Abstract: In this paper, we consider polynomials with periodic coefficients in discrete time, and we define some basic elements of periodic polynomials algebra, in particular the periodic Bezout identity and the notion of (weak and strong) coprimeness. With these notions, we treat some basic problems of periodic systems theory. Specifically, by the notion of periodic cyclic matrix, we takle the question of the existence of companion and canonical forms. Keywords: Periodic Systems, Periodic Cyclic Matrix, Companion Form, Coprimeness of Periodic Polynomials, Reachability and Observability
1. INTRODUCTION
The research activity in periodic control over more than three decades is witnessed by the papers in the Preprints of this IFAC workshop, as well as in the Proceedings of the previous and first workshop on the same subject organized in 2001 in Cernobbio-Como (Italy) (Bittanti, Colaneri, 2002). For survey papers, the interested reader is referred to (Bittanti, 2000) and (Bittanti, Colaneri, 1999). However, the polynomials approach to periodic systems is still in its early days and, in the authors’ opinion, it’s a subject deserving more attention. For a pioneer paper the reader is referred to (Colaneri, Kucera, Longhi, 2003). Herein we focus on the use of polynomials in some basic problems of periodic system theory, so providing a contribution towards the extension of the polynomial approach to control system design (see e.g. (Kucera, 1991)) from time invariant systems to periodic systems.
The main subject dealt with in this paper is the relation connecting the companion form for periodic matrices and the canonical form of periodic systems in discrete-time to the properties of certain periodic polynomials. The keys towards such connection are i) the notion of coprimenss of periodic polynomials and ii) the concept of periodic cyclic matrix. First some basic concepts on periodic polynomials are introduced. Then, the basic definitions of strong and weak coprimeness of two periodic polynomials are presented. Finally, the connection with periodic system theory is outlined.
2. ELEMENTARY ALGEBRA OF PERIODIC POLYNOMIALS Denote by pi (t), i = 0, 1, · · · n, a set of real coefficients evolving periodically in time with period T , i.e. pi (t + T ) = pi (t), ∀t, and let
243
p(σ, t) = p0 (t)σ n + p1 (t)σ n−1 + · · · pn (t) be the associated T -periodic polynomial in the symbol σ. Such a symbol denotes the one-step ahead operator. In other words, p(σ, t) is an operator transforming a time signals v(t) into a time signal w(t) as follows:
The above property can be written as: σ k p(σ, t) = p(σ, t + k)σ k
(2)
and is referred to as the pseudo-commutative property. Obviously, if k is a multiple of the period, then σ k and p(σ, t) commute.
w(t) = p(σ, t)v(t) = p0 (t)v(t + n) + + p1 (t)v(t + n − 1) + · · · + pn (t)v(t) (1) A more precise symbol could have been adopted in order to stress the operator character of p(σ, t) (for instance (p ∗ w)(t)). However, for the sake of simplicity, we will use the above notation throughout the paper. Furthermore, notice that p(σ, t) is anticausal in that σ is the one-step ahead shift operator. Remark 2.1. If w(t) is a white noise, the equation p(σ, t)v(t) = w(t) defines an Autoregressive Model. As is well known, periodic AR models or, more in general, cyclostationary models are of wide interest in seasonal time-series analysis, see e. g. (Luetkepohl, 1993), (Lund, Seymour, 2002). Regular polynomial The degree of the polynomial is defined as the T -periodic function ρ(t) given by the power associated with the maximum-power coefficient which is non zero at time t. If p0 (t) = 0, ∀t, then ρ(t) = n, ∀t, and the polynomial is called regular. A regular polynomial with p0 (t) = 1, ∀t, is said to be monic.
We are now in a position to introduce the polynomial product, as p(σ, t)q(σ, t) = r0 (t)σ 2n + r1 (t)σ 2n−1 + + r2 (t)σ 2n−1 + · · · + r2n−1 σ + r2n (t) where r0 (t) = p0 (t)q0 (t + n) r1 (t) = p0 (t)q1 (t + n) + p1 (t)q0 (t + n − 1) .. . r2n−1 = pn−1 qn (t + 1) + pn (t)qn−1 (t) r2n (t) = pn (t)qn (t) In this way the set of T -periodic polynomials forms a non commutative ring. However, the subset constituted by the polynomials in the symbol σ T is a commutative ring. Characteristic equation and zeros For time-invariant polynomials the concept of zero has been widely studied. For a T -periodic polynomial, we introduce herein the notion of zero as follows. Consider the following system of equations, for t = 0, 1, · · · , T − 1:
Sum The sum of two T -periodic polynomials, say
0 = p(λσ, t)¯ y (t) + p0 (t)λn y¯(t + n) +
p(σ, t) and q(σ, t) = q0 (t)σ n + q1 (t)σ n−1 + · · · qn (t) is still a T -periodic polynomial, given by p(σ, t) + q(σ, t) = (p0 (t) + q0 (t))σ n + +(p1 (t) + q1 (t))σ n−1 + · · · (pn (t) + qn (t)) Product First observe that, for any integer k and signal v(t), the following identity holds true: [σ k p(σ, t)]v(t) = [p(σ, t + k)σ k ]v(t) Indeed, in view of (1), [σ k p(σ, t)]v(t) = σ k w(t) = w(t + k) = p(σ, t + k)v(t + k) = [p(σ, t + k)σ k ]v(t)
y (t) + p1 (t)λn−1 y¯(t + n − 1) + · · · + pn (t)¯ For a given (real or complex) λ, this is a system of homogeneous linear equations in the unknown function y¯(·). Restrict now the attention to T periodic functions y¯(·). Suppose that a non trivial (periodic) solution of the above system exists for a given λ. Then another T -periodic non trivial solution y¯(·) exists associated with any T -roots of λT . This leads to the definition of λT as a zero of the periodic polynomial p(σ, t). Therefore a regular n-degree T -periodic polynomial admits n zeros, as in the time-invariant case. For easy reference we will refer to the equation p(λσ, t)¯ y (t) = 0 as the characteristic periodic difference equation. In the trivial case T = 1 (time-invariant polynomial) the definition of zero and characteristic equation now given reduce to the well known standard definitions.
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Unimodular polynomial Following standard algebraic terminology, a T -periodic polynomial p(σ, t) is said to be unimodular if there exists a T -periodic polynomial q(σ, t) so that for each t,
two T -periodic polynomials are said to be weakly left coprime if any left divisor is unimodular. Analogously, they are said to be weakly right coprime if any right divisor is unimodular. Strong coprimeness
p(σ, t)q(σ, t) = q(σ, t)p(σ, t) = 1 where 1 denotes the identity operator. Contrary to the case of constant polynomials, there might exist unimodular T -periodic polynomials with degree different from zero at some time point. For instance, the 2-periodic polynomial p(σ, t) = p0 (t)σ + p1 (t)
A stronger concept of coprimeness can be directly formulated in terms of the celebrated Bezout identity. Precisely, we say that two T -periodic polynomials p(σ, t) and q(σ, t) are strongly left coprime if there exists two T -periodic polynomials x(σ, t) and y(σ, t) such that the identity p(σ, t)x(σ, t) + q(σ, t)y(σ, t) = 1
with p0 (0) = 0, p0 (1) = 1, p1 (0) = 1, p1 (1) = −1, is unimodular in that
holds for each t. Analogously, they are said to be strongly right coprime if the identity
(p0 (t)σ − p1 (t + 1))(p0 (t)σ + p1 (t)) = 1 Notice that the no regular T -periodic polynomial can be unimodular.
3. THE PERIODIC BEZOUT IDENTITY AND THE NOTION OF COPRIMENESS Divisor We say that a T -periodic polynomial q(σ, t) is a left divisor of a T -periodic polynomial p(σ, t) if there exists a T -periodic polynomial r(σ, t) such that p(σ, t) = q(σ, t)r(σ, t) Analogously, q(σ, t) is said to be a right divisor of p(σ, t) if there exists a T -periodic polynomial s(σ, t) such that p(σ, t) = s(σ, t)q(σ, t) Along this route we define the notion of common left (right) divisor. To this purpose, consider two T-periodic polynomials p(σ, t) and q(σ, t). If there exists a T -periodic polynomial r(σ, t) such that p(σ, t) = r(σ, t)s1 (σ, t),
(5)
q(σ, t) = r(σ, t)s2 (σ, t)
for some T -periodic polynomials s1 (σ, t) and s2 (σ, t), then r(σ, t) is a common left divisor of p(σ, t) and q(σ, t).
x(σ, t)p(σ, t) + y(σ, t)q(σ, t) = 1
(6)
holds for some T -periodic polynomials x(σ, t) and y(σ, t). Weak coprimeness vs Strong coprimeness It is important to stress that, differently from the time-invariant case, the notions of weak and strong coprimeness are not equivalent. Strong coprimeness implies weak coprimeness, but the converse is not true in general. To see this, assume that (6) holds for some T -periodic polynomials x(σ, t) and y(σ, t). Now, let r(σ, t) be any common right divisor according to (3). Therefore it is possible to write 1 = (x(σ, t)s1 (σ, t) + y(σ, t)s2 (σ, t))r(σ, t) which shows that the generic right divisor r(σ, t) is unimodular. This means that strong right copriminess implies weak right copriminess. Analogously, strong left copriminess implies weak left copriminess. In order to see that weak coprimeness does not imply strong coprimeness, it suffices to consider the simple 2 − periodic counterexample: p(σ, t) = σ − a(t),
Analogously if
a(0) = b(0) = 0,
q(σ, t) = σ − b(t) a(1) = 1, b(1) = 2
Weak coprimeness By means of the concept of
It is apparent that the polynomials have only unimodular (zero degree) divisors, and as such they are weakly right and left coprime. However, the Bezout identity is not satisfied by any pair x(σ, t), y(σ, t), a(t) and b(t) being both zero at the same time instant t = 0.
left (right) divisor, it is possible to define the notions of weak left (right) coprimeness. Precisely,
Another difference with respect to the case of time-invariant polynomials is that it may happen
p(σ, t) = s1 (σ, t)r(σ, t)
(3)
q(σ, t) = s2 (σ, t)r(σ, t)
(4)
then r(σ, t) is a common right divisor of p(σ, t) and q(σ, t).
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that two polynomials with the same singularities are strongly coprime. An elementary example illustrating this fact can be built by considering the two 2-periodic polynomials p(σ, t) = σ + a(t) q(σ, t) = σ + a(t + 1),
a(0) = 1, a(1) = 5
It is apparent that both polynomials share the same singularity, namely λT = 5. On the other hand, the Bezout equation x(σ, t)p(σ, t) + y(σ, t)q(σ, t) = 1 with x(σ, t) = k1 (t) − σ,
y(σ, t) = k2 (t) + σ
4. CONNECTION WITH PERIODIC MATRICES AND PERIODIC SYSTEMS The use of periodic polynomials in periodic systems is dealt with in a few papers only, such as (Mrabet, Bourles, 1998) and (Colaneri, Kucera, Longhi, 2003). In the following, we will use the notion of periodic cyclic matrix to point out the connection between the coprimeness of certain periodic polynomials and the canonical forms of periodic systems. In this way, some long standing issues concerning the realization of periodic systems, partially addressed in (Bittanti, Bolzern, Guardabassi, 1985) and in (Luetkepohl, 1993), are eventually clarified.
where k1 (0) = 4.75, k1 (1) = 1.25, k2 (0) = −0.75, k2 (1) = −5.25, is satisfied. An effective way to characterize strong coprimeness is to resort to the associated characteristic periodic difference equations. Precisely, two polynomials p(σ, t) and q(σ, t) are not strongly right coprime if and only if their characteristic periodic difference equations have a common solution. i.e. there exists a complex number λ and a non identically zero T -periodic function x(·) such that p(λσ, t)x(t) = 0
(7)
q(λσ, t)x(t) = 0
(8)
Interestingly, if (7), (8) hold for some λ = 0, then the two polynomials have one common right divisor, i.e. the two polynomials are not weak right coprime. Example 3.1. Take again the 2 − periodic polynomials p(σ, t) = σ − a(t), a(0) = b(0) = 0,
q(σ, t) = σ − b(t)
Now, take any two periodic polynomials of degree one, i.e. q(σ, t) = σ − b(t)
and assume that (7) and (8) hold for some λ = 0. Then λx(t + 1) = a(t)x(t),
In this section we introduce some canonical forms for the dynamical matrix A(·). This leads to the definition of the so-called companion forms. Precisely we will focus on the following two forms. 1) The n × n matrix ⎡ 0 Ahc (t) =
λx(t + 1) = b(t)x(t)
imply that a(t) = b(t) for each t, so that the two polynomials do coincide.
1 0 .. . 0
0 1 .. . 0
⎤
··· 0 ··· 0 ⎢ 0 ⎥ ⎢ .. ⎥ .. .. ⎢ . ⎥ . . ⎣ ⎦ 0 ··· 1 −αn (t) −αn−1 (t) −αn−2 (t) · · · −α1 (t)
where αi (t), i = 1, 2, · · · , n, are T -periodic coefficients, is said to be in h-companion form 2) The n × n matrix ⎡ 0 ⎢1 ⎢ ⎢ Avc (t) = ⎢ 0 ⎢ .. ⎣.
⎤ 0 −βn (t) 0 −βn−1 (t) ⎥ ⎥ 0 −βn−2 (t) ⎥ ⎥ ⎥ .. .. ⎦ . . 0 0 · · · 1 −β1 (t)
a(1) = 1, b(1) = 2
These polynomials are not strong right coprime, since (7) and (8) hold for λ = 0 and x(1) = 0. However, they are right weak coprime, as already discussed.
p(σ, t) = σ − a(t),
4.1 Companion forms of periodic matrices
0 0 1 .. .
··· ··· ··· .. .
where βi (t), i = 1, 2, · · · , n, are T -periodic coefficients, is said to be in v-companion form We now characterize the conditions under which A(·) is algebraically equivalent to a matrix in h-companion or v-companion form. To this end, introduce the symbol ΨA (t, τ ) to denote the transition matrix of A(·) over the interval from τ to t. Then, the following definition, first introduced in (Bittanti, Colaneri, 2002), is in order. Definition 4.1. A n × n T -periodic matrix A(·) is said to be T -cyclic if there exists a T -periodic vector x(·), x(t) = 0, ∀t, such that the following n × n matrix
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R(t) = x1 (t) x2 (t) x3 (t) · · · xn (t)
(9)
where xi (t) = ΨA (t, t − i + 1)x(t),
first class we consider is the class of periodic systems in thereachable canonical form, i.e. systems described by
i = 1, 2, · · · n
⎡
0 1 0 0 1 ⎢ 0 ⎢ .. . . . . ⎢ . . . ⎣ 0 0 0 −αn (t) −αn−1 (t) −αn−2 (t)
is invertible, for all t. If such a vector x(·) exists, it is said to be a periodic cyclic generator.
A(t) =
It is not difficult to show that for a T -cyclic matrix A(·) it is possible to find a set of T -periodic coefficients αi (·), i = 1, 2, · · · n such that
B(t) = 0 0 · · · 0 1 C(t) = γ1 (t) γ2 (t) · · · γn−1 (t) γn (t)
αn (t − 1)x(t) + αn−1 (t − 2)ΨA (t, t − 1)x(t − 1) + αn−2 (t − 3)ΨA (t, t − 2)x(t − 2) + · · · + α1 (t − n)ΨA (t, t − n + 1)x(t − n + 1) + ΨA (t, t − n)x(t − n) = 0
(10)
Indeed, being matrix R(t) invertible, any vector of Rn , and, in particular ΨA (t, t − n)x(t − n), can be seen as a linear combination of the columns of R(t). The periodicity of the coefficients is a consequence of the periodicity of R(t) and ΨA (t, t − n)x(t − n). It is also important to stress that a T -periodic matrix in h-companion form or in v-companion form is indeed T -cyclic. Precisely, the cyclic generator associated with a h-companion form is xhc = 0 0 · · · 1 , whereas the one associated with a v-companion form is xvc = 1 0 · · · 0 .
where αi (·) and γi (·) are T -periodic functions, ∀i. The dynamic matrix of this form is a matrix in h-canonical form. The system in the canonical reachable form can be equivalently rewritten by resorting to the two T -periodic polynomials d(σ, t) = σ n + α1 (t)σ n−1 + α2 (t)σ n−2 + · · · + αn (t) n(σ, t) = γ1 (t)σ n−1 + γ2 (t)σ n−2 + · · · + γn (t) which completely define the system. By means of the above defined polynomials, one can represent the system given in the canonical reachable form by the input-output polynomial model y(t) = n(σ, t)z(t)
As a matter of fact, the following result can be proven, connecting cyclicity to companion forms. Theorem 4.1. With reference to an n × n T periodic matrix A(·), the following statements are equivalent each other (i) A(·) is T -cyclic. (ii) A(·) is algebraically equivalent to a T periodic matrix in v-companion form. (iii) A(·) is algebraically equivalent to a T periodic matrix in h-companion form.
4.2 Canonical forms of periodic systems Consider now a SISO periodic system described by x(t + 1) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) where B(·) and C(·) are T -periodic as well. We say that the system is cyclic whenever A(·) is a T -cyclic matrix. Among cyclic periodic systems, a
⎤
··· 0 ··· 0 ⎥ ⎥ . .. . ⎥ . . ⎦ ··· 1 · · · −α1 (t)
u(t) = d(σ, t)z(t) where, as easily checkable, the variable z(t) coincides with the first state variable x1 (t). Hence, y(t) = n(σ, t)d(σ, t)−1 u(t) is the right fractional representation of the system. The system in the canonical reachable form defined above is, by construction, a fully reachable periodic system. In addition, thanks to the structure of matrices (A(·), B(·)), the reachability interval is no longer than the system order n. Notice that this property does not hold in general since, as well known, the interval of time required to reach the states of a reachable periodic system may be as long as nT steps. Now, we want to assess the observability properties of the system, which, as already said, depends on the properties of the two polynomials above. Theorem 4.2. The system in the reachable canonical form is observable for each t if and only if the two polynomials (d(σ, t), n(σ, t)) are strongly right coprime.
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It is easy to show that a dual result can be given if the system in the canonical observable form is considered, i.e. ⎡
⎤ 0 −βn (t) 0 −βn−1 (t) ⎥ ⎥ 0 −βn−2 (t) ⎥ ⎥ ⎥ .. .. ⎦ . . 0 0 · · · 1 −β1 (t)
0 ⎢1 ⎢ ⎢ A(t) = ⎢ 0 ⎢ .. ⎣.
0 0 1 .. .
··· ··· ··· .. .
B(t) = δ1 (t) δ2 (t) · · · δn−1 (t) δn (t) C(t) = 0 0 · · · 0 1 where βi (·) and δi (·) are T -periodic functions, ∀i. Now, the fractional polynomial representation of the system in canonical observable form is d(σ, t)y(t) = z(t) n(σ, t)u(t) = z(t) where d(σ, t) = σ n + σ n−1 β1 (t) + σ n−2 β2 (t) + · · · βn (t) n(σ, t) = σ n−1 δ1 (t) + σ n−2 δ2 (t) + · · · δn (t) Hence,
y(t) = d(σ, t)−1 n(σ, t)u(t)
is the left fractional representation of the system. Notice in passing that this model corresponds to the so-called PARMA representation, widely used for prediction and identification purposes. The system in the observable canonical form is, by construction, observable for each t. Moreover, observability can be performed in n steps at most. As for the reachability properties of the system, the following result can be proven in a totally analogous way of Theorem 4.2. Theorem 4.3. The system in observable canonical form is reachable for each t if and only if the two polynomials (d(σ, t), n(σ, t)) are strongly left coprime.
5. CONCLUSIONS This paper contributes to periodic control theory with a view towards the polynomial approach (in discrete time). Some elements of periodic polynomials algebra are introduced, in particular the periodic Bezout identity and the notion of (weak and strong) coprimeness. Then, we treat two basic problems of periodic systems theory, namely the notion of periodic cyclic matrix, and the question of the existence of companion and canonical forms.
ACKNOWLEDGEMENT Research supported by the Italian National Research Project “New Techniques of Identification and Adaptive Control for Industrial Systems” and partially by CNR - IEIIT.
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