Nonlinear time-delay systems: a polynomial approach using Ore algebras Miroslav Hal´as Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology Ilkoviˇcova 3, 812 19 Bratislava, Slovakia
[email protected]
1 Introduction Algebraic approach of differential forms, originally developed for nonlinear systems without delays, [5] and [1], was recently extended to the case of timedelay systems [13, 14, 15, 18] and was shown to be effective in solving control problems like accessibility and observability, disturbance decoupling, etc. On the other side, in the case of systems without delays, there exists, in comparison to the machinery of one-forms, an alternative approach in which the system properties are described by skew polynomials from non-commutative polynomial rings. Such polynomials act as differential [19, 20] or shift [11, 12] operators on the differentials of the system inputs and outputs. This approach allows us, for instance, to introduce the accessibility condition expressed in terms of common left factors of skew polynomials derived from the inputoutput equation. Moreover, the polynomial approach, after defining quotients of skew polynomials, provides also possibility to introduce transfer functions of nonlinear systems, both continuous [6, 7] and discrete-time [8, 9]. Such transfer functions show many properties we expect from transfer functions, like the invariance to state transformations, transfer function algebra and others and was, for instance, already used in [10] to recast and solve the nonlinear model matching problem. However, a similar polynomial approach is not yet available for nonlinear time-delay systems and in what follows, it is, therefore, extended also to this case.
2 Algebraic setting The mathematical setting, to be used in this paper for dealing with nonlinear time-delay systems, was recently introduced in [13, 14, 15, 18] and will be now
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briefly reviewed. In order to avoid technicalities, we use slightly abbreviated notations and the reader is referred to those works for detailed technical constructions which are not found here. The nonlinear time-delay systems considered in this paper are objects of the form x(t) ˙ = f ({x(t − i), u(t − j); i, j ≥ 0}) y(t) = g({x(t − i), u(t − j); i, j ≥ 0})
(1)
where the entries of f and g are meromorphic functions and x ∈ Rn , u ∈ Rm and y ∈ Rp denote state, input and output to the system. Note that it is not restrictive to assume i, j ∈ N since all the delays can be considered as multiples of an elemmaentary delay h [14]. Let K be the field of meromorphic functions of {x(t − i), u(k) (t − j); i, j, k ≥ 0} and let E be the formal vector space over K given by E = spanK {dξ; ξ ∈ K} The delay operator δ is defined on K and E as δ(ξ(t)) = ξ(t − 1) δ(α(t)dξ(t)) = α(t − 1)dξ(t − 1)
(2)
for any ξ(t) ∈ K and α(t)dξ(t) ∈ E. The delay operator (2) induces the non-commutative polynomial ring K[δ] with the multiplication given by the commutation rule δa(t) = a(t − 1)δ for any a(t) ∈ K. The ring K[δ] thus represents the ring of linear shift (delay) operators. Properties of the system (1) can be now analyzed by introducing the machinery of one-forms known from systems without delays [5, 1]. This time, rather than vector spaces we introduce modules over K[δ], generally M = spanK[δ] {dξ; ξ ∈ K} Such an approach was shown to be effective in solving a number of control problems, like accessibility and observability, disturbance decoupling, to name a few. See for instance [13, 14, 15, 18]. 2.1 Pseudo-linear algebra To handle different types of linear operators, as for instance differential, shift, difference, q-shift, we, to advantage, introduce univariate skew polynomial rings, or Ore rings [2]. Such structures allow us to handle those types of operators from a uniform standpoint.
Nonlinear time-delay systems: a polynomial approach using Ore algebras
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Definition 1. Let K be a field and σ : K → K an automorphism of K. A map δ : K → K which satisfies δ(a + b) = δ(a) + δ(b) δ(ab) = σ(a)δ(b) + δ(a)b
(3)
is called a pseudo-derivation (or a σ-derivation). Definition 2. The left skew polynomial ring given by σ and δ is the ring K[x; σ, δ] of polynomials in x over K with the usual addition, and the multiplication given by the commutation rule xa = σ(a)x + δ(a)
(4)
for any a ∈ K. Elemments of such a ring are called skew polynomials or non-commutative polynomials or Ore polynomials [16, 17]. The commutation rule (4) actually represents the action of the corresponding operator on polynomials. Any skew polynomial ring K[x; σ, δ] has no zero divisors, that is, it forms non-commutative integral domain. Moreover, it satisfies the left Ore condition (existence of a common left multiple). Lemma 1 (left Ore condition). For all non-zero a, b ∈ K[x; σ, δ], there exist non-zero a1 , b1 ∈ K[x; σ, δ] such that a1 b = b1 a. So skew polynomial rings over a field are left Ore rings. Different rings of linear operators can be defined simply by choosing appropriate σ and δ. d d Example 1. If K with a derivation dt is a differential field, then K[s; 1K , dt ] is the ring of linear ordinary differential operators and we interpret (4) as a rule for differentiation d sa(t) = a(t)s + a(t) dt for any a(t) ∈ K.
Example 2. If K is a difference field and σ over K is the automorphism which takes t to t − 1, then K[δ; σ, 0] is the ring of linear ordinary shift (recurrence) operators and we interpret (4) as a rule for shifting δa(t) = a(t − 1)δ for any a(t) ∈ K.
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2.2 Ore algebras However, when one wants to handle more complicated types of operators, as for instance differential time-delay, univariate skew polynomial rings are not enough. In such cases, one has to switch from univariate case to multivariate, or in other words, from Ore rings to Ore algebras. The idea of using Ore algebras in control theory was recently developed in [3]. So, in order to treat the differential time-delay structures, we mix up the constructions of two above examples [3]. This is, in fact, possible due to the following [17]. Lemma 2. If A is a left Ore ring, so is A[x; σ, δ]. This allows us to introduce a multivariate skew polynomial ring of the form K[x1 ; σ1 , δ1 ]...[xs ; σs , δs ] Such a ring is said to be Ore algebra [4] if the σi ’s and δj ’s commute and satisfy σi (xj ) = xj , δi (xj ) = 0. Note that this does not mean that the elements of K commute with the xi ’s. An illustrative application of Ore algebras in linear time-delay systems can be found in [3] from which we carried over this idea to the nonlinear case. Naturally, to handle nonlinear time-delay systems we define Ore algebra K[δ; σ, 0][s; 1K ,
d ] dt
where σ takes t to t − 1. That is, δ and s stand for delay and, respectively, derivative operator. We will use the abbreviated notation K[δ, s]. d d Note that maps σ and dt clearly commute, δs = sδ, that is, σ( dt a(t)) = d d σ(a(t)) for any a(t) ∈ K, and satisfy σ(s) = s and (δ) = 0. So in dealing dt dt with polynomials from K[δ, s] we have to take into account the following three commutation rules sa(t) = a(t)s + a(t) ˙ δa(t) = a(t − 1)δ δs = sδ We can get a normal form of polynomials by moving δ and s on the right of each summand. For example, (δ + 1)(a(t)s + 1) = δa(t)s + a(t)s + δ + 1 = a(t − 1)δs + a(t)s + δ + 1. Finally, remark that no additional advantages appear when employing the d ring K[s; 1K , dt ][δ; σ, 0]. As δ and s commute both structures are, in fact, isomorphic.
Nonlinear time-delay systems: a polynomial approach using Ore algebras
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3 Polynomial description of the input-output equation Let K be now the field of meromorphic functions of {y (l) (t − i), u(k) (t − j); 0 ≤ l ≤ n − 1; i, j, k ≥ 0}. Consider a nonlinear time-delay single-input single-output control system described by the input-output equation y (n) (t) = φ({y (k) (t − i), u(k) (t − j); 0 ≤ k ≤ n − 1; i, j ≥ 0})
(5)
where u ∈ R and y ∈ R denote input and output to the system and φ ∈ K. The system (5) can be now represented in terms of polynomials in the Ore algebra K[δ, s]. After differentiating (5) we get dy (n) (t) −
n−1 X k=0 i≥0
n−1 X ∂φ ∂φ (k) dy (t − i) = du(k) (t − j) (k) (t − j) ∂y (k) (t − i) ∂u k=0 j≥0
which can be rewritten as a(δ, s)dy(t) = b(δ, s)du(t)
(6)
where n−1 X
a(δ, s) = sn −
k=0,i≥0 n−1 X
b(δ, s) =
k=0,j≥0
∂φ ∂y (k) (t ∂φ
∂u(k) (t
− j)
− i)
δ i sk
δ j sk
are polynomials in K[δ, s]. So the equation (6) represents the system behaviour in the terms of skew polynomials a(δ, s), b(δ, s). Consequently, in case of multi-input multi-output systems we would obtain a polynomial matrix description A(δ, s)dy(t) = B(δ, s)du(t). Example 3. Consider the system y¨(t) = y(t ˙ − 1)u(t − 1). Then d¨ y (t) − u(t − 1)dy(t ˙ − 1) = y(t ˙ − 1)du(t − 1) and
(s2 − u(t − 1)δs)dy(t) = y(t ˙ − 1)δdu(t)
3.1 Accessibility condition Accessibility of nonlinear systems, based on the concept of autonomous elements, can be dealt with by introducing the accessibility filtration [13, 18]. However, like in the case of the systems without delays [11, 19, 20], we can now express the condition for accessibility of nonlinear time-delay systems in terms of common left factors of skew polynomials derived from the inputoutput equation (5).
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Definition 3. A one-form ω ∈ spanK[δ] {dξ; ξ ∈ K} is said to be an autonomous element for the system (5) if there exists an integer ν and coefficients αi ∈ K[δ] such that α0 ω + · · · αν ω (ν) = 0
(7)
Theorem 1. The nonlinear time-delay system (5) has no autonomous elements if and only if the polynomials a(δ, s) and b(δ, s) have no common left factors. Proof. Neccessity: Suppose that (5) is not accessible; that is, there exists ω ∈ spanK[δ] {dξ; ξ ∈ K} such that (7) is satisfied. Then ω can be expressed as ω=a ˜(δ, s)dy(t) − ˜b(δ, s)du(t) and (7) as
ρ(δ, s)ω = ρ(δ, s)[˜ a(δ, s)dy(t) − ˜b(δ, s)du(t)] = 0
After matching the latter with (6), rewritten in the form a(δ, s)dy(t) − b(δ, s)du(t) = 0, we get a(δ, s) = ρ(δ, s)˜ a(δ, s) and b(δ, s) = ρ(δ, s)˜b(δ, s); that is, a(δ, s), b(δ, s) have a common left factor ρ(δ, s). Sufficiency: Suppose that a(δ, s), b(δ, s) have a (nontrivial) common left factor ρ(δ, s). Then the equation (6) can be written as ρ(δ, s)[˜ a(δ, s)dy(t) − ˜b(δ, s)du(t)] = 0 Let ω = a ˜(δ, s)dy(t) − ˜b(δ, s)du(t). Then we obtain ρ(δ, s)ω = 0 which implies the existence of an autonomous element. Clearly, one can conclude that Theorem 2. The nonlinear time-delay system (5) is accessible if the polynomials a(δ, s) and b(δ, s) have no common left factors. Example 4. Consider the system y¨(t) = y(t ˙ − 1)u(t) + y(t − 1)u(t) ˙ − y(t ˙ − 1) + y(t − 2)u(t − 1). After differentiating we get (s2 − u(t)sδ − u(t)δ ˙ + sδ − u(t − 1)δ 2 )dy(t) = = (y(t ˙ − 1) + y(t − 1)s + y(t − 2)δ)du(t) The polynomials have a common left factor. (s + δ)(s − u(t)δ)dy(t) = (s + δ)y(t − 1)du(t) The system is not accessible and can be reduced to (s − u(t)δ)dy(t) = y(t − 1)du(t) dy(t) ˙ − u(t)dy(t − 1) = y(t − 1)du(t) which after integrating yields y(t) ˙ = y(t − 1)u(t).
Nonlinear time-delay systems: a polynomial approach using Ore algebras
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4 Transfer functions of nonlinear time-delay systems The transfer function formalism for nonlinear systems without delays was recently developed in [6, 7, 8, 9]. In what follows we introduce such a formalism for time-delay case. The first step is to construct the quotient field of skew polynomials. In that respect Ore condition plays a key role. Obviously, K[δ, s] is still a left Ore ring and the left Ore condition still holds on. Hence, each two elements of K[δ, s] have a common left multiple. K[δ, s] can be, therefore, embedded to a non-commutative quotient field [16, 17] by defining quotients as a = b−1 · a b where a, b ∈ K[δ, s] and b 6= 0. Addition is defined by reducing two quotients to the same denominator a1 a2 β2 a1 + β1 a2 + = b1 b2 β2 b1
(8)
where β2 b1 = β1 b2 by Ore condition. Multiplication is defined by a1 a2 α1 a2 · = b1 b2 β2 b1
(9)
where β2 a1 = α1 b2 again by Ore condition. The resulting quotient field of skew polynomials is denoted by Khδ, si. Note that due to the non-commutative multiplication they, of course, differ from the usual rules. In particular, in case of the multiplication (9) we, in general, cannot simply multiply numerators and denominators, nor cancel them in a usual manner. We neither can commute them as the multiplication in Khδ, si is non-commutative as well. Once we have defined a fraction of two skew polynomials we can introduce transfer functions of nonlinear time-delay systems. Definition 4. An element F (δ, s) ∈ Khδ, si such that dy(t) = F (δ, s)du(t) is said to be a transfer function of the single-input single-output nonlinear time-delay system (1) (respectively (5)). In the case of multi-input multi-output system, we think of F (δ, s) as a matrix with the entries in Khδ, si and F (δ, s) is then referred to as a transfer matrix. Like in case of the input-output equation (5), where naturally F (δ, s) =
b(δ, s) a(δ, s)
also state-space representation (1) can be expressed in terms of skew polynomials. After differentiating (1) we get
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dx(t) ˙ = Adx(t) + Bdu(t) dy(t) = Cdx(t) + Ddu(t) where A, B, C, D are appropriate matrices with the entries in K[δ]. Now, we can write (sI − A)dx(t) = Bdu(t) dy(t) = Cdx(t) + Ddu(t) from which follows
F (δ, s) = C(sI − A)−1 B + D
(10)
Clearly, the elements of (sI − A) are skew polynomials from the ring K[δ, s]. Hence, inverting (sI −A) requires solving linear equations in non-commutative fields, see [16]. To find the left-hand inverse of (sI − A) one can use GaussJordan elimination considering (8) and (9), see also [6]. Note that the transfer function is defined by employing the standard algebraic formalism of differential forms, following the lines in [5] which introduces the notion of a one-form in a formal and abstract way. In particular, it is not necessary to deal here with the linearization of the system along a trajectory using the K¨ahler-type differential which leads to a time-varying linear system. Example 5. Consider the system x˙ 1 (t) = x2 (t − 1) x˙ 2 (t) = x2 (t)u(t) y(t) = x1 (t) After differentiating we get µ ¶ µ ¶ ¡ ¢ 0 δ 0 A= ,B= ,C= 1 0 0 u(t) x2 (t) Note that
à −1
(sI − A)
=
δ 1 s s2 −u(t−1)s 1 0 s−u(t)
!
Finally F (δ, s) = C(sI − A)−1 B = =
x2 (t − 1)δ s2 − u(t − 1)s
Remark that x2 (t − 1) = y(t) ˙ and
δ · x2 (t) s2 − u(t − 1)s
Nonlinear time-delay systems: a polynomial approach using Ore algebras
dy(t) =
s2
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y(t)δ ˙ du(t) − u(t − 1)s
(s2 − u(t − 1)s)dy(t) = y(t)δdu(t) ˙ d¨ y (t) − u(t − 1)dy(t) ˙ = y(t)du(t ˙ − 1) which after integrating yields y¨(t) = y(t)u(t ˙ − 1). 4.1 Invariance of the transfer functions Transfer functions of nonlinear time-delay systems have many properties we expect from transfer functions. Theorem 3. Transfer function (10) of the nonlinear time-delay system (1) is invariant with respect to the state transformation ξ(t) = φ({x(t − i); i ≥ 0}). Proof. For any state transformation ξ(t) = φ({x(t−i); i ≥ 0}) one has dξ(t) = T dx(t) where T is an appropriate unimodular (and therefore invertible) matrix from Knxn [δ]. In the new coordinates we get ˙ = (T AT −1 + T˙ T −1 )dξ(t) + T Bdu(t) dξ(t) dy(t) = CT −1 dξ(t) + Ddu(t) d d where T˙ = dt (T ) with dt applied pointwise to T . Note that the transfer function reads as
d dt (δ)
= 0. Thus,
F (s) = CT −1 (sI − T AT −1 − T˙ T −1 )T B + D = C(T −1 sT − A − T −1 T˙ )B + D After applying the commutation rule sT = T s + T˙ , we get F (s) = C(sI − A)−1 B + D which completes the proof. Example 6. Consider the system x(t) ˙ = −x2 (t)u(t − 1) y(t) = 1/x(t) which is, in fact, linear. One can easily check that y(t) ˙ = u(t − 1). The corresponding state transformation is ξ(t) = 1/x(t). After differentiating, A = (−2x(t)u(t − 1)), B = (−x2 (t)δ), C = (−1/x2 (t)), we get F (δ, s) = C(sI − A)−1 B 1 −x2 (t)δ =− 2 · x (t) s + 2x(t)u(t − 1) x2 (t)δ δ x2 (t)δ = = = (s + 2x(t)u(t − 1))x2 (t) x2 (t)s s which represents the linear system.
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4.2 Transfer function algebra We can also introduce algebra of transfer functions of nonlinear time-delay systems. Each system structure can be divided into three basic types of connections: series, parallel and feedback, see Fig. 1.
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300
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450
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100
150
200
250
300
350
400
450
500
Fig. 1. Series, parallel and feedback connection
For a series connection, it follows that dyB (t) = FB (δ, s)duB (t) = FB (δ, s)dyA (t) = FB (δ, s)FA (δ, s)duA (t). Thus F (δ, s) = FB (δ, s)FA (δ, s)
(11)
For parallel and feedback connection we get F (δ, s) = FA (δ, s) + FB (δ, s) F (δ, s) = (1− FA (δ, s)FB (δ, s))−1 · FA (δ, s)
(12)
Note that due to the non-commutative multiplication the rules (11) and (12) have to be kept exactly as they are. Following example demonstrates how to handle a series connection of two time-delay systems. It also serves as a motivation why it can be interesting to use the transfer function formalism. This idea was already used for instance in [10] to solve the nonlinear model matching problem. Example 7. Consider two systems y˙ A (t) = yA (t)uA (t − 1) Transfer functions are following
yB (t) = lnuB (t)
Nonlinear time-delay systems: a polynomial approach using Ore algebras
FA (δ, s) =
yA (t)δ s − uA (t − 1)
FB (δ, s) =
11
1 uB (t)
The systems are combined together in a series connection. For the connection A→B, when uB (t) = yA (t), the resulting transfer function is F (δ, s) = FB (δ, s)FA (δ, s) =
1 yA (t)δ · uB (t) s − uA (t − 1)
1 yA (t)δ · yA (t) s − uA (t − 1) yA (t)δ δ = = yA (t)s + y˙ A (t) − uA (t − 1)yA (t) s
=
Hence, the combination A→B is linear from an input-output point of view y˙ B (t) = uA (t − 1). However, when the systems are connected as B→A, that is uA (t) = yB (t), the result is different yA (t)δ 1 · s − uA (t − 1) uB (t) yA (t)δ = uB (t − 1)s − uB (t − 1)lnuB (t − 1)
F (δ, s) = FA (δ, s)FB (δ, s) =
This time, it does not yield a linear system.
5 Conclusions In this paper the polynomial approach to nonlinear time-delay systems was introduced. Unlike in the systems without delays the polynomials belong now to the multivariate skew polynomial ring, called Ore algebra. Polynomials from this ring act as differential time-delay operators on the differentials of the system inputs and outputs. Like in the systems without delays the condition for accessiblity of nonlinear time-delay systems was stated in terms of common left factors of the polynomials derived from the input-output equation. Introduced Ore algebra forms a non-commutative integral domain in which the left Ore condition is satisfied. This allowed us to define quotient field over the Ore algebra. Such quotients were suggested as transfer functions of nonlinear time-delay systems, as they satisfy many properties we expect from transfer functions. Firstly, their invariance to state transformations was shown and then the transfer function algebra was introduced. Mathemathical tools introduced in this paper represent thus an alternative approach to nonlinear time-delay systems and open new possibilities in analysis and feedback design.
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