Families of Reduced Order Models That Achieve ... - Semantic Scholar
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Families of reduced order models that achieve nonlinear moment matching T. C. Ionescu, A. Astolfi Abstract— In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems, affine in the input, in terms of the steady-state response of the output of a generalized signal generator driven by the nonlinear system. In addition, we define a new notion of moment matching and present the class of (nonlinear) parameterized reduced order models that achieve moment matching. Furthermore, we establish relations with existing notions of moment, showing that the families of reduced order models that achieve nonlinear moment matching are equivalent. Furthermore, we compute the reduced order model that matches moments at two sets of interpolation points, simultaneously, i.e., the number of interpolation points is twice the order of the model.
I. I NTRODUCTION In the problem of model reduction, moment matching techniques represent an efficient tool, see e.g. [1] for a complete overview for linear systems. In such techniques the (reduced order) model is obtained by constructing a lower degree rational function that approximates the original transfer function (assumed rational). The classical notion of moment has been given in [1], based on the series expansion of the transfer function of the linear system (see also [10], [18], [8]). The low degree rational function matches (some of) the terms of the original transfer function at various points in the complex plane. However, since the computation of moments is computationally costly, the reduced order models that match a prescribed number of moments are computed efficiently using e.g. Krylov subspace techniques, projections, etc. (see e.g., [7], [2], [14]). Recently, a time-domain notion of moment for nonlinear systems has been proposed. According to [4], the moments of a nonlinear system are related to the steady-state response of the nonlinear system driven by a signal generator. Furthermore, the moments are related to the solution of a nonlinear (Sylvester-like) partial differential equation (see [9], [15] for the linear system arguments). Based on this notion of moment a family of parameterized, reduced order models that achieve moment matching is computed. The free parameters can be used to enforce desired properties on the reduced order model, such as e.g., stability and passivity (see e.g. [11], [12]). T. C. Ionescu is with the Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, London, UK. E-mail: [email protected]. A. Astolfi is with the Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, London, UK and with Dipartimento di Informatica, Sistemi e Produzione, Universit`a di Roma Tor Vergata, 00133 Roma, Italy. E-mail: [email protected]. This work is supported by the EPSRC grant ”Control For Energy and Sustainability”, grant reference EP/G066477/1.
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In this paper we present a new time-domain notion of moment for a class of single-input, single-output nonlinear systems, affine in the input, by considering the swapped interconnection between the given nonlinear system and the signal generator, i.e., the nonlinear system drives a generalized signal generator. This is an extension of the linear arguments in [5]. Under some technical assumptions, the moment is in one-to-one relation with the steady-state response (provided it exists) of the output of such interconnection. Then, we define moment matching and we compute the class of parameterized reduced order models that achieve moment matching. Furthermore, we prove that the two types of moment matching are equivalent, i.e., a model from one family of models that match the moments of the system can be obtained from the ”swapped” family of models by a proper selection of parameters. In general, we compute the (subclass of) reduced order model(s) that match two moments simultaneously. The result is a nonlinear extension of the computation of a reduced order, linear model that matches a number of moments equal to twice its order. The paper represents preliminary steps towards developing nonlinear model reduction algorithms applicable in practice. In this paper we study the single-input, single-output case, although it is not difficult to extend the results to the multiple-input, multiple-output case. The paper is organized as follows. In Section II we give a brief overview of the linear notion of moment based on the steady-state response of the output of a generalized signal generator driven by the given system and the corresponding class of parameterized linear models that achieve moment matching. In Section III, we give a short presentation of the state-of-the-art notion of time-domain moments for nonlinear systems and the resulting family of reduced order models that achieve moment matching. Then, we define and analyze the output of a generalized nonlinear signal generator driven by the system. Based on this analysis we define a new notion of time-domain moment which is in a one-to-one relation with the steady-state response (provided it exists) of the output signal of the interconnection. We define the notion of moment matching, i.e., a model matches the moments of the system if the output of the signal generator driven by the former captures the behaviour of the output of the signal generator driven by the latter. Then we define the class of parameterized, reduced order, nonlinear models that achieve moment matching. We establish a relation between the notion of moment and moment matching based on the output of the system driven by a signal generator and its ”swapped” counterpart, by a proper selection of the free
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parameters. In Section IV we compute a reduced order model that matches a number of moments equal to twice the number of prescribed interpolation points for linear systems. We also give the nonlinear extension of this result and give an example to illustrate the theory. The paper is completed by some conclusions. II. L INEAR MOMENT MATCHING In this section we recall the idea of moment matching for linear, single-input, single-output systems, from a time domain point of view as presented in [5]. Let −1
G(s) = C(sI − A)
B,
the eigenvalues of Q, i.e., {s1 , s2 , ..., sν } = σ(Q). Note that the moments as in Definition 1 are equivalent to the notions depicted in Definition 2. Selecting (L, S) and (Q, R) in canonical forms, easy computations yield [η(s1 ) ... η(sν )] = φ = ϕ. Based on Definition 2, we define a family of parameterized models of order ν that achieve moment matching at the interpolation points {s1 , ..., sν } = σ(S). Theorem 1. [4], [5] 1) Let the pair (L, S) be observable and assume σ(A) ∩ σ(S) = ∅. Let ξ(t) ∈ Rν and consider the family of linear models ξ˙ = (S − GL)ξ + Gu, ΣG : (7) η = CΠξ,
(1)
G : C → C, with A ∈ Rn×n , B ∈ Rn and C ∈ R1×n . Consider the system x˙ = Ax + Bu, y = Cx,
parameterized by G ∈ Cν , where Π is the unique solution of (3). Assume σ(S − GL) ∩ σ(S) = ∅. Let φˆ ∈ C1×ν be the moments of (7) at σ(S). Then (7) describes a family of reduced order models of (2), parameterized in G and achieving moment matching ˆ at σ(S), i.e., φ = φ. 2) Let the pair (Q, R) be controllable and assume σ(A)∩ σ(Q) = ∅. Let ξ(t) ∈ Rν and consider the family of linear models ξ˙ = (Q − RH)ξ + ΥBu, ΣH : (8) η = Hξ,
(2)
with x(t) ∈ Rn , u(t) ∈ R and y(t) ∈ R. Assume that the system (2) is a minimal realization of the transfer function G(s). The moment of (1) is defined as follows. Definition 1. [4], [1] The 0-moment at s1 ∈ C of system (1) is the complex number η0 (s1 ) = C(s1 I − A)−1 B. The k−moment of system (1) at s1 is the complex number (−1)k dk [C(sI − A)−1 B] , k ≥ 1 and · ηk (s1 ) = k! dsk s=s1 integer.
Consider the linear system (2) and let the matrices S ∈ Rν×ν , L ∈ R1×ν and Q ∈ Rν×ν , R ∈ Rν be such that the pair (L, S) is observable and the pair (Q, R) is controllable, respectively. Based on Definition 1, moments can be characterized using the solution of the Sylvester equation AΠ + BL = ΠS, (3) in the unknown Π ∈ C
n×ν
parameterized by H ∈ R1×ν , where Υ is the unique solution of (4). Assume σ(Q − RH) ∩ σ(Q) = ∅. Let ϕˆ ∈ C1×ν be the moments of (8) at σ(Q). Then (8) describes a family of reduced order models of (2), parameterized in H and achieving moment matching at σ(Q), i.e., ϕ = ϕ. ˆ Remark 1. Any system ξ˙ = F ξ + Gξ, η = Hξ,
and its ”dual”
QΥ = ΥA + RC,
(4)
in the unknown Υ ∈ Cν×n . Assume that σ(A) ∩ σ(S) = ∅, where σ(A) is the spectrum of the matrix A. Since Σ is minimal, equation (3) has a unique solution Π and rank Π = ν. Assuming σ(A) ∩ σ(Q) = ∅, then equation (4) has a unique solution Υ and rank Υ = ν (see e.g., [6]). Definition 2. [4], [5] 1) Let φ = [φ1 φ2 ... φν ] ∈ C1×ν be such that φ = CΠ.
We call the moments of system (2) at σ(S) the elements φi , i = 1, ..., ν. The interpolation points are the eigenvalues of S, i.e., {s1 , s2 , ..., sν } = σ(S). 2) Let ϕ = [ϕ1 ϕ2 ... ϕν ]T ∈ Cν be such that ϕ = ΥB.
with F ∈ Rν×ν , G ∈ Rν , H ∈ R1×ν matches the moments of (2) at σ(S) if and only if it is equivalent1 to (7), i.e., HP = CΠ, where the coordinate transformation P ∈ Cν×ν is the unique solution of the Sylvester equation F P + GL = P S (see, e.g., [4] for more details). Similar arguments hold for the case of (8) (see, e.g., [5] for more details). III. N ONLINEAR MOMENT
(5)
(6)
We call the moments of system (2) at σ(Q) the elements ϕi , i = 1, ..., ν. The interpolation points are
(9)
MATCHING
Consider a single-input single-output nonlinear system described by equations of the form x˙ = f (x) + g(x)u, y = h(x),
(10)
with x(t) ∈ Rn and f, g and h smooth mappings. 1 Two minimal systems described by state-space equations are equivalent if they have the same transfer functions, i.e., the same input-output behaviour.
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A. The system driven by a signal generator
B. New approach - the signal generator driven by the system Consider a generalized signal generator defined by the equations
Consider a signal generator defined by the equations ω˙ = s(ω), θ = l(ω),
with ω(t) ∈ Rν , θ(t) ∈ R and s and l smooth. Furthermore, assume that f (0) = 0, h(0) = 0, s(0) = 0 and l(0) = 0. Let π (defined in the neighbourhood of 0) be the solution of the partial differential equation ∂π(ω) s(ω) = f (π(ω)) + g(π(ω))l(ω). ∂ω
Definition 4 (Moment matching). [4] The system ξ˙ = φ(ξ) + δ(ξ)u, η = κ(ξ), with ξ(t) ∈ Rν , matches the moment of (10) at {s(ω), l(ω)} if it has the same moment at {s(ω), l(ω)} as (10), i.e., the equation (13)
has a unique solution p(ω) such that h(π(ω)) = κ(p(ω)), where π(ω) is the unique solution of (12).
(14)
A family of reduced order models, all achieving moment matching at {s(ω), l(ω)}, is described by ( ξ˙ = s(ξ) − δ(ξ)l(ξ) + δ(ξ)u, Σδ(ξ) : (15) ψ = h(π(ξ)), where δ(·) is such that the equation s(p(ω)) + δ(p(ω))l(ω) − δ(p(ω))l(p(ω)) =
with ω(t) ∈ Rν , q : Rν → Rν a smooth mapping, q(0) = 0, r ∈ Rν , υ : Rn → Rν a smooth mapping, υ (0) = 0, w(t) ∈ R and d(t) ∈ Rν . Consider the interconnection depicted in Figure 1. α(x)
Now, we present the definition of a system that matches the moment h(π(ω)) of (10) at {s(ω), l(ω)}.
∂p(ω) s(ω), ∂ω
(17a) (17b)
(12)
Definition 3. [3], [4] Consider system (10) and the signal generator (11). Assume that the signal generator (11) is observable and that θ(t) is a persistent signal, i.e., the equilibrium ω ¯ = 0 of the system (11) is Poisson stable2 . Furthermore, assume that π(ω) is the unique solution of (12). Then, we call the function h(π(ω)) the moment of (10) at {s(ω), l(ω)} and moreover, this function is in one-to-one relation3 with the steady-state response (provided it exists) of the interconnection between the signal generator (11) and the nonlinear system (10), with u(t) = θ(t), for all t ≥ 0.
φ(p(ω)) + δ(p(ω))l(ω) =
ω˙ = q(ω) + rv, ω(0) = 0, d = ω + υ (x),
(11)
∂p(ω) s(ω), ∂ω (16)
has the unique solution p(ω) = ω.
xT
x˙ = f (x) y = w y = h(x)
Fig. 1.
ω˙ = q(ω) + rv
ωT
Diagram describing α.
Assume that the system (10) is observable4 and the system (17a) is reachable from ω(0) = 05 . Let T > 0 be such that ωT = ω(T ) and xT = x(T ). Note that such T exists by the reachability assumption. Let u(T ) = 0. Then y(t) = h(x(t)), for all t ≥ T . Letting v(t) = y(t), yields ω(t) ˙ = q(ω(t)) + rh(x(t)). Define α : Rn → Rν , ωT = α(xT ), for all T , α(0) = 0 (see Fig. 1). Hence d(T ) = α(xT ) + υ (xT ) satisfies the statements from the following result. Lemma 1. Consider the nonlinear signal generator (17a). Assume (17a) is reachable from 0 and (10) is observable. Let d(t) as in (17b) and assume it is differentiable. Then d(t) satisfies the differential equation υ (x) ∂υ ˙ d = q(ω) − q(ω − d) + g(x) u, (18) ∂x x=ρ(d−ω) υ (x)) = x, if where ρ : Rν → Rn is a mapping such that ρ(υ and only if υ (·) satisfies the equation υ (x)) = −q(−υ where υ (x) is as in (17b).
υ (x) ∂υ f (x) + rh(x), ∂x
(19)
Note that the pair (d, u) = (0, 0) is an equilibrium of (18). Remark 2. If q(ω) = Qω, Q ∈ Rν×ν , then d˙ = Qd + υ (x) ∂υ g(x) u. ∂x x=ρ(d−ω)
Throughout the rest of the paper we make the following working assumption. Assumption 1. The mapping υ in (17b) is the unique solution of the nonlinear partial differential equation (19).
2 An equilibrium point ω ¯ is said to be Poisson stable if the trajectory ω(t), solution of the equation ω˙ = s(ω), passes close to ω ¯ for arbitrarily large times, in forward and backward direction. Hence, if every point in a neighbourhood of ω ¯ is Poisson stable, no trajectory of (11) can decay to zero as time tends to infinity, see e.g., [13, Chapter 8]. 3 By one-to-one relation between a set of ν moments η(s ), i = 1, ..., ν i and the well-defined steady state response of the signal y(t) we mean that the moments η uniquely determine the steady-state response of y(t).
4 System (10) is not observable if, for any pair of initial conditions x1 (0) 6= x2 (0), the corresponding output trajectories h(x1 (t)) and h(x2 (t)) are such that h(x1 (t)) = h(x2 (t)), for all t ≥ 0 (see e.g, [4]). 5 System (17a) is called reachable from ω(0) if for all ω there exists an input v and T ≥ 0 such that ω = ψ(T, 0, ω(0), v), where ψ(t, 0, ω(0), v) denotes the solution of (17a) for a particular control w and initial condition ω(0) (see e.g., [17], [16]).
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υ (x)) = Furthermore, there exists ρ : Rν → Rn such that ρ(υ x. We are now ready to define a notion of moment for the nonlinear system (10). Definition 5. Consider the nonlinear system (10) and the generalized nonlinear signal generator (17a). Suppose that Assumption 1 holds. We call the moment of (10) at {q(ω), r} υ (x) ∂υ the function g(x) , with the signal d as in ∂x x=ρ(d−ω) (17b).
Theorem 2. Consider the system (10) and the signal generator (17a). Suppose Assumption 1 holds. Assume that the zero equilibrium of (10) is locally exponentially stable, the system (10) is observable, the system (17a) is reachable and its response for ω(0) = 0 is Poisson stable. Then the moment of (10) at {q(ω), r} is in one-to-one relation with the welldefined steady state of d(t) for u = δ0 (t) (the Dirac function) with d(t) as in (17b). We now give the definition of a model that achieves moment matching. Consider the nonlinear system ξ˙ = ϕ(ξ) + γ(ξ)u, η = ψ(ξ),
(20)
where ξ(t) ∈ Rν , u(t) ∈ R and η(t) ∈ R, with ϕ, γ and ψ smooth. Consider the interconnection w(t) = η(t), between (20) and the signal generator defined by (17a) with the output ζ(t) ∈ Rν (see Figure 2). u
ξ˙ = ϕ(ξ) + γ(ξ)u η = w η = ψ(ξ)
ω˙ = q(ω) + rv
ζ
Fig. 2. Interconnection between the nonlinear reduced order model and the generalized signal generator.
Hence, the equations υ (x) ξ˙ = −q(−ξ) − rψ(ξ) + ∂υ g(x) u, ∂x Σψ(ξ) : x=ρ(ξ) η = ψ(ξ), (21) with ξ(t) ∈ Rν define a class of models of order ν, parameterized in ψ(ξ) ∈ R, that match the moments of (10) at {q(ω), r}. Remark 3. If q(ω) is linear, i.e., q(ω) = Qω, then the model (20) of order ν, that matches the moments of the nonlinear system (10) at σ(Q) if and only if ζ˙ = υ (x) ∂υ g(x) u which is equivalent to ϕ(ξ) = Qζ + ∂x x=ρ(ξ) υ (x) ∂υ g(x) , where υ Qξ − rψ(ξ) and γ(ξ) = ∂x x=ρ(ξ)
is the (unique) solution of (19) with the left side term υ (x)) = Qυ υ (x). −q(−υ
Moment matching in the sense of Definition 6 is related to moment matching in the sense of Definition 4, i.e., a model Σψ(ξ) matches the moments of (10) at {q(ω), r}, in the sense of Definition 6 if and only if Σψ(ξ) is yielded from a model Σδ(ξ) , by a particular selection of δ(ξ). The converse statement also holds, i.e., a model Σδ(ξ) matches the moments of (10) at {−q(−ω), rT }, in the sense of Definition 4 if and only if Σδ(ξ) is yielded from a model Σψ(ξ) , by a particular selection of ψ(ξ). Theorem 4. Consider a nonlinear observable system (10). Let the pair {q(ω), r} define a generalized signal generator (17a) and assume that (17a) is reachable from 0. Furthermore, assume that the pair {−q(−ω), rT } defines an observable signal generator (11). Consider the systems Σψ(ξ) as in (21) and Σδ(ξ) as in (15) and assume that Σδ(ξ) is observable. Suppose that Assumption 1 holds. The following statements are equivalent.
Definition 6. Consider the nonlinear system (10) and the nonlinear signal generator (17a). Assume (17a) is reachable from 0. A system (20) matches the moments of the system (10) at {q(ω), r} if the steady-state response (provided it exists) of the output signal ζ(t) of the interconnection between (20) and (17a) matches the steady-state response (provided it exists) of the signal d(t) in (17b). Based on Definition 6 we define a class of parameterized models of order ν, that achieve moment matching. Theorem 3. Consider the nonlinear system (20) and the signal generator (17a). Suppose Assumption 1 holds. Assume (17a) is reachable from 0 and (20) is observable. Let ζ = ω + ξ. Then the system (20) matches the moments of (10) at {q(ω), r} (in the sense of Definition 6) if and only if υ (x) ∂υ ϕ(ξ) = −q(−ξ) − rψ(ξ) and γ(ξ) = g(x) , ∂x x=ρ(ξ) for all ψ(ξ) ∈ R. 5521
1) Σψ(ξ) , described by equations (21), matches the moments of (10) at {q(ω), r}, in the sense of Definition 6. υ (x) 2) rT ψ(ξ) = ∂υ∂x , ψ(ω) = h(π(ω)), where x=ρ(ξ)
ψ(ξ) is as in (21), υ and ρ are such that Assumption 1 holds and π(ω) is the unique solution of (12). 3) Σψ(ξ) , described by equations (21), matches the moments of (10) at {−q(−ω), rT }, in the sense of Definition 4. 4) Σδ(ξ) , described by equations (15), matches the moments of (10) at {−q(−ω), rT }, in the sense of Definition 4. υ (x) , where 5) rT h(π(ξ)) = δ T (ξ), δ T (ξ) = ∂υ∂x x=ρ(ξ)
δ(ξ) is as in (15), υ and ρ are such that Assumption 1 holds and π(ω) is the unique solution of (12). 6) Σδ(ξ) , described by equations (15), matches the moments of (10) at {q(ω), r}, in the sense of Definition 6.
ω˙ = −q(−ω) θ = rT ω Fig. 3.
θ=u
Σψ(ξ) at {q(ω), r}
at {q(ω), r}, of a given nonlinear system, simultaneously. The arguments yielding the next result follow the proof of Proposition 1.
η
Diagram illustrating statements 1), 2) and 3) of Theorem 4.
IV. M ATCHING
FURTHER MOMENTS
In this section we compute the reduced order model(s) that matches a number of moments equal to twice its order. This is done by a proper selection of the free parameters, i.e., from the classes of models that achieve moment matching we identify the one matching two sets of moments, simultaneously.
Proposition 2. Consider the nonlinear system (10) and the signal generators (11) and (17a). Let π ∈ Rν be the unique solution of (12), υ ∈ Rn be the unique solution of (19) and ρ be such that Assumption 1 holds. Then the following statements hold. 1) There exists a subclass of models Σδ(ξ) as in (15) that match the moments of (10) at {s(ω), l(ω)} and {q(ω), r} simultaneously, if and only if δ(ξ) satisfies the equation υ (x) ∂υ ∂p(ξ) δ(ξ) = g(x) , (22) ∂ξ ∂x
A. Linear case
x=ρ(ξ)
ν
where p : R → R satisfies the equation υ (x) ∂p(ξ) ∂υ q(−p(ξ)) + s(ξ) = g(x) ∂ξ ∂x
We begin with a simple example illustrating the problem and the idea. Example 1. Let η0 ∈ C and η1 ∈ C be such that η0 6= η1 . By η0 g Theorem 1, the transfer function Kg (s) = , with s − s0 + g g ∈ C, defines a class of first order models, parameterized in g, that match the moment η0 at s0 ∈ C, i.e., Kg (s0 ) = η0 . Let q1 ∈ C be such that q1 6= s0 . We want to compute the first order model Kg that also matches the moment η1 at q1 , i.e., compute the parameter g, such that Kg (q1 ) = η1 . This last condition is equivalent to η0 g = [(s0 − q1 ) + g]η1 ⇔ g(η0 − η1 ) = (s0 − q1 )η1 . Since we assume η0 6= η1 , s0 − q1 . Let p = we obtain the unique parameter g = η1 η0 − η1 η0 − η1 . Note that since we assume that s0 6= q1 , p is well s0 − q1 defined. Hence g = η/p. Since g is unique, then p is also unique and it satisfies the equation ps0 − q1 p = η0 − η1 . Proposition 1. Consider a linear system (2) and let S ∈ Cν×ν and Q ∈ Cν×ν be such that σ(S) ∩ σ(Q) ∩ σ(A) = ∅. Let LT , R ∈ Cν be such that the pair (L, S) is observable and the pair (Q, R) is controllable. Let Π ∈ Cν×n be the unique solution of (3) and Υ ∈ Cn×ν be the unique solution of (4). Then the following statements hold and are equivalent. 1) A model ΣG as in (7) matches the moments of (2) at σ(S) and σ(Q) simultaneously, if and only if G = P −1 ΥB, where P ∈ Cν×ν is the unique solution of the equation P S − QP = ΥBL − RCΠ. Furthermore, ΣP −1 ΥB is the unique model, from the class ΣG , that matches 2ν moments of (2). 2) A model ΣH as in (8) matches the moments of (2) at σ(S) and σ(Q) simultaneously, if and only if H = CΠP −1 , where P ∈ Cν×ν is the unique solution of the equation P S − QP = ΥBL − RCΠ. Furthermore, ΣCΠP −1 is the unique model, from the class ΣH , that matches 2ν moments of (2).
l(ξ)
x=ρ(ξ)
− rh(π(ξ)).
(23)
2) There exists a subclass of models Σψ(ξ) as in (21) that match the moments of (10) at {s(ω), l(ω)} and {q(ω), r} simultaneously, if and only if ψ(p(ω)) = h(π(ω)),
(24)
where p : Rν → Rν satisfies the equation υ (x) ∂υ ∂p(ω) s(ω) = g(x) q(−p(ω)) + ∂ω ∂x
l(ω)
x=ρ(p(ω))
− rh(π(ω)).
(25)
Remark 4. Assuming that p uniquely satisfies equation (23), there exists a unique model Σδ(ξ) that matches the moments of (10) at {s(ω), l(ω)} and {q(ω), r}. Furthermore, if p(ξ) is a diffeomorphism, then Σδ(ξ) = Σ ∂p(ξ) −1 ∂υυ(x) g(x) . ( ∂ξ ) |x=ρ(ξ) ∂x Similarly, assuming that p uniquely satisfies equation (25), there exists a unique model Σψ(ξ) that matches the moments of (10) at {s(ω), l(ω)} and {q(ω), r}. Finally, if we assume that p(·) is the identity, then statement 1) and statement 2) from Proposition 2 are equivalent and moreover Σδ(ξ) = Σ ∂υυ(x) g(x)| ∂x
= Σh(π(ξ)) = Σψ(ξ) x=ρ(ξ)
is the unique model of order ν that matches the moments of (10) at {s(ω), l(ω)} and {q(ω), r}, simultaneously. Example 2. Consider a first order system (10), described by the equations x˙ = −x − x3 + u, y = x,
B. Nonlinear case In this section we give the nonlinear counterpart of Proposition 1, i.e., we compute the subclass of models of order ν, that match the moments at {s(ω), l(ω)} and the moments
ν
(26)
and let the signal generator (11) be ω˙ = 0, θ = ω. Consider the equation π 3 (ω) + π(ω) − ω = 0, which is of the form
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(12). This equation has the unique (real) solution q p 1 3 π(ω) = 108 ω + 12 12 + 81 ω 2 6 1 −2 p , π(0) = 0. √ 3 108 ω + 12 12 + 81 ω 2
(27)
A class of first order models, parameterized in δ(ξ) that match the moment π(ω), as in (27), of (26) at {0, ω} is ξ˙ = δ(ξ)(u − ξ), q p 1 3 η = 108 ξ + 12 12 + 81 ξ 2 6 1 , − 2q p 3 108 ξ + 12 12 + 81 ξ 2
(28)
where δ(ξ) satisfies (13). Consider now the generalized signal generator ω˙ = rw, υ (x) 3 ∂υ r ∈ R. Equation (19) becomes (x + x) = rx. For ∂x the initial condition υ (0) = 0, the (unique) solution of this differential equation is υ (x) = r arctan x. Since arctan is υ (x)) = x, given invertible, there exists ρ that satisfies ρ(υ x by ρ(x) = tan (Assumption 1 holds). Hence the moment r υ (x) ∂υ of (26) at {0, r} as in Definition 5 is g(x) = ∂x x=ρ(ξ) r r ξ = . A class of first = r cos2 1 + x2 r 1 + tan2 ξ x=ρ(ξ)
r
order models parameterized by ψ(ξ) that match the moment 2 ξ r cos r of (26) at {0, r}, are given by the equations ξ u − ψ(ξ) , ξ˙ = r cos2 r η = ψ(ξ). (29) Let r = 1. Consider equation (25), i.e., π(ω(ξ)) − cos2 (ξ) = 0,
(30)
with π(ω) as in (27). Equation (30) has a unique solution ω(ξ) = cos6 ξ+cos2 ξ. By Proposition 2, there exists a model (29) with ψ(ξ) = π(ω(ξ)) = cos2 ξ that matches both the moment π(ξ) as in (27) and the moment cos2 ξ of (26), i.e., ξ˙ = cos2 ξ(u − ξ), η = cos2 ξ. V. C ONCLUSIONS We have presented a new time-domain notion of moments for a class of nonlinear systems by considering the swapped interconnection between the nonlinear system and a generalized signal generator, i.e., the nonlinear system
drives a generalized signal generator which contains the interpolation information. We have stated that the moments are in one-to-one relation with the steady-state response of such interconnection. Based on this notion we have defined moment matching and we have computed the family of parameterized reduced order models that achieve moment matching. Furthermore, we have shown that the two types of nonlinear moment matching are equivalent in the sense that a model from one family that matches the moments of the given system can be obtained from the other family of models by a proper selection of the parameter. Finally, we have computed the reduced order model(s) that matches moments of the system at two sets of interpolation points, simultaneously, i.e., the number of interpolation points is twice the order of the model. R EFERENCES [1] A. C. Antoulas. Approximation of large-scale dynamical systems. SIAM, Philadelphia, 2005. [2] A. C. Antoulas, J. A. Ball, J. Kang, and J. C. Willems. On the solution of the minimal rational interpolation problem. Linear Algebra & its Applications, 137/138:511–573, 1990. [3] A. Astolfi. Model reduction by moment matching for nonlinear systems. In Proc. 47th IEEE Conf. on Decision and Control, pages 4873–4878, 2008. [4] A. Astolfi. Model reduction by moment matching for linear and nonlinear systems. IEEE Trans. Autom. Contr., 50(10):2321–2336, 2010. [5] A. Astolfi. Model reduction by moment matching, steady-state response and projections. In Proc. 49th IEEE Conf. on Decision and Control, pages 5344 – 5349, 2010. [6] E. de Souza and S. P. Bhattacharyya. Controllability, observability and the solution of AX − XB = C. Linear Algebra & Its App., 39:167–188, 1981. [7] C. de Villemagne and R. E. Skelton. Model reductions using a projection formulation. Int. J. Control, 46:2141–2169, 1987. [8] K. Gallivan and P. Van Dooren. Rational approximations of pre-filtered transfer functions via the Lanczos algorithm. Numerical Algorithms, 20:331–342, 1999. [9] K. Gallivan, A. Vandendorpe, and P. Van Dooren. Sylvester equations and projection based model reduction. J. Comp. Appl. Math., 162:213– 229, 2004. [10] W. B. Gragg and A. Lindquist. On the partial realization problem. Linear Algebra & its Applications, 50:277–319, 1983. [11] T. C. Ionescu and A. Astolfi. On moment matching with preservation of passivity and stability. In Proc. 49th IEEE Conf. on Decision and Control, pages 6189 – 6194, 2010. [12] T. C. Ionescu and A. Astolfi. Moment matching for linear port Hamiltonian systems. In Proc. 50th IEEE Conf. Decision & Control - European Control Conf., pages 7164–7169, 2011. [13] A. Isidori. Nonlinear control systems. Springer-Verlag, New York, Third edition, 1995. [14] I. M. Jaimoukha and E. M. Kasenally. Implicitly restarted Krylov subspace methods for stable partial realizations. SIAM J. Matrix Anal. Appl., 18:633–652, 1997. [15] A. J. Mayo and A. C. Antoulas. A framework for the solution of the generalized realization problem. Linear Algebra & Its App., 425:634– 662, 2007. [16] H. Nijmeijer and A. J. van der Schaft. Nonlinear dynamical control systems. Springer-Verlag, New York, 1990. [17] A. J. van der Schaft. L2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control. IEEE Trans. Autom. Contr., 37(6):770–784, 1992. [18] P. van Dooren. The Lanczos algorithm and Pad´e approximation. Benelux Meeting on Systems and Control, 1995. Minicourse.
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Families of reduced order models that achieve nonlinear moment matching T. C. Ionescu, A. Astolfi Abstract— In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems, affine in the input, in terms of the steady-state response of the output of a generalized signal generator driven by the nonlinear system. In addition, we define a new notion of moment matching and present the class of (nonlinear) parameterized reduced order models that achieve moment matching. Furthermore, we establish relations with existing notions of moment, showing that the families of reduced order models that achieve nonlinear moment matching are equivalent. Furthermore, we compute the reduced order model that matches moments at two sets of interpolation points, simultaneously, i.e., the number of interpolation points is twice the order of the model.
I. I NTRODUCTION In the problem of model reduction, moment matching techniques represent an efficient tool, see e.g. [1] for a complete overview for linear systems. In such techniques the (reduced order) model is obtained by constructing a lower degree rational function that approximates the original transfer function (assumed rational). The classical notion of moment has been given in [1], based on the series expansion of the transfer function of the linear system (see also [10], [18], [8]). The low degree rational function matches (some of) the terms of the original transfer function at various points in the complex plane. However, since the computation of moments is computationally costly, the reduced order models that match a prescribed number of moments are computed efficiently using e.g. Krylov subspace techniques, projections, etc. (see e.g., [7], [2], [14]). Recently, a time-domain notion of moment for nonlinear systems has been proposed. According to [4], the moments of a nonlinear system are related to the steady-state response of the nonlinear system driven by a signal generator. Furthermore, the moments are related to the solution of a nonlinear (Sylvester-like) partial differential equation (see [9], [15] for the linear system arguments). Based on this notion of moment a family of parameterized, reduced order models that achieve moment matching is computed. The free parameters can be used to enforce desired properties on the reduced order model, such as e.g., stability and passivity (see e.g. [11], [12]). T. C. Ionescu is with the Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, London, UK. E-mail: [email protected]. A. Astolfi is with the Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, London, UK and with Dipartimento di Informatica, Sistemi e Produzione, Universit`a di Roma Tor Vergata, 00133 Roma, Italy. E-mail: [email protected]. This work is supported by the EPSRC grant ”Control For Energy and Sustainability”, grant reference EP/G066477/1.
978-1-4799-0178-4/$31.00 ©2013 AACC
In this paper we present a new time-domain notion of moment for a class of single-input, single-output nonlinear systems, affine in the input, by considering the swapped interconnection between the given nonlinear system and the signal generator, i.e., the nonlinear system drives a generalized signal generator. This is an extension of the linear arguments in [5]. Under some technical assumptions, the moment is in one-to-one relation with the steady-state response (provided it exists) of the output of such interconnection. Then, we define moment matching and we compute the class of parameterized reduced order models that achieve moment matching. Furthermore, we prove that the two types of moment matching are equivalent, i.e., a model from one family of models that match the moments of the system can be obtained from the ”swapped” family of models by a proper selection of parameters. In general, we compute the (subclass of) reduced order model(s) that match two moments simultaneously. The result is a nonlinear extension of the computation of a reduced order, linear model that matches a number of moments equal to twice its order. The paper represents preliminary steps towards developing nonlinear model reduction algorithms applicable in practice. In this paper we study the single-input, single-output case, although it is not difficult to extend the results to the multiple-input, multiple-output case. The paper is organized as follows. In Section II we give a brief overview of the linear notion of moment based on the steady-state response of the output of a generalized signal generator driven by the given system and the corresponding class of parameterized linear models that achieve moment matching. In Section III, we give a short presentation of the state-of-the-art notion of time-domain moments for nonlinear systems and the resulting family of reduced order models that achieve moment matching. Then, we define and analyze the output of a generalized nonlinear signal generator driven by the system. Based on this analysis we define a new notion of time-domain moment which is in a one-to-one relation with the steady-state response (provided it exists) of the output signal of the interconnection. We define the notion of moment matching, i.e., a model matches the moments of the system if the output of the signal generator driven by the former captures the behaviour of the output of the signal generator driven by the latter. Then we define the class of parameterized, reduced order, nonlinear models that achieve moment matching. We establish a relation between the notion of moment and moment matching based on the output of the system driven by a signal generator and its ”swapped” counterpart, by a proper selection of the free
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parameters. In Section IV we compute a reduced order model that matches a number of moments equal to twice the number of prescribed interpolation points for linear systems. We also give the nonlinear extension of this result and give an example to illustrate the theory. The paper is completed by some conclusions. II. L INEAR MOMENT MATCHING In this section we recall the idea of moment matching for linear, single-input, single-output systems, from a time domain point of view as presented in [5]. Let −1
G(s) = C(sI − A)
B,
the eigenvalues of Q, i.e., {s1 , s2 , ..., sν } = σ(Q). Note that the moments as in Definition 1 are equivalent to the notions depicted in Definition 2. Selecting (L, S) and (Q, R) in canonical forms, easy computations yield [η(s1 ) ... η(sν )] = φ = ϕ. Based on Definition 2, we define a family of parameterized models of order ν that achieve moment matching at the interpolation points {s1 , ..., sν } = σ(S). Theorem 1. [4], [5] 1) Let the pair (L, S) be observable and assume σ(A) ∩ σ(S) = ∅. Let ξ(t) ∈ Rν and consider the family of linear models ξ˙ = (S − GL)ξ + Gu, ΣG : (7) η = CΠξ,
(1)
G : C → C, with A ∈ Rn×n , B ∈ Rn and C ∈ R1×n . Consider the system x˙ = Ax + Bu, y = Cx,
parameterized by G ∈ Cν , where Π is the unique solution of (3). Assume σ(S − GL) ∩ σ(S) = ∅. Let φˆ ∈ C1×ν be the moments of (7) at σ(S). Then (7) describes a family of reduced order models of (2), parameterized in G and achieving moment matching ˆ at σ(S), i.e., φ = φ. 2) Let the pair (Q, R) be controllable and assume σ(A)∩ σ(Q) = ∅. Let ξ(t) ∈ Rν and consider the family of linear models ξ˙ = (Q − RH)ξ + ΥBu, ΣH : (8) η = Hξ,
(2)
with x(t) ∈ Rn , u(t) ∈ R and y(t) ∈ R. Assume that the system (2) is a minimal realization of the transfer function G(s). The moment of (1) is defined as follows. Definition 1. [4], [1] The 0-moment at s1 ∈ C of system (1) is the complex number η0 (s1 ) = C(s1 I − A)−1 B. The k−moment of system (1) at s1 is the complex number (−1)k dk [C(sI − A)−1 B] , k ≥ 1 and · ηk (s1 ) = k! dsk s=s1 integer.
Consider the linear system (2) and let the matrices S ∈ Rν×ν , L ∈ R1×ν and Q ∈ Rν×ν , R ∈ Rν be such that the pair (L, S) is observable and the pair (Q, R) is controllable, respectively. Based on Definition 1, moments can be characterized using the solution of the Sylvester equation AΠ + BL = ΠS, (3) in the unknown Π ∈ C
n×ν
parameterized by H ∈ R1×ν , where Υ is the unique solution of (4). Assume σ(Q − RH) ∩ σ(Q) = ∅. Let ϕˆ ∈ C1×ν be the moments of (8) at σ(Q). Then (8) describes a family of reduced order models of (2), parameterized in H and achieving moment matching at σ(Q), i.e., ϕ = ϕ. ˆ Remark 1. Any system ξ˙ = F ξ + Gξ, η = Hξ,
and its ”dual”
QΥ = ΥA + RC,
(4)
in the unknown Υ ∈ Cν×n . Assume that σ(A) ∩ σ(S) = ∅, where σ(A) is the spectrum of the matrix A. Since Σ is minimal, equation (3) has a unique solution Π and rank Π = ν. Assuming σ(A) ∩ σ(Q) = ∅, then equation (4) has a unique solution Υ and rank Υ = ν (see e.g., [6]). Definition 2. [4], [5] 1) Let φ = [φ1 φ2 ... φν ] ∈ C1×ν be such that φ = CΠ.
We call the moments of system (2) at σ(S) the elements φi , i = 1, ..., ν. The interpolation points are the eigenvalues of S, i.e., {s1 , s2 , ..., sν } = σ(S). 2) Let ϕ = [ϕ1 ϕ2 ... ϕν ]T ∈ Cν be such that ϕ = ΥB.
with F ∈ Rν×ν , G ∈ Rν , H ∈ R1×ν matches the moments of (2) at σ(S) if and only if it is equivalent1 to (7), i.e., HP = CΠ, where the coordinate transformation P ∈ Cν×ν is the unique solution of the Sylvester equation F P + GL = P S (see, e.g., [4] for more details). Similar arguments hold for the case of (8) (see, e.g., [5] for more details). III. N ONLINEAR MOMENT
(5)
(6)
We call the moments of system (2) at σ(Q) the elements ϕi , i = 1, ..., ν. The interpolation points are
(9)
MATCHING
Consider a single-input single-output nonlinear system described by equations of the form x˙ = f (x) + g(x)u, y = h(x),
(10)
with x(t) ∈ Rn and f, g and h smooth mappings. 1 Two minimal systems described by state-space equations are equivalent if they have the same transfer functions, i.e., the same input-output behaviour.
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A. The system driven by a signal generator
B. New approach - the signal generator driven by the system Consider a generalized signal generator defined by the equations
Consider a signal generator defined by the equations ω˙ = s(ω), θ = l(ω),
with ω(t) ∈ Rν , θ(t) ∈ R and s and l smooth. Furthermore, assume that f (0) = 0, h(0) = 0, s(0) = 0 and l(0) = 0. Let π (defined in the neighbourhood of 0) be the solution of the partial differential equation ∂π(ω) s(ω) = f (π(ω)) + g(π(ω))l(ω). ∂ω
Definition 4 (Moment matching). [4] The system ξ˙ = φ(ξ) + δ(ξ)u, η = κ(ξ), with ξ(t) ∈ Rν , matches the moment of (10) at {s(ω), l(ω)} if it has the same moment at {s(ω), l(ω)} as (10), i.e., the equation (13)
has a unique solution p(ω) such that h(π(ω)) = κ(p(ω)), where π(ω) is the unique solution of (12).
(14)
A family of reduced order models, all achieving moment matching at {s(ω), l(ω)}, is described by ( ξ˙ = s(ξ) − δ(ξ)l(ξ) + δ(ξ)u, Σδ(ξ) : (15) ψ = h(π(ξ)), where δ(·) is such that the equation s(p(ω)) + δ(p(ω))l(ω) − δ(p(ω))l(p(ω)) =
with ω(t) ∈ Rν , q : Rν → Rν a smooth mapping, q(0) = 0, r ∈ Rν , υ : Rn → Rν a smooth mapping, υ (0) = 0, w(t) ∈ R and d(t) ∈ Rν . Consider the interconnection depicted in Figure 1. α(x)
Now, we present the definition of a system that matches the moment h(π(ω)) of (10) at {s(ω), l(ω)}.
∂p(ω) s(ω), ∂ω
(17a) (17b)
(12)
Definition 3. [3], [4] Consider system (10) and the signal generator (11). Assume that the signal generator (11) is observable and that θ(t) is a persistent signal, i.e., the equilibrium ω ¯ = 0 of the system (11) is Poisson stable2 . Furthermore, assume that π(ω) is the unique solution of (12). Then, we call the function h(π(ω)) the moment of (10) at {s(ω), l(ω)} and moreover, this function is in one-to-one relation3 with the steady-state response (provided it exists) of the interconnection between the signal generator (11) and the nonlinear system (10), with u(t) = θ(t), for all t ≥ 0.
φ(p(ω)) + δ(p(ω))l(ω) =
ω˙ = q(ω) + rv, ω(0) = 0, d = ω + υ (x),
(11)
∂p(ω) s(ω), ∂ω (16)
has the unique solution p(ω) = ω.
xT
x˙ = f (x) y = w y = h(x)
Fig. 1.
ω˙ = q(ω) + rv
ωT
Diagram describing α.
Assume that the system (10) is observable4 and the system (17a) is reachable from ω(0) = 05 . Let T > 0 be such that ωT = ω(T ) and xT = x(T ). Note that such T exists by the reachability assumption. Let u(T ) = 0. Then y(t) = h(x(t)), for all t ≥ T . Letting v(t) = y(t), yields ω(t) ˙ = q(ω(t)) + rh(x(t)). Define α : Rn → Rν , ωT = α(xT ), for all T , α(0) = 0 (see Fig. 1). Hence d(T ) = α(xT ) + υ (xT ) satisfies the statements from the following result. Lemma 1. Consider the nonlinear signal generator (17a). Assume (17a) is reachable from 0 and (10) is observable. Let d(t) as in (17b) and assume it is differentiable. Then d(t) satisfies the differential equation υ (x) ∂υ ˙ d = q(ω) − q(ω − d) + g(x) u, (18) ∂x x=ρ(d−ω) υ (x)) = x, if where ρ : Rν → Rn is a mapping such that ρ(υ and only if υ (·) satisfies the equation υ (x)) = −q(−υ where υ (x) is as in (17b).
υ (x) ∂υ f (x) + rh(x), ∂x
(19)
Note that the pair (d, u) = (0, 0) is an equilibrium of (18). Remark 2. If q(ω) = Qω, Q ∈ Rν×ν , then d˙ = Qd + υ (x) ∂υ g(x) u. ∂x x=ρ(d−ω)
Throughout the rest of the paper we make the following working assumption. Assumption 1. The mapping υ in (17b) is the unique solution of the nonlinear partial differential equation (19).
2 An equilibrium point ω ¯ is said to be Poisson stable if the trajectory ω(t), solution of the equation ω˙ = s(ω), passes close to ω ¯ for arbitrarily large times, in forward and backward direction. Hence, if every point in a neighbourhood of ω ¯ is Poisson stable, no trajectory of (11) can decay to zero as time tends to infinity, see e.g., [13, Chapter 8]. 3 By one-to-one relation between a set of ν moments η(s ), i = 1, ..., ν i and the well-defined steady state response of the signal y(t) we mean that the moments η uniquely determine the steady-state response of y(t).
4 System (10) is not observable if, for any pair of initial conditions x1 (0) 6= x2 (0), the corresponding output trajectories h(x1 (t)) and h(x2 (t)) are such that h(x1 (t)) = h(x2 (t)), for all t ≥ 0 (see e.g, [4]). 5 System (17a) is called reachable from ω(0) if for all ω there exists an input v and T ≥ 0 such that ω = ψ(T, 0, ω(0), v), where ψ(t, 0, ω(0), v) denotes the solution of (17a) for a particular control w and initial condition ω(0) (see e.g., [17], [16]).
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υ (x)) = Furthermore, there exists ρ : Rν → Rn such that ρ(υ x. We are now ready to define a notion of moment for the nonlinear system (10). Definition 5. Consider the nonlinear system (10) and the generalized nonlinear signal generator (17a). Suppose that Assumption 1 holds. We call the moment of (10) at {q(ω), r} υ (x) ∂υ the function g(x) , with the signal d as in ∂x x=ρ(d−ω) (17b).
Theorem 2. Consider the system (10) and the signal generator (17a). Suppose Assumption 1 holds. Assume that the zero equilibrium of (10) is locally exponentially stable, the system (10) is observable, the system (17a) is reachable and its response for ω(0) = 0 is Poisson stable. Then the moment of (10) at {q(ω), r} is in one-to-one relation with the welldefined steady state of d(t) for u = δ0 (t) (the Dirac function) with d(t) as in (17b). We now give the definition of a model that achieves moment matching. Consider the nonlinear system ξ˙ = ϕ(ξ) + γ(ξ)u, η = ψ(ξ),
(20)
where ξ(t) ∈ Rν , u(t) ∈ R and η(t) ∈ R, with ϕ, γ and ψ smooth. Consider the interconnection w(t) = η(t), between (20) and the signal generator defined by (17a) with the output ζ(t) ∈ Rν (see Figure 2). u
ξ˙ = ϕ(ξ) + γ(ξ)u η = w η = ψ(ξ)
ω˙ = q(ω) + rv
ζ
Fig. 2. Interconnection between the nonlinear reduced order model and the generalized signal generator.
Hence, the equations υ (x) ξ˙ = −q(−ξ) − rψ(ξ) + ∂υ g(x) u, ∂x Σψ(ξ) : x=ρ(ξ) η = ψ(ξ), (21) with ξ(t) ∈ Rν define a class of models of order ν, parameterized in ψ(ξ) ∈ R, that match the moments of (10) at {q(ω), r}. Remark 3. If q(ω) is linear, i.e., q(ω) = Qω, then the model (20) of order ν, that matches the moments of the nonlinear system (10) at σ(Q) if and only if ζ˙ = υ (x) ∂υ g(x) u which is equivalent to ϕ(ξ) = Qζ + ∂x x=ρ(ξ) υ (x) ∂υ g(x) , where υ Qξ − rψ(ξ) and γ(ξ) = ∂x x=ρ(ξ)
is the (unique) solution of (19) with the left side term υ (x)) = Qυ υ (x). −q(−υ
Moment matching in the sense of Definition 6 is related to moment matching in the sense of Definition 4, i.e., a model Σψ(ξ) matches the moments of (10) at {q(ω), r}, in the sense of Definition 6 if and only if Σψ(ξ) is yielded from a model Σδ(ξ) , by a particular selection of δ(ξ). The converse statement also holds, i.e., a model Σδ(ξ) matches the moments of (10) at {−q(−ω), rT }, in the sense of Definition 4 if and only if Σδ(ξ) is yielded from a model Σψ(ξ) , by a particular selection of ψ(ξ). Theorem 4. Consider a nonlinear observable system (10). Let the pair {q(ω), r} define a generalized signal generator (17a) and assume that (17a) is reachable from 0. Furthermore, assume that the pair {−q(−ω), rT } defines an observable signal generator (11). Consider the systems Σψ(ξ) as in (21) and Σδ(ξ) as in (15) and assume that Σδ(ξ) is observable. Suppose that Assumption 1 holds. The following statements are equivalent.
Definition 6. Consider the nonlinear system (10) and the nonlinear signal generator (17a). Assume (17a) is reachable from 0. A system (20) matches the moments of the system (10) at {q(ω), r} if the steady-state response (provided it exists) of the output signal ζ(t) of the interconnection between (20) and (17a) matches the steady-state response (provided it exists) of the signal d(t) in (17b). Based on Definition 6 we define a class of parameterized models of order ν, that achieve moment matching. Theorem 3. Consider the nonlinear system (20) and the signal generator (17a). Suppose Assumption 1 holds. Assume (17a) is reachable from 0 and (20) is observable. Let ζ = ω + ξ. Then the system (20) matches the moments of (10) at {q(ω), r} (in the sense of Definition 6) if and only if υ (x) ∂υ ϕ(ξ) = −q(−ξ) − rψ(ξ) and γ(ξ) = g(x) , ∂x x=ρ(ξ) for all ψ(ξ) ∈ R. 5521
1) Σψ(ξ) , described by equations (21), matches the moments of (10) at {q(ω), r}, in the sense of Definition 6. υ (x) 2) rT ψ(ξ) = ∂υ∂x , ψ(ω) = h(π(ω)), where x=ρ(ξ)
ψ(ξ) is as in (21), υ and ρ are such that Assumption 1 holds and π(ω) is the unique solution of (12). 3) Σψ(ξ) , described by equations (21), matches the moments of (10) at {−q(−ω), rT }, in the sense of Definition 4. 4) Σδ(ξ) , described by equations (15), matches the moments of (10) at {−q(−ω), rT }, in the sense of Definition 4. υ (x) , where 5) rT h(π(ξ)) = δ T (ξ), δ T (ξ) = ∂υ∂x x=ρ(ξ)
δ(ξ) is as in (15), υ and ρ are such that Assumption 1 holds and π(ω) is the unique solution of (12). 6) Σδ(ξ) , described by equations (15), matches the moments of (10) at {q(ω), r}, in the sense of Definition 6.
ω˙ = −q(−ω) θ = rT ω Fig. 3.
θ=u
Σψ(ξ) at {q(ω), r}
at {q(ω), r}, of a given nonlinear system, simultaneously. The arguments yielding the next result follow the proof of Proposition 1.
η
Diagram illustrating statements 1), 2) and 3) of Theorem 4.
IV. M ATCHING
FURTHER MOMENTS
In this section we compute the reduced order model(s) that matches a number of moments equal to twice its order. This is done by a proper selection of the free parameters, i.e., from the classes of models that achieve moment matching we identify the one matching two sets of moments, simultaneously.
Proposition 2. Consider the nonlinear system (10) and the signal generators (11) and (17a). Let π ∈ Rν be the unique solution of (12), υ ∈ Rn be the unique solution of (19) and ρ be such that Assumption 1 holds. Then the following statements hold. 1) There exists a subclass of models Σδ(ξ) as in (15) that match the moments of (10) at {s(ω), l(ω)} and {q(ω), r} simultaneously, if and only if δ(ξ) satisfies the equation υ (x) ∂υ ∂p(ξ) δ(ξ) = g(x) , (22) ∂ξ ∂x
A. Linear case
x=ρ(ξ)
ν
where p : R → R satisfies the equation υ (x) ∂p(ξ) ∂υ q(−p(ξ)) + s(ξ) = g(x) ∂ξ ∂x
We begin with a simple example illustrating the problem and the idea. Example 1. Let η0 ∈ C and η1 ∈ C be such that η0 6= η1 . By η0 g Theorem 1, the transfer function Kg (s) = , with s − s0 + g g ∈ C, defines a class of first order models, parameterized in g, that match the moment η0 at s0 ∈ C, i.e., Kg (s0 ) = η0 . Let q1 ∈ C be such that q1 6= s0 . We want to compute the first order model Kg that also matches the moment η1 at q1 , i.e., compute the parameter g, such that Kg (q1 ) = η1 . This last condition is equivalent to η0 g = [(s0 − q1 ) + g]η1 ⇔ g(η0 − η1 ) = (s0 − q1 )η1 . Since we assume η0 6= η1 , s0 − q1 . Let p = we obtain the unique parameter g = η1 η0 − η1 η0 − η1 . Note that since we assume that s0 6= q1 , p is well s0 − q1 defined. Hence g = η/p. Since g is unique, then p is also unique and it satisfies the equation ps0 − q1 p = η0 − η1 . Proposition 1. Consider a linear system (2) and let S ∈ Cν×ν and Q ∈ Cν×ν be such that σ(S) ∩ σ(Q) ∩ σ(A) = ∅. Let LT , R ∈ Cν be such that the pair (L, S) is observable and the pair (Q, R) is controllable. Let Π ∈ Cν×n be the unique solution of (3) and Υ ∈ Cn×ν be the unique solution of (4). Then the following statements hold and are equivalent. 1) A model ΣG as in (7) matches the moments of (2) at σ(S) and σ(Q) simultaneously, if and only if G = P −1 ΥB, where P ∈ Cν×ν is the unique solution of the equation P S − QP = ΥBL − RCΠ. Furthermore, ΣP −1 ΥB is the unique model, from the class ΣG , that matches 2ν moments of (2). 2) A model ΣH as in (8) matches the moments of (2) at σ(S) and σ(Q) simultaneously, if and only if H = CΠP −1 , where P ∈ Cν×ν is the unique solution of the equation P S − QP = ΥBL − RCΠ. Furthermore, ΣCΠP −1 is the unique model, from the class ΣH , that matches 2ν moments of (2).
l(ξ)
x=ρ(ξ)
− rh(π(ξ)).
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2) There exists a subclass of models Σψ(ξ) as in (21) that match the moments of (10) at {s(ω), l(ω)} and {q(ω), r} simultaneously, if and only if ψ(p(ω)) = h(π(ω)),
(24)
where p : Rν → Rν satisfies the equation υ (x) ∂υ ∂p(ω) s(ω) = g(x) q(−p(ω)) + ∂ω ∂x
l(ω)
x=ρ(p(ω))
− rh(π(ω)).
(25)
Remark 4. Assuming that p uniquely satisfies equation (23), there exists a unique model Σδ(ξ) that matches the moments of (10) at {s(ω), l(ω)} and {q(ω), r}. Furthermore, if p(ξ) is a diffeomorphism, then Σδ(ξ) = Σ ∂p(ξ) −1 ∂υυ(x) g(x) . ( ∂ξ ) |x=ρ(ξ) ∂x Similarly, assuming that p uniquely satisfies equation (25), there exists a unique model Σψ(ξ) that matches the moments of (10) at {s(ω), l(ω)} and {q(ω), r}. Finally, if we assume that p(·) is the identity, then statement 1) and statement 2) from Proposition 2 are equivalent and moreover Σδ(ξ) = Σ ∂υυ(x) g(x)| ∂x
= Σh(π(ξ)) = Σψ(ξ) x=ρ(ξ)
is the unique model of order ν that matches the moments of (10) at {s(ω), l(ω)} and {q(ω), r}, simultaneously. Example 2. Consider a first order system (10), described by the equations x˙ = −x − x3 + u, y = x,
B. Nonlinear case In this section we give the nonlinear counterpart of Proposition 1, i.e., we compute the subclass of models of order ν, that match the moments at {s(ω), l(ω)} and the moments
ν
(26)
and let the signal generator (11) be ω˙ = 0, θ = ω. Consider the equation π 3 (ω) + π(ω) − ω = 0, which is of the form
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(12). This equation has the unique (real) solution q p 1 3 π(ω) = 108 ω + 12 12 + 81 ω 2 6 1 −2 p , π(0) = 0. √ 3 108 ω + 12 12 + 81 ω 2
(27)
A class of first order models, parameterized in δ(ξ) that match the moment π(ω), as in (27), of (26) at {0, ω} is ξ˙ = δ(ξ)(u − ξ), q p 1 3 η = 108 ξ + 12 12 + 81 ξ 2 6 1 , − 2q p 3 108 ξ + 12 12 + 81 ξ 2
(28)
where δ(ξ) satisfies (13). Consider now the generalized signal generator ω˙ = rw, υ (x) 3 ∂υ r ∈ R. Equation (19) becomes (x + x) = rx. For ∂x the initial condition υ (0) = 0, the (unique) solution of this differential equation is υ (x) = r arctan x. Since arctan is υ (x)) = x, given invertible, there exists ρ that satisfies ρ(υ x by ρ(x) = tan (Assumption 1 holds). Hence the moment r υ (x) ∂υ of (26) at {0, r} as in Definition 5 is g(x) = ∂x x=ρ(ξ) r r ξ = . A class of first = r cos2 1 + x2 r 1 + tan2 ξ x=ρ(ξ)
r
order models parameterized by ψ(ξ) that match the moment 2 ξ r cos r of (26) at {0, r}, are given by the equations ξ u − ψ(ξ) , ξ˙ = r cos2 r η = ψ(ξ). (29) Let r = 1. Consider equation (25), i.e., π(ω(ξ)) − cos2 (ξ) = 0,
(30)
with π(ω) as in (27). Equation (30) has a unique solution ω(ξ) = cos6 ξ+cos2 ξ. By Proposition 2, there exists a model (29) with ψ(ξ) = π(ω(ξ)) = cos2 ξ that matches both the moment π(ξ) as in (27) and the moment cos2 ξ of (26), i.e., ξ˙ = cos2 ξ(u − ξ), η = cos2 ξ. V. C ONCLUSIONS We have presented a new time-domain notion of moments for a class of nonlinear systems by considering the swapped interconnection between the nonlinear system and a generalized signal generator, i.e., the nonlinear system
drives a generalized signal generator which contains the interpolation information. We have stated that the moments are in one-to-one relation with the steady-state response of such interconnection. Based on this notion we have defined moment matching and we have computed the family of parameterized reduced order models that achieve moment matching. Furthermore, we have shown that the two types of nonlinear moment matching are equivalent in the sense that a model from one family that matches the moments of the given system can be obtained from the other family of models by a proper selection of the parameter. Finally, we have computed the reduced order model(s) that matches moments of the system at two sets of interpolation points, simultaneously, i.e., the number of interpolation points is twice the order of the model. R EFERENCES [1] A. C. Antoulas. Approximation of large-scale dynamical systems. SIAM, Philadelphia, 2005. [2] A. C. Antoulas, J. A. Ball, J. Kang, and J. C. Willems. On the solution of the minimal rational interpolation problem. Linear Algebra & its Applications, 137/138:511–573, 1990. [3] A. Astolfi. Model reduction by moment matching for nonlinear systems. In Proc. 47th IEEE Conf. on Decision and Control, pages 4873–4878, 2008. [4] A. Astolfi. Model reduction by moment matching for linear and nonlinear systems. IEEE Trans. Autom. Contr., 50(10):2321–2336, 2010. [5] A. Astolfi. Model reduction by moment matching, steady-state response and projections. In Proc. 49th IEEE Conf. on Decision and Control, pages 5344 – 5349, 2010. [6] E. de Souza and S. P. Bhattacharyya. Controllability, observability and the solution of AX − XB = C. Linear Algebra & Its App., 39:167–188, 1981. [7] C. de Villemagne and R. E. Skelton. Model reductions using a projection formulation. Int. J. Control, 46:2141–2169, 1987. [8] K. Gallivan and P. Van Dooren. Rational approximations of pre-filtered transfer functions via the Lanczos algorithm. Numerical Algorithms, 20:331–342, 1999. [9] K. Gallivan, A. Vandendorpe, and P. Van Dooren. Sylvester equations and projection based model reduction. J. Comp. Appl. Math., 162:213– 229, 2004. [10] W. B. Gragg and A. Lindquist. On the partial realization problem. Linear Algebra & its Applications, 50:277–319, 1983. [11] T. C. Ionescu and A. Astolfi. On moment matching with preservation of passivity and stability. In Proc. 49th IEEE Conf. on Decision and Control, pages 6189 – 6194, 2010. [12] T. C. Ionescu and A. Astolfi. Moment matching for linear port Hamiltonian systems. In Proc. 50th IEEE Conf. Decision & Control - European Control Conf., pages 7164–7169, 2011. [13] A. Isidori. Nonlinear control systems. Springer-Verlag, New York, Third edition, 1995. [14] I. M. Jaimoukha and E. M. Kasenally. Implicitly restarted Krylov subspace methods for stable partial realizations. SIAM J. Matrix Anal. Appl., 18:633–652, 1997. [15] A. J. Mayo and A. C. Antoulas. A framework for the solution of the generalized realization problem. Linear Algebra & Its App., 425:634– 662, 2007. [16] H. Nijmeijer and A. J. van der Schaft. Nonlinear dynamical control systems. Springer-Verlag, New York, 1990. [17] A. J. van der Schaft. L2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control. IEEE Trans. Autom. Contr., 37(6):770–784, 1992. [18] P. van Dooren. The Lanczos algorithm and Pad´e approximation. Benelux Meeting on Systems and Control, 1995. Minicourse.
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