- Email: [email protected]
A note on model reduction by moment matching for nonlinear systems
8th IFAC Symposium on Nonlinear Control Systems University of Bologna, Italy, September 1-3, 2010
A note on model reduction by moment matching for nonlinear systems A. Astolfi ∗ ∗
EEE Dept, Imperial College London, London, UK, and DISP, Universit` a di Roma “Tor Vergata”, Rome, Italy [email protected]
Abstract: The model reduction problem by moment matching for (single-input, single-output) nonlinear systems is studied. A new family of reduced order models, achieving moment matching, is presented. It is shown that this new family is a natural nonlinear enhancement of the family of models obtained, in the linear case, using Krylov projections. Finally, connections between this novel family of models and the family proposed in Astolfi [2010] are discussed. The theory c 2010 IFAC is illustrated by means of a simple example. Copyright Keywords: Model reduction, moment matching, projection 1. INTRODUCTION The model reduction problem, for linear and nonlinear systems, has been widely studied in recent years. While for linear systems the theory is well-understood and several successful applications of model reductions have been reported, the nonlinear theory (and in particular the one relying on the notion of moment) has only recently been developed.
theory. To this end, we revisit the linear theory providing novel interpretations which allow the development of nonlinear counterparts. We also note that the nonlinear enhancement of the notion of Krylov projection provides a natural nonlinear version of the so-called Petrov-Galerkin projection method, see Antoulas [2005]. The paper is organized as follows. Section 2 provides some background material. In Section 3 the equivalence between two families of reduced order models for linear systems is established. This result is extended to nonlinear systems in Section 4, in which also a nonlinear enhancement of the so-called projection-based model reduction theory is established. Finally, Section 5 and Section 6 contain an illustrative example and some summarizing comments.
Model reduction is a fundamentally important problem in the analysis and design of control systems, and reduced order models constitute the starting point of several design and analysis procedures. The problem has been addressed from several perspectives exploiting the notions of Hankel operators, the theory of balanced realizations, interpolation theory and the notion of projections, see for example Scherpen [1993], Scherpen and van der Schaft [1994], Scherpen [1996], Scherpen and Gray [2000], Gray and Scherpen [2001], Willcox and Peraire [2002], Lall et al. [2003], Lall and Beck [2003], Fujimoto and Scherpen [2005], Gray and Scherpen [2005], Krener [2006], Verriest and Gray [2006], Fujimoto and Tsubakino [2008], Nillson and Rantzer [2010]. In this paper we focus on model reduction problems rooted in interpolation theory. In particular we consider model reduction problems based on the notion of moment, i.e. the reduced order model is required to interpolate the system to be reduced at some specific points: the moments of the model and of the system have to coincide at the selected points.
Notation. Throughout the paper we use standard notation. IR, IRn and IRn×m denote the set of real numbers, of n-dimensional vectors with real components, and of n × m-dimensional matrices with real entries, respectively. C I denotes the set of complex numbers. σ(A) denotes the spectrum of the matrix A ∈ IRn×n . Finally, ∅ denotes the empty set, I the identity matrix and 1 a column vector with all entries equal to one.
Model reduction theory by moment matching for linear systems has been widely studied. An excellent overview of existing results is given in Antoulas [2005] (see also the references therein). A nonlinear counterpart of model reduction theory by moment matching has been given in the recent papers Astolfi [2007, 2010].
Consider a linear, single-input, single-output, minimal, continuous-time system described by equations of the form x˙ = Ax + Bu, (1) y = Cx,
In what follows we intend to contribute to the nonlinear theory, and in particular to provide a nonlinear enhancement of the so-called projection-based model reduction
with x(t) ∈ IRn , u(t) ∈ IR, y(t) ∈ IR, A ∈ IRn×n , B ∈ IRn and C ∈ IR1×n constant matrices, and the associated transfer function
978-3-902661-80-7/10/$20.00 © 2010 IFAC
2. PRELIMINARIES This section is based upon the results in Antoulas [2005] and Astolfi [2010]. 2.1 Linear systems
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W (s) = C(sI − A)−1 B. (2) Definition 1. (Antoulas [2005]) The 0-moment of system (1) at s⋆ ∈ C I is the complex number η0 (s⋆ ) = C(s⋆ I − A)−1 B. Note that moments are associated to the transfer function of the system (1), thus the minimality assumption is without loss of generality. The moment of system (1) at s⋆ can be also characterized as follows (see also [Antoulas, 2005, Chapter 6] and Gallivan et al. [2006]). Lemma 1. (Astolfi [2010]) Consider system (1) and s⋆ ∈ C. I Suppose s⋆ 6∈ σ(A). Then η0 (s⋆ ) = CΠ, where Π is the unique solution of the Sylvester equation AΠ + B = Πs⋆ . (3) The point s⋆ , in which the moment of the system is computed, is often referred to as interpolation point. If one is interested in moments at several, non-coincident, interpolation points, these can be characterized as follows. Lemma 2. Consider system (1) and ν interpolation points s⋆1 , . . . , s⋆ν , with s⋆i ∈ C, I for all i ∈ [1, ν] and s⋆i 6= s⋆j , for [1, ν] ∋ i 6= j ∈ [1, ν]. Then the moments η0 (s⋆1 ), . . . , η(s⋆ν ) are in one-to-one relation with the matrix CΠ, where Π is the unique solution of the Sylvester equation AΠ + BL = ΠS, (4) {s⋆1 , · · · , s⋆ν }
with S any real matrix such that σ(S) = and L any matrix such that the pair (L, S) is observable. As a direct consequence of Lemma 2 and of the results in Astolfi [2010], a family of reduced order models, achieving moment matching at σ(S) = {s⋆1 , · · · , s⋆ν }, with si 6= sj , for i 6= j, is described by ξ˙ = (S − ∆L)ξ + ∆u, (5) ψ = CΠξ, with ∆ any matrix such that σ(S) ∩ σ(S − ∆L) = ∅. Reduced order models can also be obtained using the so-called Krylov projection method (see [Antoulas, 2005, Chapter 9]). In particular, a family of reduced order models achieving moment matching at {s⋆1 , · · · , s⋆ν }, with si 6= sj , for i 6= j, is given by ξ˙ = W ′ AV ξ + W ′ Bu, (6) ψ = CV ξ, where
V = (s1 I − A)−1 B · · · (sν I − A)−1 B ,
and W is any matrix such that W ′ V = I.
(7) (8)
2.2 Nonlinear systems Consider a nonlinear, single-input, single-output, continuous-time system described by equations of the form x˙ = f (x, u), (9) y = h(x),
with x(t) ∈ IRn , u(t) ∈ IR, y(t) ∈ IR and f (·, ·) and h(·) smooth mappings, a signal generator described by the equations ω˙ = s(ω), (10) θ = l(ω), with ω(t) ∈ IRκ , θ(t) ∈ IR and s(·) and l(·) smooth mappings, and the interconnected system ω˙ = s(ω), x˙ = f (x, l(ω)),
(11)
y = h(x). Suppose, in addition, that f (0, 0) = 0, s(0) = 0, h(0) = 0 and l(0) = 0. In what follows all statements hold locally around the origin which is, by assumption, an equilibrium for the unforced system (9), of the signal generator (10) and of the interconnected system (11). The signal generator captures the requirement that one is interested in studying the behavior of system (9) only in specific circumstances. Consider now the following assumptions and definitions. Assumption 1. There is a unique mapping π(·), locally defined in a neighborhood of ω = 0, which solves the partial differential equation ∂π f (π(ω), l(ω)) = s(ω). (12) ∂ω Assumption 1 implies that the interconnected system (11) possesses an invariant manifold, described by the equation x = π(ω). Note that the (well-defined) dynamics of the interconnected system (11) restricted to the invariant manifold are described by ω˙ = s(ω), i.e. are a copy of the dynamics of the signal generator (10). Assumption 2. The signal generator (10) is observable, i.e. for any pair of initial conditions ωa (0) and ωb (0), such that ωa (0) 6= ωb (0), the corresponding output trajectories l(ωa (t)) and l(ωb (t)) are such that l(ωa (t)) − l(ωb (t))) 6≡ 0. Definition 2. Consider system (9) and the signal generator (10). Suppose Assumptions 1 and 2 hold. The function h(π(ω)), with π(·) solution of equation (12), is the moment of system (9) at {s(ω), l(ω)}. Finally, as detailed in Astolfi [2010], a family of reduced order models, all achieving moment matching, is described by ξ˙ = s(ξ) − δ(ξ)l(ξ) + δ(ξ)u, (13) ψ = h(π(ξ)), where δ(·) is any mapping such that the equation ∂p s(p(ω)) − δ(p(ω))l(p(ω)) + δ(p(ω))l(ω) = s(ω) (14) ∂ω has the unique solution p(ω) = ω. Remark 1. The family of models described by equations (13) is affine by construction. A family of reduced order models achieving moment matching, and not necessarily affine in the input, is described by the equations ξ˙ = s˜(ξ, u − l(ξ)), (15) ψ = h(π(ξ)),
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the projector V , defined in equation (7), are related by a change of cordinates. In turn, in Astolfi [2010] it has been shown that the matrix Π is related to the existence of an invariant subspace for the interconnected system ω˙ = Sω,
with s˜(·) any function such that s˜(ξ, 0) = s(ξ), provided that the equation s˜(p(ω), (lω) − l(p(ω))) =
∂p s(ω) ∂ω
x˙ = Ax + BLω,
has the unique solution p(ω) = ω.
(17)
y = Cx.
2.3 Petrov-Galerkin projection For completeness, we briefly recall the notion of PetrovGalerkin projection. Consider the nonlinear system (9) and a pair of matrices V ∈ IRn×ν and W ∈ IRn×ν , with ν < n, such that equation (8) holds. Then a reduced order model, of dimension ν, is given by ξ˙ = W ′ f (V ξ, u), (16) ψ = h(V ξ), and the matrix V W ′ is called the Petrov-Galerkin projection. 3. ON THE PARAMETERIZATION OF REDUCED ORDER MODELS FOR LINEAR SYSTEMS
These two observations yield a geometric interpretation of the projector V , which will be exploited in Section 4 to derive a nonlinear counterpart of such projectors, and hence a nonlinear version of the family of reduced order models described by equations (6). 4. NONLINEAR SYSTEMS In this section a new family of reduced order models achieving moment matching is presented. This family is the nonlinear counterpart of the family of systems described by equations (6). In addition, we provide the nonlinear counterpart of Lemma 3. 4.1 A new family of reduced order models
To develop one of the main results of Section 4 we now establish the equivalence between the two families of reduced order models given in Section 2.1. This result completes the discussion in Astolfi [2010] on the equivalence between the parameterization (5) and the Georgiou-Kimura parameterization. To provide concise statements let R∆ (s) be the family of transfer functions, parameterized by ∆, with state space realization (5) and RW (s) be the family of transfer functions, parameterized by W , with state space realization (6). Lemma 3. Consider the families of reduced order models (5) and (6), with ∆ any matrix such that σ(S) ∩ σ(S − ∆L) = ∅, and W any matrix such that equation (8) holds. Then the following statements hold. (a) For any W there exists a (unique) ∆ such that R∆ (s) = RW (s). (b) For any ∆ there exists a W such that R∆ (s) = RW (s). Proof. (a). Pick any W such that equation (8) holds. Let ∆ = T −1 W ′ B, with T the (unique) matrix such that T S = diag(s1 , · · · , sν )T, L = 1′ T. The definitions of Π, by means of equation (4), and of V in equation (7) yield Π = V T and W ′ Π = T . As a result (W ′ AV )T = T (S − ∆L), W ′ B = T ∆, and CV T = CΠ, which proves the claim. (b). Pick any ∆ such that σ(S) ∩ σ(S − ∆L) = ∅. Select W such that W ′V = I W ′ B = T ∆, with T as in the proof of (a). Note that such a matrix W always exists by controllability of system (1). The proof of the claim follows now from considerations similar to those in the proof of (a). 2 Remark 2. The proof of Lemma 3 has shown that the matrix Π, which solves the Sylvester equation (4), and
Theorem 1. Consider system (9) and the signal generator (10). Suppose Assumptions 1 and 2 hold. Consider the family of models ˙ξ = ∂ρ f (x, u) , ∂x x=π(ξ)
(18)
ψ = h(π(ξ)), where π(·) is the solution of equation (12) and ρ(·) is any mapping such that ρ ◦ π(·) = Id. (19) Suppose that the zero equilibrium of the system (18) with u = 0 is locally exponentially stable and the system (10) is Poisson stable. Then all models in the family (18) have the same moment of system (9) at {s(ω), l(ω)}, hence equations (18) yield a family of reduced order models all achieving moment matching at {s(ω), l(ω)}. Remark 3. The local exponential stability assumption can be replaced by the assumption that the zero equilibrium is hyperbolic. Proof. To prove the claim it is sufficient to show that the moment at {s(ω), l(ω)} of all models described by equations (18) coincides with the moment at {s(ω), l(ω)} of the system (9), which is given by h(π(ω)), where π(·) is the (unique, by Assumption 1) solution of equation (12). To this end, consider the partial differential equation " # ∂ρ ∂p f (x, l(ω)) = s(ω) (20) ∂x ∂ω x=π(ξ) ξ=p(ω)
and note that, by equation (12) and the property ∂π ∂ρ = I, ∂x x=π(ξ) ∂ξ
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resulting from equation (19), the function p(ω) = ω is a solution of equation (20). This solution is unique by the center manifold theory, see Carr [1981]. As a result, the moment at {s(ω), l(ω)} of all models in the family (18) is given by h(π(ω)), which proves the claim. 2 Remark 4. Equations (18) may be regarded as a nonlinear version of the Petrov-Galerkin method, in which the projector is given by the mapping π ◦ ρ : IRn → IRn . In fact, if π(·) and ρ(·) are linear mapping, i.e. π(ξ) = V ξ and ρ(x) = W ′ x, then equation (19) is equivalent to equation (8) and equations (18) coincide with equations (16). The family of models described by equations (18) provides the nonlinear counterpart of the family of models described by equations (6). Similarly to what discussed for linear systems in Lemma 3, these families of reduced order models are strongly related. Theorem 2. Consider the families of reduced order models (13) and (18), with δ(·) any mapping such that equation (14) has a unique solution, and ρ(·) any function such that equation (19) holds. Assume the mapping f (x, u) in equation (9) is affine in u, i.e. with some abuse of notation, f (x, u) = f (x) + g(x)u. Then the following statements hold. (a)’ For any ρ(·) there exists a (unique) δ(·) such that the resulting models (locally) coincide. (b)’ Suppose that ∂π g(π(ξ)) = ν + 1. rank (22) ∂ξ
and
∂π δ(ξ) P , g(π(ξ)) = γ(ξ) ∂ξ for some mapping γ(·). Note that the matrix ∂π P =n ∂ξ ˆ = π(ξ) + P ξ, ˆ where is the Jacobian of the function π ˆ (ξ, ξ) n−ν ˆ ξ ∈ IR is an auxiliary variable, and that the matrix P and the mapping γ(·) exist, by condition (22), which also implies that γ(·) is not identically zero. By the implicit function theorem the function π ˆ (·) is locally invertible, and the inverse ρˆ(·) has, by construction, the following properties. Let ρ(·) denote the first ν components of the mapping ρˆ(·). Then condition (21) holds and ∂ρ g(π(ξ)) = δ(ξ). ∂x x=π(ξ) With the above equality in place, the claim follows using arguments similar to those exploited in the proof of claim (a)’. 2 Remark 5. The statement (a)’ can be proved without the assumption that the mapping f (·) is affine in u, provided the family of reduced order models described by equations (15) is used in place of the family described by equations (13). 5. AN ILLUSTRATIVE EXAMPLE ´ The averaged model of the DC–to–DC Cuk converter is given by the equations (see Rodriguez et al. [2005]) d i1 dt d C2 v 2 dt d L 3 i3 dt d C4 v 4 dt y L1
Then for any δ(·) there exists a ρ(·) such that the resulting models coincide. Proof. (a)’. To prove the claim we show that for any ρ(·) such that equation (19) holds there exists a δ(·) such that s(ξ) − δ(ξ)l(ξ) + δ(ξ)u = ∂ρ (f (x) + g(x)u) , ∂x x=π(ξ)
(23)
= −(1 − u) v2 + E, = (1 − u) i1 + u i3 , = −u v2 − v4 ,
(24)
= i3 − G v 4 , = v4 ,
where i1 (t) ∈ IR+ and i3 (t) ∈ IR− describe currents, v2 (t) ∈ IR+ and v4 (t) ∈ IR− voltages, L1 , C2 , L3 , C4 , E and G positive parameters and u(t) ∈ (0, 1) a continuous control signal which represents the slew rate of a PWM circuit used to control the switch position in the converter.
for all ξ and all u. To this end, let ∂ρ g(x) , δ(ξ) = ∂x x=π(ξ) and note that this selection is such that the terms linear in u in equation (23) coincide. Recall now equation (12). Using this equality in equation (23), with u = 0, yields s(ξ) − δ(ξ)l(ξ) = ∂ρ ∂π s(ξ) − g(π(ξ))l(ξ) , ∂x x=π(ξ) ∂ξ which is an identity by the definition of δ(·) and equation (21). (b)’. Pick any δ(·) such that equation (14) has a unique solution and let P be any constant matrix such that ∂π P =n rank ∂ξ
The solution π(·) of the equation (12), with s(ω) = 0 and l(ω) = ω, is ω2 GE (1 − ω)2 1 E , 1 − ω π(ω) = GE ω ω−1 ω E ω−1 and the 0-moment of the system at {s(ω), l(ω)} = {0, ω} ω is h(π(ω)) = ω−1 E. As detailed in Astolfi [2010], a locally asymptotically stable reduced order model, defined for
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all ξ 6= 1, achieving moment matching at {s(ω), l(ω)} = {0, ω} is ξ˙ = −δ(ξ)(ξ − u), ψ = E
ξ , ξ−1
(25)
with δ(0) > 0. An alternative reduced order model can be obtained exploiting the result of Theorem 1. Note that, while the free parameter of the family of reduced order models (13) is the mapping δ(·), the free parameter of the family of reduced order models (18) is the mapping ρ(·), which should satisfy the condition (19). A family of such mappings ρ(·) is given by αx3 + (1 − α)x4 ρ(x) = αx3G , + (1 − α)x4 − E G where α can be either a constant or a function of x. For example, selecting α = 0 yields the reduced order model G ξ˙ = − ξ(ξ − u) C2 (26) ξ , ψ = E ξ−1 whereas selecting α = α(x3 , x4 ) yields the family of reduced order models ˜ ξ˙ = −δ(ξ)(ξ − u) ψ = E
ξ , ξ−1
(27)
where
α(x3 , x4 )(ξ − 1) Gξ(α(x3 , x4 ) − 1) ˜ . + δ(ξ) = L3 G C2 x=π(ξ)
It is worth noting that, consistently with Theorem 2, the family (25), parameterized by δ(·) and the family (27), parameterized by α(·), coincide. In fact, selecting Gξ − δ(ξ)C2 α(x)x=π(ξ) = L3 G C2 (ξ − 1) + G2 L3 ξ in (27) yields (25). 6. CONCLUSIONS The problem of model reduction by moment matching for nonlinear systems has been discussed. A nonlinear counterpart of the model reduction theory exploiting the notion of Krylov subspaces and Krylov projections has been derived. It has been also shown that the nonlinear theory provides a natural generalization of the so-called Petrov-Galerkin method. The relation between projectionbased reduced order models and the class of reduced order model derived in Astolfi [2010] has been established in the linear and in the nonlinear cases.
A. Astolfi. Model reduction by moment matching (semiplenary presentation). In IFAC Symposium on Nonlinear Control System Design, Pretoria, S. Africa, pages 95–102, 2007. A. Astolfi. Model reduction by moment matching for linear and nonlinear systems. IEEE Trans. Autom. Control, 55, 2010. J. Carr. Applications of Center Manifold Theory. Springer Verlag, 1981. K. Fujimoto and J.M.A. Scherpen. Nonlinear inputnormal realizations based on the differential eigenstructure of Hankel operators. IEEE Trans. Autom. Control, 50:2–18, 2005. K. Fujimoto and D. Tsubakino. Computation of nonlinear balanced realization and model reduction based on Taylor series expansion. Systems and Control Letters, 57:283–289, 2008. K.A. Gallivan, A. Vandendorpe, and P. Van Dooren. Model reduction and the solution of Sylvester equations. In MTNS, Kyoto, 2006. W.S. Gray and J.M.A. Scherpen. On the nonuniqueness of singular value functions and balanced nonlinear realizations. Systems and Control Letters, 44:219–232, 2001. W.S. Gray and J.M.A. Scherpen. Hankel singular value functions from Schmidt pairs for nonlinear input-output systems. Systems and Control Letters, 54:135–144, 2005. A.J. Krener. Model reduction for linear and nonlinear control systems. In 45th Conference on Decision and Control, San Diego, CA, 2006. Bode Lecture. S. Lall and C. Beck. Error bounds for balanced model reduction of linear time-varying systems. IEEE Trans. Autom. Control, 48(6):946–956, 2003. S. Lall, P. Krysl, and J.E. Marsden. Structure-preserving model reduction for mechanical systems. Physica D, 184 (1–4):304–318, 2003. O. Nillson and A. Rantzer. A novel approach to balanced truncation of nonlinear systems. In European Control Conference, Budapest, Hungary, 2010. H. Rodriguez, R. Ortega, and A. Astolfi. Adaptive partial ´ state feedback control of the DC-to-DC Cuk converter. In American Control Conference, Portland, OR, pages 5121–5126, 2005. J.M.A. Scherpen. Balancing for nonlinear systems. Systems and Control Letters, 21:143–153, 1993. J.M.A. Scherpen. H∞ balancing for nonlinear systems. Int. J. Rob. Nonl. Contr., 6:645–668, 1996. J.M.A. Scherpen and W.S. Gray. Minimality and local state decompositions of a nonlinear state space realization using energy functions. IEEE Trans. Autom. Control, 45(11):2079–2086, 2000. J.M.A. Scherpen and A.J. van der Schaft. Normalized coprime factorizations and balancing for unstable nonlinear systems. Int. Journal of Control, 60:1193–1222, 1994. E.I. Verriest and W.S. Gray. Geometry and topology of the state space via balancing. In Mathematical Theory of Networks and Systems, Kyoto, Japan, 2006. K. Willcox and J. Peraire. Balanced model reduction via the proper orthogonal decomposition. AIAA Journal, 40(11):2323–2330, 2002.
REFERENCES A.C. Antoulas. Approximation of Large-Scale Dynamical Systems. SIAM Advances in Design and Control, 2005. 1248
A note on model reduction by moment matching for nonlinear systems A. Astolfi ∗ ∗
EEE Dept, Imperial College London, London, UK, and DISP, Universit` a di Roma “Tor Vergata”, Rome, Italy [email protected]
Abstract: The model reduction problem by moment matching for (single-input, single-output) nonlinear systems is studied. A new family of reduced order models, achieving moment matching, is presented. It is shown that this new family is a natural nonlinear enhancement of the family of models obtained, in the linear case, using Krylov projections. Finally, connections between this novel family of models and the family proposed in Astolfi [2010] are discussed. The theory c 2010 IFAC is illustrated by means of a simple example. Copyright Keywords: Model reduction, moment matching, projection 1. INTRODUCTION The model reduction problem, for linear and nonlinear systems, has been widely studied in recent years. While for linear systems the theory is well-understood and several successful applications of model reductions have been reported, the nonlinear theory (and in particular the one relying on the notion of moment) has only recently been developed.
theory. To this end, we revisit the linear theory providing novel interpretations which allow the development of nonlinear counterparts. We also note that the nonlinear enhancement of the notion of Krylov projection provides a natural nonlinear version of the so-called Petrov-Galerkin projection method, see Antoulas [2005]. The paper is organized as follows. Section 2 provides some background material. In Section 3 the equivalence between two families of reduced order models for linear systems is established. This result is extended to nonlinear systems in Section 4, in which also a nonlinear enhancement of the so-called projection-based model reduction theory is established. Finally, Section 5 and Section 6 contain an illustrative example and some summarizing comments.
Model reduction is a fundamentally important problem in the analysis and design of control systems, and reduced order models constitute the starting point of several design and analysis procedures. The problem has been addressed from several perspectives exploiting the notions of Hankel operators, the theory of balanced realizations, interpolation theory and the notion of projections, see for example Scherpen [1993], Scherpen and van der Schaft [1994], Scherpen [1996], Scherpen and Gray [2000], Gray and Scherpen [2001], Willcox and Peraire [2002], Lall et al. [2003], Lall and Beck [2003], Fujimoto and Scherpen [2005], Gray and Scherpen [2005], Krener [2006], Verriest and Gray [2006], Fujimoto and Tsubakino [2008], Nillson and Rantzer [2010]. In this paper we focus on model reduction problems rooted in interpolation theory. In particular we consider model reduction problems based on the notion of moment, i.e. the reduced order model is required to interpolate the system to be reduced at some specific points: the moments of the model and of the system have to coincide at the selected points.
Notation. Throughout the paper we use standard notation. IR, IRn and IRn×m denote the set of real numbers, of n-dimensional vectors with real components, and of n × m-dimensional matrices with real entries, respectively. C I denotes the set of complex numbers. σ(A) denotes the spectrum of the matrix A ∈ IRn×n . Finally, ∅ denotes the empty set, I the identity matrix and 1 a column vector with all entries equal to one.
Model reduction theory by moment matching for linear systems has been widely studied. An excellent overview of existing results is given in Antoulas [2005] (see also the references therein). A nonlinear counterpart of model reduction theory by moment matching has been given in the recent papers Astolfi [2007, 2010].
Consider a linear, single-input, single-output, minimal, continuous-time system described by equations of the form x˙ = Ax + Bu, (1) y = Cx,
In what follows we intend to contribute to the nonlinear theory, and in particular to provide a nonlinear enhancement of the so-called projection-based model reduction
with x(t) ∈ IRn , u(t) ∈ IR, y(t) ∈ IR, A ∈ IRn×n , B ∈ IRn and C ∈ IR1×n constant matrices, and the associated transfer function
978-3-902661-80-7/10/$20.00 © 2010 IFAC
2. PRELIMINARIES This section is based upon the results in Antoulas [2005] and Astolfi [2010]. 2.1 Linear systems
1244
10.3182/20100901-3-IT-2016.00186
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W (s) = C(sI − A)−1 B. (2) Definition 1. (Antoulas [2005]) The 0-moment of system (1) at s⋆ ∈ C I is the complex number η0 (s⋆ ) = C(s⋆ I − A)−1 B. Note that moments are associated to the transfer function of the system (1), thus the minimality assumption is without loss of generality. The moment of system (1) at s⋆ can be also characterized as follows (see also [Antoulas, 2005, Chapter 6] and Gallivan et al. [2006]). Lemma 1. (Astolfi [2010]) Consider system (1) and s⋆ ∈ C. I Suppose s⋆ 6∈ σ(A). Then η0 (s⋆ ) = CΠ, where Π is the unique solution of the Sylvester equation AΠ + B = Πs⋆ . (3) The point s⋆ , in which the moment of the system is computed, is often referred to as interpolation point. If one is interested in moments at several, non-coincident, interpolation points, these can be characterized as follows. Lemma 2. Consider system (1) and ν interpolation points s⋆1 , . . . , s⋆ν , with s⋆i ∈ C, I for all i ∈ [1, ν] and s⋆i 6= s⋆j , for [1, ν] ∋ i 6= j ∈ [1, ν]. Then the moments η0 (s⋆1 ), . . . , η(s⋆ν ) are in one-to-one relation with the matrix CΠ, where Π is the unique solution of the Sylvester equation AΠ + BL = ΠS, (4) {s⋆1 , · · · , s⋆ν }
with S any real matrix such that σ(S) = and L any matrix such that the pair (L, S) is observable. As a direct consequence of Lemma 2 and of the results in Astolfi [2010], a family of reduced order models, achieving moment matching at σ(S) = {s⋆1 , · · · , s⋆ν }, with si 6= sj , for i 6= j, is described by ξ˙ = (S − ∆L)ξ + ∆u, (5) ψ = CΠξ, with ∆ any matrix such that σ(S) ∩ σ(S − ∆L) = ∅. Reduced order models can also be obtained using the so-called Krylov projection method (see [Antoulas, 2005, Chapter 9]). In particular, a family of reduced order models achieving moment matching at {s⋆1 , · · · , s⋆ν }, with si 6= sj , for i 6= j, is given by ξ˙ = W ′ AV ξ + W ′ Bu, (6) ψ = CV ξ, where
V = (s1 I − A)−1 B · · · (sν I − A)−1 B ,
and W is any matrix such that W ′ V = I.
(7) (8)
2.2 Nonlinear systems Consider a nonlinear, single-input, single-output, continuous-time system described by equations of the form x˙ = f (x, u), (9) y = h(x),
with x(t) ∈ IRn , u(t) ∈ IR, y(t) ∈ IR and f (·, ·) and h(·) smooth mappings, a signal generator described by the equations ω˙ = s(ω), (10) θ = l(ω), with ω(t) ∈ IRκ , θ(t) ∈ IR and s(·) and l(·) smooth mappings, and the interconnected system ω˙ = s(ω), x˙ = f (x, l(ω)),
(11)
y = h(x). Suppose, in addition, that f (0, 0) = 0, s(0) = 0, h(0) = 0 and l(0) = 0. In what follows all statements hold locally around the origin which is, by assumption, an equilibrium for the unforced system (9), of the signal generator (10) and of the interconnected system (11). The signal generator captures the requirement that one is interested in studying the behavior of system (9) only in specific circumstances. Consider now the following assumptions and definitions. Assumption 1. There is a unique mapping π(·), locally defined in a neighborhood of ω = 0, which solves the partial differential equation ∂π f (π(ω), l(ω)) = s(ω). (12) ∂ω Assumption 1 implies that the interconnected system (11) possesses an invariant manifold, described by the equation x = π(ω). Note that the (well-defined) dynamics of the interconnected system (11) restricted to the invariant manifold are described by ω˙ = s(ω), i.e. are a copy of the dynamics of the signal generator (10). Assumption 2. The signal generator (10) is observable, i.e. for any pair of initial conditions ωa (0) and ωb (0), such that ωa (0) 6= ωb (0), the corresponding output trajectories l(ωa (t)) and l(ωb (t)) are such that l(ωa (t)) − l(ωb (t))) 6≡ 0. Definition 2. Consider system (9) and the signal generator (10). Suppose Assumptions 1 and 2 hold. The function h(π(ω)), with π(·) solution of equation (12), is the moment of system (9) at {s(ω), l(ω)}. Finally, as detailed in Astolfi [2010], a family of reduced order models, all achieving moment matching, is described by ξ˙ = s(ξ) − δ(ξ)l(ξ) + δ(ξ)u, (13) ψ = h(π(ξ)), where δ(·) is any mapping such that the equation ∂p s(p(ω)) − δ(p(ω))l(p(ω)) + δ(p(ω))l(ω) = s(ω) (14) ∂ω has the unique solution p(ω) = ω. Remark 1. The family of models described by equations (13) is affine by construction. A family of reduced order models achieving moment matching, and not necessarily affine in the input, is described by the equations ξ˙ = s˜(ξ, u − l(ξ)), (15) ψ = h(π(ξ)),
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the projector V , defined in equation (7), are related by a change of cordinates. In turn, in Astolfi [2010] it has been shown that the matrix Π is related to the existence of an invariant subspace for the interconnected system ω˙ = Sω,
with s˜(·) any function such that s˜(ξ, 0) = s(ξ), provided that the equation s˜(p(ω), (lω) − l(p(ω))) =
∂p s(ω) ∂ω
x˙ = Ax + BLω,
has the unique solution p(ω) = ω.
(17)
y = Cx.
2.3 Petrov-Galerkin projection For completeness, we briefly recall the notion of PetrovGalerkin projection. Consider the nonlinear system (9) and a pair of matrices V ∈ IRn×ν and W ∈ IRn×ν , with ν < n, such that equation (8) holds. Then a reduced order model, of dimension ν, is given by ξ˙ = W ′ f (V ξ, u), (16) ψ = h(V ξ), and the matrix V W ′ is called the Petrov-Galerkin projection. 3. ON THE PARAMETERIZATION OF REDUCED ORDER MODELS FOR LINEAR SYSTEMS
These two observations yield a geometric interpretation of the projector V , which will be exploited in Section 4 to derive a nonlinear counterpart of such projectors, and hence a nonlinear version of the family of reduced order models described by equations (6). 4. NONLINEAR SYSTEMS In this section a new family of reduced order models achieving moment matching is presented. This family is the nonlinear counterpart of the family of systems described by equations (6). In addition, we provide the nonlinear counterpart of Lemma 3. 4.1 A new family of reduced order models
To develop one of the main results of Section 4 we now establish the equivalence between the two families of reduced order models given in Section 2.1. This result completes the discussion in Astolfi [2010] on the equivalence between the parameterization (5) and the Georgiou-Kimura parameterization. To provide concise statements let R∆ (s) be the family of transfer functions, parameterized by ∆, with state space realization (5) and RW (s) be the family of transfer functions, parameterized by W , with state space realization (6). Lemma 3. Consider the families of reduced order models (5) and (6), with ∆ any matrix such that σ(S) ∩ σ(S − ∆L) = ∅, and W any matrix such that equation (8) holds. Then the following statements hold. (a) For any W there exists a (unique) ∆ such that R∆ (s) = RW (s). (b) For any ∆ there exists a W such that R∆ (s) = RW (s). Proof. (a). Pick any W such that equation (8) holds. Let ∆ = T −1 W ′ B, with T the (unique) matrix such that T S = diag(s1 , · · · , sν )T, L = 1′ T. The definitions of Π, by means of equation (4), and of V in equation (7) yield Π = V T and W ′ Π = T . As a result (W ′ AV )T = T (S − ∆L), W ′ B = T ∆, and CV T = CΠ, which proves the claim. (b). Pick any ∆ such that σ(S) ∩ σ(S − ∆L) = ∅. Select W such that W ′V = I W ′ B = T ∆, with T as in the proof of (a). Note that such a matrix W always exists by controllability of system (1). The proof of the claim follows now from considerations similar to those in the proof of (a). 2 Remark 2. The proof of Lemma 3 has shown that the matrix Π, which solves the Sylvester equation (4), and
Theorem 1. Consider system (9) and the signal generator (10). Suppose Assumptions 1 and 2 hold. Consider the family of models ˙ξ = ∂ρ f (x, u) , ∂x x=π(ξ)
(18)
ψ = h(π(ξ)), where π(·) is the solution of equation (12) and ρ(·) is any mapping such that ρ ◦ π(·) = Id. (19) Suppose that the zero equilibrium of the system (18) with u = 0 is locally exponentially stable and the system (10) is Poisson stable. Then all models in the family (18) have the same moment of system (9) at {s(ω), l(ω)}, hence equations (18) yield a family of reduced order models all achieving moment matching at {s(ω), l(ω)}. Remark 3. The local exponential stability assumption can be replaced by the assumption that the zero equilibrium is hyperbolic. Proof. To prove the claim it is sufficient to show that the moment at {s(ω), l(ω)} of all models described by equations (18) coincides with the moment at {s(ω), l(ω)} of the system (9), which is given by h(π(ω)), where π(·) is the (unique, by Assumption 1) solution of equation (12). To this end, consider the partial differential equation " # ∂ρ ∂p f (x, l(ω)) = s(ω) (20) ∂x ∂ω x=π(ξ) ξ=p(ω)
and note that, by equation (12) and the property ∂π ∂ρ = I, ∂x x=π(ξ) ∂ξ
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resulting from equation (19), the function p(ω) = ω is a solution of equation (20). This solution is unique by the center manifold theory, see Carr [1981]. As a result, the moment at {s(ω), l(ω)} of all models in the family (18) is given by h(π(ω)), which proves the claim. 2 Remark 4. Equations (18) may be regarded as a nonlinear version of the Petrov-Galerkin method, in which the projector is given by the mapping π ◦ ρ : IRn → IRn . In fact, if π(·) and ρ(·) are linear mapping, i.e. π(ξ) = V ξ and ρ(x) = W ′ x, then equation (19) is equivalent to equation (8) and equations (18) coincide with equations (16). The family of models described by equations (18) provides the nonlinear counterpart of the family of models described by equations (6). Similarly to what discussed for linear systems in Lemma 3, these families of reduced order models are strongly related. Theorem 2. Consider the families of reduced order models (13) and (18), with δ(·) any mapping such that equation (14) has a unique solution, and ρ(·) any function such that equation (19) holds. Assume the mapping f (x, u) in equation (9) is affine in u, i.e. with some abuse of notation, f (x, u) = f (x) + g(x)u. Then the following statements hold. (a)’ For any ρ(·) there exists a (unique) δ(·) such that the resulting models (locally) coincide. (b)’ Suppose that ∂π g(π(ξ)) = ν + 1. rank (22) ∂ξ
and
∂π δ(ξ) P , g(π(ξ)) = γ(ξ) ∂ξ for some mapping γ(·). Note that the matrix ∂π P =n ∂ξ ˆ = π(ξ) + P ξ, ˆ where is the Jacobian of the function π ˆ (ξ, ξ) n−ν ˆ ξ ∈ IR is an auxiliary variable, and that the matrix P and the mapping γ(·) exist, by condition (22), which also implies that γ(·) is not identically zero. By the implicit function theorem the function π ˆ (·) is locally invertible, and the inverse ρˆ(·) has, by construction, the following properties. Let ρ(·) denote the first ν components of the mapping ρˆ(·). Then condition (21) holds and ∂ρ g(π(ξ)) = δ(ξ). ∂x x=π(ξ) With the above equality in place, the claim follows using arguments similar to those exploited in the proof of claim (a)’. 2 Remark 5. The statement (a)’ can be proved without the assumption that the mapping f (·) is affine in u, provided the family of reduced order models described by equations (15) is used in place of the family described by equations (13). 5. AN ILLUSTRATIVE EXAMPLE ´ The averaged model of the DC–to–DC Cuk converter is given by the equations (see Rodriguez et al. [2005]) d i1 dt d C2 v 2 dt d L 3 i3 dt d C4 v 4 dt y L1
Then for any δ(·) there exists a ρ(·) such that the resulting models coincide. Proof. (a)’. To prove the claim we show that for any ρ(·) such that equation (19) holds there exists a δ(·) such that s(ξ) − δ(ξ)l(ξ) + δ(ξ)u = ∂ρ (f (x) + g(x)u) , ∂x x=π(ξ)
(23)
= −(1 − u) v2 + E, = (1 − u) i1 + u i3 , = −u v2 − v4 ,
(24)
= i3 − G v 4 , = v4 ,
where i1 (t) ∈ IR+ and i3 (t) ∈ IR− describe currents, v2 (t) ∈ IR+ and v4 (t) ∈ IR− voltages, L1 , C2 , L3 , C4 , E and G positive parameters and u(t) ∈ (0, 1) a continuous control signal which represents the slew rate of a PWM circuit used to control the switch position in the converter.
for all ξ and all u. To this end, let ∂ρ g(x) , δ(ξ) = ∂x x=π(ξ) and note that this selection is such that the terms linear in u in equation (23) coincide. Recall now equation (12). Using this equality in equation (23), with u = 0, yields s(ξ) − δ(ξ)l(ξ) = ∂ρ ∂π s(ξ) − g(π(ξ))l(ξ) , ∂x x=π(ξ) ∂ξ which is an identity by the definition of δ(·) and equation (21). (b)’. Pick any δ(·) such that equation (14) has a unique solution and let P be any constant matrix such that ∂π P =n rank ∂ξ
The solution π(·) of the equation (12), with s(ω) = 0 and l(ω) = ω, is ω2 GE (1 − ω)2 1 E , 1 − ω π(ω) = GE ω ω−1 ω E ω−1 and the 0-moment of the system at {s(ω), l(ω)} = {0, ω} ω is h(π(ω)) = ω−1 E. As detailed in Astolfi [2010], a locally asymptotically stable reduced order model, defined for
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all ξ 6= 1, achieving moment matching at {s(ω), l(ω)} = {0, ω} is ξ˙ = −δ(ξ)(ξ − u), ψ = E
ξ , ξ−1
(25)
with δ(0) > 0. An alternative reduced order model can be obtained exploiting the result of Theorem 1. Note that, while the free parameter of the family of reduced order models (13) is the mapping δ(·), the free parameter of the family of reduced order models (18) is the mapping ρ(·), which should satisfy the condition (19). A family of such mappings ρ(·) is given by αx3 + (1 − α)x4 ρ(x) = αx3G , + (1 − α)x4 − E G where α can be either a constant or a function of x. For example, selecting α = 0 yields the reduced order model G ξ˙ = − ξ(ξ − u) C2 (26) ξ , ψ = E ξ−1 whereas selecting α = α(x3 , x4 ) yields the family of reduced order models ˜ ξ˙ = −δ(ξ)(ξ − u) ψ = E
ξ , ξ−1
(27)
where
α(x3 , x4 )(ξ − 1) Gξ(α(x3 , x4 ) − 1) ˜ . + δ(ξ) = L3 G C2 x=π(ξ)
It is worth noting that, consistently with Theorem 2, the family (25), parameterized by δ(·) and the family (27), parameterized by α(·), coincide. In fact, selecting Gξ − δ(ξ)C2 α(x)x=π(ξ) = L3 G C2 (ξ − 1) + G2 L3 ξ in (27) yields (25). 6. CONCLUSIONS The problem of model reduction by moment matching for nonlinear systems has been discussed. A nonlinear counterpart of the model reduction theory exploiting the notion of Krylov subspaces and Krylov projections has been derived. It has been also shown that the nonlinear theory provides a natural generalization of the so-called Petrov-Galerkin method. The relation between projectionbased reduced order models and the class of reduced order model derived in Astolfi [2010] has been established in the linear and in the nonlinear cases.
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