evalu-5.2. Quadratic-bilinear DAEs 137
138 Nonlinear Systems the form
˙
x(t) =Hx(t)⊗x(t), with H=
a b c d
e f g h
.
Writing down the dynamics explicitly, we obtain
˙
x1(t) = ax1(t)2+bx1(t)x2(t) +cx2(t)x1(t) +dx2(t)2,
˙
x2(t) = ex1(t)2+f x1(t)x2(t) +gx2(t)x1(t) +hx2(t)2.
Using j = b+c2 and k = f+g2 , the above system is equivalent to
˙
x1(t) = ax1(t)2+jx1(t)x2(t) +jx2(t)x1(t) +dx2(t)2,
˙
x2(t) = ex1(t)2+kx1(t)x2(t) +kx2(t)x1(t) +hx2(t)2.
Hence, we can replace H by H˜ =
a j j d
e k k h
. One now easily observes that for arbi-trary u,v∈R2, it holds that
H˜ (u⊗v) = ˜H(v⊗u) =
au1v1+ju1v2+ju2v1+ku2v2
eu1v1+ku1v2+ku2v1+f u2v2
.
Obviously, the above also holds true for n >2.
5.2.1 Quadratic-bilinerization of nonlinear systems
As we have already mentioned, a large class of smooth nonlinear control affine systems can be transformed into a system of QBDAEs. This is done via introducing new state variables for the occurring nonlinearities of the underlying control system. The new dynamics then can be derived by symbolic differentiation or adding algebraic constraints.
For example, if the dynamics of a nonlinear system are given via
˙
x1 = exp(−x2)· q
x21+ 1,
˙
x2 =−x2+u,
5.2. Quadratic-bilinear DAEs 139 we can introduce two new state variables x3 := exp(−x2) and x4 := p
x21+ 1. As a result, we can transform the above system as follows
˙
x1 =x3x4,
˙
x2 =−x2+u,
˙
x3 =−exp(−x2) ˙x2 =−x3x2+x3u,
˙
x4 = 1
2(p
x21+ 1)2x1x˙1 =x1x3.
Hence, we have found a quadratic bilinear system of dimension 4 whose solution is also a solution of the original nonlinear system. In a similar way, we may proceed for common nonlinear functions such as sin(x),cos(x), xβ,k+xx ,see [72]. In general, the transformation is done in two steps. First, one tries to polynomialize the system by suitable variable changes before in the second step, the polynomial system is iteratively simplified to a quadratic bilinear system. As it has been discussed in [72], instead of computing the Lie derivative of the artificially introduced state variables, in special situations it might be advantageous to add the algebraic constraints resulting from the introduction of the variables. However, for our purposes this is not of particular interest and we thus refer to [72] for a discussion on this topic.
Of course, the transformation to a set of QBDAEs is not unique in general and the question arises if there exists a minimal transformation. So far, this issue has not been considered and there does not seem to be a trivial answer to that question. Moreover, note that for the transformation of the original system it is desirable to have nonlinear-ities that are given by (a composition of) uni-variable functions. Especially for systems of ODEs that result from the semi-discretization of an underlying PDE, this is often fulfilled, rendering this approach quite promising.
It should be mentioned that the idea of the above transformation has already been known as McCormick-Relaxation for several years, see [98]. The fact that the idea has not been used for model reduction purposes might be surprising. On the other hand, at a first glance it seems counterintuitive to first increase the state dimension of a control system which actually is to be reduced.
Before we proceed with the concepts of variational analysis for these systems, we mention some differences to the theory discussed in [72]. There, the author includes a term of the form
L(x(t)⊗x(t))u(t), L∈Rn×n
2.
Although it might further increase the state dimension of a transformed system, it should be emphasized that by introducing a new state variable z(t) :=x(t)⊗x(t), the nonlinearity becomes purely bilinear, i.e. Lz(t)u(t). Since this simplifies the structure of the transfer functions that we introduce in the following, we always assume that the system under consideration does not contain multiplicative couplings of quadratic and
140 Nonlinear Systems bilinear variables. Moreover, in [72], the systems are denoted as quadratic-linear since the state variable x(t) appears quadratically while the input variable appears linearly.
Since one can interpret system (5.3) as a combination of a purely quadratic system and a bilinear control system we use the notation QBDAE.
5.2.2 Variational analysis for nonlinear systems
Let us now turn our attention to the analysis of QBDAEs. The key idea in the analysis of nonlinear systems is to express the solution by means of a Volterra series analog to the one for bilinear systems in (4.4). Although this concept can also be applied for more general linear-analytic systems (see [115, Section 3.4]), we illustrate the approach for the special case of QBDAEs. Here, we follow the discussion in [115, Section 3.4] and present the variational analysis approach. As a first step, we want to assume that the system (5.3) is forced by an input of the form αu(t).Due to the fact that a system of QBDAEs belongs to the class of linear-analytic systems, we may assume that the solution x(t) of (5.3) exhibits an analytic representation and thus can be written as
x(t) =αx1(t) +α2x2(t) +α3x3(t) +. . . (5.4) wherexi ∈Rn.Next, we insert the above expressions for input and response, respectively, into the state space representation ofΣQ. Hence, for the state equation (5.3), we obtain
E αx˙1(t) +α2x˙2(t) +. . .
=A αx1(t) +α2x2(t) +. . . +H αx1(t) +α2x2(t) +. . .
⊗ αx1(t) +α2x2(t) +. . . +N αx1(t) +α2x2(t) +. . .
αu(t) +bαu(t).
Finally, if we collect all terms αi corresponding to powers of α, we obtain dynamical equations for each of the state variables, i.e.,
Ex˙1(t) =Ax1(t) +bu(t),
Ex˙2(t) =Ax2(t) +Hx1(t)⊗x1(t) +Nx1(t)u(t),
Ex˙3(t) =Ax3(t) +H(x1(t)⊗x2(t) +x2(t) +x1(t)) +Nx2(t)u(t), ...
The advantage of this approach is that the solution x(t) can be derived by solving a series of nonlinearly coupled linear systems. In particular, this means that we can start
5.2. Quadratic-bilinear DAEs 141 by integrating the first subsystem in order to get
x1(t) = Z t
0
eA(t−τ)bu(τ)dτ.
If we consider this expression as a pseudo-input for the second equation, we can easily derive an expression forx2(t).Continuing in this manner, we finally arrive at the desired Volterra series representation forΣQ.As already mentioned, the basic idea is well-known and its origin goes back to works by Euler, Cauchy and Poincar´e, see [115]. Another derivation of the variational expansion approach is discussed in detail in [63].
5.2.3 Generalized transfer functions of QBDAEs
A similar technique allows an input-output characterization in the frequency domain.
Again, we just recapitulate the presentation from [115, Section 3.5]. Here, the essential idea is motivated by the following property of a stable linear continuous time-invariant control system. Let us assume that such a system is driven by an input signal u(t) = eλt, λ >0. Due to the explicit solution formula, we know that it holds
y(t) = Z ∞
0
h(σ)e(t−σ)λdσ = Z ∞
0
h(σ)e−λσdσ eλt=H(λ)eλt,
whereH(λ) = cT(λI−A)−1bdenotes the transfer function of the linear system. Hence, the output signal to agrowing exponential signal is simply scaled by the transfer function.
Moreover, for a linear combination of growing exponentials, i.e.,
u(t) =
p
X
i=1
αieλit, λ1, . . . , λp >0,
the output is given by
y(t) =
p
X
i=1
αiH(λi)eλit.
In order to keep the derivations clear, in the following we restrict ourselves to the compu-tation of the first two transfer functions for the systemΣQ.Since we denoted the Hessian of the system by H, we use a slightly different notation for the resulting transfer func-tions G1(s1) and G2(s1, s2). Let us now consider an input of the form u(t) =es1t+es2t which is supposed to yield a transient response
x(t) =G10es1t+G01es2t+G20e2s1t+G02e2s2t+G11e(s1+s2)t.
142 Nonlinear Systems Inserting this expression into the state equation (5.3) and comparing the coefficients then leads to the first two generalized symmetric transfer functions
G1(s1) = cT(s1E−A)−1
| {z }
F(s1)
b,
G2(s1, s2) = 1
2cT ((s1+s2)E−A)−1H(F(s1)⊗F(s2) +F(s2)⊗F(s1)) +1
2cT ((s1+s2)E−A)−1N(F(s1) +F(s2))
=cT((s1+s2)E−A)−1H(F(s1)⊗F(s2)) +1
2cT ((s1+s2)E−A)−1N(F(s1) +F(s2)).
Similarly, one can derive higher order transfer functions, see e.g. [72, 115]. As we can see, the first two transfer functions ofΣQgeneralize the theory for linear control systems.
However, similarly to the case of bilinear control systems, a meaningful interpretation of the frequency variabless1 and s2 cannot be given. The approach thus should rather be considered as an abstract theoretical tool for interpolation-based model order reduction.
Especially, it is important to realize that G1(s1) and G2(s1, s2) formally describe the input-output relationship of ΣQ in the frequency domain.