Related Problems

As detailed in the previous section, there are two key issues with the state-of-the-art method:

1) the computational limitations associated with solving the LMIs in order to determine a transformation and 2) the additional rate dependence caused by parameter-varying transformations which in practice restricts the search space to parameter-independent Gramians.

Gramian Approximations

Remedies to avoid the computational complexity of the LMI solution usually make use of a “local” approximation, i. e., they evaluate the LPV system for a fixed parameter value and apply methods for LTI systems. For a frozen parameterρ≡ρk, the solution of the optimization problem (3.31) simplifies to solving the two Lyapunov equations

A(ρk)X_c+X_cA^T(ρk) +B(ρk)B^T(ρk) = 0, (3.34a) A^T(ρk)Xo+XoA(ρk) +C^T(ρk)C(ρk) = 0. (3.34b) This is an immediate consequence of the fact that Y −X ≺ 0 for all X andY that satisfyA X+XA^T+B B^T ≺0 andA Y +Y A^T +B B^T = 0 [Dullerud & Paganini 2005, Proposition 4.4, p. 134]. That is, for a single point in the parameter domain it is easy to calculate a projection that represents balancing and truncation. This projection can then be applied to the LPV system with the assumption that it is similar to the (parameter-independent) projection that would balance the state space representation over the whole parameter space. For some applications this approach can be successfully applied [e. g., Balas 2002a]. If the dynamics vary substantially over the parameter space, such a constant projection is usually insufficient. It is then tempting to calculate local approximations of the Gramians at various parameter values and to interpolate between grid points.

This interpolation is guaranteed to be smooth for a sufficiently dense grid, sinceX_c(ρ)

andX_o(ρ) which satisfy (3.22) are continuous functions ofρ[Wood 1995, Cha. 7]. The problem of additional rate dependence nevertheless remains, as the interpolated Gramian approximations are necessarily varying and therefore result in a parameter-varying projection. In case either of these local approximations is used, any stability guarantees and error bounds are lost since the local solutions only satisfy the LMI constraints at single points in the parameter domain and not necessarily anywhere else.

For large-scale systems with several thousands of state variables, even the Lyapunov equations (3.34) become intractable. In this case, iterative low-rank approximations of the solutions to (3.34) can be used [e. g., Jaimoukha & Kasenally 1994, Li & White 2002, Benner et al. 2008]. Alternatively, empirical Gramian approximations [Willcox & Peraire 2002, Lall et al. 2002] can be constructed from impulse response data and the integral expressions

Xc=Z ∞

0 e^{A t}B B^Te^ATtdt, (3.35a)

Xo=Z ∞

0 e^ATtC^TCe^Atdt. (3.35b)

The required matrix of impulse responsese^{A t}B in Equation (3.35a) can be obtained by simulating the full-order autonomous system ˙x= A x with initial conditionsx0=Bi, i = 1, . . . , nu, where Bi denotes the i^th column of the matrixB. Each of the nu sim-ulations results in a state trajectoryxi(t), i= 1, . . . , nu which is collected in a matrix X(t) := [x1(t) · · · xnu(t)] =e^{A t}B. As the full-order system is stable, these trajectories decay for a sufficiently long simulation timeT_sim and can be calculated in a sampled form by standard numerical integration. Using a fixed step size ∆t, the integral (3.35a) can be approximated as

X_c≈

Tsim/∆t

X

k=0

X(k∆t)X^T(k∆t) ∆t. (3.36a)

An equivalent representation of Equation (3.36a) is

Xc≈XsampleX_sample^T ∆twithXsample= [X(0)X(∆t)X(2∆t)· · ·X(Tsim)] (3.36b) The number of columns inX_sample(and hence the rank of the estimate ofX_c) is limited by the number of samplesn_sample:=^Tsim/∆t. Equation (3.36b) further shows that√

∆t Xsample

provides a factorization that resembles the Cholesky factorization. As the projection (3.33) requires only the Cholesky factor ofXc, this means that the multiplication of the two matrices never has to be carried out and that only thenx-by-nsamplenumatrixXsamplehas to be stored, not thenx-by-nxmatrixXc. An approximation for the integral (3.35b) can be obtained using standard duality results from simulating the adjoint system ˙x=A^Tx with initial conditions corresponding to thei^th row of the matrix C[e. g. Antoulas 2005, Sec. 9.1.3]. The computational effort to obtain empirical Gramian approximation scales almost linearly with the state dimension and is hence also applicable for large-scale systems.

Modal State Space Decomposition

The rate dependence introduced by parameter-varying state transformations does not only complicate balanced model order reduction, but also prohibits a modal decomposition.

For LTI systems, modal decomposition is a very powerful and widely used tool for model reduction. It allows dynamics below or above the frequency range of interest to be removed without affecting other parts of the system. Therefore, it is most useful when physical insight into the problem is available. Another important application is the decomposition of a system into a stable and an unstable part.

An LTI system ˙x=A x+B u,y=C x+D uin modal form is described by representation with state variablesξ, one possible transformation to transform the system into the modal form (3.37) is

x=Re(v1) Im(v1) · · · Re(vm) Im(vm)v_2m+1v_2m+2 · · · vn−1

ξ,

where vi are the eigenvectors corresponding to the eigenvaluesλi. The basis vectors in the modal coordinates are thus the eigenvectors of the original system.

For LPV systems, eigenvalues and eigenvectors of the parameter-dependent matrix A(ρ) are also parameter-dependent and hence a transformation into modal coordinates would necessarily be itself parameter-dependent [e. g., Wood 1995, Sec. 7.2]. As detailed in Section 3.1.1, such a parameter-dependent transformation inevitably leads to a rate-dependent term in the transformed model, which may produce large off-diagonal elements for non-zero rates. Consequently, the decoupled structure of Equation (3.37) can, in general, not be attained. The only remedy for this problem is again to apply a parameter-independent transformation. Such a transformation is usually calculated such that a modal

form is attained at a single grid point in the parameter domain. The modal coupling at the other grid points is then accepted when state variables are removed from the LPV system [e. g., Moreno et al. 2014, Sec. 4.3.2]. This can severely diminish the usefulness of a modal form, where achieving decoupling is the primary goal.