Numerics.DimRed

Dimension reduction: From linear to bilinear control systems

Introduction

Before discussing the dimension reduction itself, we shall first shed some light on the transfer function/matrix (and its H₂ norm). The transfer function/matrix describes how for every input/control u(t), the linear/bilinear control systems responds with an output/observation y(t). It is the task of dimension reduction to find lower-dimensional (reduced) systems which approximate this input-output behavior as closely as possible on any compact time interval [0;T]. We note that this problem is closely connected to the concepts of controllability and observability.

Transfer function/matrix and H₂ norm

In linear time-invariant (LTI) system theory, the input equation of a control system describes the evolution of a high-dimensional system in terms of its state vector x(t)∈Cⁿ. The field-free evolution is described by linear ODEs with real symmetric matrix A∈R^n×n (or complex Hermitian matrix A∈C^n×n) with simple eigenvalue 0. The (linear!) interaction with a low-dimensional control u(t)∈R^m, m<<n is given by input matrix B∈R^n×m. The output equation defines the low-dimensional observables y(t)∈R^p, p<<n in terms of output matrix C ∈R^p×n.

where we have dropped the imaginary unit from our previous control equations because it does not affect the calculations of the Gramians discussed below. Upon Legendre transform, the input-output behavior of the system is simply given by a linear mapping

where U(s) and Y(s) are the Legendre transforms of input u(t) and y(t), respectively, and where the transfer function/matrix

is the Legendre transform of matrix g(t) consisting of elements

which can be considered as output y_i component generated by a δ-like (impulse) input u_j component.

Then the H₂ norm of the control system can be obtained as

where tr denote the trace of a matrix and where the Gramians W_c and W_o are explained in the following.

Controllability and observability

Ttwo important properties of control systems are the controllability (aka reachability) and the observability which can be regarded as dual aspects of the same problem. For the case of linear control systems, the corresponding controllability Gramian W_C (often referred to as P in the literature) and the observability Gramian W_O (often referred to as Q in the literature) are defined as

If the pair (A,B) is controllable, i.e., if the controllability matrix [ B AB A²B ... A^n-1B ] has full rank n, then the controllability Gramian W_C is positive definite. Qualitatively, the quadratic form x^* W_Cx is a measure for the "input energy" (in terms of the integral over |u(t)|²) that is needed to drive the system to a state x. For a generalization to bilinear systems (involving multiple nested time integrals), see Ref. [1,2]

Conversely, if the pair (A,C) is observable, i.e., if the observability matrix [ C^T A^TC^T (A^T)²C^T ... (A^T)^n-1C^T ] has full rank n, then the observability Gramian W_O is positive definite. Qualitatively, the quadratic form x^* W_Ox is a measure for the "output" energy (time integral over |y(t)|²) that can be extracted from the system when u=0 and the system has been initialized at state x.

In practice, evaluation of the above integrals is avoided; instead the Gramians W_C and W_O for the linear case are computed as the symmetric positive semi-definite solutions of the Lyapunov equations

which have unique solutions if A is stable. The effort for numerical solution scales as O(n³). For the case of a bilinear input relation, this has to be replaced by the following generalized Lyapunov equations [1,2]

While direct methods for solving such equations have a numerical complexity of O(n⁶), the iterative method introduced in [3] can be used instead which requires the solution of a standard Lyapunov equation in each step.

While for linear control systems the Lyapunov equations can always be solved if matrix A is stable (see here), for the generalized Lyapunov equations we recall that a system is called stable when the system matrix A has only eigenvalues in the open left half complex plane (i.e., excluding the imaginary axis). Stability thus means that there are constants λ , a > 0 such that ||exp(At)|| ≤ λ exp(−at), where ||•|| is a suitable matrix norm. If moreover

then controllability and observability Gramians exist. This can be achieved by a suitable scaling u→ηu, N→N/η, B →B/η with real η>1 which leaves the equations of motion invariant but, clearly, not the Gramians. Hence, by increasing η, we drive the system to its linear counterpart. For the limit η→∞, the system matrix N vanishes and we obtain a linear system. For this reason, η should not be chosen too large. Note that also the shift of the spectrum of A may serve to render the generalized Lyapunov equations solvable.

Dimension reduction

The need for efficient numerical treatment of control problems leads to the problem of model order reduction, i.e. the approximation of large-scale systems by significantly smaller ones. While model reduction of linear systems has been studied for several years now, and a well established theory including error bounds and structure-preserving properties fulfilled by a reduced-order model exists, the situation is less favorable for non-linear systems (which occur, e.g. in control of quantum problems). There, the mathematical techniques developed to linear systems are more difficult and much less general. Here we will deal with following two classes of methods, representing generalizations of their linear counterparts:

• Balanced truncation
• H2 norm interpolation

References

1. L. Zhang, J. Lam: Automatica 38, 205 (1988)

2. Z. Bai, D. Skoogh: Lin. Alg. Appl. 415, 406 (2006) (dx.doi.org)

3. E. Wachspress: Appl. Math. Lett. 1, 87 (1988) (dx.doi.org)

For a review of the theory of linear control systems, see e.g. the excellent lecture notes of Umea university and/or the books by Kemin Zhou.