control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...

, the cross Gramian (

W_X

, also referred to by

W_

) is a

Gramian In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...

matrix used to determine how

controllable Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. Controllability and observabi ...

and

observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...

a linear system is. For the stable

time-invariant In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...

linear system :

\dot = A x + B u \,

y = C x \,

the cross Gramian is defined as: :

W_X := \int_0^\infty e^ BC e^ dt \,

and thus also given by the solution to the

Sylvester equation In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: :A X + X B = C. Then given matrices ''A'', ''B'', and ''C'', the problem is to find the possible matrices ''X'' that obey this equation. All m ...

: :

A W_X + W_X A = -BC \,

This means the cross Gramian is not strictly a

Gramian matrix In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...

, since it is generally neither positive semi-definite nor symmetric. The triple

(A,B,C)

and

, and hence minimal, if and only if the matrix

W_X

nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...

, (i.e.

W_X

has full rank, for any

t > 0

). If the associated system

(A,B,C)

is furthermore symmetric, such that there exists a transformation

J

with :

AJ = JA^T \,

B = JC^T \,

then the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

of the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s of the cross Gramian equal

Hankel singular value In control theory, Hankel singular values, named after Hermann Hankel, provide a measure of energy for each state in a system. They are the basis for balanced model reduction, in which high energy states are retained while low energy states are dis ...

s: :

, \lambda(W_X),  = \sqrt. \,

Thus the direct truncation of the

Eigendecomposition In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matr ...

of the cross Gramian allows

model order reduction Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical model ...

(se

without a balancing procedure as opposed to balanced truncation. The cross Gramian has also applications in decentralized control,

sensitivity analysis Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty anal ...

, and the

inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to so ...

References

Control theory Systems theory Matrices Determinants Analytic geometry {{systemstheory-stub

See also

References