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Lecture

Balanced Realization: SISO Case

In course

EE-611: Linear system theory

The course covers control theory and design for linear time-invariant systems : (i) Mathematical descriptions of systems (ii) Multivariables realizations; (iii) Stability ; (iv) Controllability and Observability; (v) Minimal realizations and coprime fracti ...

Description

This lecture covers the concept of balanced realization in the SISO case, focusing on balancing Gramians, Markov parameters, and constructing right coprime matrix fraction descriptions. It explains the process of achieving a balanced realization and the significance of symmetric matrices in the context of system observability and controllability.

Instructor

Philippe Müllhaupt

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V.

QR decomposition

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).

State-space representation

In control engineering, model based fault detection and system identification a state-space representation is a mathematical model of a physical system specified as a set of input, output and variables related by first-order (not involving second derivatives) differential equations or difference equations. Such variables, called state variables, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables.

LU decomposition

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

Polar decomposition

In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix ( is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size. Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes.

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