For more than half a century, spectral factorization is encountered in various fields of science and engineering. It is a useful tool in robust and optimal control and filtering and many other areas. It is also a nice control-theoretical concept closely related to Riccati equation. As a quadratic equation in polynomials, it is a challenging algebraic task.
KeywordsController design, H2-optimal control, H∞-optimal control, J-spectral factorization, Linear systems, Polynomial, Polynomial matrix, Polynomial equation, Polynomial methods, Spectral factorization
Polynomial Spectral Factorization
As a mathematical tool, the spectral factorization was invented by Wiener in the 1940s to find a frequency domain solution of optimal filtering problems. Since then, this technique has turned up numberless applications in system, network and communication theory, robust and optimal control, filtration, prediction, and state reconstruction. Spectral factorization of scalar polynomials is naturally encountered in the area of single-input single-output systems.
Obviously, (4) is a quadratic equation in polynomials and its stable solution is the desired spectral factor.
When the right-hand side b(z) possesses some roots on the unit circle, this problem turns out to be unsolvable as (8) fails. If necessary, a less restrictive formulation can then be applied replacing (8) by b(e i ω ) ≥ 0 and with x(z) ≠ 0 only for | z | > 1 instead of (10). Clearly, the unit-circle roots of b(z) must then appear both in x(z) and x∗(z).
When formulated as above, the spectral factorization problem is always solvable and its solution is unique up to the change of sign (if x is a solution, so is − x, and no other solutions exist).
Polynomial Matrix Spectral Factorization
As in the scalar case, less restrictive definitions are sometimes used where the given right-hand side matrix B(s) is only nonnegative definite on the imaginary axis and so the spectral factor is free of zeros in the open right half plain Re s > 0 only.
In robust control, game theory, and several other fields, the symmetric right-hand side in the matrix spectral factorization may have a general signature. With such a right-hand side, standard (positive or nonnegative definite) factorization becomes impossible. Here, a similar yet different J-spectral factorization takes its role.
The J-spectral factorization problem is quite general having standard (either positive or nonnegative) spectral factorization as a particular case. No necessary and sufficient existence conditions appear to be known for J-spectral factorization. A sufficient condition by Jakubovič (1970) states that the problem is solvable if the multiplicity of the zeros on the imaginary axis of each of the invariant polynomials of the right-hand side matrix is even. In particular, this condition is satisfied whenever det B(s) has no zeros on the imaginary axis. In turn, the condition is violated if any of the invariant factors is not factorable by itself. An example of a nonfactorizable polynomial is 1 + s2.
Nonsymmetric Spectral Factorization
Algorithms and Software
Spectral factorization is a crucial step in the solution of various control, estimation, filtration, and other problems. It is no wonder that a variety of methods have been developed over the years for the computation of spectral factors. The most popular ones are briefly mentioned here. For details on particular algorithms, the reader is referred to the papers recommended for further reading.
Factor Extraction Method
If all roots of the right-hand side polynomial are known, the factorization becomes trivial. Just write the right-hand side as a product of first- and second-order factors and then collect the stable ones to create the stable factor. If the roots are not known, one can first enumerate them and then proceed as above. Somewhat surprisingly, a similar procedure can be used for the matrix case. To every zero, a proper matrix factor must be extracted. For further details, see Callier (1985) or Henrion and Sebek (2000).
This procedure is an iterative scheme with linear rate of convergence. It relies on equivalence between the polynomial spectral factorization and the Cholesky factorization of a related infinite-dimensional Toeplitz matrix. For further details, see Youla and Kazanjian (1978).
An iterative algorithm with quadratic convergence rate based on consecutive solutions of symmetric linear polynomial Diophantine equations. It is inspired by the classical Newton’s method for finding a root of a function. To learn more, read Davis (1963), Ježek and Kučera (1985), and Vostrý (1975).
Factorization via Riccati Equation
In state-space solution of various problems, an algebraic Riccati equation plays the role of spectral factorization. It is therefore not surprising that the spectral factor itself can directly be calculated by solution of a Riccati equation. For further info, see, e.g., Šebek (1992).
This is the most efficient and accurate procedure for factorization of scalar polynomials with very high degrees (in orders of hundreds or thousands). Such polynomials appear in some special problems of signal processing in advanced audio applications involving inversions of dynamics of loudspeakers or room acoustics. The algorithm is based on the fact that logarithm of a product (such as the spectral factorization equation) turns into a sum of logarithms of particular entries. For details, see Hromčík and Šebek (2007)
All the procedures above are either directly programmed or can be easily composed from the functions of Polynomial Toolbox for Matlab, which is a third-party Matlab toolbox for polynomials, polynomial matrices, and their applications in systems, signals, and control. For more details on the toolbox, visit www.polyx.com.
Consequences and Comments
Polynomial and polynomial matrix spectral factorization is an important step when frequency domain (polynomial) methods are used for optimal and robust control, filtering, estimation, or prediction. Numerous particular examples can be found throughout this encyclopedia as well as in the textbooks and papers recommended for further reading below.
Spectral factorization of rational functions and matrices is an equally important topic, but it is omitted here due to lack of space. Inquiring readers are referred to the papers of Oara and Varga (2000) and Zhong (2005).
The concept of spectral factorization was introduced by Wiener (1949), for further information see later original papers Wilson (1972) or Kwakernaak and Šebek (1994) as well as survey papers Kwakernaak (1991), Sayed and Kailath (2001) or Kučera (2007).
Nice applications of spectral factorization in control problems can be found e.g., in Green et al. (1990), Henrion et al. (2003) or Zhou and Doyle (1998). For its use of in other engineering problems see e.g., Sternad and Ahlén (1993).
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