Masaaki Kanno

Parametric polynomial spectral factorization using the sum of roots and its application to a control design problem

This paper presents an algebraic approach to polynomial spectral factorization, an important mathematical tool in signal processing and control. The approach exploits an intriguing relationship between the theory of Grobner bases and polynomial spectral factorization which can be observed through the sum of roots, and allows us to perform polynomial spectral factorization in the presence of real parameters. It is discussed that parametric polynomial spectral factorization enables us to express quantities such as the optimal cost in terms of parameters and the sum of roots. Furthermore an optimization method over parameters is suggested that makes use of the results from parametric polynomial spectral factorization and also employs two quantifier elimination techniques. This proposed approach is demonstrated in a numerical example of a particular control problem.

Parametric optimization in control using the sum of roots for parametric polynomial spectral factorization

This paper proposes an algebraic approach for parametric optimization which can be utilized for various problems in signal processing and control.The approach exploits the relationship between the sum of roots and polynomial spectral factorization and solves parametric polynomial spectral factorization by means of the sum of roots and the theory of Gröbner basis. This enables us to express quantities such as the optimal cost in terms of parameters and the sum of roots.Furthermore an optimization method over parameters is suggested that makes use of the results from parametric polynomial spectral factorization and also employs quantifier elimination.The proposed approach is demonstrated on a numerical example of a particular control problem.

Optimizing a Particular Real Root of a Polynomial by a Special Cylindrical Algebraic Decomposition

We study the problem of optimizing over parameters a particular real root of a polynomial with parametric coefficients. We propose an efficient symbolic method for solving the optimization problem based on a special cylindrical algebraic decomposition algorithm, which asks for a semi-algebraic decomposition into cells in terms of number-of-roots-invariance.

Sum of roots, polynomial spectral factorization, and control performance limitations

It has been pointed out that the notion of the sum of roots can be utilized for solving (parametric) polynomial spectral factorization. This paper reveals another aspect of the sum of roots as an indicator of achievable performance limitations. It is shown that performance limitations for some H2 control problems can be expressed as the difference of two sums of roots. An optimization approach combining the two aspects of the sum of roots is also demonstrated that minimizes the achievable performance limitation over plant parameters.

On the relationship between the sum of roots with positive real parts and polynomial spectral factorization

This paper is concerned with the relationship between the sum of roots with positive real parts (SORPRP) of an even polynomial and the polynomial spectral factor of the even polynomial. The SORPRP and its relationship to Grobner bases are firstly reviewed. Then it is shown that the system of equations satisfied by the coefficients of the polynomial spectral factor is directly related to a Grobner basis. It is then demonstrated by means of an H2 optimal control problem that the above fact can be used to facilitate guaranteed accuracy computation.

The Best Achievable ℋ2 Tracking Performances for SIMO Feedback Control Systems

This paper is concerned with the inherent ℋ2 tracking performance limitation of single-input and multiple-output (SIMO) linear time-invariant (LTI) feedback control systems. The performance is measured by the tracking error between a step reference input and the plant output with additional penalty on control input. We employ the plant augmentation strategy, which enables us to derive analytical closed-form expressions of the best achievable performance not only for discrete-time system, but also for continuous-time system by exploiting the delta domain version of the expressions.

Characterization of Easily Controllable Plants Based on the Finite Frequency Phase/Gain Property: A Magic Number 4+22 in H Loop Shaping Design

This paper introduces a phase/gain condition for (marginally) stable systems for characterization of easily controllable systems, and investigates the relationship between the condition and the optimal performance gammaopt in Hinfin loop shaping design. More specifically it is shown that there is a close relationship between the condition and a magic number radic(4+2(radic2)), for both continuous-time and discrete-time systems. Furthermore a simple design procedure for robust control based on the obtained knowledge is proposed.

H 2 Model Reduction Using LMIs

This paper considers the problem of H 2 reduced order approximation for both continuous and discrete time MIMO systems. A heuristic algorithm is proposed that utilizes necessary and sufficient conditions expressed in terms of a set of LMIs and a matrix rank constraint, and the alternating projection method. Also a method of finding starting points is suggested. Three numerical examples are employed to show the effectiveness of the choice of starting points and the capability of the algorithm to find at least as good approximants as other methods.

Cooperative gain output feedback stabilization for multi-agent dynamical systems

This paper is concerned with cooperative stabilization for LTI homogeneous multi-agent dynamical systems. We first formulate the cooperative stabilization problem by constant output feedback and show that it can be reduced to a stabilization problem with complex gain feedback. We then present several classes of systems in which the system is cooperatively stabilizable if and only if it can be stabilized alone. We also show a multi-agent system with even number of agents whose dynamics is represented by a 4th order transfer function, which can be stabilized by cooperation even if any single agent alone is not stabilizable.

Symbolic optimization of algebraic functions

This paper attempts to establish a new framework of symbolic optimization of algebraic functions that is relevant to possibly a wide variety of practical application areas. The crucial aspects of the framework are (i) the suitable use of algebraic methods coupled with the discovery and exploitation of structural properties of the problem in the conversion process into the framework, and (ii) the feasibility of algebraic methods when performing the optimization. As an example an algebraic approach is developed for the discrete-time polynomial spectral factorization problem that illustrates the significance and relevance of the proposed framework. A numerical example of a particular control problem is also included to demonstrate the development.