Using the Sum of Roots and Its Application to a Control Design Problem

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A symbolic-numeric method for the parametric H∞\infty loop-shaping design problem

International audienceIn this paper, we present a symbolic-numeric method for solving the H_infinity loop-shaping design problem for low order single-input single-output systems with parameters. Due to the system parameters, no purely numerical algorithm can indeed solve the problem. Using Gröbner basis techniques and the Rational Univariate Representation of zero-dimensional algebraic varieties, we first give a parametrization of all the solutions of the two Algebraic Riccati Equations associated with the H_infinity-control problem. Then, following the works H. Anai, S. Hara, M. Kanno, K. Yokoyama, Parametric polynomial factorization using the sum of roots and its application to a control design problem, J. Symb. Comp., 44 (2009), 703-725, and M. Kanno, S. Hara, Symbolic-numeric hybrid optimization for plant/controller integrated design in H_infinity loop-shaping design, Journal of Math-for-Industry, 4 (2012), 135-140, on the spectral factorization problem, a certified symbolic-numeric algorithm is obtained for the computation of the positive definite solutions of these two Algebraic Riccati Equations. Finally, we present a certified symbolic-numeric algorithm which solves the H_infinity loop-shaping design problem for the above class of systems. This algorithm is illustrated with a standard example

A symbolic-numeric method for the parametric H∞\infty loop-shaping design problem

International audienceIn this paper, we present a symbolic-numeric method for solving the H_infinity loop-shaping design problem for low order single-input single-output systems with parameters. Due to the system parameters, no purely numerical algorithm can indeed solve the problem. Using Gröbner basis techniques and the Rational Univariate Representation of zero-dimensional algebraic varieties, we first give a parametrization of all the solutions of the two Algebraic Riccati Equations associated with the H_infinity-control problem. Then, following the works H. Anai, S. Hara, M. Kanno, K. Yokoyama, Parametric polynomial factorization using the sum of roots and its application to a control design problem, J. Symb. Comp., 44 (2009), 703-725, and M. Kanno, S. Hara, Symbolic-numeric hybrid optimization for plant/controller integrated design in H_infinity loop-shaping design, Journal of Math-for-Industry, 4 (2012), 135-140, on the spectral factorization problem, a certified symbolic-numeric algorithm is obtained for the computation of the positive definite solutions of these two Algebraic Riccati Equations. Finally, we present a certified symbolic-numeric algorithm which solves the H_infinity loop-shaping design problem for the above class of systems. This algorithm is illustrated with a standard example

Autonomous frequency domain identification: Theory and experiment

The analysis, design, and on-orbit tuning of robust controllers require more information about the plant than simply a nominal estimate of the plant transfer function. Information is also required concerning the uncertainty in the nominal estimate, or more generally, the identification of a model set within which the true plant is known to lie. The identification methodology that was developed and experimentally demonstrated makes use of a simple but useful characterization of the model uncertainty based on the output error. This is a characterization of the additive uncertainty in the plant model, which has found considerable use in many robust control analysis and synthesis techniques. The identification process is initiated by a stochastic input u which is applied to the plant p giving rise to the output. Spectral estimation (h = P sub uy/P sub uu) is used as an estimate of p and the model order is estimated using the produce moment matrix (PMM) method. A parametric model unit direction vector p is then determined by curve fitting the spectral estimate to a rational transfer function. The additive uncertainty delta sub m = p - unit direction vector p is then estimated by the cross spectral estimate delta = P sub ue/P sub uu where e = y - unit direction vectory y is the output error, and unit direction vector y = unit direction vector pu is the computed output of the parametric model subjected to the actual input u. The experimental results demonstrate the curve fitting algorithm produces the reduced-order plant model which minimizes the additive uncertainty. The nominal transfer function estimate unit direction vector p and the estimate delta of the additive uncertainty delta sub m are subsequently available to be used for optimization of robust controller performance and stability

The method of spectral analysis of the determination of random digital signals

The methods of spectral analysis based on the use of any model to describe the signal are considered in the article, while using them some assumptions about the behavior of the signal outside the observation interval are made.  The task of spectral analysis or evaluation, in this case, is to find the parameters of the model used, which is selected based on the available a priori information about the studied process.A new spectral analysis method is proposed, which uses the partially classical Prony method, and this method has been improved by replacing the damping sine wave with the use of damped sine wave.Replacing damped sine wave with the use of non-damped sine wave, allows you to very accurately isolate the signal and determine its characteristics against the background of very rich in interference with air space, against the background of radio devices that work legally. For the first time, a fast conversion algorithm was applied to solve the normal equations for finding variables to sequentially determine the parameters of random short-term signals such as amplitude, frequency, and phase.  It is suggested to determine not only static parameters but also the rate of change of these parameters.  Speed of change of parameters allows determining more carefully the signal of the means of silent receiving of information.Modeling of processes of determination of random short-term pulses simulating digital signals of the means of silent receiving of information is carried out, on the basis of the proposed method of spectral analysis, the simulation results are presented in the form of three-dimensional graphs.The main difference is the use not only of the analysis of the amplitude, frequency, phase, and spectrum of the signal but the main analysis of the spectral density of the signal.Analyzes of the simulation result fully confirm the advantages of the proposed method for the determination of random short-term pulses

Are Spectral Estimators Useful for Implementing Long-Run Restrictions in SVARs?

No, not really. Responding to lingering concerns about the reliability of SVARs, Christiano et al (NBER Macro Annual, 2006, "CEV") propose to combine OLS estimates of a VAR with a spectral estimate of long-run variance. In principle, this could help alleviate specification problems of SVARs in identifying long-run shocks. But in practice, spectral estimators suffer from small sample biases similar to those from VARs. Moreover, the spectral estimates contain information about serial correlation in VAR residuals and the VAR dynamics must be adjusted accordingly. Otherwise, a naive application of the CEV procedure would misrepresent the data's variance.

Rational Covariance Extension, Multivariate Spectral Estimation, and Related Moment Problems: Further Results and Applications

This dissertation concerns the problem of spectral estimation subject to moment constraints. Its scalar counterpart is well-known under the name of rational covariance extension which has been extensively studied in past decades. The classical covariance extension problem can be reformulated as a truncated trigonometric moment problem, which in general admits infinitely many solutions. In order to achieve positivity and rationality, optimization with entropy-like functionals has been exploited in the literature to select one solution with a fixed zero structure. Thus spectral zeros serve as an additional degree of freedom and in this way a complete parametrization of rational solutions with bounded degree can be obtained. New theoretical and numerical results are provided in this problem area of systems and control and are summarized in the following. First, a new algorithm for the scalar covariance extension problem formulated in terms of periodic ARMA models is given and its local convergence is demonstrated. The algorithm is formally extended for vector processes and applied to finite-interval model approximation and smoothing problems. Secondly, a general existence result is established for a multivariate spectral estimation problem formulated in a parametric fashion. Efforts are also made to attack the difficult uniqueness question and some preliminary results are obtained. Moreover, well-posedness in a special case is studied throughly, based on which a numerical continuation solver is developed with a provable convergence property. In addition, it is shown that solution to the spectral estimation problem is generally not unique in another parametric family of rational spectra that is advocated in the literature. Thirdly, the problem of image deblurring is formulated and solved in the framework of the multidimensional moment theory with a quadratic penalty as regularization