Polynomial Solutions to the Matrix Equation X

Solutions are constructed for the Kalman-Yakubovich-transpose equation X−AXTB=C. The solutions are stated as a polynomial of parameters of the matrix equation. One of the polynomial solutions is expressed by the symmetric operator matrix, controllability matrix, and observability matrix. Moreover, the explicit solution is proposed when the Kalman-Yakubovich-transpose matrix equation has a unique solution. The provided approach does not require the coefficient matrices to be in canonical form. In addition, the numerical example is given to illustrate the effectiveness of the derived method. Some applications in control theory are discussed at the end of this paper

Efficient Approaches for Enclosing the United Solution Set of the Interval Generalized Sylvester Matrix Equation

In this work, we investigate the interval generalized Sylvester matrix equation ${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$ and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of ${\bf{A}}$, ${\bf{B}}$, ${\bf{C}}$ and ${\bf{D}}$ and in both of them we suppose that the midpoints of ${\bf{A}}$ and ${\bf{C}}$ are simultaneously diagonalizable as well as for the midpoints of the matrices ${\bf{B}}$ and ${\bf{D}}$. Some numerical experiments are given to illustrate the performance of the proposed methods

Signal estimation using H [infinity sign] criteria

In many signal processing and communication (SPC) applications we require to estimate signal corrupted by channel and additive noise. Optimal linear filters and predictors are used to recover signal from given observed (corrupted) signal. Kalman and Wiener filters are commonly used as optimal filters. These filters minimize the mean square error (MSE) or variance of the output error. The minimization require exact knowledge of input signal and noise power spectral density (PSD). Therefore, the performance of Kalman or Wiener filters degrade if the input signal and noise statistics is changing with time and is not known a priori. In many SPC applications there is no exact knowledge of the input signal and noise Statistics and Probability; One solution to this is to use the filters which minimizes MSE and adapt to changing input signals and noise Statistics and Probability; This solution falls into a general category of adaptive filters. Often, convergence speed of the adaptive filter algorithm determines the performance as it is assumed that the convergence speed is fast enough to track the changes in the input signal and noise Statistics and Probability; If the convergence speed is not able to track the input signal and noise statistics one can expect large variation in the output error power. Another approach to overcome unknown input signal and noise statistics is to use the mini-max estimation. One approach towards mini-max estimation is to minimize the error using H[infinity] criteria to obtain H[infinity] filters. This will lead to a conservative (minimize over the worst case input signals) design that is more robust to changes in the input signal and noise Statistics and Probability;;In this dissertation, interpretation of H[infinity] filters for zero mean stationary signals is discussed. From this H[infinity] filters are represented in the time and frequency domain. Performance benefits of H[infinity] filters over minimum variance filters are derived from this representation. Mathematical solutions to compute sub-optimal H[infinity] filters in time and frequency domain are discussed. Finally, performance benefits of H[infinity] filters for the code division multiple access (CDMA) system, signal estimation problems, and adaptive filters are shown through simulation results

Robust port-Hamiltonian representations of passive systems

We discuss the problem of robust representations of stable and passive transfer functions in particular coordinate systems, and focus in particular on the so-called port-Hamiltonian representations. Such representations are typically far from unique and the degrees of freedom are related to the solution set of the so-called Kalman-Yakubovich-Popov linear matrix inequality (LMI). In this paper we analyze robustness measures for the different possible representations and relate it to quality functions defined in terms of the eigenvalues of the matrix associated with the LMI. In particular, we look at the analytic center of this LMI. From this, we then derive inequalities for the passivity radius of the given model representation

Robust Quasi-LPV Controller Design via Integral Quadratic Constraint Analysis

Reduced cost of sensors and increased computing power is enabling the development and implementation of control systems that can simultaneously regulate multiple variables and handle conflicting objectives while maintaining stringent performance objectives. To make this a reality, practical analysis and design tools must be developed that allow the designer to trade-off conflicting objectives and guarantee performance in the presence of uncertain system dynamics, an uncertain environment, and over a wide range of operating conditions. As a first step towards this goal, we organize and streamline a promising robust control approach, Robust Linear Parameter Varying control, which integrates three fields of control theory: Integral Quadratic Constraints (IQC) to characterize uncertainty and nonlinearities, Linear Parameter Varying systems (LPV) that formalizes gain-scheduling, and convex optimization to solve the resulting robust control Linear Matrix Inequalities (LMI). To demonstrate the potential of this approach, it was applied to the design of a robust linear parametrically varying controller for an ecosystem with nonlinear predator-prey-hunter dynamics

Convex Identifcation of Stable Dynamical Systems

This thesis concerns the scalable application of convex optimization to data-driven modeling of dynamical systems, termed system identi cation in the control community. Two problems commonly arising in system identi cation are model instability (e.g. unreliability of long-term, open-loop predictions), and nonconvexity of quality-of- t criteria, such as simulation error (a.k.a. output error). To address these problems, this thesis presents convex parametrizations of stable dynamical systems, convex quality-of- t criteria, and e cient algorithms to optimize the latter over the former. In particular, this thesis makes extensive use of Lagrangian relaxation, a technique for generating convex approximations to nonconvex optimization problems. Recently, Lagrangian relaxation has been used to approximate simulation error and guarantee nonlinear model stability via semide nite programming (SDP), however, the resulting SDPs have large dimension, limiting their practical utility. The rst contribution of this thesis is a custom interior point algorithm that exploits structure in the problem to signi cantly reduce computational complexity. The new algorithm enables empirical comparisons to established methods including Nonlinear ARX, in which superior generalization to new data is demonstrated. Equipped with this algorithmic machinery, the second contribution of this thesis is the incorporation of model stability constraints into the maximum likelihood framework. Speci - cally, Lagrangian relaxation is combined with the expectation maximization (EM) algorithm to derive tight bounds on the likelihood function, that can be optimized over a convex parametrization of all stable linear dynamical systems. Two di erent formulations are presented, one of which gives higher delity bounds when disturbances (a.k.a. process noise) dominate measurement noise, and vice versa. Finally, identi cation of positive systems is considered. Such systems enjoy substantially simpler stability and performance analysis compared to the general linear time-invariant iv Abstract (LTI) case, and appear frequently in applications where physical constraints imply nonnegativity of the quantities of interest. Lagrangian relaxation is used to derive new convex parametrizations of stable positive systems and quality-of- t criteria, and substantial improvements in accuracy of the identi ed models, compared to existing approaches based on weighted equation error, are demonstrated. Furthermore, the convex parametrizations of stable systems based on linear Lyapunov functions are shown to be amenable to distributed optimization, which is useful for identi cation of large-scale networked dynamical systems