Browsing by Subject "Linear systems."

Now showing 1 - 3 of 3

Open Access
Exact and approximate decoupling and noninteracting control problems
(1989) Akar, Nail
In this thesis, we consider “exact” and “approximate” versions of the disturbance decoupling problem and the noninteracting control problem for linear, time-invariant systems. In the exact versions of these problems, we obtain necessary and sufficient conditions for the existence of an internally stabilizing dynamic output feedback controller such that prespecified interactions between certain sets of inputs and certain sets of outputs are annihilated in the closed-loop system. In the approximate version of these problems we require these interactions to be quenched in the ‘Hoo sense, up to any degree of accuracy. The solvability of the noninteracting control problems are shown to be equivalent to the existence of a common solution to two linear matrix equations over a principal ideal domain. A common solution to these equations exists if and only if the equations each have a solution and a bilateral matrix equation is solvable. This yields a system theoretical interpretation for the solvability of the original noninteracting control problem.
Open Access
Robust fault detection by simultaneous observers
(2000) Ammar, Nejib
This thesis addresses the problem of fault detection and isolation in linear systems based on unknown input observers. Functional disturbance decoupled observers which estimate specified or unspecified linear functions of system states regardless of the disturbances are first studied. Necessary and sufficient condition for the existence of such observers are presented. The investigation is extended to simultaneous disturbance decoupled observers where multiple systems are observed by a single disturbance decoupled observer. The application of disturbance decoupled observers to fault detection and diagnosis are explicitly outlined, and a new scheme that is based on simultaneous unknown input observers is proposed to enhance the already existing schemes. Finally, a detailed simulation example is carried out to examine the utility of the proposed scheme.
Open Access
Uncertain linear equations
(2010) Pilancı, Mert
In this thesis, new theoretical and practical results on linear equations with various types of uncertainties and their applications are presented. In the first part, the case in which there are more equations than unknowns (overdetermined case) is considered. A novel approach is proposed to provide robust and accurate estimates of the solution of the linear equations when both the measurement vector and the coefficient matrix are subject to uncertainty. A new analytic formulation is developed in terms of the gradient flow to analyze and provide estimates to the solution. The presented analysis enables us to study and compare existing methods in literature. We derive theoretical bounds for the performance of our estimator and show that if the signal-to-noise ratio is low than a treshold, a significant improvement is made compared to the conventional estimator. Numerical results in applications such as blind identification, multiple frequency estimation and deconvolution show that the proposed technique outperforms alternative methods in mean-squared error for a significant range of signal-to-noise ratio values. The second type of uncertainty analyzed in the overdetermined case is where uncertainty is sparse in some basis. We show that this type of uncertainty on the coefficient matrix can be recovered exactly for a large class of structures, if we have sufficiently many equations. We propose and solve an optimization criterion and its convex relaxation to recover the uncertainty and the solution to the linear system. We derive sufficiency conditions for exact and stable recovery. Then we demonstrate with numerical examples that the proposed method is able to recover unknowns exactly with high probability. The performance of the proposed technique is compared in estimation and tracking of sparse multipath wireless channels. The second part of the thesis deals with the case where there are more unknowns than equations (underdetermined case). We extend the theory of polarization of Arikan for random variables with continuous distributions. We show that the Hadamard Transform and the Discrete Fourier Transform, polarizes the information content of independent identically distributed copies of compressible random variables, where compressibility is measured by Shannon’s differential entropy. Using these results we show that, the solution of the linear system can be recovered even if there are more unknowns than equations if the number of equations is sufficient to capture the entropy of the uncertainty. This approach is applied to sampling compressible signals below the Nyquist rate and coined ”Polar Sampling”. This result generalizes and unifies the sparse recovery theory of Compressed Sensing by extending it to general low entropy signals with an information theoretical analysis. We demonstrate the effectiveness of Polar Sampling approach on a numerical sub-Nyquist sampling example.