penrose pseudoinverse
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This paper presents modeling and infinite-time suboptimal control of a quadcopter device using the state-dependent Riccati equation (SDRE) method. It establishes a solution to the control problem using SDRE and proposes a new procedure for solving the problem. As a new contribution, the paper proposes a modified SDRE-based suboptimal control technique for affine nonlinear systems. The method uses a pseudolinearization of the closed-loop system employing Moore–Penrose pseudoinverse. Then, the algebraic Riccati equation (ARE), related to the feedback compensator gain, is reduced to state-independent form, and the solution can be computed only once in the whole control process. The ARE equation is applied to the problem reported in this study that provides general formulation and stability analysis. The effectiveness of the proposed control technique is demonstrated through the use of simulation results for a quadrotor device.

This paper describes the use of spherical wave expansion (SWE) to model the embedded element patterns of the LOFAR low-band array. The goal is to reduce the amount of data needed to store the embedded element patterns. The coefficients are calculated using the Moore–Penrose pseudoinverse. The Fast Fourier Transform (FFT) is used to interpolate the coefficients in the frequency domain. It turned out that the embedded element patterns can be described by only 41.8% of the data needed to describe them directly if sampled at the Nyquist rate. The presented results show that a frequency resolution of 1 MHz is needed for proper interpolation of the spherical wave coefficients over the 80 MHz operating frequency band of the LOFAR low-band array. It is also shown that the error due to interpolation using the FFT is less than the error due to linear interpolation or cubic spline interpolation.

Abstract Using the Moore–Penrose pseudoinverse this work generalizes the gradient approximation technique called the centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the generalized centred simplex gradient. We develop error bounds and, under a full-rank condition, show that the error bounds have ${\mathcal O}(\varDelta ^2)$, where $\varDelta $ is the radius of the sample set of points used. We establish calculus rules for generalized centred simplex gradients, introduce a calculus-based generalized centred simplex gradient and confirm that error bounds for this new approach are also ${\mathcal O}(\varDelta ^2)$. We provide several examples to illustrate the results and some benefits of these new methods.

Computing the pseudoinverse of a matrix is an essential component of many computational methods. It arises in statistics, graphics, robotics, numerical modeling, and many more areas. Therefore, it is desirable to select reliable algorithms that can perform this operation efficiently and robustly. A demanding benchmark test for the pseudoinverse computation was introduced. The stiffness matrices for higher order approximation turned out to be such tough problems and therefore can serve as good benchmarks for algorithms of the pseudoinverse computation. It was found out that only one algorithm, out of five known from literature, enabled us to obtain acceptable results for the pseudoinverse of the proposed benchmark test.