Minimal unknown-input functional observers for multi-input multi-output LTI systems

Designing minimum possible order (minimal) disturbance-decoupled proper functional observers for multi-input multi-output (MIMO) linear time-invariant (LTI) systems is studied. It is not necessary that a minimum-order unknown-input functional observer (UIFO) exists in our proposed design procedure. If the minimum-order observer cannot be attained, the observer's order is increased sequentially through a recursive algorithm, so that the minimal order UIFO can be obtained. To the best of our knowledge, this is the first time that this specific problem is addressed. It is assumed that the system is unknown-input functional detectable, which is the least requirement for the existence of a stable UIFO. This condition also is a certificate for the convergence of our observer's order-increase algorithm. Two methodologies are demonstrated to solve the observer design equations. The second presented scheme, is a new design method that based on our observations has a better numerical performance than the first conventional one. Numerical examples and simulation results in the MATLAB/Simulink environment describe the overall observer design procedure, and highlight the efficacy of our new methodology to solve the observer equations in comparison to the conventional one.

On unknown-input functional observability of linear systems

Finding the least possible order of a stable Unknown-Input Functional Observer (UIFO) has always been a challenge in observer design theory. A practical recursive algorithm is proposed in this technical note to design a minimal multi-functional observer for multi-input multi-output (MIMO) linear time-invariant (LTI) systems with unknown-inputs. The concept of unknown-input functional observability is introduced,and it is used as a certificate of the convergence of our algorithm. The proposed procedure looks for a number of additional auxiliary functions to be augmented to the original functions desired for reconstruction. The resulting UIFO is proper, and minimal (of minimum possible order). Moreover, the algorithm does not need the system to be unknown-input observable. A numerical example shows the procedure as well as the effectiveness of the proposed algorithm.

On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics

The use of the fast Fourier transform (FFT) accelerates Lanczos tridiagonalisation method for Hankel and Toeplitz matrices by reducing the complexity of matrix-vector multiplication. In multiprecision arithmetics, the FFT has overheads that make it less competitive compared with alternative methods when the accuracy is over 10000 decimal places. We studied two alternative Hankel matrix-vector multiplication methods based on multiprecision number decomposition and recursive Karatsuba-like multiplication, respectively. The first method was uncompetitive because of huge precision losses, while the second turned out to be five to 14 times faster than FFT in the ranges of matrix sizes up to n = 8192 and working precision of b = 32768 bits we were interested in. We successfully applied our approach to eigenvalues calculations to studies of spectra of matrices that arise in research on Riemann zeta function. The recursive matrix-vector multiplication significantly outperformed both the FFT and the traditional multiplication in these studies.

Robust adaptive backstepping controller design for DC-DC buck converters with external disturbances

In modern power electronic systems, DC-DC converter is one of the main controlled power sources for driving DC systems. But the inherent nonlinear and time-varying characteristics often result in some difficulties mostly related to the control issue. This paper presents a robust nonlinear adaptive controller design with a recursive methodology based on the pulse width modulation (PWM) to drive a DC-DC buck converter. The proposed controller is designed based on the dynamical model of the buck converter where all parameters within the model are assumed as unknown. These unknown parameters are estimated through the adaptation laws and the stability of these laws are ensured by formulating suitable control Lyapunov functions (CLFs) at different stages. The proposed control scheme also provides robustness against external disturbances as these disturbances are considered within the model. One of the main features of the proposed scheme is that it overcomes the over-parameterization problems of unknown parameters which usually appear in some conventional adaptive methods. Finally, the effectiveness of the proposed control scheme is verified through the simulation results and compared to that of an existing adaptive backstepping controller. Simulation results clearly indicate the performance improvement in terms of a faster output voltage tracking response.