Numerical methods for the analysis of sampled-data systems and for the computation of system zeros

MARSYAS is a computer-aided control system design package for the simulation and analysis of dynamic systems. In the summer of 1991 MARSYAS was updated to allow for the analysis of sampled-data systems in terms of frequency response, stability, etc. This update was continued during the summer of 1992 in order to extend further MARSYAS commands to the study of sampled-data systems. Further work was done to examine the computation of OPENAT transfer functions, root-locii and w-plane frequency response plots

Octave: A MARSYAS post-processor for computer-aideed control system design

MARSYAS is a computer-aided control system analysis package for the simulation and analysis of dynamic systems. In the summer of 1991 MARSYAS was updated to allow for the analysis of sampled-data systems in terms of frequency response, stability, etc. This update was continued during the summer of 1992 in order to extend further MARSYAS commands to the study of sampled data systems. Further work was done to examine the computation of openat transfer functions, root-locii and omega-plane frequency response plots. At the conclusion of the summer of 1992 work, it was proposed that control-system design capability be incorporated into the MARSYAS package. It was decided at that time to develop a separate 'stand-alone' computer-aided control system design (CACSD) package. This report is a brief description of such a package

A Singular Perturbation Approach for Time-Domain Assessment of Phase Margin

This paper considers the problem of time-domain assessment of the Phase Margin (PM) of a Single Input Single Output (SISO) Linear Time-Invariant (LTI) system using a singular perturbation approach, where a SISO LTI fast loop system, whose phase lag increases monotonically with frequency, is introduced into the loop as a singular perturbation with a singular perturbation (time-scale separation) parameter Epsilon. First, a bijective relationship between the Singular Perturbation Margin (SPM) max and the PM of the nominal (slow) system is established with an approximation error on the order of Epsilon(exp 2). In proving this result, relationships between the singular perturbation parameter Epsilon, PM of the perturbed system, PM and SPM of the nominal system, and the (monotonically increasing) phase of the fast system are also revealed. These results make it possible to assess the PM of the nominal system in the time-domain for SISO LTI systems using the SPM with a standardized testing system called "PM-gauge," as demonstrated by examples. PM is a widely used stability margin for LTI control system design and certification. Unfortunately, it is not applicable to Linear Time-Varying (LTV) and Nonlinear Time-Varying (NLTV) systems. The approach developed here can be used to establish a theoretical as well as practical metric of stability margin for LTV and NLTV systems using a standardized SPM that is backward compatible with PM

Least-Squares Approximate Solution of Overdetermined Sylvester Equations

We address the problem of computing a low-rank estimate Y of the solution X of the Lyapunov equation AX + XA\u27 + Q = O without computing the matrix X itself. This problem has applications in both the reduced-order modeling and the control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternating direction implicit method. While no known methods for low-rank approximate solution provide the two-norm optimal rank k estimate Xk of the exact solution X of the Lyapunov equation, our iterative algorithms provide an effective method for estimating the matrix X(k) by minimizing the error AY + YA\u27+ Q(F)

Partial Pivoting in the Computation of Krylov Subspaces of Large Sparse Systems

The use of Krylov subspace approaches, based on the Arnoldi iteration, has become a preferred technique for the solution of several medium to high order matrix equations. These include linear algebraic systems of equations as well as Riccati, Lyapunov and Sylvester equations encountered in control systems. In this paper it is shown that existing implementations of Arnoldi iteration for computation of orthogonal basis of Krylov subspace can lead to erroneous conclusions. A partial pivoting strategy is proposed that overcomes the pitfall in implementations currently in use

Least-Squares Approximate Solution of Overdetermined Sylvester Equations

We address the problem of computing a low-rank estimate Y of the solution X of the Lyapunov equation AX + XA\u27 + Q = O without computing the matrix X itself. This problem has applications in both the reduced-order modeling and the control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternating direction implicit method. While no known methods for low-rank approximate solution provide the two-norm optimal rank k estimate Xk of the exact solution X of the Lyapunov equation, our iterative algorithms provide an effective method for estimating the matrix X(k) by minimizing the error AY + YA\u27+ Q(F)

Solution of underdetermined Sylvester equations in sensor array signal processing

AbstractWe propose efficient solution of an underdetermined matrix equation. The matrix equations studied here appear in the study of subspace-based high-resolution multiparameter direction-of-arrival estimation. The proposed gradient-based technique is at least an order-of-magnitude computational improvement over the existing approaches. Simulation results are provided to illustrated the proposed technique

Underactuated robot control: comparing LQR, subspace stabilization, and combined error metric approaches

In this paper, three techniques. for robust control of underactuated robots are experimentally compared on the classical ball and beam system. An adaptive tracking controller is first designed and implemented to identify the nominal friction characteristic. Then, designs for a linear quadratic regulator (LQR), subspace stabilization controller, and combined error metric controller are presented. Step response tests confirm that both nonlinear approaches exhibit better stability properties than the standard LQR design. In addition, the subspace stabilization approach permits a much more aggressive beam motion, resulting in shorter settling time with excellent control of overshoot