Liyu Cao

Complementary results on the stability bounds of singularly perturbed systems

In this note, we will provide a new systematic approach to characterize and compute the stability bound of a singularly perturbed linear system. The approach is based on the feedback system representation of an additional matrix perturbation problem. The idea is to change the stability bound problem to the stability problem of an underlying feedback system. This approach allows multiple choices in formulating the underlying feedback system and thus has the potential of characterizing and computing the stability bound in a number of different ways. By formulating two kinds of different feedback systems, some existing and new results on the stability bounds are derived based on the feedback system approach. The new results complement the existing frequency-domain based stability criteria and make the frequency-domain technique more applicable and useful to the stability bound problem. An example is provided to show the new stability criterion is effective and useful in determining the stability bound.

Brief A directional forgetting algorithm based on the decomposition of the information matrix

A novel algorithm for directional forgetting is proposed based on a matrix decomposition method, which is developed in this paper. This algorithm performs exponential forgetting according to the direction of the data vector, thus preventing the problem known as estimator windup which is a characteristic of the standard exponential forgetting algorithm.

Analysis of the Kalman filter based estimation algorithm: an orthogonal decomposition approach

In this paper we shall provide new analysis on some fundamental properties of the Kalman filter based parameter estimation algorithms using an orthogonal decomposition approach based on the excited subspace. A theoretical analytical framework is established based on the decomposition of the covariance matrix, which appears to be very useful and effective in the analysis of a parameter estimation algorithm with the existence of an unexcited subspace. The sufficient and necessary condition for the boundedness of the covariance matrix in the Kalman filter is established. The idea of directional tracking is proposed to develop a new class of algorithms to overcome the windup problem. Based on the orthogonal decomposition approach two kinds of directional tracking algorithms are proposed. These algorithms utilize a time-varying covariance matrix and can keep stable even in the case of unsufficient and/or unbounded excitation.

Oscillation, Instability and Control of Stepper Motors

A novel approach to analyzing instability in permanent-magnet stepper motors is presented. It is shown that there are two kinds of unstable phenomena in this kind of motor: mid-frequency oscillation and high-frequency instability. Nonlinear bifurcation theory is used to illustrate the relationship between local instability and mid-frequency oscillatory motion. A novel analysis is presented to analyze the loss of synchronism phenomenon, which is identified as high-frequency instability. The concepts of separatrices and attractors in phase-space are used to derive a quantity to evaluate the high-frequency instability. By using this quantity one can easily estimate the stability for high supply frequencies. Furthermore, a stabilization method is presented. A generalized approach to analyze the stabilization problem based on feedback theory is given. It is shown that the mid-frequency stability and the high-frequency stability can be improved by state feedback.

Stick-slip friction compensation for PID position control

A simple compensation method for stick-slip limit cycle in PID controlled position systems is proposed in the paper. In order to analyze the property of the proposed compensation method, an iterative analysis approach is developed to determine and analyze the possible limit cycles in this kind of system with sufficient static friction. Based on the iterative analysis method, it is shown that the compensation method can reduce the amplitude of the limit cycles effectively. Computer simulation is used to validate the iterative analysis and the effectiveness of the proposed compensation strategy.

A novel recursive algorithm for directional forgetting

An algorithm for directional forgetting is developed. This algorithm performs exponential forgetting according to the direction of data vector, thus preventing the problem known as estimator windup which is a characteristic of the standard exponential forgetting algorithm. This algorithm is able to track fast parameter changes and is similar in complexity to the standard least square algorithm. The superior performance of the algorithm is verified via theoretical and simulation studies.

Output feedback stabilization of linear systems with a singular perturbation model

The problem of output feedback stabilization of linear systems with a singular perturbation model is addressed in this paper. A design method based on an accurate reduced-order model is proposed for this kind of system. This approach can overcome the difficulties in the existing design method using the so-called zero-order approximation model, whose validity is highly dependent on the value of the perturbation parameter. The closed-loop stability of the design method is analyzed and robust stability condition is given.

Oscillation control in non-linear systems using a first-order filter

In this paper we shall address the oscillation control problems in certain classes of non-linear systems whose outputs are required to follow their inputs. It is assumed that the non-linear systems can be well represented by a set of state-space equations and undergo Hopf bifurcation at some particular value of their parameters or their inputs. A simple first-order output feedback controller is proposed for oscillation control in these non-linear systems. First it is shown that in most cases the first-order controller is effective in locally stabilizing a second-order non-linear system which is undergoing Hopf bifurcation. Then a state separation method based on the solution of the associated Riccati equation is applied to the oscillation control of higher-order non-linear systems and a second-order approximated model is developed for the purpose of designing an oscillation controller. The closed-loop stability of the reduced-order model based design is analysed and some sufficient stability conditions are provided. Finally, a detailed application example of a stepper motor is given to show how the controller design method developed in this paper is applied to practical oscillation control problems.

A Decomposition Method for Positive Semidefinite Matrices and Its Application to Recursive Parameter Estimation

A matrix decomposition method for positive semidefinite matrices based on a given subspace is proposed in this paper. It is shown that any positive semidefinite matrix can be decomposed uniquely into two positive semidefinite parts with specified rank one of which is orthogonal to the subspace. This method is then compared with the rank-additivity decomposition, and the difference as well as the close connection between these two decompositions are given. Finally, the proposed decomposition method is used to develop a new recursive parameter estimation algorithm for linear systems.