Bernard Friedland

Control Systems Design: An Introduction to State-Space Methods

From the Publisher: One thesis of this book is that state-space methods can be presented in a style that can be grasped by the engineer who is more interested in using the results than in proving them. Another thesis is that results are useful. The book is addressed not only to students but also to a general audience of engineers and scientists who are interested in becoming familiar with state-space methods either for direct application to control system design or as a background for reading the periodical literature. The author has tried to keep the chapters reasonably independent and to use customary symbols wherever practical. He has also selected fifteen or so examples and weaved them into the fabric of the text and the homework problems.

Treatment of bias in recursive filtering

The problem of estimating the state x of a linear process in the presence of a constant but unknown bias vector b is considered. This bias vector influences the dynamics and/or the observations. It is shown that the optimum estimate \hat{x} of the state can be expressed as \hat{x} = x + V_{x}\hat{b} (1) where \tilde{x} is the bias-free estimate, computed as if no bias were present, \hat{b} is the optimum estimate of the bias, and V x is a matrix which can be interpreted as the ratio of the covariance of \tilde{x} and \hat{b} to the variance of \hat{b} . Moreover, \hat{b} can be computed in terms of the residuals in the bias-free estimate, and the matrix V x depends only on matrices which arise in the computation of the bias-free estimates. As a result, the computation of the optimum estimate \tilde{x} is effectively decoupled from the estimate of the bias \hat{b} , except for the final addition indicated by (1).

Optimum Steady-State Position and Velocity Estimation Using Noisy Sampled Position Data

The Kalman filtering technique is used to obtain analytical expressions for the optimum position and velocity accuracy that can be achieved in a navigation system that measures position at uniform sampling intervals of T seconds through random noise with an rms value of σx. A one-dimensional dynamic model, with piecewise-constant acceleration assumed, is used in the analysis, in which analytic expressions for position and velocity accuracy (mean square), before and after observations, are obtained. The errors are maximum immediately before position measurements are made. The maximum position error, however, can be bounded by the inherent sensor error by use of a sufficiently high sampling rate, which depends on the sensor accuracy and acceleration level. The steady-state Kalman filter for realizing the optimum estimates consists of a double integrator, the initial conditions of which are reset at each observation.

Implementation of a friction estimation and compensation technique

A friction estimation and compensation technique was implemented on a laboratory apparatus designed to permit the direct measurement of friction. Experimental results are reported for a friction observer which estimates total friction present assuming it to be a constant times the sign of velocity. A second observer is used to estimate the velocity using the measured position of the rotating shaft in the apparatus, when velocity is not measurable. Experimental results show that the friction estimate is consistent with the measured friction, displaying the theoretical hysteresis phenomenon. Moreover, the performance of the system is substantially improved by the use of the estimated friction to compensate the system, especially at very low velocity.

Analysis Strapdown Navigation Using Quaternions

A brief review of the theory of strapdown inertial navigation is presented in which the attitude of the sensor box with respect to inertial space is represented by the four-parameter quaternion vector. . A 4X4 matrix R is defined which aids in relating quaternions to direction cosines and facilitates interpretation of the equations for error propagation, which are also derived. In the interpretation, it is shown that the error in the quaternion vector causes aor-(correctable) scale factor error and an equivalent tilt vector error that propagates the same way as the platform tilt vector in a gimbaled system.

Modern control theory for design of autopilots for bank-to-turn missiles

The state-space techniques of modern control theory are used to develop a methodology for the design of autopilots for bank-to-turn missiles. The methodology accommodates the gyroscopic and coriolis cross-coupling between the pitch and the yaw axes that result due to the high roll rates that can be present. The design uses the assumption that the roll rate is constant, but not zero, and results in an autopilot structure in which there are cross-couplings between the pitch and yaw channels that are dependent on the roll rate. The autopilot gains are also scheduled as functions of the dynamic pressure. A reduced-order extended Kalman filter, with fixed gains, is used to estimate the actuator states and the commanded acceleration. The performance of an autopilot designed by this methodology was evaluated in a six-degree of freedom simulation using the dynamics of a typical high-performance tactical missile. Excellent performance was obtained in teres of low miss distance and small side-slip.

Theory and error analysis of vibrating-member gyroscope

There is a general class of gyroscopic instruments that use a vibrating member as the sensitive element. Instruments in this class are governed by the differential equations \ddot{x} + \Omega^{2}x - c\omega\dot{y} = 0 \ddot{y} + \Omega^{2}y - c\omega\dot{x} = 0 which describe the position of a reference point on the vibrating member relative to a coordinate system fixed in the instrument case. The point (x,y) moves in an elliptical orbit with poriod 2\pi/\Omega . The orbit precesses at a rate -c\omega/2 (with c \leq 2 ) relative to the coordinate system and, hence, tends to remain fixed in inertial space. The differential equations for the orbital elements ( a =semimajor axis b =semiminor axis, φ=inclination, and \theta =orbital angle) are derived for a nonideal gyroscope with damping and anisoelasticity present. The difforential equations for the long-term effects are obtained by averaging the coefficients over the approximate period of oscillation. These equations can he transformed into a fourth-order system of linear differential equations with constant coefficients in the average energy and angular momentum, and their derivatives. These equations are solved explicitly for several cases of practical interest and the results are interpreted physically.

On adaptive friction compensation without velocity measurement

The estimation and compensation of friction that may be present in a control system in which the velocity cannot be directly measured are considered. The compensator uses two observers: one to estimate the velocity and another which uses the velocity estimate of the first to estimate the friction coefficient. Local stability of the system is shown, and favorable simulation results are presented.<<ETX>>

Separate-bias estimation with reduced-order Kalman filters

This paper presents the optimal two-stage Kalman filter for systems that involve noise-free observations and constant but unknown bias. Like the full-order separate-bias Kalman filter, this new filter provides an alternative to state vector augmentation and offers the same potential for improved numerical accuracy and reduced computational burden. When dealing with systems involving accurate, essentially noise-free measurements, this new filter offers an additional advantage, a reduction in filter order. The optimal separate-bias reduced order estimator involves a reduced order filter for estimating the state, the order equalling the number of states less the number of observations.

A nonlinear observer for estimating parameters in dynamic systems

A new algorithm for estimating constant parameters in a dynamic system is presented. A generalization of a recently developed method for estimating the coefficient of friction in a dynamic system; the algorithm is a reduced-order observer having two nonlinear functions, one being the Jacobian of the other. If the dynamics of the system are affine in the parameters to be estimated, the error in estimation of these parameters satisfies a linear, time-varying homogeneous differential equation. By proper choice of the nonlinear function in the observer, it is possible to achieve asymptotic stability of the estimation error. Although the algorithm is derived on the assumption that the process state can be measured, it can be used to estimate the parameters concurrently with the state. Example applications, including one of adaptive control, are presented.