Cornelius T. Leondes

On the design of linear time invariant systems by periodic output feedback Part I. Discrete-time pole assignment†

Abstract In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for discrete-time pole assignment by output feedback for linear time-invariant systems. The resultant output feedback gain is a periodic function of time with a period equal to the sampling interval. It is also shown that periodic discrete-output feedback gains can be used in stabilizing a linear time-invariant system if and only if its unstable modes are controllable and observable.

Nonlinear Smoothing Theory

Differential equations are developed for the smoothing density function and for the smoothed expectation of an arbitrary function of the state. The exact equations developed herein are difficult to solve except in trivially simple cases. Approximations to these equations are developed for the smoothed expectation of the state and the smoothing covariance matrix. For linear systems these equations are shown to reduce to previously derived results. An iterative technique is suggested for even greater accuracy in approximations for severely nonlinear systems.

On the finite time control of linear systems by piecewise constant output feedback

The problem of finite time controllability of linear time invariant systems by piecewise constant output feedback is considered. It is shown that complete controllability and complete observability are both necessary and sufficient conditions for output feedback controllability by piecewise constant gain. Dead-beat output feedback controllers that drives the system to the origin in at most (μd × vd) steps, are also presented, where μd and vg are the indices of controllability and observability.

1972 IFAC congress paper: Optimal minimal-order observers for discrete-time systems-A unified theory

Luenbergers minimal-order observer is considered as an alternate to the Kalman filter for obtaining state estimates in linear discrete-time stochastic systems. The general solution to the problem of constructing the optimal minimal-order observer is presented for systems having white noise disturbances. In the special case of no measurement noise the observer estimation errors are shown to be identical with those of the corresponding Kalman filter. Estimation errors comparable with the Kalman filter are obtained when measurement noise is not excessive. The observer solution is extended to systems for which the noise disturbances are time-wise correlated processes of the Markov type. In considering correlated noise inputs, the system state equations are not augmented as is done in the usual Kalman filtering theory. The observer solution, modified appropriately to account for the time-wise correlation of the noise inputs, yields minimum mean-square estimates of the state vector. Application of the theory to the design of a radar tracking system shows that the performance obtained using a minimal-order observer may be comparable to that achieved with a Kalman filter.Luenbergers minimal-order observer is considered as an alternate to the Kalman filter for obtaining state estimates in linear discrete-time stochastic systems. The general solution to the problem of constructing the optimal minimal-order observer is presented for systems having white noise disturbances. In the special case of no measurement noise the observer estimation errors are shown to be identical with those of the corresponding Kalman filter. Estimation errors comparable with the Kalman filter are obtained when measurement noise is not excessive. The observer solution is extended to systems for which the noise disturbances are time-wise correlated processes of the Markov type. In considering correlated noise inputs, the system state equations are not augmented as is done in the usual Kalman filtering theory. The observer solution, modified appropriately to account for the time-wise correlation of the noise inputs, yields minimum mean-square estimates of the state vector. Application of the theory to the design of a radar tracking system shows that the performance obtained using a minimal-order observer may be comparable to that achieved with a Kalman filter.

In-Flight Alignment and Calibration of Inertial Measurement Units - Part I: General Formulation

This is the first part of a two-part paper which summarizes work pursued by the author in 1966 [1]. The paper describes the application of minimum-variance estimation techniques for in-flight alignment and calibration of an inertial measurement unit (IMU) relative to another IMU and/or some other reference. The first part formulates the problem, and the second part [2] reports numerical results and analyses. The approach taken is to cast the problem into the framework of Kalman-Bucy estimation theory, where velocity and position differences between the two IMUs are used as observations and the IMU parameters of interest become part of the state vector. Instrument quantization and computer roundoff errors are considered as measurement noise, and environmental induced random accelerations are considered as state noise. Typical applications of the technique presented might include the alignment and calibration of IMUs on aircraft carriers, the initialization of rockets or rocket airplanes which are launched from the wing of a mother ship, the alignment and calibration of IMUs which are only used in the latter phases of rocket flight, and for the initialization/updating of SST guidance systems.

The multiple linear quadratic gaussian problem

In this paper we define and solve the multiple linear quadratic gaussian (LQG) problem for discrete-time systems using Salukvadzes ideal point method which treats dynamic, but deterministic, multicriteria optimization problems. The solution consists in designing a scalar regulator where the weighting matrices are replaced by the sums of their counterparts in the individual criteria. The resulting control is optimal, in the sense that it minimizes the difference between the cost of the implemented policy from the cost that would occur if we optimized each criterion separately and added the outcomes. All the nice properties of the scalar LQG regulator are enjoyed by the multicriteria regulator as well. Extension to the continuous-time case is straightforward, yielding identical results.

On the design of linear time-invariant systems by-periodic output feedback Part II. Output feedback controllability†

Abstract In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for the complete controllability of linear time-invariant systems by discrete output feedback. In the case where the desired final state is the origin, the resultant, output feedback gain is periodic and the transfer is accomplished in at most vd periods, where vd is the discrete-time index of observability. In the more general case where the desired final state is non-zero, the transfer is accomplished using at most vd output measurements.

Statistically Linearized Estimation of Reentry Trajectories

Several filters are applied to the problem of state estimation from inertial measurements of reentry drag. This is a highly nonlinear problem of practical significance. It is found that a filter based on the technique of statistical linearization performs better than the extended Kalman in this application. This is believed to be the first application of the statistically linearized filter to a practical dynamics problem. A sensitivity analysis is performed to demonstrate the relative insensitivity of this filter to modeling errors and approximations.

A minimax filter for systems with large plant uncertainties

A minimax filter is derived in order to estimate the state of a system when large uncertainties in the plant dynamics and process noise are present. If the system dynamics and measurements are uncoupled and the noise covariance matrices are diagonal, simple results occur.

Application of the Volterra Series to the Analysis and Design of an Angle Track Loop

A typical function of an angle tracking loop is to keep a radar antenna pointed at a target. The error in pointing is directly related to successful operation of the tracking device; therefore, its behavior is of interest. For a tracker with a general polynomial nonlinearity, an arbitrary initial pointing error, and a bounded deterministic input, a method is developed for finding upper bounds on the magnitude of the tracking error using Volterra series techniques. Convergence regions of the Volterra series are also obtained. Applications of these results are made to a second-order tracking device.