Optimal Linear State Research Articles

Optimal Linear State Estimator With Multiple Packet Dropouts

This technical note is concerned with the optimal linear estimation problem in the presence of multiple packet dropouts in both sensor-to-controller and controller-to-actuator channels. The considered stochastic packet dropouts are independent and Bernoulli distributed. Based on a linear packet dropout model with stochastic parameters, a linear-minimum-variance (LMV) filter is proposed using the orthogonality principle. A sufficient condition for convergence of the steady-state LMV filter under a dc controller output is also given. Simulation results show the effectiveness of the proposed filter.

Cramer–Rao Bounds for Hybrid TOA/DOA-Based Location Estimation in Sensor Networks

This letter derives the Cramer-Rao bound (CRB) for source location estimation using joint time-/direction-of-arrival (TOA/DOA) measurements collected in a wireless sensor network setup. Each sensor is capable of measuring both TOA and DOA from a signal source, which enables it to estimate the source's location individually. Data fusion is then employed to reach a global position estimate. For both optimal measurement fusion and linear state fusion, we derive the CRBs to assess the attainable positioning accuracy, which shed light on the impact of network topology and sensor selection on localization accuracy.

Linear and non‐linear simulations of feedback control in plane Poiseuille flow

AbstractThis paper examines the performance of optimal linear quadratic state and output feedback controllers in stabilizing two‐dimensional perturbations in a plane Poiseuille flow. The synthesis of the controllers is based on a linearized model of the flow using a new set of interpolating polynomials in the wall‐normal direction, which automatically satisfy the homogeneous Dirichlet and Neumann boundary conditions at the walls and eliminate spurious eigenvalues. The controllers are implemented into a non‐linear Navier–Stokes solver, which is modified to compute the evolution of the flow perturbations. Two cases are examined, one with small initial disturbances that do not violate the linearity assumptions and the other with much larger disturbances that trigger the non‐linear convection terms. For the smallest disturbances, the solver accurately reproduced the results of the linear simulations of open‐ and closed‐loop systems. The simulations for the larger disturbances without control showed a rapid initial growth but the flow soon reached a saturated state in agreement with previous findings in the literature. The large initial growth is a consequence of the non‐normal nature of the system dynamics. The state feedback and output feedback controllers were able to reduce significantly the perturbation energy. For the larger disturbances, the energy calculated from the state variables is well below the energy evaluated by direct integration of the velocity field. This is probably due to the non‐linear terms transferring energy to harmonics of the considered wavenumber, which are not sensed by the linear controller. Copyright © 2008 John Wiley & Sons, Ltd.

High bandwidth tilt measurement using low-cost sensors

In this paper, a state estimation technique is developed for sensing inclination angles using low-cost sensors. A low-bandwidth tilt sensor is used along with an inaccurate rate gyro and a low-cost accelerometer to obtain the measurement. The rate gyro has an inherent bias along with sensor noise. The tilt sensor uses an internal pendulum and therefore has its own slow dynamics. These sensor dynamics were identified experimentally and combined to achieve high-bandwidth measurements using an optimal linear state estimator. Potential uses of the measurement technique range from robotics, to rehabilitation, to vehicle control.

Exploratory assessment of historical recruitment patterns using relative abundance and catch data

Historical apparent recruitment rates can be estimated from time series of relative abundance and catch data using a simple difference equation in conjunction with assumed catchability coefficient and natural survival rate. Simple smoothing and optimal linear state estimation procedures can remove much of the variance in such estimates due to measurement error in relative abundance. The apparent patterns can be helpful to reveal important recruitment changes prior to availability of data for more complex stock assessments, provide checks on recruitments estimated by more complex assessment procedures, and provide warnings about biases in other assessment procedures that might result from using the relative abundance data for model tuning.

Optimal control of precision paraboloidal shell structronic systems

Abstract Paraboloidal shells of revolution are commonly used in advanced aerospace, civil and telecommunication structures, e.g., antennas, reflectors, mirrors, rocket fairings, nozzles, solar collectors, dome structures, etc. A structronic shell system is defined as an elastic shell embedded, bonded or laminated with distributed piezoelectric sensors and actuators and it is governed by either in situ or external control electronics. A closed-loop control system of paraboloidal shell structronic system consists of distributed sensors/actuators and controller coupled with an elastic paraboloidal shell. State equation for the paraboloidal shell structronic system is derived and optimal linear quadratic state feedback control is implemented, such that the “best” shell control performance with the least control cost can be achieved. The gain matrix is estimated based on minimizing a performance criterion function. Optimal control effects are compared with controlled responses with other non-optimal control parameters. Control effects of identical-sized sensor/actuator patches at different locations are studied and compared. Modal control effects for different natural modes are also investigated.

On static feedback for the ℓ 1 and other optimal control problems

Although Ê 1 -optimal linear state feedback controllers are known to be dynamic, it has been shown that suboptimal performance arbitrarily closed to optimal can be achieved by using a static non-linear feedback law. In this paper, this fact is established by using a novel approach which shows that the result is a natural consequence of elementary state-space theory. The approach is motivated by recent works in active vision systems, which have considered a state-feedback problem tightly connected with Ê 1 optimization. This problem, which has independent interest, is discussed in some detail. The new formulation of the problem provides additional insight in Ê 1 state-feedback. In particular, it leads naturally to some extensions which do not follow in a straightforward manner from previous works on the subject.

Guaranteed cost control of discrete time uncertain systems

Guaranteed cost control problem is considered and the conventional linear quadratic optimal state feedback design method is applied to find the robust regulator for linear discrete time systems with structured uncertainties in the kind of linear combination in the parameters. The obtained control is a robust optimal linear state feedback which guarantees global asymptotic stability of the closed loop system and minimizes the maximal performance bound corresponding to a used quadratic criterion.

Optimal feedback control of a nonlinear system - Wing rock example

A procedure is presented for optimizing a state feedback control law for a nonlinear system with respect to a positive performance index. The Hamilton-Jacobi-Bellman equation is employed to derive the optimality equations wherein this performance index is minimized. The closed-loop Lyapunov function is assumed to have the same matrix form of state variables as the performance index. The constant interpolated terms of these matrix forms are easily determined so as to guarantee their positive definitenesses. The optimal nonlinear system is asymptotically stable in the large, as both the closed-loop Lyapunov function and performance index are positive definite. An unstable wing rock equation of motion is employed to illustrate this method. It is shown that the wing rock model using nonlinear state feedback is asymptotically stable in the large. Both optimal linear and nonlinear state feedback cases are evaluated.

Game theory approach to optimal linear state estimation and its relation to the minimum H/sub infinity /-norm estimation

A possible game theory approach to optimal state estimation is presented. It is found that in a certain differential game, the minimizer's policy is identical to the one obtained by optimal estimation in the minimum H/sub infinity /-norm sense. This interpretation of H/sub infinity /-optimal state estimation provides better insight into the mechanism of H/sub infinity /-optimal filtering, especially in the case where the exogenous signals are not energy bounded.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On dynamic output feedback compensator design using the Riccati equation

A simple formula for the design of dynamic output feedback compensators so as to reduce closed-loop system sensitivity is given. It is obtained by reformulating the compensator design problem as a deterministic linear optimal state regulator problem (i.e., an LQG problem), and an appropriate sensitivity reduction interpretation. The result is given for SISO linear systems. >

On the Control of the Flexible Manipulator ARM

Abstract The paper considers the control of a class of robots whose arm flexibility, due to light construction and the demand for fast robot motion, is not negligible. The mathematical model of dynamics of one translational joint and one flexible arm manipulator with the payload on its end is developed. On the basis of the mathematical model developed, an optimal linear quadratic state regulator is designed. In the paper the influence of additional control action, using the states of vibration modes of the manipulator arm, is analysed.

A transfer function approach to the problems of discrete-time systems: H/sub infinity /-optimal linear control and filtering

A solution is derived to the problems of H/sub infinity /-optimal linear state regulation and filtering. The solution method for both problems is based on a transfer function approach which applies standard spectral factorization. Return difference relations are given which are extensions of the relations associated with the linear quadratic problem. >

Output feedback controllers of suboptimality degree beta

A misconception in a recent paper by D.Z. Zheng (1989) about suboptimal feedback control is highlighted. A novel synthesis method for designing output feedback controllers of suboptimality degree beta is developed. These controllers yield a cost value equal to or less than a scalar multiple ( beta ) times the cost value obtained by optimal linear quadratic state feedback control. The output feedback control gains are explicit in terms of the standard Riccati equation when certain conditions are satisfied. >

Design of a systolic array system for linear state equations

A systolic array system for linear state equations is presented. The Gauss-Jordan algorithm and the matrix-vector multiplicationaccumulation are chosen to solve this problem. For systematically designing the systolic arrays system, the dependence graph (DG) approach has to be extended. The two dependence graphs which represent two different but data-dependent process algorithms are first linked together. Tag bits are added to index nodes in this linked DG which will use to indicate the different functions to be executed in a single processor element (PE). By applying the conventional time-scheduling and node-assignment procedures on this DG with a tag, the interfacing communication problem of a systolic arrays system will be well solved and the control of the diferent functions of the PE can be distributed throughout the array. Based on this method, an optimal linear state solver has been designed successfully.

Realization of a high speed biped using modern control theory

The purpose of this paper is to propose a new controlling method of the biped locomotion composed of seven links and to show simulated and experimental data of the locomotive motion of the biped called CW-1 built in our laboratory. The control method depends mainly on the linear optimal state feedback regulator, which stabilizes the stance supported by both legs as a commanded attitude. As a result the biped successfully walks any pre-determined steps such that one step takes only one second.

Reduced order state estimation for linear systems with exact measurements

Necessary and sufficient conditions are derived for the existence of reduced order, steady state, optimal linear state estimators for continuous, linear, dynamical systems when some output measurements are noise free. These conditions are shown to be dual to the generalized, strengthened Legendre-Clebsch conditions for the extremal controls of a dual singular regulator when the order of singularity is at most two. The optimal estimators are designed without the use of Luenberger observers.

Optimal estimation of linear discrete-time systems with stochastic parameters

This paper considers optimal linear state estimation in the general case of linear discrete-time systems with stochastic parameters which are statistically independent with respect to time. The estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise. It is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence, uniqueness and stability of the time invariant estimator.

Reduced Order State Estimation for Linear Systems with Exact Measurements

Necessary and sufficient conditions are derived for the existence of reduced order, steady state, optimal linear state estimators for continuous, linear, dynamical systems when some output measurements are noise free. These conditions are dual to the generalized, strengthened Legendre-Clebsch conditions for the extremal controls of a dual singular regulator. The optimal estimators are designed without the use of Luenberger observers.

AN ELECTROMAGNETICALLY LEVITATED VEHICLE DYNAMIC ANALYSIS OF AN ELECTROMAGNETICALLY LEVITATED VEHICLE USING LINEAR SYNCHRONOUS MOTORS

ABSTRACT The problem of stabilizing the airgap clearance of an electromagnetically levitated vehicle equippedwith linear synchronous motors is considered. The motors provide both propulsion and suspension. The dynamics of such a vehicle is expressed by a nonlinear state space model which takes into account the vertical and longitudinal translational motions. Expressions for both thrust and suspension forces are derived. An optimal linear state regulator is considered to stabilize the system and improve its dynamic behaviour. It acts simultaneously on the armature and field voltages of the motors. The response of a vehicle equipped with such a controller to some track defects is simulated. The effects of the introduction of an auxiliary suspension are studied.