This technical brief shows that given a system and its abstraction, the evolution of uncertain initial conditions in the original system is, in some sense, matched by the evolution of the uncertainty in the abstracted system. In other words, it is shown that the concept of Φ-related vector fields extends to the case of stochastic initial conditions where the probability density function (pdf) for the initial conditions is known. In the deterministic case, the Φ mapping commutes with the system dynamics. In this paper, we show that in the case of stochastic initial conditions, the induced mapping Φpdf commutes with the evolution of the pdf according to the Liouville equation.

This paper studies the relationship between the evolutions of uncertain initial conditions in Φ-related control systems. It is shown that a control system abstraction can capture the time evolution of the uncertainty in the original system by an appropriate choice of control input. Φ-related control systems with stochastic initial conditions show the same behaviour as systems with deterministic initial conditions. A conservation law is applied to the probability density function (pdf) requiring that the area under it be unity. Application of the conservation law results in a partial differential equation known as the Liouville equation, for which a closed form solution is known. The solution provides the time evolution of the initial pdf which can be followed by the abstracted system.

This technical brief shows that given a system and its abstraction, the evolution of uncertain initial conditions in the original system is, in some sense, matched by the evolution of the uncertainty in the abstracted system. In other words, it is shown that the concept of Φ-related vector fields extends to the case of stochastic initial conditions where the probability density function (pdf) for the initial conditions is known. In the deterministic case, the Φ mapping commutes with the system dynamics. In this paper, we show that in the case of stochastic initial conditions, the induced mapping Φpdf commutes with the evolution of the pdf according to the Liouville equation.

This article reviews five published “second-order” risk comparisons from the past four decades that implied precise understanding, and hence clear relationships or orderings, of the underlying risks. “Second order” here refers to efforts that extract information from original sources with the goal of relating diverse findings. All five of these publications have frequently been cited in the peer-reviewed literature and/or in risk regulatory debate in the United States. Each is associated with at least one contemporaneous critique that the findings were excessively precise. None of these critiques suggested that an alternative relationship or ordering of the risks evaluated was more appropriate. Instead, each critique concluded that alternative, contradictory relationships were at least as plausible given data and/or analytical limitations. In one case, the critique led to the withdrawal of the original publication. The original findings have been propagated or used uncritically in subsequent literature, including political support for cost-effectiveness analysis. In other cases, the critiques have been used to discredit quantitative risk analysis in general, especially in the cases of nuclear power and cost-benefit analysis. Both of these outcomes are undesirable. Future risk comparisons should avoid excessive precision, include explicit discussion of uncertainty, and differentiate between plausible estimates and expected values.

The forms and contexts in which risk analytical methods can provide useful inputs to policy decisions remains an open question. This paper assesses the role of uncertainty in costeffectiveness estimation using the example of life-saving interventions available to regulators in the United States, as calculated by Tengs et al. (Risk Analysis 15(3), 369–90, 1995). It identifies ‘equally plausible’ values for those interventions, based on alternative assumptions about costs and benefits. These alternative values suggest that in no case can credible point estimates indicate more than order of magnitude precision, and in the worst case, plausible point estimates range from infinite cost to net benefit for a single intervention. Some but not all of this uncertainty is irreducible. This suggests that decisions based on point-estimate costeffectiveness calculations can give a false impression of rational, evidence-based policy. In such cases, risk assessment based decisions are ‘systematically arbitrary.’ The analysis nonetheless suggests a role for cost effectiveness analysis in rejecting interventions that under all assumptions appear extremely costly, and promoting those that in all cases appear extremely inexpensive. Finally, it affirms that risk analysis is useful and appropriate as a tool for understanding complex problems, a role that could be undermined by excessive attention to calculating point estimates.

Comparative risk projects can provide broad policy guidance but they rarely have adequate scientific foundations to support precise risk rankings. Many extant projects report rankings anyway, with limited attention to uncertainty. Stochastic uncertainty, structural uncertainty, and ignorance are types of incertitude that afflict risk comparisons. The recently completed New Jersey Comparative Risk Project was innovative in trying to acknowledge and accommodate some historically ignored uncertainties in a substantive manner. This article examines the methods used and lessons learned from the New Jersey project. Monte Carlo techniques were used to characterize stochastic uncertainty, and sensitivity analysis helped to manage structural uncertainty. A deliberative process and a sorting technique helped manage ignorance. Key findings are that stochastic rankings can be calculated but they reveal such an alarming degree of imprecision that the rankings are no longer useful, whereas sorting techniques are helpful in spite of uncertainty. A deliberative process is helpful to counter analytical overreaching.

The paper presents an approach to end point trajectory control of elastic manipulators based on the nonlinear predictive control theory. Although this approach is applicable to manipulators of general configuration, only planar flexible multilink manipulators are considered. A predictive control law is derived by minimizing a quadratic function of the trajectory error of the end points of each link, elastic modes, and control torques. This approach avoids the instability of the zero dynamics encountered in the controller design using feedback linearization and variable structure control techniques. Furthermore, the derived predictive controller is robust to uncertainty in the system parameters. Simulation results are presented for a one link flexible manipulator to show that in the closed-loop system accurate end point trajectory control and vibration damping can be accomplished.

Women and men respond differently to mock news stories about new developments in science and technology, with women associating more risk (p ≤ .05) and less benefit (p ≤ .05) than do men with reported developments overall. Interview data were used to construct a survey instrument designed to probe for differences in underlying attitudes that might explain this outcome. Results from administration of the questionnaire reveal that women are more likely than men to agree with "antiscience" statements. The assertion that women and men can be thought of as members of distinct cultures is invoked to provide a theoretical explanation for the data.

The question of control of the end effector trajectory and stabilization of a two-link flexible robotic arm is considered. A control law based on the inversion of an input-output map is obtained. The outputs are chosen as the sum of the joint angle and tip elastic deformation times a constant factor for each link. The stable maneuver of the arm critically depends on the stability of the zero dynamics of the system. Stability of the zero dynamics is shown to be sensitive to the choice of the constant multiplying factor, which explains the difficulty in controlling the tip position. A critical value of the constant factor for control is obtained and this corresponds to a coordinate in the neighbourhood of the actual tip position. Although the inverse controller accomplishes output control, this excites the rigid and elastic modes. A linear stabilizer is designed for final capture of the terminal state and stabilization of the elastic modes. Simulation results are presented to show that in the closed-loop system, large maneuvers can be performed in the presence of payload uncertainty.

The trajectory control of aircraft in rapid, nonlinear maneuvers is discussed. Based on nonlinear invertibility theory, a control law is derived to independently control roll, pitch, and sideslip angles using rudder, elevator, and aileron. Integral feedback is introduced in order to obtain robustness in the control system to parameter uncertainty. The stability of the zero dynamics is examined. Simulation results are presented to show that in a closed-loop system, precise simultaneous lateral and longitudinal maneuvers can be performed despite the presence of uncertainty in the stability derivatives.

This paper treats the question of control of a class of nonlinear systems which can be decoupled by state variable feedback. Based on variable structure system (VSS) theory, a discontinuous control law is derived which accomplishes asymptotic decoupled output trajectory following in the presence of uncertainty in the system. In the closed-loop system, the trajectories are attracted towards a chosen hypersurface in the state space and then slide along it. During the "sliding phase" the motion is insensitive to parameter variations. Based on this result, a control law for asymptotically decoupled control of roll angle, angle of attack and sideslip in rapid, nonlinear maneuvers is derived. Simulation results are presented to show that large, simultaneous lateral and longitudinal maneuvers can be performed in spite of uncertainty in the stability derivatives.

The question of control of a class of nonlinear systems that can be decoupled by state-variable feedback is considered. Based on variable-structure system theory, a discontinuous control law is derived that accomplishes asymptotic decoupled output trajectory-following in the presence of uncertainty in the system. In the closed-loop system, the trajectories are attracted toward a chosen hypersurface in the state space and then slide along it. During the sliding phase the motion is insensitive to parameter variations. Based on this result, a control law for asymptotically decoupled control of roll angle, angle of attack, and sideslip in rapid, nonlinear maneuvers is derived. Simulation results are presented to show that large, simultaneous lateral and longitudinal maneuvers can be performed in spite of uncertainty in the stability derivatives.

This paper considers control of a class of uncertain nonlinear systems which can be decoupled by state variable feedback. A variable structure control (VSC) law is derived such that in the closed-loop system the output variable asymptotically tracks a given output trajectory in spite of the uncertainty in the system. Based on this result, a control law is derived for the attitude control of an orbiting spacecraft in the presence of uncertainty using reaction jets. The controlled outputs are the three Euler angles which describe the orientation of the spacecraft relative to an orbital frame. Simulation results are presented to show that in the closed-loop system precise attitude control is accomplished in spite of the uncertainty in the system.

An analysis is presented of the motion of a single axis rate gyroscope mounted in a space vehicle, which is simultaneously accelerating and spinning about the output and spin axis, respectively, of the gyro. The time-varying acceleration and deceleration Â¿x(t) and the spin rate Â¿z(t) of the vehicle are unknown but bounded functions of time t. It is shown that the motion of the gyro remains bounded if the bound Â¿*2 on the uncertainty in Â¿z(t) does not exceed some threshold Â¿*2 and that this threshold depends on the various parameters of the gyro. Furthermore, by a proper selection of the gyro parameters, its motion can be forced to remain in a small neighborhood (called region of ultimate boundedness) of the origin in Â¿-Â¿ plane after a certain finite interval of time for any bounded uncertain Â¿x(t) and Â¿z(t). Analytical relations are derived for the selection of gyro parameters to keep the error caused in the measurement of the input rate due to Â¿x and Â¿z within any given limit.

A model reference adaptive control law is presented for large angle rotational maneuvers of spacecraft using reaction jets. It is assumed that the various parameters of the spacecraft are completely unknown, and unknown but bounded disturbance torques are acting on the spacecraft. The controller includes a dynamic system in the feedback path. Simulation results are presented to show that fast, large angle rotational maneuvers can be performed using the adaptive controller in spite of uncertainty in the system.

Control of a class of uncertain nonlinear systems which can be decoupled by state-variable feedback is considered. A variable-structure-control (VSC) law is derived so that in the closed-loop system the output variables asymptotically track given output trajectories in spite of any uncertainty in the system. On the basis of this result, a control law is derived for the attitude control of an orbiting spacecraft in the presence of uncertainty using reaction jets. The controlled outputs are the three Euler angles which describe the orientation of the spacecraft relative to an orbital frame. Simulation results are presented to show that, in the closed-loop system, precise attitude control is accomplished in spite of the uncertainty in the system.