Worst-case Disturbance Research Articles

A local approach to solving the inverse minimax control problem for discrete-time systems

A local approach is used to solve the inverse problems of minimax control and of worst-case disturbance for linear discrete-time systems. It is shown that minimax control and worst-case disturbance with respect to a quadratic performance index are always those for some local criterion defined by a one-step increment of a state quadratic function along a system trajectory and current values of the control and the disturbance, while the inverse statement generally speaking is not true. Using a solution to the inverse problem of worst-case disturbance, which is of interest itself, the necessary and sufficient condition expressed in a frequency domain are derived under which a given local worst-case disturbance and a given local minimax control will be the worst-case disturbance and the minimax control respectively for some quadratic performance index with a suitable non-negative weight on the state. In addition, the set of all linear state feedbacks corresponding to minimax controls is revealed to be a subset of the set of all stable feedbacks corresponding to optimal controls in the absence of disturbances.

The sampled-data H problem: A unified framework for discretizationbased methods and Riccati equation solution

H A new discretization-based solution to the sampled-data control problem is given. In contrast to previous solution procedures, the method is not based on the lifting technique. Instead, an equivalent finite-dimensional discrete problem representation is derived directly from a description of the sampled-data system. This is achieved via a closed-loop expression of the worst-case intersample disturbance and an associated variable transformation. In this way the solution is obtained completely in terms of classical linear-quadratic optimal control theory. The discretization procedure described here is closely related to both the liftingH based technique and a two-Riccati equation solution of the sampled-data problem, in which the solution is obtained in terms of two coupled Riccati differential equations with jumps. In this way the method makes the connections between the various solution procedures transparent. In particular, it can be shown that the lifting approach and the two-Riccati equation solution lead to identical H synthesis equations for the-optimal sampled-data controller.

Observer-based compensators for fixed-order H∞control

This paper present a two-Riccati inequality approach for the design of fixed-order H∞ suboptimal controllers with an observer-based structure. The two Riccati inequalities are only one-way coupled and can be solved consecutively. The fixed-order controller has the structure of a fixed-order estimator of the state feedback control law in the presence of the worst-case disturbances. It retains the observed-based compensator structure of the full-order central controller of the standard H∞ design but adds a fixed-order dynamic and mappings between the full-order and the fixed-order state spaces. In the full-order special case, this controller recovers to the minimal entropy central controller of the standard H∞ design.

Analysis of torques in large steam turbine driven induction generator shafts following disturbances on the system supply

The paper first summarises the advantages of steam turbine-driven induction generators over conventional generators such as low cost, less maintenance, rugged and brushless rotors (squirrel cage type, no need for synchronisation, etc.), together with problems concerning excitation (VAr compensation at loads etc). A mathematical model of the induction generator simulated in direct-phase quantities where saturation of the magnetising reactances is simulated and saturation of stator and rotor leakage reactances is ignored is developed and employed for detailed simulation of the machine. Discrete-mass models of the machine shaft where both steam and electrical viscous damping is simulated are employed in comparing transient shaft torsional response evaluated by time domain simulation and frequency domain analysis following incidence and clearance of severe system faults. The paper then investigates torsional response following incidence and clearance of severe supply system disturbances, when the rotor is stationary and when running at close to synchronous speed unexcited, and following malsynchronisation when excited by a controlled VAr source, together with torsional response following bolted stator-terminal short-circuits at full-load and no-load following switching in of the induction generator onto the system supply. It examines precision of predicting torque in turbine-generator shafts by frequency domain analysis not analyzed for induction generators in the literature heretofore following incidence and clearance of worst-case disturbances on the supply. Effect of steam and electrical damping on maximum shaft torques predicted by frequency domain analysis is also illustrated. The results illustrate there is no tendency for shaft torques to become more onerous as the fault clearing time is increased as is the case for shaft torques in large synchronous machines. Three large two-pole machines of rating of up to a few hundred MWs are analysed.

Global optimization for the Biaffine Matrix Inequality problem

It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI feasibility problem is the bilinear version of the Linear (Affine) Matrix Inequality (LMI) feasibility problem, and may also be viewed as a bilinear extension to the Semidefinite Programming (SDP) problem. The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. This paper presents a branch and bound global optimization algorithm for the BMI. A simple numerical example is included. The robust control problem, i.e., the synthesis of a controller for a dynamic physical system which guarantees stability and performance in the face of significant modelling error and worst-case disturbance inputs, is frequently encountered in a variety of complex engineering applications including the design of aircraft, satellites, chemical plants, and other precision positioning and tracking systems.

An optimal thruster configuration design and evaluation for quick step

A procedure to calculate the margin of safety of a thruster system is presented. The margin of safety is the excess control authority of a thruster system available to overcome the worst case disturbances and is based on calculating the minimum control authority of the thruster system normalized by the worst case disturbances. Various controllers are presented for both one-sided and two-sided thrusters and their minimum control authorities are compared. Finally, a procedure to optimize a thruster system configuration in the sense of maximizing the margin of safety is presented and applied to the design of a thruster system for the Quick STEP spacecraft, an orbiting gravitational physics experiment. The techniques presented in this article are generally applicable to many other multi-degree of freedom actuation systems like the primary flight control effectors on a highly maneuverable aircraft for example.

TURBINE-GENERATOR SHAFT TORQUES AND FATIGUE LIFE EXPENDITURE PREDICTED BY FREQUENCY DOMAIN AND TIME DOMAIN METHODS FOLLOWING SEVERE DISTURBANCES ON THE SYSTEM SUPPLY

ABSTRACT This paper examines precision of predicting time responses for torque in turbine-generator-exciter shafts together with fatigue life expenditure of shafts by frequency domain analysis not analysed in the literature heretofore following incidence and clearance of worst-case disturbances on the electrical supply system. Effect of both governor and avr action following a range of disturbances is also illustrated. The effect of steam and electrical viscous damping on maximum and minimum shaft torques predicted by frequency domain analysis for a range of disturbances as a function of the fault clearing time is also depicted The analysis is based on Fourier analysis of generator airgap torque following incidence and clearance of a severe supply network disturbance or following mal-synchronisation to obtain torque excitation which acts on the generator rotor corresponding to each modal vibration. Amplitude and phase of each modal vibration is thereby determined. Using appropriate damping, time responses...

Minimum control authority plot - A tool for designing thruster systems

A new performance measure for thruster systems called the minimum control authority is defined. It is one of the most important performance measures since it corresponds to the worst case output of the thruster system. Various thruster system designs can easily be evaluated based on this performance measure. Several techniques are derived to calculate the minimum control authority as a function of the thruster configuration, the way in which the thrusters are controlled, and the physical mechanism that limits the thrusters' outputs. A technique to generate a plot of the worst case moment vs force-generating capability of a thruster system is also derived. This so-called minimum control authority plot can be used to evaluate the worst case output of various thruster system designs against each other and against the worst case disturbance forces they have to overcome. To illustrate the concept and demonstrate its usefulness, the minimum control authority plot is applied to evaluate various thruster system designs for the Gravity Probe B (GP-B) spacecraft. The baseline thruster for GP-B was selected based on the minimum control authority plot. Although the focus of this article is on thruster systems, the techniques developed are applicable in general to any system of actuators of any type, for example, the primary flight control effectors of highly maneuverable aircraft.

Structural properties of minimax controllers for a class of differential games arising in nonlinear [formula omitted] control

This paper introduces, in precise mathematical terms, two properties (named, certainty equivalence and generalized certainty equivalence) that nonlinear minimax controller problems might possess. The certainty equivalence is a generalization of the one introduced earlier by Başar and Bernhard (1991) and Bernhard (1990), which applies to problems where the ‘worst-case disturbance’ may not be unique (but the worst-case state trajectory is). The generalized certainty equivalance, on the other hand, extends this to accomodate nonunique worst-case state trajectories, and leads to the constrollers that guarantee a bounded upper value for the underlying game. The paper also shows that for a large class of games (and under certain conditions), certainty-equivalent (as well as generalized certainty-equivalent) controllers admit (infinite-dimensional) estimator (Kalman-filter) structures, where the estimator gain depends on the state of the estimator.

A dynamic games approach to controller design: disturbance rejection in discrete-time

It is shown that the discrete-time disturbance rejection problem, formulated in finite and infinite horizons, and under perfect state measurements, can be solved by making direct use of some results on linear-quadratic zero-sum dynamic games. For the finite-horizon problem an optimal (minimax) controller exists, and can be expressed in terms of a generalized (time-varying) discrete-time Riccati equation. The existence of an optimum also holds in the infinite-horizon case, under an appropriate observability condition, with the optimal control, given in terms of a generalized algebraic Riccati equation, also being stabilizing. In both cases, the corresponding worst-case disturbances turn out to be correlated random sequences with discrete distributions, which means that the problem (viewed as a dynamic game between the controller and the disturbance) does not admit a pure-strategy saddle point. Results for the delayed state measurement and the nonzero initial state cases are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Minimax Controllers for LTI Plants Under l1-Bounded Disturbances

A currently popular controller design technique for linear time-invariant (LTI) systems is H∞-optimization, which involves the minimization of the H∞-norm of a certain transfer function matrix. When formulated in the time domain, using state space description, this optimum design problem can be shown to be equivalent to a linear-quadratic differential game, and this formulation provides a convenient set-up for also studying finite-horizon versions of the original infinite-horizon problem. In this differential game, the minimizer is the controller and the maximizer is the unknown disturbance which is subject to an L2energy constraint. The performance of interest is the upper value of the differential game, with the controller achieving that value called the minimax controller.In this paper, we formulate such a differential game problem which arises in the context of disturbance rejection, but in the discrete time and under l1-bounded disturbances. We study the derivation of the minimax controller associated with the game, as well as the characterization of the worst-case disturbance. We show, in the context of a two-stage design problem, that the saddle point of the game involves a random disturbance, unless the initial state exceeds a certain threshold. Another feature of the solution is that the minimax controller is generally not unique, with the linear feedback controller being outperformed by nonlinear and/or memory controllers, locally or regionally.

Disturbance attenuation in LTI plants with finite horizon: Optimality of nonlinear controllers

It is an established fact that search for an optimum in the problem of disturbance attenuation in linear time-invariant (LTI) plants can be restricted to linear designs when the sole criterion is the minimization of the H ∞- norm of an appropriate input-output operator. What is not known, however, is whether there exist also optimal nonlinear controllers which would lead to uniformly better performance (than linear controllers) in a neighborhood of the worst-case disturbance. It is shown in this paper that for the finite-horizon version, and with the disturbance energy fixed, there do exist optimal (minimax) controllers outside the linear class, which can oout-perform the best linear controller locally. It is also argued that a differential (dynamic) games set-up provides a natural framework for the time-domain derivation of optimal (minimax) controllers for tracking, model matching and disturbance rejection problems, in both finite and infinite horizons.

Method for Error Budgeting of Inertial Navigation Systems

a and b. If the value of Q for the system has the same algebraic sign as the boundary value of Q in that region, and if the absolute value of Q is greater than the boundary value plotted in Fig. 3, then a double-firing instability will occur if the worst-case disturbance torque is applied. If the absolute value of Q is less than the boundary value, then no doublefiring instability can occur. For a given structural damping and bending frequency, AT// will increase with increases in rate gyro damping, filter time constant, or valve delays, or with a decrease in rate gyro natural frequency. A decrease in A^ occurs for the reverse trends. Instability in the left-hand negative Q region, therefore, will tend to require a high-frequency lightly damped rate gyro with short filter time constant and valve delays. Instability in this region may be of practical concern for some systems, but for the systems studied by the author the left-hand region is of no interest because the values of Q are usually positive and Ai// is always greater than 90 deg. In the right-hand negative Q region, instability requires relatively low rate gyro frequencies and long filter and valve delays; however, the values of Q required for instability in the right-hand region are so large that instability in this region is probably not a matter of practical concern for any real system. The middle positive Q region is, however, of practical interest for the systems studied by the author. It is seen that the difference between on and off delays affects the curves significantly, and that the use of average delay gives an overly conservative boundary value (Q smaller than necessary) if the off delay is less than the on delay, as it usually is for real solenoid valves.

On control of linear systems with time lags

The terminal control of a general time invariant linear system with time lags has been studied in a variety of situations by means of the technique of dynamic programming. Optimum control laws, satisfying a minimum energy constraint are explicitly derived under each of the following assumptions concerning the noise in the system: noise free, noise generated by a Markov source, noise whose statistics are partially unknown, and worst case or competitive disturbances.