Open-loop Control Law Research Articles

Approximate Tracking for Systems on Three-Dimensional Matrix Lie Groups Via Feedback Nilpotentization

Abstract A wide range of dynamical systems from fields as diverse as mechanics, electrical networks and molecular chemistry can be modeled by invariant systems on matrix Lie groups. This paper extends the concept of approximate tracking in the high-frequency limit to non-nilpotent three-dimensional matrix Lie groups by making use of feedback nilpotentization for the local representations of these systems. Further it is shown how to convert these tracking controls involving a state-feedback to an open-loop control law, which can be interpreted as an approximate inverse of the original system.

Aspects of Passive Control of Structural Vibrations

A preliminary attempt to organize innovative strategies such as Base Isolation, Supplemental Energy Dissipation, Tuned Mass Damping and other combined approaches recently proposed by the authors, in a more general theory of Structural Control, is presented. To visualize the relationships between dynamic variables and subsystems, the block-diagram representations are used in order to obtain the regulation properties of the systems. It has been proven that the Base Isolation technique applies an open-loop control law and that Tuned Mass Damping and Supplemental Energy Dissipation realize closed-loop control laws. The block-diagram of the new combined strategy Base Isolation and Tuned Mass Damping is presented. The objectives of the regulating criteria are also discussed in order to shape the dynamic response in the frequency domain. The H 2 and H ∞ methods are considered as optimal control algorithms.

Global Dynamics of a Rapidly Forced Cart and Pendulum

In this paper, we study emergent behaviors elicited by applying open-loop, high-frequency oscillatory forcing to nonlinear control systems. First, we study hovering motions, which are periodic orbits associated with stable fixed points of the averaged system which are not fixed points of the forced system. We use the method of successive approximations to establish the existence of hovering motions, as well as compute analytical approximations of their locations, for the cart and pendulum on an inclined plane. Moreover, when small-amplitude dissipation is added, we show that the hovering motions are asymptotically stable. We compare the results for all of the local analysis with results of simulating Poincare maps. Second, we perform a complete global analysis on this cart and pendulum system. Toward this end, the same iteration scheme we use to establish the existence of the hovering periodic orbits also yields the existence of periodic orbits near saddle equilibria of the averaged system. These latter periodic orbits are shown to be saddle periodic orbits, and in turn they have stable and unstable manifolds that form homoclinic tangles. A quantitative global analysis of these tangles is carried out. Three distinguished limiting cases are analyzed. Melnikov theory is applied in one case, and an extension of a recent result about exponentially small splitting of separatrices is developed and applied in another case. Finally, the influence of small damping is studied. This global analysis is useful in the design of open-loop control laws.

Non-linear compensation control of a manipulator tip on a flexible body

The objective of this paper is to apply the compensation method to control the tip position of a rigid manipulator on a class of main flexible uniform beams. Here, it is proposed to add a compensator at the connection joint between the manipulator and the beam, so that the manipulator can be rotated and also programmed to change its length during the period of the two-dimensional system maneuvers. The non-linear open-loop compensation control law for the space manipulator presented here is extended from the concept of linear compensation control. The exact compensatory correction matches both the linear and non-linear range of system maneuvers. The control law of the system is synthesized based on linear quadratic regulator techniques, in which the inertial torque and force generated by the compensator are considered. The significant results of numerical simulations show that the tip co-ordinates of the manipulator can be made to very closely follow the rigid beam motion which is assumed a desired motion, and that the tip vibration can be successfully suppressed.

Effect of the deformation in the inertia forces on the inverse dynamics of planar flexible mechanical systems

The inverse dynamics problem for articulated structural systems such as robotic manipulators is the problem of the determination of the joint actuator forces and motor torques such that the system components follow specified motion trajectories. In many of the previous investigations, the open loop control law was established using an inverse dynamics procedure in which the centrifugal and Coriolis inertia forces are linearized such that these forces in the flexible model are the same as those in the rigid body model. In some other investigations, the effect of the nonlinear centrifugal and Coriolis forces is neglected in the analysis and control system design of articulated structural systems. It is the objective of this investigation to study the effect of the linearization of the centrifugal and Coriolis forces on the nonlinear dynamics of constrained flexible mechanical systems. The virtual work of the inertia forces is used to define the complete nonlinear centrifugal and Coriolis force model. This nonlinear model that depends on the rate of the finite rotation and the elastic deformation of the deformable bodies is used to obtain the solution of the inverse dynamics problem, thus defining the joint torques that produce the desired motion trajectories. The effect of the linearization of the mass matrix as well as the centrifugal and Coriolis forces on the obtained feedforward control law is examined numerically. The results presented in this investigation are obtained using a slider crank mechanism with a flexible connecting rod.

Optimal open/closed-loop control of a rayleigh beam

Optimal open/closed-loop control of a Rayleigh beam relative to a specified performance measure is investigated. A method to suppress the undesired transient beam vibrations is proposed and utilized to obtain numerical results for different problem parameters. With the assumption of a prescribed closed-loop control law, the Fourier-type open-loop optimal control law is derived in closed form through minimization of the performance measure via a calculus-of-variations technique. Numerical results presented in graphical form indicate that a feedback control mechanism may be effectively utilized in the presence of an open-loop control.

Open-loop versus closed-loop control of processor loading

In many processing systems, the goodput (the number of succesfully completed tasks per time unit) may decrease as the offered load increases. Such systems need some form of overload control. We distinguish between open-loop control that limits the number of tasks in the system with a fixed maximum and closed-loop control that rejects new tasks if the observed processor loading is too high. In most control systems closed-loop control performs better than open-loop control. Here we present a remarkable illustration of the converse: we show that higher goodput can be obtained with an open-loop control-law. The results are obtained with a simplified analytical model of a multi-tasking processing system, such as the central processor of a telephone exchange. Compared to open-loop control, closed-loop control is more robust with respect to a change in task idle-time. Under conditions of overload, the mentioned differences between the two types of control law become more pronounced. The computed results are confirmed by the outcome of simulation runs. In the simulations we used a more realistic definition of goodput than was used in the analytical model.

Energy Balances in Dynamic Systems Under Unmodelled Dynamics

This paper presents a control law synthesis technique to achieve robustness against uncertainties arising from the presence of unmodeled dynamics. The basic tool is the knowledge of typical specifications for control systems in the frequency domain like bandwidth, resonance peak and static gain for the nominal system. Only “a priori” (worst case) absolute upperbounds of the same above quantities are required to be known for the unmodelled dynamics frequency response. The philosophy is to synthesize the time open-loop control law, or the compensator, in such a way that the above specifications are within a prescribed tolerance with respect to the nominal situation when unmodelled dynamics is present. The analytic tool to develop the above philosophy is to fix the admitted tolerance against unmodelled dynamics of the input-output energetic balances. Stability of both the perfectly modeled nominal and current (namely that which involves unmodelled dynamics) systems is an “a priori” requirement in the formulation.

Hipparcos Astrometry Satellite: Closed-Loop Digital Attitude Control using Gas Jets

The design of a closed-loop control for the ESA Hipparcos satellite is presented in this paper. The control law, actuated by gas jets, is requested to operate with large sampling period and to keep the satellite in its nominal motion in spite of existing torque disturbances.The main design problem, solved in original way, is to avoid the effect of the aliasing errors due to the sampling with large period the system response caused by the applied impulsive torque control.The closed-loop performances, evaluated by computer simulations, are compared with the ones of an open-loop optimal control law developed in the assumption that the torque disturbances are known. From the obtained results, it is deduced that the present closed loop control law is close to the best achievable one.

A Multiple Shooting Method for Numerical Computation of Open and Closed Loop Controls in Nonlinear Systems

In recent years, the computation of optimal open loop control laws by multiple shooting has found very much attraction due to stability and efficiency of the method on the one hand and the flexibility and wide applicability of the indirect approach on the other hand.The present paper introduces an extension of this method which allows a very convenient computation of first order approximations of feedback control laws, which adapt the nominal solution to perturbations of state variables and process parameters (and thus admits a direct treatment of specific perturbations in boundary conditions or differential equations).The new method is numericallv stable and does not need any on - line computations. Moreover, it reauires only first order derivatives to be provided by the user.

Controller Design for a Gas Turbine Engine

An aircraft gas turbine engine is modelled as a multivariable system of nonlinear dynamic equations which is subject to a number of operating restrictions. This paper considers the specific control problem of transferring the states of the engine from an operating point of low thrust to high thrust output in an optimal manner (transient mode) and then maintaining the demanded high thrust output (regulation mode). An open-loop suboptimal control law is proposed as a solution for the transient mode and a closed-loop feedback law proposed for the regulation mode. Parameters of both control strategies are determined using a common method of gradient minimisation of a nonquadratic performance index in which penalty functions may be used to limit control input behaviour. A composite controller is then applied to the specific control problem and the results are compared to those obtained for the open-loop step input.

Decentralized control in packet switched satellite communication

The problem of access control to a packet switched multi-access satellite broadcast channel is considered in the framework of nonclassical control theory. This viewpoint, with the simple model presented, gives us some interesting results. It is shown under certain assumptions, that because of the inherent delay in information, the optimal control law will be within the class of open loop control laws. By a result in Bayesion decision theory it is shown that randomized decisions will not be necessary. Given an additional minor restriction we find that for "new" pockets only two modes of operation of the channel could be optimal, namely, All Together (every station sends whenever it has a packet) or Round Robin (every station sends whenever it is turn and it has a packet), and an equation for the crossover arrival probability P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> the channel would switch from All Together <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(P&lt;P_{0})</tex> to Round Robin <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(P&gt;P_{0})</tex> is given. For collided packets we find that it is optimal to have predetermined assignments of groups of stations to time slots for retransmissions, and a dynamic programming algorithm for determining the optimal grouping is given.

Performance of the a priori stochastic-optimal feedback control law

For a stochastic control system with noisy observations, some interest has been shown in mechanizing the optimal open-loop control law as a suboptimal feedback control law. Examples have appeared in the literature to show that the performance of this feedback control law is better than the performance of the optimal control function without feedback. It is shown here that in general this unproven observation is true.

Control of linear dynamic systems with set constrained disturbances

A linear dynamic system with input and observation uncertainties is studied. The uncertainties are constrained to be contained in specified sets. No probabilistic structure is assumed. The problem of keeping the state of the system in a specified region is investigated. Necessary and sufficient conditions for a solution to the problem and an algorithm that constructs the control are derived for open-loop and closed-loop control laws. The algorithm is also approximated by a bounding ellipsoid algorithm. Two special control laws, linear and "linear-plus-dead-band," are studied, and the regions that contain the state and control are characterized. Ellipsoids that bound the state and control are also derived, and a simple example of linear and linear-plus-dead-band control laws is presented.

Suboptimal controls that maximize the probability of entering a target manifold

A suboptimal control scheme is developed for a linear time-dependent process subject to additive random disiturbances. An open-loop control law is chosen from a one-parameter class to optimize the tradeoff between control energy expenditure and the probability of entering a given target set. The computation is done by means of a digital computer simulation.

On the comparison of the sensitivities of open-loop and closed-loop optimal control systems

This paper derives, via Pontryagin's minimum principle, the invariance of the sensitivity of cost for optimal systems with respect to open-loop and closed-loop control law implementation when system parameters undergo small variations. The result obtained here is more general than that of Pagurek[1] and somewhat easier to apply than that of Witsenhausen.[2] The cost of the sensitivity vector is simply related to the gradient of the Hamiltonian function.