Showing papers on "Feedback linearization published in 1985"

Nonlinear Control Systems

Abstract: : The principal goal of this three years research effort was to enhance the research base which would support efforts to systematically control, or take advantage of, dominant nonlinear or distributed parameter effects in the evolution of complex dynamical systems. Such an enhancement is intended to support the development of flight controllers for increasing the high angle of attack or high agility capabilities of existing and future generations of aircraft and missiles. The principal investigating team has succeeded in the development of a systematic methodology for designing feedback control laws solving the problems of asymptotic tracking and disturbance rejection for nonlinear systems with unknown, or uncertain, real parameters. Another successful research project was the development of a systematic feedback design theory for solving the problems of asymptotic tracking and disturbance rejection for linear distributed parameter systems. The technical details which needed to be overcome are discussed more fully in this final report.

The structure of nonlinear control systems possessing symmetries

Abstract: A concept of symmetry is defined for general nonlinear control systems. It is shown, under various technical conditions, that nonlinear control systems with symmetries admit local and/or global decompositions in terms of lower dimensional subsystems and feedback loops. The structure of the individual subsystems is dependent on the structure of the symmetry group; for example, if the symmetry group is Abelian, one of the subsystems is a quadrature. An additional feature of the decomposition is that the state-space dimensions of the subsystems sum to the state-space dimension of the original system.

An exothermic continuous stirred tank reactor is feedback equivalent to a linear system

Abstract: This brief paper demonstrates the concept of linear feedback equivalence for an exothermic eontinu-ous stirred tank reactor with first order kinetics. Feedback control is achieved by finding a transformation for the nonlinear system which carries this system into a linear controllable system in Brunovsky canonical form. A linear state feedback controller is then designed which achieves control over a broad range of operating conditions. This example demonstrates how recent developments in nonlinear control theory can be applied to chemical systems without relying on the usual methods of local linearization.

Global feedback stabilization of nonlinear systems

Abstract: In this paper we continue our development of the analogues for nonlinear systems of those frequency domain notions so important in classical control. One of our long-term goals, about which we can now say quite a bit in a reasonably broad framework (see 4), is to develop a design philosophy for the construction of (globally) stabilizing compensators for nonlinear systems, based on seemingly familiar notions such as the (strong) relative degree of a nonlinear system or knowledge that the system is "minimum phase." Aside from the development of a basic, "frequency domain package" for nonlinear systems, this paper contains applications to system invertibility, (global) stabilization by dynamic compensation, and global linearization by state feedback for nonlinear systems with relative degree or minimum phase properties.

Global external linearization of nonlinear systems via feedback

Abstract: This note presents necessary conditions and sufficient conditions for an affine nonlinear system to be globally feedback equivalent to a controllable linear system over an open subset V of Rn. When V equals Rn, necessary and sufficient conditions are obtained.

The matching of a prescribed linear input-output behavior in a nonlinear system

Abstract: In this paper we deal with the problem of finding a feedback under which the input-output behavior of a nonlinear system becomes exactly the same as that of a specified linear model. The solvability of this problem is shown to depend on a formal infinite zero structure associated with the system.

Controlled invariance for discrete-time nonlinear systems with an application to the disturbance decoupling problems

Abstract: Invariant distributions are defined for discrete-time nonlinear control systems, and necessary and sufficient conditions are given for their controlled invariance. This extends to discrete-time systems the basic tool which has been so important in solving the various synthesis problems for continuous-time systems. To indicate their utility in the discrete-time setting, they are used to locally solve the disturbance decoupling problem.

Magnetic suspension and pointing system with flux feedback linearization

Abstract: A magnetic actuator for providing a single-axis, noncontacting force capability with unconstrained cross-axis mobility utilizes flux feedback force linearization. Electromagnets cooperating with an armature suspended therebetween are energized by a bias flux and a signal derived from flux densities sensed in the air gaps. The bias flux density may be optimized to minimize the peak power required by the actuator. The flux density signals are applied in a closed force loop to provide a net force directly proportional to the commanded flux.

A Slow Manifold Approach to Feedback Control of Nonlinear Flexible Systems

Abstract: In this paper we consider the control problem for a class of coupled, second-order singularly perturbed nonlinear dynamical systems. The problem has important application to flexible mechanical systems including robot manipulators with flexible joinra, where the singular perturbation parameter ? is the inverse of the joint stiffness. For this class of systems it is known that the reduced order model corresponding to the mechanical system under the assumption of perfect rigidity is globally linearizable via nonlinear state feedback, but that the full order flexible system is not, in general, linearizable. We utilize the concept of integral manifold to represent the dynamics of the slow subsystem, which reduces to the rigid model as the perturbation parameter tends to zero. We show that linearizability of the rigid model implies linearizability of the flexible system restricted to the integral manifold. Based on a power series expansion of the integral manifold around ? = 0 we show how to approximate the feedback linearizing control to any order in ?. The result is a nonlinear feedback control scheme to "stiffen" the nonlinear flexible system. That is, the behavior of the closed loop flexible system is nearly that of the controlled rigid system.

Nonlinear versus linear control in the direct output feedback stabilization of linear systems

Abstract: This paper investigates the use of nonlinear direct output feedback in the control of linear time-invariant systems. Inparticular, it is concerned with those nonlinear controllers which result in a closed loop system which is quadratically stable. That is, the closed loop nonlinear system is asymptotically stable and furthermore, a quadratic Lyapunov function can be used to establish this stability. The main result of this paper can be stated roughly as follows: Any linear system which can be made quadratically stable using nonlinear direct output feedback control can also be stabilized using linear direct output feedback control.

About the local linearization of nonlinear systems

Abstract: In this paper, we show that for any locally controllable multi-input nonlinear system of the form x=f(x,u), u ? Rn, there exists an implicit feedback v=S(x,u) assigning the poles of the closed loop linearized system at specified values independent of the operating point. This result generalizes the work of Baumann and Rugh [1] to the multi-input case by reconsidering the problem in its natural geometrical framework.

On the robustness of nonlinear regulators and its application to nonlinear systems stabilization

Abstract: In this note, we develop a new robustness property of optimal regulators of nonlinear dynamic systems. The stability condition we establish presents a great advantage when compared to other stability conditions already available in the literature. It does not depend explicitly on the optimal control problem solution, allowing then a very simple and a priori analysis of the closed-loop system stability. Based on that, we propose a methodology for determining linear decentralized controllers which stabilize asymptotically a wide class of nonlinear dynamic systems. As an application example we present and discuss in detail the control design for a two-pendulum system [4].

Pseudolinearization of nonlinear systems by dynamic precompensation

Abstract: For general nonlinear system of the form x = f(x,u) a "generic" condition of pseudolinearizability (in the sense of [1], [2]) is set out. This result is used to show how a large class of nonlinear systems can be pseudolinearized by using a dynamic precompensator.

Global linearization by feedback and state transformation

Abstract: Differential geometric conditions equivalent to the existence of a solution to the global feedback linearization problem are given If global feedback for linearization is obtained with an atlas of local state space transformations, the resulting closed loop system still has almost all the important features of a linear system Existence of a compact leaf in any of the standard foliations arising in the local feedback linearization problem is shown to represent nontrivial linear holonomy and hence an obstruction to the existence of global feedback We prove that in the analytic case, if the state space is simply-connected, this obstruction does not occur We show that in the two dimensional (C?) case, if the manifold is simply connected, then the local conditions and controllability are sufficient for global feedback linearization

Single step optimization of feedback-decoupled spacecraft attitude manuevers

Abstract: Recent research on nonlinear guidance of either manipulators or spacecraft concerns the use of feedback linearization and decoupling to generate exact nominal commands. The advantage of such transformation is that the otherwise highly nonlinear controller can be implemented with parallel, systolic or pipelined microprocessor architectures. Such global linearization based on quaternion kinematics leads to a possible feedback singularity so that Gibbs vector kinematics is used here instead. In practice limits on available actuators could still lead to unsatisfactory or unstable response. Current investigation on the analytical side includes correction for saturation effects and oscillation prevention during saturated operating regimes. The desired controller is one that would fully utilize the available actuators while smoothly approaching the target state as well as avoiding oscillations. In the case of spacecraft maneuvered by momentum transfer devices, real time command generation is possible by pointwise minimization of the sum of the squares of the norms of the "next state error" of the equivalent system and the linear system input. This is possible by the use of a suitable discretization of the linear system. It is also possible to track a nominal trajectory, such as a critically damped harmonic oscillator response, by minimizing the square of the norm of the error between the actual and the tracked states. The required solution of a sequence of state-dependent constrained quadratic minimization problems could also be done instantaneously by currently experimental sensor signal-biased nonlinear analog circuits.

Manipulator maneuvering by feedback linearization with saturating inputs

Abstract: Recent research on fast exact maneuvering strategies for manipulators has employed acceleration commands as control variables. The forces and torques can then be synthesized, either in software or with dedicated hard-wired interfaces. Among the difficulties that occur when such maneuver techniques are employed is the fact that actuator saturation constraints are related to acceleration bounds in a state-dependent way. Recent work in the literature relies on generating a correction to the acceleration inputs, by pointwise constrained acceleration error minimization. This technique works best for infinite time horizons and highly coupled manipulator geometries, but not for terminal control or when the forces and torques enter the dynamics multiplied by a diagonal control influence matrix. An alternative technique discussed in the present paper consists of running actuators at saturation levels between sampling instants, then re-initializing an exact optimal regulator algorithm whenever the inputs drop below saturation range. Examples of such a maneuver strategy are given, both for asymptotic regulation and terminal control.

A new approach to the absolute stability of nonlinear feedback systems with slope-restricted nonlinearity

Abstract: A novel frequency-domain criterion is presented for the absolute stability of nonlinear feedback systems. The criterion requires sector and slope informations of the nonlinearity. The result is derived by artificially increasing the system order by one and applying a Lure type Lyapunov function to the resulting equivalent system.

An algorithm for computing state feedback in multi-input linear systems

Abstract: A new algorithm is proposed in this paper to compute the state feedback for arbitrary pole placement in multi-input linear systems. The algorithm has good numerical behavior. It also allows the user to adjust the eigenstructure of the resulting feedback system and therefore improve the condition of the problem and computation.

The self-oscillation determination to a category of nonlinear closed-loop systems

Abstract: A geapho-analytical method is presented for the determination of self-oscillations in nonlinear closed-loop systems with two nonlinearities without memory. The method relies on the graphical solution of the system equations, derived by harmonic linearization of the nonlinear functions that describe the two nonlinearities.

Single step optimization strategies for constrained retargeting maneuvers

Abstract: The present consideration of recent advancements in the nonlinear guidance of spacecraft, using feedback linearization and decoupling to generate exact nominal commands, gives attention to the correction of saturation effects and oscillation prevention in saturated operating regimes. In the case of spacecraft maneuvered by momentum-transfer devices, real time command generation is possible by means of pointwise minimization of the sum of the squares of the norms of the 'next state error' for the equivalent system and the linear system input. It is also possible to track a nominal trajectory, such as a critically damped harmonic oscillator response, by minimizing the square of the norm of the error between the actual and the tracked states.

Linearization of nonlinear systems about constant operating points

Abstract: For bilinear-realizable nonlinear systems described by a Volterra series, an input-output definition of the linearized system about a constant operating point is given. This definition is then compared to the usual notion of linearization in terms of a bilinear realization of the nonlinear system. Properties of the linearized system with respect to the original nonlinear system are developed, and implications for analysis and design of nonlinear systems are discussed.