Brad Paden

Nonlinear inversion-based output tracking

An inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory. In the case of minimum phase systems, the trajectory produced agrees with that generated by Hirschorns inverse dynamic system; however, the preimage trajectory is noncausal (rather than unstable) in the nonminimum phase case. In addition, the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case. With the addition of stabilizing feedback to the preimage trajectory, asymptotically exact output tracking is achieved. Tracking is demonstrated with a numerical example and compared to the well-known Byrnes-Isidori regulator. Rather than solving a partial differential equation to construct a regulator, the inverse is calculated using a Picard-like interaction. When preactuation is not possible, noncausal inverse trajectories can be truncated resulting in the tracking-error transients found in other control schemes.

Globally asymptotically stable ‘PD+’ controller for robot manipulators

We describe a globally stable tracking controller for robot manipulators. The controller is an extension of Takegaki and Arimotos position controller to the tracking case where a theorem of Matrosov is used to prove its stability. An attractive feature of this controller is its resemblance to the computed torque controller with the inertia matrix outside the position and velocity feedback loops. Thus, our controller is decomposed into an inner PD loop and an outer dynamic compensation loop. This structure allows the simple PD computations to be run at a higher speed than the dynamic compensation loop in digital implementations.

Stability of learning control with disturbances and uncertain initial conditions

The authors investigate the effects of state disturbances, output noise, and errors in initial conditions on a class of learning control algorithms. They present a simple learning algorithm and exhibit, via a concise proof, bounds on the asymptotic trajectory errors for the learned input and the corresponding state and output trajectories. Furthermore, these bounds are continuous functions of the bounds on the initial condition errors, state disturbances, and output noise, and the bounds are zero in the absence of these disturbances. >

Stable inversion of nonlinear non-minimum phase systems

Output tracking control of non-minimum phase systems is a highly challenging problem encountered in the control of flexible manipulators, space structures, and elsewhere. Classical inversion provides exact output tracking but leads to internal instability, while recent nonlinear regulation provides stable asymptotic tracking but admits large transient errors. As a first step to solve this problem, this paper addresses the stable inversion of non-minimum phase nonlinear systems. Using the notions of zero dynamics and stable/unstable manifolds, an invertibility condition is established for a class of systems. A stable but non-causal inverse is obtained offline that can be incorporated into a stabilizing controller for dead-beat output tracking. This inverse contrasts with the causal inverse proposed by Hirschorn where unstable zero dynamics result in unbounded inverse solutions. Our results reduce to those of Hirschorn for minimum phase systems, however. In a numerical example, the stable inverse has achiev...

Adaptive identification and control for manipulators without using joint accelerations

We present a new scheme for the adaptive control of mechanical manipulators along with proof of convergence. This work is an extension of our earlier work [Craig, Hsu and Sastry] [1]. The new scheme does not require the measurement of joint accelerations and needs less computation. We illustrate the theory with some simulations.

Nonlinear repetitive control

Repetitive controllers are generally applied to reject periodic disturbances and to track periodic reference signals with a known period. Their design is based on the internal model principle, proposed by Francis and Wonham (1975). This paper describes a new finite-dimensional SISO repetitive controller for two different classes of nonlinear plants. Simulation results show asymptotic tracking of the periodic reference signal by the proposed repetitive controller in closed loop up to the Nth harmonic frequency. A proof of robustness of the repetitive control system to small nonlinearities, like actuator nonlinearities, is provided.

Stable inversion for nonlinear nonminimum-phase time-varying systems

We extend stable inversion to nonlinear time-varying systems and study computational issues-the technique is applicable to minimum-phase as well as nonminimum-phase systems. The inversion technique is new, even in the linear time-varying case, and relies on partitioning (the dichotomic split of) the linearized system dynamics into time-varying, stable, and unstable, submanifolds. This dichotomic split is used to build time-varying filters which are, in turn, the basis of a contraction used to find a bounded inverse input-state trajectory. Finding the inverse input-state trajectory allows the development of exact-output tracking controllers. The method is local to the time-varying trajectory and requires that the internal dynamics vary slowly; however, the method represents a significant advance relative to presently available tracking controllers. Present techniques are restricted to time-invariant nonlinear systems and, in the general case, track only asymptotically.

Adaptive linearization of hybrid step motors: stability analysis

An adaptive linearization scheme for torque-ripple cancellation is presented, and the stability and robustness are established. By taking a new approach in parameterizing the motor dynamics, the number of adapted parameters is reduced by a factor of two relative to the standard approach. This parameterization and the unique periodic property of the motor enable the authors to find conditions on exogenous signals which guarantee persistency of excitation. The authors develop a robustness result which, roughly speaking, shows that the allowable model perturbation does not decrease in size as the adaptation rate is slowed. This is accomplished with a unique dual-Lyapunov-function technique. The kinds of perturbation considered include nonlinear dependence on state and parameter error. This nonlinear adaptive control scheme has been successfully implemented. Experimental results demonstrate over 30 db reduction in torque ripple. >

Exponentially Stable Tracking Control for Multijoint Flexible-Link Manipulators

In this paper we describe a new tracking control law for multijoint flexible-link manipulators. The scheme is a synthesis of the inverse dynamics solution for flexible manipulators developed by Bayo at UCSB and tracking control theory for rigid-link manipulators put forth by Bayard, Wen and others. We show that passive joint controllers, together with the feedforward of nominal joint torques corresponding to a desired end-effector trajectory, results in exponentially stable tracking control

Nonlinear adaptive torque-ripple cancellation for step motors

The modeling of torque-ripple in hybrid step motors and its cancellation using adaptive linearization control are discussed. Although the nonlinear adaptive control of this problem can fit into a general framework, a representation of the torque-ripple which reduces the number of adapted parameters per torque-ripple harmonic by half is used. By doing so, it is possible to prove conditions on exogenous signals to guarantee the persistency of excitation of the regressor, and hence the exponential stability of the unperturbed system. It is shown that the adaptive system is robust to a class of state- and parameter-dependent modeling errors and disturbances even when the adaptation gain and convergence rate of the unperturbed system become small. The adapted parameter errors are proved to converge to a neighborhood of zero whose radius can be made small by slow adaptation. The proposed control scheme is verified in an experiment in which a 32-dB reduction in torque-ripple component at the rotor pole frequency is observed.<<ETX>>