Jer-Nan Juang

An eigensystem realization algorithm for modal parameter identification and model reduction

A method, called the Eigensystem Realization Algorithm (ERA), is developed for modal parameter identification and model reduction of dynamic systems from test data. A new approach is introduced in conjunction with the singular value decomposition technique to derive the basic formulation of minimum order realization which is an extended version of the Ho-Kalman algorithm. The basic formulation is then transformed into modal space for modal parameter identification. Two accuracy indicators are developed to quantitatively identify the system modes and noise modes. For illustration of the algorithm, examples are shown using simulation data and experimental data for a rectangular grid structure.

Identification of observer/Kalman filter Markov parameters - Theory and experiments

This paper discusses an algorithm to compute the Markov parameters of an observer or Kalman filter from experimental input and output data. The Markov parameters can then be used for identification of a state-space representation, with associated Kalman or observer gain, for the purpose of controller design. The algorithm is a nonrecursive matrix version of two recursive algorithms developed in previous works for different purposes, and the relationship between these other algorithms is developed. The new matrix formulation here gives insight into the existence and uniqueness of solutions of certain equations and offers bounds on the proper choice of observer order. It is shown that if one uses data containing noise and seeks the fastest possible deterministic observer, the deadbeat observer, one instead obtains the Kalman filter, which is the fastest possible observer in the stochastic environment. The results of the paper are demonstrated in numerical studies and experiments on the Bubble space telescope.

Simple learning control made practical by zero-phase filtering: applications to robotics

Iterative learning control (ILC) applies to control systems that perform the same finite-time tracking command repeatedly. It iteratively adjusts the command from one repetition to the next in order to reduce the tracking error. This creates a two-dimensional (2-D) system, with time step and repetition number as independent variables. The simplest form of ILC uses only one gain times one error in the previous repetition, and can be shown to converge to the zero-tracking error independent of the system dynamics. Hence, it appears very effective from a mathematical perspective. However, in practice, there are unacceptable learning transients. A zero-phase low-pass filter is introduced here to eliminate the worst transients. The main purpose of this paper is to supply a presentation of experiments on a commercial robot that demonstrate the effectiveness of this approach, improving the tracking accuracy of the robot performing a high speed maneuver by a factor of 100 in six repetitions. Experiments using a two-gain ILC reaches this error level in only three iterations. It is suggested that these two simple ILC laws are the equivalent for learning control of proportional and PD control in classical control system design. Thus, what was an impractical approach, becomes practical, easy to apply, and effective.

Model reduction in limited time and frequency intervals

The controllability and observability gramians in limited time and frequency intervals are studied, and used for model reduction. In balanced and modal coordinates, a near – optimal reduction procedure is used, vielding the reduction error (norm of the different between the output of the orginal system and the reduced model) almost minimal. Several examples are given to illustrate the concept of model reduction in limited time or/and frequency intervals, for continuous- and discrete-time systems, as well as stable and unstable systems. In modal coordinates, the reduced model obtained from a stable system is always stable. In balanced coordinates it is not necessarily true, and stability conditions for the balanced reduced model are presented. Finally, model reduction is applied to advanced supersonic transport and a flexible truss structure.

A slewing control experiment for flexible structures

A hardware setup has been developed to study slewing control for flexible structures including a steel beam and a solar panel. The linear optimal terminal control law is used to design active controllers that are implemented in an analog computer. The objective of this experiment is to demonstrate and verify the dynamics and optimal terminal control laws as applied to flexible structures for large-angle maneuver. Actuation is provided by an electric motor while sensing is given by strain gages and angle potentiometers. Experimental measurements are compared with analytical predictions in terms of modal parameters of the system stability matrix, and sufficient agreement is achieved to validate the theory.

Robust eigensystem assignment for flexible structures

An improved method is developed for eigenvalue and eigenvector placement of a closed-loop control system using either state or output feedback. The method basically consists of three steps. First, the singular value or QR decomposition is used to generate an orthonormal basis that spans admissible eigenvector space corresponding to each assigned eigenvalue. Second, given a unitary matrix, the eigenvector set that best approximates the given matrix in the least-square sense and still satisfies eigenvalue constraints is determined. Third, a unitary matrix is sought to minimize the error between the unitary matrix and the assignable eigenvector matrix. For use as the desired eigenvector set, two matrices, namely, the open-loop eigenvector matrix and its closest unitary matrix, are proposed. The latter matrix generally encourages both minimum conditioning and control gains. In addition, the algorithm is formulated in real arithmetic for efficient implementation. To illustrate the basic concepts, numerical examples are included.

Model reduction for flexible structures

Several conditions for a near-optimal reduction of general dynamic systems are presented focusing on the reduction in balanced and modal coordinates. It is shown that model and balanced reductions give very different results for the flexible structure with closely-spaced natural frequencies. In general, balanced reduction is found to give better results. A robust model reduction technique was developed to study the sensitivity of modeling error to variations in the damping of a structure. New concepts of grammians defined over a finite time and/or a frequency interval are proposed including computational procedures for evaluating them. Application of the model reduction technique to these grammians is considered to lead to a near-optimal reduced model which closely reproduces the full system output in the time and/or frequency interval.

Predictive feedback and feedforward control for systems with unknown disturbances

This paper presents the theory and implementation of a hybrid controller for general linear systems by incorporating a feedforward path in the feedback control. The generalized predictive control is extended to include a feedfoward path in the multi‐input multi‐output cases. There are cases in acoustic‐induced vibration where the disturbance signal is not available to be used by the hybrid controller, but a disturbance model is available. In this case the disturbance model may be used in the feedback controller to enhance performance. In practice, however, neither the disturbance signal nor the disturbance model is available. This paper presents the theory of identifying and incorporating the noise model into the feedback controller. Implementations are performed on a test plant and regulation improvements over the case where no noise model is used are demonstrated.

Improvement of Observer/Kalman Filter Identification (OKID) by Residual Whitening

This paper presents a time-domain method to identify a state space model of a linear system and its corresponding observer/Kalman filter from a given set of general input-output data. The identified filter has the properties that its residual is minimized in the least squares sense, orthogonal to the time-shifted versions of itself, and to the given input-output data sequence. The connection between the state space model and a particular auto-regressive moving average description of a linear system is made in terms of the Kalman filter and a deadbeat gain matrix. The procedure first identifies the Markov parameters of an observer system, from which a state space model of the system and the filter gain are computed. The developed procedure is shown to improve results obtained by an existing observer/Kalman filter identification method, which is based on an auto-regressive model without the moving average terms. Numerical and experimental results are presented to illustrate the proposed method.

Discrete frequency based learning control for precision motion control

Concerns MIMO learning control design with well behaved transients during the learning process. The method allows dynamic and inverse dynamic control laws. The theory gives a unifying understanding of the stability boundary for convergence to zero tracking error, and of a stability condition obtained by using frequency response arguments. The former is easy to satisfy, making learning control converge with little knowledge of the system. The much more restrictive frequency response condition is interpreted as a robustness condition, representing the robustness relative to good transient behavior during learning. This ensures that the amplitudes of the frequency components of the error signal decay in a monotonic and geometric manner with each successive repetition. Noncausal zero phase filtering is used both to facilitate the generation of learning controllers having this convergence at important frequencies, and to ensure that the learning controllers maintain this property in the presence of unmodeled dynamics. The approach is in discrete time. Experiments are performed on a 7 degree-of-freedom robot, demonstrating the effectiveness of the design process for producing precision motion control.<<ETX>>