Optimization-based design and implementation of multidimensional zero-phase IIR filters

This paper considers multidimensional infinite-impulse response (IIR) filters that are iteratively implemented. The focus is on zero-phase filters with symmetric polynomials in the numerator and denominator of the multivariable transfer function. A rigorous optimization-based design of the filter is considered. Transfer function magnitude specifications, convergence speed requirements for the iterative implementation, and spatial decay of the filter impulse response (which defines the boundary condition influence in the spatial domain of the filtered signal) are all formulated as optimization constraints. When the denominator of the zero-phase IIR filter is strictly positive, these frequency domain specifications can be cast as a linear program and then efficiently solved. The method is illustrated with two two-dimensional IIR filter design examples.     

Design of Low-Bandwidth Spatially Distributed Feedback

The paper considers a family of linear time-invariant and spatially invariant (LTSI) systems that are both distributed and localized. The spatial responses of the distributed plant are localized in spatial neighborhoods of each location. The feedback computations are also distributed and the information flow is localized in a spatial neighborhood of each location. The feedback is aimed at controlling spatial distributions of variables in the systems with a relatively low bandwidth in the time direction. Such systems have many important applications including industrial processes, imaging systems, signal and image processing, and others. We describe a new method for designing (tuning) a certain family of low-bandwidth controllers for such plants. We consider LTSI controllers with a fixed structure, which is a PID or a similar low-bandwidth feedback in time and local in spatial coordinates. Two spatial feedback filters, symmetric and with finite spatial response, modify the local PID control signal by mixing in the error and control signals at nearby nodes. These two filters provide loopshaping and regularization of the spatial feedback loop. Like an ordinary PID controller, this controller structure is simple, but provides adequate performance in many practical settings. We cast a variety of specifications on the steady-state spatial response of the controller and its time response as a set of linear inequalities on the design variables, and so can carry out the design of the spatial filters using linear programming. The method handles steady-state limits on actuator signals, error signals, and several constraints related to robustness to plant and controller variation. The method allows handling the effects of boundary conditions and guaranteed closed-loop spatial or time decay. It does appear to work very well for low-bandwidth controllers, and so is applicable in a variety of practical situations.      

Feedback control of one-link flexible manipulator with gear train

This article reports experimental results in control of a one-link flexible manipulator. A d.c. drive with gear train rotates the link with a tip-mass in the horizontal plane. A “stiff” hardware servo-system tracks the commanded drive velocity. We describe the experimental setup and develop a dynamic model for the one-mode flexible vibration control. We design and experimentally test controllers for active damping of the vibrations and stabilization of the link-tip position. Results for continuous control implemented with an analog computer and a sampled-data, digital-computer control are reported. A high control performance is achieved despite the gear-train friction influence. In addition, sampled-data digital controllers for tracking control of the link reference motion are designed and tested.     

Two-dimensional loop shaping

This article considers the design of practical feedback controllers for a class of spatially distributed processes where the state of a uniform physical substrate is influenced by an array of evenly distributed, identically constructed actuators and measured by an array of evenly distributed, identical sensors. A constructive procedure is introduced for designing spatially distributed feedback controllers with the objective of practical implementation. The full controller design cycle is addressed; including specification, synthesis, and implementation. The proposed design procedure has been field-tested and forms the basis of a software tool that has recently been implemented in a commercial product for the design of industrial paper machine controllers.     

On the persistency of excitation in radial basis function networkidentification of nonlinear systems

Considers radial basis function (RBF) network approximation of a multivariate nonlinear mapping as a linear parametric regression problem. Linear recursive identification algorithms applied to this problem are known to converge, provided the regressor vector sequence has the persistency of excitation (PE) property. The main contribution of this paper is formulation and proof of PE conditions on the input variables. In the RBF network identification, the regressor vector is a nonlinear function of these input variables. According to the formulated condition, the inputs provide PE, if they belong to domains around the network node centers. For a two-input network with Gaussian RBF that have typical width and are centered on a regular mesh, these domains cover about 25% of the input domain volume. The authors further generalize the proposed solution of the standard RBF network identification problem and study affine RBF network identification that is important for affine nonlinear system control. For the affine RBF network, the author formulates and proves a PE condition on both the system state parameters and control inputs.     

System analysis of power transients in advanced WDM networks

This paper considers dynamical transient effects in the physical layer of an optical circuit-switched wavelength-division-multiplexed network. These transients of the average transmission power have millisecond time scales. Instead of studying detailed nonlinear dynamics of the network elements, such as optical line amplifiers, a linearized model of the dynamics around a given steady state is considered. System-level analysis in this paper uses modern control theory methods and handles nonlinearity as uncertainty. The analysis translates requirements on the network performance into the requirements to the network elements. These requirements involve a few gross measures of performance for network elements and do not depend on the circuit switching state. One such performance measure is the worst amplification gain for all harmonic disturbances of the average transmission power. Another is cross-coupling of the wavelength channel power variations. The derived requirements guarantee system-level performance for all network configurations and can be used for specifying optical components and subsystems.     

An approach to parametric nonlinear least square optimization andapplication to task-level learning control

This paper considers a parametric nonlinear least square (NLS) optimization problem. Unlike a classical NLS problem statement, we assume that a nonlinear optimized system depends on two arguments: an input vector and a parameter vector. The input vector can be modified to optimize the system, while the parameter vector changes from one optimization iteration to another and is not controlled. The optimization process goal is to find a dependence of the optimal input vector on the parameter vector, where the optimal input vector minimizes a quadratic performance index. The paper proposes an extension of the Levenberg-Marquardt algorithm for a numerical solution of the formulated problem. The proposed algorithm approximates the nonlinear system in a vicinity of the optimum by expanding it into a series of parameter vector functions, affine in the input vector. In particular, a radial basis function network expansion is considered. The convergence proof for the algorithm is presented. The proposed approach is applied to task-level learning control of a two-link flexible arm. Each evaluation of the system in the optimization process means completing a controlled motion of the arm     

Semi-intrusive multivariable model invalidation

This paper introduces a mechanism for testing multivariable models employed by model-based controllers. Although external excitation is not necessary, the data collection includes a stage where the controller is switched to open-loop operation (manual mode). The main idea is to measure a certain “distance” between the closed-loop and the open-loop signals, and then trigger a flag if this “distance” is larger than a threshold level. Moreover, a provision is made for accommodating model uncertainty. Since no hard bounds are assumed with respect to the noise amplitude, the model invalidation mechanism works in a probabilistic framework.     

Structured uncertainty analysis of robust stability for multidimensional array systems

This paper considers one of the fundamental issues in design and analysis of sampled multidimensional systems - that of uncertainty modeling and robust stability analysis. Methods of structured uncertainty analysis (/spl mu/-analysis) are extended toward systems with dynamical and noncausal spatial coordinates. The stability is understood in a broad sense and includes decay (localization) of system response along the noncausal spatial coordinates. Robustness of dynamical stability and spatial localization of response and boundary effects are addressed in a unified way. The main technical condition enabling the technical results of the paper is that the feedback loop including a multidimensional plant and controller does not have a feedthrough in the dynamical (time) coordinate sense. As an example, this paper applies the multidimensional structured uncertainty analysis to closed-loop control of a cross-directional paper machine process. The paper formulates multidimensional models of the process, its controller, and a structured uncertainty. The uncertainty corresponds to a combination of errors in the actuator mapping, the cross-directional response gain, and the response width.     

Performance-optimized applied identification of separabledistributed-parameter processes

Studies practical algorithms for parametric identification of cross-directional processes from input/output data. Instead of working directly with the original two-dimensional array of the high-resolution profile scans, the proposed algorithms use separation properties of the problem. It is demonstrated that by estimating and identifying in turn cross directional and time responses of the process, it is possible to obtain unbiased least-square error estimates of the model parameters. At each step, a single data sequence is used for identification which ensures high computational performance of the proposed algorithm. A theoretical proof of algorithm convergence is presented. The discussed algorithms are implemented in an industrial identification tool and the note includes a real-life example using paper machine data