Abhishek Dhanda

An Improved 2-DOF Proximate Time Optimal Servomechanism

The proximate time optimal servomechanism (PTOS) is a well established algorithm for the control of linear systems where control bounds and the speed of response are important. The PTOS approach modifies the time-optimal switching curve by including an unsaturated ldquoslabrdquo region that leads to a linear control regime for small errors. The control parameters are required to satisfy the continuity and the stability constraints resulting in only a single independent design parameter. In this note, the PTOS theory is reviewed and the stability conditions are derived. An improved version of the PTOS algorithm, called the MPTOS, is presented that results in a performance improvement over the original PTOS by providing two independent control parameters in the design. As a result, the system response can be shaped in a more efficient way in both continuous and discrete time domains. The stability conditions are derived for the proposed MPTOS scheme for continuous and discrete time implementations. Further extensions are discussed including a new discrete algorithm suitable for systems with slow sampling and a model reference tracking method also known as a feedforward/feedback structure for reducing the response of flexible modes. The latter method can be extended to dual actuator systems as are being introduced to the disk drive read/write head assembly fabrication and control. An approach is also presented for combining the vibration reduction filters with the model reference structure to further enhance the performance.

Optimal Control Formulations of Vibration Reduction Problems

Design of controls to move a flexible body such as a robot arm while minimizing unwanted vibrations has been described in many papers and presented in many forms. For the vibration reduction issue alone, it is shown that almost all the proposed designs can be formulated as optimal controls of either the fixed final time or the minimum time type. Furthermore, it is shown that under reasonable assumptions the two types have the same solution and are thus equivalent. Continuous time, tapped-delay-line input shaping filters, and discrete controls are considered. It is shown that the discrete equivalent of the general vibration reduction problem is a convex problem for the fixed final time case and quasi-convex for the free final time problem. The two formulations are compared in terms of computation complexity as well as practical implementation issues.

Direct Verification of Parametric Solution for Vibration Reduction Control Problems

For the optimal control problems formulated with limited actuator authority, the optimal solution is given by Pontryagins minimum principle that provides the necessary and sufficient conditions of optimality for systems that are normal. If the resulting solution is a switched control, an alternate computation methodology exists where the problem is formulated as a parametric optimization problem with the switching times as the variables. For vibration reduction problems with constrained endpoints, the parametric formulation is not convex and the Karush-Kuhn-Tucker conditions can only guarantee the local optimality of the solution. In this paper an approach is presented to verify the optimality of the parametric solution for optimal control problems with terminal state inequalities. The verification conditions are derived using the switching function, Karush-Kuhn-Tucker, and the transversality equations. The resulting problem is formulated as a linear program that provides a very efficient test of optimality. Example problems are given to demonstrate the application of this algorithm.

Minimum Move-Vibration Control for Flexible Structures

In the rest-to-rest motion of a flexible structure with limited torque or force, it is sometimes important to control the vibrations during the move trajectory as well as at the final time. In order to achieve this control, the duration of the move must be set to be longer than the theoretical minimum time. In this paper, three definitions of trajectory vibration performance are given and a new method for input design is described, which is based on switched control. An analytical expression is obtained for the solution for lightly damped fourth-order systems, which enables a very efficient control implementation of approximately time-optimal moves. The switched control approach is generalized for presenting simplified approximate solutions for the tracking problem and other optimal control problems with singular arcs.

Vibration control via pre-loading

Control of unwanted vibrations in mechanisms is a ubiquitous and long studied problem. A class of such problems occurs in point-to-point motion common to robotics and hard-disk seek motions. A new approach to this class of problems is described which appears to achieve a faster response time than published alternatives with little or no residual vibration.

Equivalent Representations of Vibration Reduction Control Problems

Control strategies for vibration reduction problems are discussed. The problem is represented as an optimal control problem, and a generalized vibration cost functional is given. Two formulations are presented where either the cost objective is minimized with a given settling time, or the settling time is minimized with a constraint on the required objective value. The selection criterion depends upon the designers prior knowledge of the acceptable settling time, or the allowable objective limits. The current work shows the equivalence of these two formulations under some conditions on the cost functional. The resulting optimal solution is the same for both fixed-time and the free-time control problems. Discretization of the generalized vibration reduction problem is shown to be a convex problem for the fixed-time formulation and a quasi-convex problem for the free-time formulation. The two formulations are compared in terms of the solution method complexity and practical implementation.

Optimal input shaping filters for non-zero initial states

This paper presents an approach to design optimal vibration reduction input shapers for systems with non-zero initial conditions. The problem is first formulated as an optimal control problem and the optimal solution is shown to be bang-bang. Once the structure of the optimal shaper is known, a parametric problem formulation is presented for the computation of the switching times. For digital implementation, discrete time approximate solutions are derived by solving a quasi convex Linear Program. Simulation results are shown for closed-form implementation of these filters on flexible structures. The digital solutions are experimentally verified on a portable bridge crane.

Reduction of CO Poisoning in PEM Fuel Cell by Application of Optimal Pulse Control

Hydrogen obtained from fuel reforming offers an attractive solution to the hydrogen generation and distribution problem until a hydrogen based economy can be implemented. Reformed hydrogen, however, contains trace amount of contaminants such as CO, S, and NH3. Among these, CO adsorb strongly on Pt catalyst surfaces thereby poisoning the cell by blocking the adsorption sites for H2. As a result, the hydrogen oxidation reaction, that generates protons and electrons at the anode, is prevented causing excessive power loss. CO concentrations as small as 5-10 ppm can be detrimental to the performance of a PEM fuel cell [1, 2, 3, 4]. Various methods for making fuel cells more tolerant to CO have been explored as a more practical and economical alternative to more expensive gas clean-up systems [1, 2, 4]. One possible approach is to promote electro-oxidation of CO that is very facile at high anode potentials [1] as compared to CO adsorption kinetics which is relatively slow. In other words, increasing the anode overvoltage momentarily frees the surface from adsorbed CO, which leads to higher activity of the catalyst until the surface is again blocked by the slowly adsorbing CO species. Then the voltage pulsing can again be applied to repeat the cycle. Thus, the pulsing approach can provide a clean, fast, and reliable technique of reducing CO poisoning [1, 2, 3]. In this paper the pulsing approach is further extended to improve the computation and practical implementation of the time varying control signal. In the traditional pulsing approach, the control signal is assumed to be periodic where the frequency, amplitude, and duty ratio of the pulse are treated as variables [1]. The operating values of these parameters are determined empirically based on experimental observations. However, depending upon the relative reaction kinetics and coverage dynamics, the true optimal control signal might not be a pulse. With the advancement in digital controllers, more efficient control signals can be computed and applied to improve the performance and to reduce CO poisoning in PEM fuel cells. Further, the control input is usually computed for a single MEA. A fuel cell stack consists of several MEAs connected in series and parallel arrangements. As a result, small variations in the MEA voltage gets amplified in the output stack voltage. The pulsing approach is based on fluctuations in the electrode potential where adsorbed CO is electro-oxidized periodically. Such large fluctuations are, however, not desirable for load and other electronics. A power converter is usually integrated with the fuel cell stack in order to maintain a constant output voltage [6]. A more efficient derivation of optimal control signal for a fuel cell stack can be obtained by considering the dynamics of the DC-DC power converter in the computation process. In this work, an algorithm is presented to derive more efficient optimal control signals for reducing the CO poisoning that can be implemented on fuel cell stacks. The approach is based on formulating the problem in terms of optimal control theory by deriving a suitable objective function and corresponding constraints in terms of state dynamics [7]. The states correspond to the electrochemistry occurring at the electrodes, the nonfaradaic process of charging and discharging of the double layer capacitance, and the nonlinear switching dynamics of the DC-DC boost converter. A detailed electrochemistry model is used for the hydrogen oxidation on Pt catalyst in the presence of CO as suggested in [3, 5]. The time-average fuel efficiency, which is related to the anode overvoltage, is maximized with a constraint on maintaining the time-average output power above a specified threshold. The control signal can be either a time varying current (TVC) or a time varying load (TVL). The respective control strategies are defined as time-varying current control (TVCC) and time-varying load control (TVLC). The deviation of output stack voltage from the steady state value is further constrained to lie within allowable limits. The optimal control problem is solved using variational calculus [7]. For implementation on digital controllers, approximate solutions are derived by solving an equivalent nonlinear parametric optimization problem using gradient based optimization techniques [7]. An example of such a solution is shown in Fig. 1. The resulting control satisfies all the bounds while still proving improvement in performance as evident from a decrease in the average anode overvoltage from 0.2V to 0.03V. Based on the simulation results, further theoretical analysis is presented for the proposed control.

Vibration reduction using time-optimal shaping filters with reduced higher-mode excitations

Vibration reduction in flexible systems can be achieved by shaping the input command using time-delay filters designed for time-optimality, vibration cancellation of characteristic modes, or control robustness. Fast settling time with limited control requires application of bang-bang control which often excite the higher modes of unmodeled dynamics. In this paper an approach is presented to improve the performance by better utilization of the actuator authority. Optimal control formulations are given where the mode excitations in specified bands of frequencies are limited by terminal state inequality constraints while minimizing the settling time. Simulation results are presented comparing the proposed design with other reported solutions.

An improved implementation scheme for time-optimal commands using symmetric filters

This note presents a new approach of generating time-optimal commands for reducing vibrations in flexible systems with lightly damped modes. Existing schemes for real-time implementation of time-optimal commands requires expensive computations where non-convex parametric optimization problems are solved for different reference moves. For undamped systems, the time-optimal command is known to be symmetric. Based on this, a new design of symmetric filters is proposed that can be computed offline and tabulated for quick lookup. This reduces the on-line computational burden of deriving time-optimal profiles for undamped multi-mode flexible systems. The proposed approach is compared with existing input shaping based schemes in terms of computational requirement and system performance.