Polynomial interpolation for inversion-based control

To help to achieve high performances in the regulation of linear scalar (SISO) nonminimum-phase systems, an inversion-based (feedforward) control method is proposed. The aim is designing an inverse input to smoothly switch from a current, arbitrary, steady-state regime to a new, future, desired steady-state output. A new-found polynomial basis solves the related interpolation problem to join the current output to the future one while ensuring the necessary or desired smoothness. The (interpolation) transition time can be minimized in order to optimally reduce the delay with which the desired output occurs. By applying a behavioral stable inversion formula to the overall smoothed output, detailed expressions of the inverse input are finally derived. A simulation of a flexible arm rotating in the horizontal plane exemplifies the presented method

Feedforward control approach to precision trajectory design and tracking : Theory and application to nano-mechanical property mapping using Scanning Probe Microscope

The output tracking problem has been extensively studied. The linear system case has been addressed by B. A. Francis. (1976) by converting the tracking problem to a regulator problem. Such an approach was later extended to nonlinear systems by A. Isidori. et al. (1990). On the feedforward control side, the stable inversion theory solved the challenging output tracking problem and achieved exact tracking of a given desired output trajectory for nonminimum phase systems (linear and nonlinear). The obtained solution is noncausal and requires the entire desired trajectory to be known a priori. This noncausality constraint has been alleviated through the development of the preview-based inversion approach, which showed the precision tracking can be achieved with a finite preview of the future desired trajectory, and the effect of the limited future trajectory information on output tracking can be quantified. Moreover, optimal scan trajectory design and control method provided a systematic approach to the optimal output-trajectory-design problem, where the output trajectory is repetitive and composed of pre-specified trajectory and unspecified trajectory for transition that returns from ending point to starting point in a given time duration. This dissertation focuses on the development of novel inversion-based feedforward control technique, with applications to output tracking problem with tracking and transition switchings, possibly non-repetitive. The motivate application examples come from atomic force microscope (AFM) imaging and material property measurements. The raster scanning process of AFM and optimal scan trajectory design and control method inspired the repetitive output trajectory tracking problem and attempt to solve in frequency domain. For the output tracking problem, especially for the AFM, there are several issues that have to be addressed. At first, the shape of the desired trajectory must be designed and optimized. Optimal output-trajectory-design problem provided a systematic approach to design the desired trajectory by minimizing the total input energy. However, the drawback is that the desired trajectory becomes very oscillatory when the system dynamics such as the dynamics of the piezoelectric actuator in AFM is lightly damped. Output oscillations need to be small in scanning operations of the AFM. In this dissertation, this problem is addressed through the pre-filter design in the optimal scan trajectory design and tracking framework, so that the trade off between the input energy and the output energy in the optimization is achieved. Secondly, the dissertation addressed the adverse effect of modeling error on the performance of feedforward control. For example, modeling errors can be caused in process of curve fitting. The contribution of this dissertation is the development of novel inversion based feedforward control techniques. Based on the inversion-based iterative learning control (S. Tien. et al. (2005)) technique, the dissertation developed enhanced inversion-based iterative control and the model-less inversion-based iterative control. The convergence of the iterative control law is discussed, and the frequency range of the convergence as well as the effect of the disturbance/noise to signal ratio is quantified. The proposed approach is illustrated by implementing them to high-speed force-distance curve measurements by using atomic force microscope (AFM). Then the control approach is extended to high-speed force-volume mapping. In high-speed force-volume mapping, the proposed approach utilizes the concept of signal decoupling-superimposition and the recently-developed model-less inversion-based iterative control (MIIC) technique. Experiment of force volume mapping on a Polydimethylsiloxane (PDMS) sample is presented to illustrate the proposed approach. The experimental results show that the mapping speed can be increased by over 20 times

A Robust Offline Precomputed Optimal Feedforward Control Action for the Real Time Feedback/Feedforward Control of Double Pendulum Gantry Cranes

none1openvalentina orsiniOrsini, Valentin

Inverse modelling and inverse simulation for system engineering and control applications

Following extensive development over the past two decades, techniques of inverse simulation have led to a range of successful applications, mainly in the fields of helicopter flight mechanics, aircraft handling qualities and associated issues in terms of model validation. However, the available methods still have some well-known limitations. The traditional methods based on the Newton-Raphson algorithm suffer from numerical problems such as high-frequency oscillations and can have limitations in their applicability due to problems of input-output redundancy. The existing approaches may also show a phenomenon which has been termed “constraint oscillations” which leads to low-frequency oscillatory behaviour in the inverse solutions. Moreover, the need for derivative information may limit their applicability for situations involving manoeuvre discontinuities, model discontinuities or input constraints. Two new methods are developed to overcome these issues. The first one, based on sensitivity-analysis theory, allows the Jacobian matrix to be calculated by solving a sensitivity equation and also overcomes problems of input-output redundancy. In addition, it can improve the accuracy of results compared with conventional methods and can deal with the problem of high-frequency oscillations to some extent. The second one, based on a constrained Nelder-Mead search-based optimisation algorithm, is completely derivative-free algorithm for inverse simulation. This approach eliminates problems which make traditional inverse simulation techniques difficult to apply in control applications involving discontinuous issues such as actuator amplitude or rate limits. This thesis also offers new insight into the relationship between mathematically based techniques of model inversion and the inverse simulation approach. The similarities and shortcomings of both these methodologies are explored. The findings point to the possibility that inverse simulation can be used successfully within the control system design process for feedforward controllers for model-based output-tracking control system structures. This avoids the more complicated and relatively tedious techniques of model inversion which have been used in the past for feedforward controller design. The methods of inverse simulation presented in this thesis have been applied to a number of problems which are concerned mainly with helicopter and ship control problems and include cases involving systems having nonminimum-phase characteristics. The analysis of results for these practical applications shows that the approaches developed and presented in this thesis are of practical importance. It is believed that these developments form a useful step in moving inverse simulation methods from the status of an academic research topic to a practical and robust set of tools for engineering system design

Synthesis and Analysis of Design Methods in Linear Repetitive, Iterative Learning and Model Predictive Control

Repetitive Control (RC) seeks to converge to zero tracking error of a feedback control system performing periodic command as time progresses, or to cancel the influence of a periodic disturbance as time progresses, by observing the error in the previous period. Iterative Learning Control (ILC) is similar, it aims to converge to zero tracking error of system repeatedly performing the same task, and also adjusting the command to the feedback controller each repetition based on the error in the previous repetition. Compared to the conventional feedback control design methods, RC and ILC improve the performance over repetitions, and both aiming at zero tracking error in the real world instead of in a mathematical model. Linear Model Predictive Control (LMPC) normally does not aim for zero tracking error following a desired trajectory, but aims to minimize a quadratic cost function to the prediction horizon, and then apply the first control action. Then repeat the process each time step. The usual quadratic cost is a trade-off function between tracking accuracy and control effort and hence is not asking for zero error. It is also not specialized to periodic command or periodic disturbance as RC is, but does require that one knows the future desired command up to the prediction horizon. The objective of this dissertation is to present various design schemes of improving the tracking performance in a control system based on ILC, RC and LMPC. The dissertation contains four major chapters. The first chapter studies the optimization of the design parameters, in particular as related to measurement noise, and the need of a cutoff filter when dealing with actuator limitations, robustness to model error. The results aim to guide the user in tuning the design parameters available when creating a repetitive control system. In the second chapter, we investigate how ILC laws can be converted for use in RC to improve performance. And robustification by adding control penalty in cost function is compared to use a frequency cutoff filter. The third chapter develops a method to create desired trajectories with a zero tracking interval without involving an unstable inverse solution. An easily implementable feedback version is created to optimize the same cost every time step from the current measured position. An ILC algorithm is also created to iteratively learn to give local zero error in the real world while using an imperfect model. This approach also gives a method to apply ILC to endpoint problem without specifying an arbitrary trajectory to follow to reach the endpoint. This creates a method for ILC to apply to such problems without asking for accurate tracking of a somewhat arbitrary trajectory to accomplish learning to reach the desired endpoint. The last chapter outlines a set of uses for a stable inverse in control applications, including Linear Model Predictive Control (LMPC), and LMPC applied to Repetitive Control (RC-LMPC), and a generalized form of a one-step ahead control. An important characteristic is that this approach has the property of converging to zero tracking error in a small number of time steps, which is finite time convergence instead of asymptotic convergence as time tends to infinity

Energy-Optimal Control of Over-Actuated Systems - with Application to a Hybrid Feed Drive

Over-actuated (or input-redundant) systems are characterized by the use of more actuators than the degrees of freedom to be controlled. They are widely used in modern mechanical systems to satisfy various control requirements, such as precision, motion range, fault tolerance, and energy efficiency. This thesis is particularly motivated by an over-actuated hybrid feed drive (HFD) which combines two complementary actuators with the aim to reduce energy consumption without sacrificing positioning accuracy in precision manufacturing. This work addresses the control challenges in achieving energy optimality without sacrificing control performance in so-called weakly input-redundant systems, which characterize the HFD and most other over-actuated systems used in practice. Using calculus of variations, an optimal control ratio/subspace is derived to specify the optimal relationship among the redundant actuators irrespective of external disturbances, leading to a new technique termed optimal control subspace-based (OCS) control allocation. It is shown that the optimal control ratio/subspace is non-causal; accordingly, a causal approximation is proposed and employed in energy-efficient structured controller design for the HFD. Moreover, the concept of control proxy is proposed as an accurate causal measurement of the deviation from the optimal control ratio/subspace. The proxy enables control allocation for weakly redundant systems to be converted into regulation problems, which can be tackled using standard controller design methodologies. Compared to an existing allocation technique, proxy-based control allocation is shown to dynamically allocate control efforts optimally without sacrificing control performance. The relationship between the proposed OCS control allocation and the traditional linear quadratic control approach is discussed for weakly input redundant systems. The two approaches are shown to be equivalent given perfect knowledge of disturbances; however, the OCS control allocation approach is shown to be more desirable for practical applications like the HFD, where disturbances are typically unknown. The OCS control allocation approach is validated in simulations and machining experiments on the HFD; significant reductions in control energy without sacrificing positioning accuracy are achieved.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146104/1/molong_1.pd