Input-output Inversion Research Articles

On the equivalence of inversion-based control architectures

Inversion-based control architectures are feedforward-feedback schemes in which the feedforward control is determined by stable input-output inversion. The main architectures of this type are the plant and the closed-loop inversion architectures. For scalar, linear, nonminimum-phase systems these architectures are shown to be equivalent for any disturbance and any plant perturbation for which the feedback controller ensures closed-loop stability. This equivalence — deduced by using a behavioral approach — also holds in practice when a careful truncation of the preaction control occurs. New output-error-based rules to set the preaction and postaction times of a stable inverse input are introduced. Still holding the equivalence, these rules help in choosing the preferable architecture at the implementation stage. An example with simulation comparisons highlights the paper findings.

Simplified input-output inversion control of a double pendulum overhead crane for residual oscillations reduction

In this paper we present the application of an input-output inversion technique for the open-loop control of an overhead crane modelled as a double pendulum. The method is mathematically derived, obtaining a parametric trajectory that ensures reduced residual oscillations. Then, it is shown that the postactuation can be neglected so that the method can be implemented with standard industrial drives. The robustness of the method is evaluated by means of simulations, and the performance of the method is experimentally compared with the well-known input shaping technique. The advantages of using a double pendulum model instead of a simple pendulum one are also shown.

A New Prediction Scheme for Input Delay Compensation in Restricted-Feedback Linearizable Systems

The input-output inversion of a system under the effect of input delays typically relies on the ability to predict the future of the system's state. Indeed, if the latter is known ahead of time, one can cope with the input delay by using a prediction of the state instead of the state itself. Such methods are efficient when the plant is stable but become numerically unstable otherwise. We present a new method to compensate input delays; our approach relies on imposing a desired error dynamics which is designed to be linear and asymptotically stable at the origin. Then, the state prediction is computed from the state reference trajectory and the predicted error dynamics. In this technical note, we concentrate on the case study of systems in strict feedback form and present a simple backstepping procedure.

Set-point filter design for a two-degree-of-freedom fractional control system

This paper focuses on a new approach to design U+0028 possibly fractional U+0029 set-point filters for fractional control systems. After designing a smooth and monotonic desired output signal, the necessary command signal is obtained via fractional input-output inversion. Then, a set-point filter is determined based on the synthesized command signal. The filter is computed by minimizing the 2-norm of the difference between the command signal and the filter step response. The proposed methodology allows the designer to synthesize both integer and fractional set-point filters. The pros and cons of both solutions are discussed in details. This approach is suitable for the design of two degree-of-freedom controllers capable to make the set-point tracking performance almost independent from the feedback part of the controller. Simulation results show the effectiveness of the proposed methodology.

Inversion-based propofol dosing for intravenous induction of hypnosis

In this paper we propose an inversion-based methodology for the computation of a feedforward action for the propofol intravenous administration during the induction of hypnosis in general anesthesia. In particular, the typical initial bolus is substituted with a command signal that is obtained by predefining a desired output and by applying an input–output inversion procedure. The robustness of the method has been tested by considering a set of patients with different model parameters, which is representative of a large population.

Inversion-based feedforward and reference signal design for fractional constrained control systems

In this paper we propose a solution for the synthesis of a feedforward control action for a fractional control system. In particular, an input–output inversion based methodology is devised in order to determine the open-loop signal that provides a predefined process variable transition from a steady-state value to another. The transition time is then minimized subject to constraints on the process and control variables and their derivatives. Simulation results show the effectiveness of the technique.

A Methodology for Integrated System Identification, PID Controller Tuning and Noncausal Feedforward Control Design

In this paper we propose a systematic methodology that integrates the three main phases of the design of an industrial control system, namely, the identification phase, the tuning of the (PID) controller and the design of a (noncausal) feedforward action. In particular, the tuning of the controller is based on frequency loop shaping where the modelling uncertainty is explicitly considered and the noncausal feedforward command input synthesis is performed by applying a stable input-output inversion procedure. In this context, a parameter that allows to handle the trade-off between aggressiveness and robustness (and control effort) is given to the user. Simulation and experimental results show the effectiveness of the methodology.

Iterative Feedforward Tuning for Residual Vibration Reduction

An iterative approach for the determination of an input-output inversion feedforward control law for residual vibration reduction is proposed in this paper. In particular, point-to-point motion planning of vibratory servosystems is considered. The method aims at estimating recursively the parameters of the system in order to determine the exact command input to be applied to the control system in order to achieve a predefined motion without oscillations. In this context, a gradient based minimisation of the integrated square error cost function is performed. Simulation results show the effectiveness of the methodology.

Non-linear control of a series direct current motor via flatness and decomposition in the bond graph domain

A bond graph (BG) based methodology for non-linear control system synthesis is presented through its application to a speed-tracking problem stated on a series direct current motor. After a global flatness analysis of the motor BG model, a two-loop cascade control structure is decided and developed on the basis of a physical system decomposition in electrical, mechanical, and coupling submodels. Each loop of the cascade tracks a reference for a flat output that is local to a subsystem of the decomposition. Bond graph techniques are given for the three main components of the design methodology: system decomposition, flatness analysis, and tracking controller design. Theoretical and practical properties of the resulting control system are discussed, and its performance is demonstrated through simulation experiments. The methodology is applicable to the broader class of non-linear BG models where input-output system inversion is well defined.

PARETO OPTIMAL FEEDFORWARD CONSTRAINED REGULATION OF MIMO LINEAR SYSTEMS

In this paper we present a methodology for the feedforward minimum-time regulation of Multiple-Input-Multiple-Output (MIMO) square linear systems. The aim is to synthesize bounded smooth input functions, subject to non-saturating constraints on the inputs and their derivatives until an arbitrary predefined order, in order to provide given transitions of the outputs. The posed problem yields to the construction of a Pareto set of output transition times, which is determined by employing an ad hoc parametrized family of output transition functions and a suitably devised input-output inversion procedure. Illustrative examples are given in order to show the effectiveness of the methodology.

STABLE DYNAMIC INVERSION OF NONMINIMUM-PHASE SCALAR LINEAR SYSTEMS

The paper presents a transfer function approach to the stable dynamic inversion of nonminimum-phase scalar linear systems. The technique is based on the study of the structure of the causal unstable input obtained with the standard inversion technique. It is shown that the unbounded term of this input can be decomposed as the sum of a linear combination of unstable zero modes for which new formulae are provided. Then, a closed-form expression of the bounded noncausal solution of the input-output inversion problem is proposed. An automatic inversion scheme built by exploiting the new inversion formula is also presented.

Using stable input–output inversion for minimum-time feedforward constrained regulation of scalar systems

The paper approaches the feedforward minimum-time smooth control of non-minimum-phase linear scalar systems for set-point regulation. The aim is to synthesize a bounded input subject to non-saturating constraints on the input and its derivatives till a prespecified arbitrary order. The provided solution relies on a stable input–output dynamic inversion technique and an ad hoc parameterized family of output transition functions. The synthesized minimum-time input that exhibits pre- and post-actuation is determined by means of easily-implementable closed-form expressions. Examples are given to illustrate the methodology.

Une interprétation algébrique de la transformation de laplace et des matrices de transfert

The tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where, even in the time-varying case, a (formal) Laplace transform and the transfer matrix are most naturally defined. Several classic problems are examined in this algebraic setting: the relationship between left (right) coprime matrix decomposition and controllability (observability), the state-variable canonical realization, the transfer algebra with respect to parallel and series connections, the input-output inversion, and model matching.

An Algebraic Definition of Time-Varying Transfer Matrices

Abstract Tensoring the module of a linear system with the quotient field of the ring of linear differential operators yields an algebraic framework where transfer matrices most naturally appear and which easily applies to the time-varying case. Several basic structural properties are examined : transfer algebra, matrix fraction decompositions and their connections with controllability and observability, realization, input-output inversion, model-matching.

Trajectory control of a non-linear one-link flexible arm

Abstract The trajectory-tracking control problem is considered for a one-link flexible arm described by a non-linear model. Two meaningful system outputs are chosen; namely, the joint angle and the angular position of a suitable point along the link. The common goal is to stiffen the behaviour of the flexible link with respect to the chosen output. Based on the input-output inversion algorithm, a state-feedback control law is designed that enables exact tracking of any smooth trajectory specified for the output. In the closed loop an unobservable dynamics naturally arises, related to the variables describing the arm's distributed flexibility. Joint-based design is shown to be always stable, whereas in the link-point design the closed-loop dynamics may become unstable depending on the location of the output along the link. Open- versus closed-loop strategies are developed and compared. Extensive simulation results are included.

Nonlinear Control Theory and Differential Algebra: Some Illustrative Examples

Differential algebraic techniques, which were recently introduced for solving the input-output inversion problem, are here employed to give a precise definition of dynamic feedbacks. Several feedback synthesis problems are solved: input-output decoupling, perturbation rejection, and exact linearization, for which the weakest possible conditions are given.