Raffaele Romagnoli

Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems

Abstract A new method to achieve an accurate output tracking for nonminimum phase linear systems with nonhyperbolic and near nonhyperbolic internal dynamics is presented. For the classical methods, developed in the framework of the preview based stable inversion, the nonhyperbolicity represents a rather serious inconvenience because the required preactuation time tends to be unacceptably large. Different stable inversion techniques, developed for SISO systems, are based on a proper redefinition of the desired output trajectory with the aim of canceling the undesired effects of unstable zeros. The main purpose of the new approach is to alleviate some theoretical and practical limitations inherent in the above methods. The desired output is partitioned into the transient and steady-state components. The transient input is “a priori” assumed to be given by a spline function. Once the desired output trajectory has been specified, this allows the computation of the unknown transient input as the approximate least-squares solution of Fredholm׳s integral equation corresponding to the explicit formula of the output forced response. The steady-state input is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of the desired steady-state output. The main advantage of the resulting technique is that its generality does not require “ad hoc” procedures depending on the particular plant to be controlled. This allows the designer to freely specify the desired output trajectory without requiring it to depend on the unstable zeros of the plant or it to be null over an initial time interval.

Almost perfect tracking through mixed numerical-analytical stable pseudo-inversion of non minimum phase plants

This paper considers the problem of computing the input u(t) of an internally asymptotically stable, possibly non minimum phase, linear, continuous-time system Σ yielding a very accurate tracking of a pre-specified desired output trajectory ỹ(t). The main purpose of the new approach proposed here is to alleviate some limitations inherent the classical methods developed in the framework of the preview based stable inversion, which represents an important reference context for this class of control problems. In particular the new method allows one to deal with arbitrary and possibly uncertain initial conditions and does not require a pre-actuation. The desired output ỹs(t) to be exactly tracked in steady-state is here assumed to belong to the set of polynomials, exponential and sinusoidal time functions. The desired transient response ỹt(t) is specified to obtain a fast and smooth transition towards the steady-state trajectory ỹs(t), without under and/or overshoot in the case of a set point reset. The transient control input ut(t) is “a priori” assumed to be given by a piecewise polynomial function. Once ỹ(t) has been specified, this allows the computation of the unknown ut(t) as the approximate least-squares solution of the Fredholms integral equation corresponding to the explicit formula of the output forced response. The steady-state input us(t) is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of ỹs(t).

A spline-based technique for optimal set point regulation through pseudo-inversion of nonminimum phase linear systems

This paper considers the optimal output set-point regulation for MIMO, non minimum phase sampled data systems. The usually proposed methods are based on stable model inversion whose exact solution is approximated through preview based implementation schemes. The new approach proposed here considers the meaningful practical situation of plants with a given, possibly uncertain, initial state, that can not be modified through pre-actuation. The structure of the optimal control input is ”a priori” assumed to be given by a smoothing spline function. In this way a twofold objective is achieved: a smooth behavior of the control input and its derivatives can be imposed, a very accurate tracking performance can be obtained by reducing the mesh size of spline [1]. Given the desired transient output response between two fixed set points, the spline coefficients are determined as the least-squares solution of the over determined system of linear equations obtained imposing that the spline function assumed as control input yields the specified output. In this way an optimal least square approximation of the desired output trajectory is obtained avoiding the stable explicit model inversion. Rather, this operation is implicitly approximately performed solving for the spline coefficients, the over-determined system of linear equations carrying the information on the model to be inverted and on the desired output. An interesting feature of this new method is that it also works for linear systems which are not required to be either square or right invertible.

Output-transition optimization through a multi-objective least square procedure

The purpose of this paper is to propose a new method for the optimization of the transient step response of a linear continuous-time system. The approach situates in the framework of model pseudo-inversion [1]-[3] because the input reference is computed starting from some desired features of the transient output. A significant feature of the new method is that the transition trajectory is not “ad hoc” exactly pre-specified by the designer. Rather, it is implicitly defined by the procedure for the minimization of a suitably multi-objective quadratic cost functional.

The quadratic stabilization problem for LTV plants with arbitrary mode-switch dynamics and non uniformly bounded parametric uncertainties

This paper considers the quadratic stabilization of linear time-varying (LTV) continuous-time plants with a switch-mode dynamics and non uniformly bounded parametric uncertainties. To this purpose, the trajectories of the time varying uncertain parameters are assumed to be described by polynomial functions of arbitrary degree. This extra information on the parameters dynamics allows the parametric uncertainty to be transferred in a polytopic region of the coefficient space. At some isolated time instants the parameters trajectories can exhibit some first kind discontinuities due e.g. to sharply varying operating conditions. The coefficients of the polynomials are unknown and time varying inside known bounded intervals. Using a parameter independent Lyapunov function, a quadratically stabilizing dynamic output controller is directly obtained by the solution of some LMIs.

BMI-Based Stabilization of Linear Uncertain Plants With Polynomially Time Varying Parameters

This technical note considers the robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics. Each mode is characterized by a theoretically unbounded dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. Using a parameter dependent Lyapunov function polynomially depending on time, the stabilizing controller for each single mode is directly obtained by the solution of some BMIs, which become LMIs by fixing two positive scalars. The stability conditions of the switching closed loop system are derived defining a switched Lyapunov function and involving the permanence time interval of the switching plant over each single mode. A salient feature of the technical note is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.

Robust stabilization of linear uncertain plants with polynomially time varying parameters

Abstract The robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics is considered. Each mode is characterized by a dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. The stability conditions of the switching closed-loop system are derived defining a switched Lyapunov function and involving the dwell time of the system over each single mode. An important feature of the paper is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.

A B-spline-based pseudo-inversion approach for constrained optimal output transition

ABSTRACT The contribution of this paper is to propose a new method for the continuous-time optimal output transition problem under constraints on the control effort. The new method situates in the recently proposed framework of pseudo-inversion, which proved to be very effective to achieve an almost perfect tracking. The presented method assumes a B-spline function as the external input reference r(t) forcing a given stable closed-loop system. The actual control input u(t) yielded by r(t) and forcing the plant is optimally approximated (in the least square sense) by a B-spline . The control points of the B-spline are chosen in such a way to satisfy the saturation constraints on u(t). The parameters of interest to define the solution of the considered problem are given by the control points of the B-splines r(t) and . These parameters are simultaneously estimated as elements of the vector solving the constrained least-squares minimisation of a suitably defined multi-objective cost functional. If is a sufficiently accurate approximation of u(t), the exact fulfilment of saturation constraints by is transferred to u(t). The simulations on a practical case of a relevant interest show excellent results.

B-splines and pseudo-inversion as tools for handling saturation constraints in the optimal set-point regulation

This paper deals with the optimal output transition problem under saturation constraints on the control effort. The recently proposed pseudo inversion approach is here adopted to define a feedforward action optimizing the set-point following response of a given stable closed-loop system Σf,c. To take into account saturating actuators, the optimal external input reference r(t) forcing Σf,c is assumed to be given by a B-spline function. The actual control input u(t) yielded by r(t) and forcing the plant is optimally approximated by a B-spline û(t), whose control points û(t) are chosen in such a way to satisfy the saturation constraints on u(t). If û(t) is a sufficiently accurate approximation of u(t), the exact fulfillment of saturation constraints by û(t) are transferred to u(t). The simulations on a practical case show excellent results.

Reference-tracking feedforward control design for linear dynamical systems through signal decomposition

In this work, we study a novel approach towards the reference-tracking feedforward control design for linear dynamical systems. By utilizing the superposition property and exploiting signal decomposition together with a quadratic optimization process, we obtain a feedforward design procedure for arbitrary linear multi-input and multi-output (MIMO) systems with arbitrary time/parameter-varying characteristics. In other words, the proposed algorithm is applicable to the broad class of linear systems, i.e. linear-time-invariant (LTI), linear- time-varying (LTV) and linear-parameter-varying (LPV) systems. The interplay between the initial state, feedforward and feedback actions are elaborated in detail. The potential of the presented methodology is demonstrated through simulation examples.