Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems
Introduction
Inherent limitations on transient performance make it difficult to obtain an accurate tracking for nonminimum phase systems. The theory of model stable inversion [5], [9], [15] provides a valid framework to deal with the above problem and allows exact tracking through an infinite preactuation interval. The practical unfeasibility of this solution motivated the use of approximated preview-based techniques with preactuation (see e.g. [7], [14], [15], [17], [20], [24], [25], [26]). Nevertheless, the preactuation interval tends to increase for systems with nonminimum phase zeros near the imaginary axis (near nonhyperbolic systems) and tends to be infinite for purely imaginary zeros (nonhyperbolic systems). See [6] for more details. The solution proposed in [6] for linear systems realizes a compromise between classical stable inversion and approximation based inversion. To reduce the preactuation time, the internal dynamics is modified with the purpose of removing nonhyperbolicity. The tradeoff between exact tracking and reduction of the preactuation time is illustrated by an example of helicopter hover control. Different approaches have been proposed in [1], [22] with reference to the SISO linear model of one-link flexible manipulator. In these papers the output trajectory is planned in such a way to cancel the undesired effects of unstable zeros. The extension of this approach to MIMO systems seems to be rather involved.
The purpose of this paper is to alleviate some theoretical and practical limitations inherent in the above methods.
Reference is here made to a linear nonhyperbolic (or near nonhyperbolic) system Σf given by the unitary feedback connection of a nonhyperbolic (or near nonhyperbolic) plant with a proper stabilizing controller which also provides the internal model of the desired steady-state output trajectory . This is useful in view of possible parametric uncertainties on the plant and makes it straightforward for the calculation of the steady-state input us(t).
The desired is here assumed to belong to the set of polynomials, exponentials and sinusoidal time functions. This class of functions is generally enough to embrace almost all trajectories of a practical interest in control applications. The desired transient response is here freely pre-specified by the designer without any constraint imposed by the particular plant. The desired is only defined on the basis of the common desired requirements, like a fast and smooth transition towards the steady-state trajectory , without under and/or overshoot in the case of a set point reset.
The present method computes the transient (ut(t)) and steady state (us(t)) components of the whole input u(t) following the lines of the approach recently proposed in [11], [12]. A significant difference is that, to enhance the smoothness of the transient input, ut(t) is here “a priori” assumed to be given by a spline function [2]. As shown in Section 3, this gives rise to a rather involved hybrid least square estimation problem where both continuous and discrete time observations have to be considered.
As splines of a fixed order m are continuously differentiable up to order , the above assumption guarantees a sufficient smoothness of the transient input signal, which is also required to smoothly converge towards the steady state behavior us(t). In this way the undesired ripples on the controlled output are strongly attenuated. This feature is particularly useful for mechanical systems, whose components should not be affected by an excessive stress. Once the desired has been specified, the explicit formula of the forced output response is given by Fredholm׳s integral equation of the first kind which has to be solved with respect to ut(t). As an exact solution is practically impossible, the transient input ut(t) is numerically computed imposing a transient tracking error with minimum L2 norm. Namely ut(t) is obtained as the least square solution of Fredholm׳s integral equation over a sufficiently long time interval in order to guarantee the practical attainment of steady-state. Once the desired steady-state output trajectory has also been specified, the corresponding steady-state input us(t) is directly obtained by exploiting the classical formulas on steady-state response.
The main merit of the new approach is its generality. It is not based on “ad hoc” procedures which need to be modified according to the particular dynamics of the plant to be controlled. As no pre-actuation is needed, the desired can be freely assigned without requiring it to be null over an initial time interval, or that it depends on the unstable zeros. Additional theoretical and practical advantages of the proposed approach are: (1) it allows one to deal with the case of any given, arbitrary initial state of Σf because the unforced output response originating from is automatically compensated by the forced transient output response yielded by ut(t); (2) nonexactly known initial conditions can also be considered; (3) the method can be directly extended to deal with the more general tracking problem of a nonperiodic switching desired output with only a limited preview information [11], [12].
To show the effectiveness of the proposed approach, the method has been tested on two significant examples. The first one is the same helicopter hover control problem with near nonhyperbolic internal dynamics considered in [6]. In the light of the above point (3), the second example deals with a nonperiodic set-point reset, which is a control problem of a particular importance for large-scale flexible manipulators [17]. Reference is here made to the same nonhyperbolic robot model considered in [1], [22].
The paper is organized in the following way. The problem statement is given in Section 2. The new approach is explained in 3 Transient solution, 4 Steady-state solution. The extension to the tracking of a nonperiodically switching desired output is explained in Section 5. Two examples of application to the helicopter hovering and to one flexible link manipulator is reported in Section 6. The concluding remarks of Section 7 end the paper. Appendix reports some preliminaries on spline functions.
Section snippets
Problem statement
Let Σf be the continuous time nonhyperbolic (or near nonhyperbolic) system given by the feedback connection of a LTI nonhyperbolic (or near nonhyperbolic) plant with a proper LTI stabilizing controller. The controller is also required to provide the internal model of the desired steady-state output trajectory. Without any loss of generality the feedback connection is assumed to be unitary, namely the gain matrix of the feedback connection is the identity matrix.
Let the triplet be a
Transient solution
The transient problem is solved through a least-square procedure providing an optimal estimate of all the spline coefficients giving ut(t), . The purpose of next calculations is to show how to formalize this procedure in the general form of (7), (8), (9).
The observation vector of the least-square problem is given by , , where , and is the pseudo-observation vector relative to all the continuity constraints which must be satisfied by the
Steady-state solution
In steady-state condition, Eq. (6) assumes the formFor the considered class of , it is possible to find an input function us(t) exactly giving by simply exploiting classical results on the steady-state response of LTI systems. As the system under consideration Σf is given by the unitary feedback connection of a nonhyperbolic plant with a stabilizing controller which also provides the internal model of , it directly follows that the
Nonperiodic switching desired output
The presented approach can be easily extended to deal with a more general tracking problem where, according to the particular application, the desired output trajectory is nonperiodically changed at some switching instants . A similar problem has been recently investigated in [23] with reference to hyperbolic systems.
Over each interval with , the method described in 3 Transient solution, 4 Steady-state solution can be applied provided that (i) is long enough to
Numerical results
Two numerical examples are proposed to prove the effectiveness of the new method with respect to the existing literature. The first one considers a nonminimum phase near nonhyperbolic MIMO system. The comparison has been made with the classical stable inversion techniques based on the preactuation, where it is mandatory to modify the internal dynamics for removing nonhyperbolicity. A simulation starting from nonnull initial conditions is also reported to prove the generality of the proposed
Conclusions
A new approach to achieve a very accurate output tracking for nonminimum phase linear systems with nonhyperbolic and near nonhyperbolic internal dynamics has been presented. Reference has been made to the very general AEOTP formulated in Section 2. The main merits of the presented method are in terms of generality of the underlying theoretical framework. No preactuation is needed and the desired output trajectory can be freely assigned independently of the unstable zeros of the particular plant
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