Mathematical Problems in Engineering

Volume 2015, Article ID 319761, 13 pages

http://dx.doi.org/10.1155/2015/319761

## A Novel Dissipativity-Based Control for Inexact Nonlinearity Cancellation Problems

^{1}Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Firenze, Via S. Marta 3, 50139 Firenze, Italy^{2}School of Engineering, University of Liverpool, Liverpool L69 3GH, UK

Received 1 October 2014; Accepted 8 January 2015

Academic Editor: Victor Sreeram

Copyright © 2015 Giacomo Innocenti and Paolo Paoletti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

When dealing with linear systems feedback interconnected with memoryless nonlinearities, a natural control strategy is making the overall dynamics linear at first and then designing a linear controller for the remaining linear dynamics. By canceling the original nonlinearity via a first feedback loop, global linearization can be achieved. However, when the controller is not capable of exactly canceling the nonlinearity, such control strategy may provide unsatisfactory performance or even induce instability. Here, the interplay between accuracy of nonlinearity approximation, quality of state estimation, and robustness of linear controller is investigated and explicit conditions for stability are derived. An alternative controller design based on such conditions is proposed and its effectiveness is compared with standard methods on a benchmark system.

#### 1. Introduction and Motivations

One of the most prolific areas of interest in the nonlinear control theory deals with the existence of* coordinate changes* and* nonlinear inputs* which are capable of making the complete system linear, so that the mathematical tools of the linear control framework can be successfully exploited. The idea of suppressing the process nonlinearities by means of a properly designed controller dates back to the early stages of the control theory [1], and it can be well illustrated by considering a system in the* Lur’e form* [2, ch. 7], which features a linear dynamic part feedback interconnected through a static (or memoryless) nonlinear operator:
Such a system, indeed, can be* globally feedback linearized* by exploiting the control input
that transforms the original nonlinear model into the new linear system
which can be controlled through using standard linear control techniques [3]. Observe that (3) is qualitatively different from the direct linearization of (1) around the desired nominal solution [2, ch. 4] as it has global validity. Despite this noteworthy result, the nonlinear input (2) can only be used under two very strict conditions which often prevent its application to real cases: (i) the nonlinearity must be* exactly known a priori* and (ii) the controller must be capable of* exactly reproducing* it.

The extension to generic systems, for which only an output is measurable, is known as* output feedback linearization* and it exploits a linearizing control input of the form to make the relationship between the output and the new input linear. In his pioneering work, Krener was the first providing sufficient conditions for the exact cancellation of the nonlinearity [4], while the complete solution of the output feedback problem was found later by Isidori [5]. The* state feedback linearization*, instead, aims to design a suitable change of coordinates such that the transformed system turns out to be locally diffeomorphic to a linear system. Necessary and sufficient conditions for the single input case were already presented in the work of Krener [4], while the multiple inputs problem has been independently solved by Su [6] and Hunt et al. [7]. Moreover, a large number of techniques have been developed to* partially feedback linearize* the system when it does not satisfy the previous scenarios (see, e.g., [8, 9] and the references therein).

Despite the rigorous results available in the literature, all the feedback linearization methods suffer from the lack of robustness with respect to* inexact nonlinearity cancellations* when applied to real world cases. Indeed, the inexact cancellations give rise to unmodeled dynamics, which usually have detrimental effects on the controlled system performance, and which can even lead to instability (see, e.g., [10–12] and the references therein). Unfortunately, such an issue is quite common when dealing with real problems, where the exact cancellation of the nonlinearity may not be possible for several complications. The detrimental effect of inexact nonlinearity estimation has been highlighted in autopilot design [13], aeroelastic systems [14, 15] and DC motor control [16], just to cite a few examples. With reference to system (1)-(2) the following negative scenarios can occur.(1)The nonlinear operator cannot be completely and exactly known a priori. For instance, only a limited set of sample points are available, or just its lower and upper bound can be experimentally investigated.(2)The system state cannot be directly observed and must be reconstructed. The use of a state observer negatively affects the cancellation, since the state estimates may deviate from the true values during transients.(3)The control input can be designed by using only a finite set of basis functions, preventing a perfect reproduction of the nonlinear operator . These issues can be well illustrated by considering a Lur’e system of the form
where is the state, the control input, the measured output, and the actual argument of the nonlinear operator that does not necessarily coincide with . Here, for the sake of the simplicity, let us assume that , although the general case is straightforward by considering multidimensional signals for and the signals defined in Section 2.1. Finally, without any loss of generality, suppose that so that the desired equilibrium sits on the origin of the state space.

Let us assume that the pair is observable. The standard theory then recommends the use of a state observer
to recover the missing information on and [17]. Hence, if is also controllable, the control input can be designed as
where the function is the approximation of the nonlinearity . Here the controller might not be able to exactly reproduce the actual nonlinear operator via , potentially causing an inexact nonlinearity cancellation. However, is still supposed to satisfy at least the condition , in order to preserve the equilibrium in the origin. By defining the* cancellation residual*
and the* state estimation error*
the controlled system in the traditional framework can be written as
In Figure 1 the corresponding block diagram is reported. It is worth noticing that when the cancellation residual is the null signal, that is, , the* separation principle* holds and therefore the linear dynamics of the controlled system can be arbitrarily set by separately choosing and [18, ch. 16]. Unfortunately, such condition is hardly met in any real world scenario, since it requires to achieve both exact nonlinearity cancellation and perfect state estimation. Hence, in the most common situations the cancellation residual plays the role of a disturbance that simultaneously affects both the original system state and its estimation, thus introducing a mutual interplay between the controller and the observer that prevents one to independently design and . The stability of the system (7)–(9) under the interference of the cancellation residual can be investigated through standard approaches by observing that it is in (extended) Lur’e form; that is, it consists of a linear subsystem feedback interconnected with nonlinearity (7). For the sake of completeness a short review of traditional methods is presented hereafter, highlighting the pros and cons of each solution. (i)*Stability through the Analysis of the Linearized Dynamics around the Equilibrium [2, ch. 4].* This category groups all of the methods which aim to set up the poles of the system linearized around the equilibrium. Therefore, they assume that the local behaviour of at the fixed point, that is, the value of its derivative there, is known. On one hand, when and have already been fixed the computation of the equilibrium eigenvalues can be efficiently carried out via numerical techniques. On the other hand, the complex mutual dependence between the controller and the observer prevents deriving explicit formulas for designing the gains and so that all the eigenvalues at the equilibrium have negative real parts. This problem is usually overcome by enforcing a sufficient degree of separation between the observer and the controller, bringing on a very high degree of conservatism in the solution [17].(ii)*Closed Loop Methods Based on the Lur’e Problem Formulation [2, ch. 7].* These techniques analyze the equilibrium stability by exploiting the closed loop form that features the feedback interconnection between a linear subsystem and a static nonlinearity satisfying a sector condition. The originating method of this family is the circle criterion. Since these techniques explicitly take into account the nonlinearity , that is, the cancellation residual, their results are partially based on the knowledge of the interplay between controller and observer. Unfortunately, this advantage vanishes by the very complicated dependence of the linear subsystem properties from and that prevents their explicit computation.(iii)*Closed Loop Methods Based on the Input-to-Output Properties of the Loop Branches [2, ch. 6].* This family counts approaches inspired by the passivity theorem for feedback systems. The basic idea is based in representing the system as a loop of subsystems, whose input-to-output properties prevent the existence of self-sustained nonzero signals. Suitable ad hoc mathematical manipulations allow one to decouple the model so to enclose and in different subsystems, but this causes the presence of nonlinear dynamic subsystems, whose input-to-output features are quite formidable to be explicitly designed using and . As highlighted above, designing a stabilizing controller with guaranteed performance in presence of state estimation and inexact nonlinearity cancellation is a challenging problem. In the rest of the paper a simple effective strategy based on the dissipativity theory [19, ch. 9] will be presented along with a comparative example with the traditional approaches.