Elsevier

Systems & Control Letters

Volume 85, November 2015, Pages 109-117
Systems & Control Letters

Model recovery anti-windup for output saturated SISO linear closed loops

https://doi.org/10.1016/j.sysconle.2015.09.005Get rights and content

Abstract

We define model recovery anti-windup for SISO linear control systems with output saturation. We address the problem by relying on a hybrid modification of the linear closed loop which employs a suitable logic variable to activate/deactivate various components of a control scheme. The scheme relies on a finite-time observation law, an open-loop observer and an open-loop input generator which is capable of driving the plant output within the saturation limits. Then the control scheme is based on suitable (hybrid) resetting laws allowing the controller to operate on the artificial output signal generated by the open-loop observer when the actual plant output is outside the saturation limits. Unlike existing results, not only we prove uniform global asymptotic stability of the closed loop, but we also prove the local preservation and global recovery properties, typical of model recovery anti-windup paradigms. We also illustrate the proposed technique on an example study.

Introduction

In practical applications the input and/or output signals generated by the controller or by the plant itself are subject to a saturation bound. Input saturation, in particular, has received increasing attention in the past few decades and is often tackled by employing the so-called anti-windup techniques  [1], whose main objective consists in the design of auxiliary compensators to be combined with standard controllers designed by disregarding input saturation. Anti-windup for input-saturated linear systems can be divided into two big families (see  [2]): one called Direct Linear Anti-Windup (DLAW) which mainly focuses on proving asymptotic stability of the origin (possibly global or local with guaranteed regions of attraction), the other one called Model Recovery Anti-Windup (MRAW) focusing on recovering the so-called “unconstrained response” in the presence of non-zero inputs, namely the response that one would have experienced in the absence of saturation. Both approaches guarantee the so-called local preservation property, that is the unconstrained linear response is left unchanged by the compensation scheme as long as it never exceeds the saturation limits.

The parallel problem of sensor or output saturation has not been addressed as extensively as the input saturation problem, even though it is an important problem in several applications (see, e.g.,  [3] where it is related to Wiener-type nonlinear models or the edited book  [4]). Despite its reduced popularity, some relevant works can be found in the literature and are very clearly surveyed in the excellent introduction of [5]. While apparently similar, the two anti-windup problems are radically different as output saturation poses a nontrivial observability problem, which is nonexistent in the input saturation case. Conversely certain global stabilizability issues appearing in the input saturation case (see, e.g.,  [6]) are absent in the output saturation problem, where global stabilization of the origin can be achieved in SISO  [7] and MIMO [8] output saturated linear plants. Finally, the same problem is approached in  [9] for a, possibly unstable, planar system in the presence of quantized and saturated input and output. Anti-windup design for output saturated systems is somewhat more challenging than mere stabilization because one has also to take into account the above mentioned local preservation constraint induced by a pre-existing controller. So far, all existing anti-windup methods only focus on asymptotic stability and follow the DLAW paradigm. In particular, the schemes in  [10] provide global results for exponentially stable MIMO plants (see  [10, Remark 4]), while the work in  [11] addresses local results for any type of MIMO plant. Perhaps the most powerful existing approach is that in [5] (see also [12] where a similar construction is given without semiglobal results), where semiglobal results are obtained for any MIMO controllable and detectable plant (see [5, Prop. 2]). However, it is admitted in  [5, Section IV.B] that large domains of attraction lead to unacceptably large controller gains.

In this paper we adopt a different paradigm closer in spirit to the MRAW solution to input saturated anti-windup design. First of all, as compared to the works above, we establish global asymptotic stability results of our scheme without needing increasingly large gains. More importantly, beyond stability, we establish a suitable unconstrained response recovery property that has never been previously considered in this field. This property enables us to track references that cause the plant output to spend most of the time outside the saturation limits, as long as they come back within the limits once in a while (this is made more precise in Section  2). The price to be paid for this important extra feature is the complexity of the scheme, which is presented for SISO output saturated systems. Indeed, for effective unconstrained response recovery, suitable switching logics need to activate when the plant output exceeds the saturation limits in such a way that the controller is well behaved in spite of output saturation. We describe these switching logics using the hybrid systems formalism in  [13]. We give formal proofs of the main results and we propose a modified scheme for which we can prove Uniform Global Asymptotic Stability (UGAS) of a compact attractor by relying on a proposition pertaining general hybrid systems, which may be of separate interest and is stated in our Appendix.

The rest of the paper is organized as follows. Section  2 formalizes the design goal, together with some basic notation and preliminary results. The proposed control architecture is described in Section  3 by detailing its building blocks and their properties. The statement and the proof of the main result are provided in Section  4. The effectiveness of the control scheme is validated by means of a numerical example in Section  5, before drawing some conclusions and future outlook in Section  6. Finally, Appendix contains a stability result for hybrid dynamical systems necessary for the proof of the main theorem and also of independent interest.

Section snippets

Problem statement

Consider a single-input single output linear plant P described by equations of the form ẋ=Ax+Bu+Bdd,y=Cx, where xRnp denotes the plant state, uR denotes its control input, yR denotes its measurement output and dRp is a vector disturbance. In the paper, we denote by αR the maximum real part among all the eigenvalues of matrix A. Assume that the following controller C has been designed for plant (1)ẋc=Acxc+Bcuc+Brr,yc=Ccxc+Dcuc+Drr, where xcRnc is the controller state, rRq denotes a

Proposed architecture

In this section we propose an anti-windup architecture corresponding to the scheme represented in Fig. 1, while the statement of the main result is postponed to Section  4. To suitably describe its dynamical behavior, we split the scheme in the following three main components, which are described in details in Sections  3.1, 3.2 Finite-time/open-loop observer, 3.3 Recovery filter, respectively:

  • 1.

    a switching logic, described by a hybrid system and a logic variable q{0,1}, which governs the

Anti-windup closed-loop system and its properties

Before stating our main result, we summarize next the overall closed loop arising from the interconnection of the plant–controller pair to the proposed anti-windup compensation scheme. With reference to the block diagram of Fig. 1, the output saturation anti-windup architecture consists of the following state-space components:

  • 1.

    (xp,xc)Rnp×nc arising from the continuous-time plant (1) and the continuous-time controller (2), represented by blocks P and C of Fig. 1, respectively, in addition to

Numerical example

We consider a simple example corresponding to the following input/output relation for plant (1), y(s)=1s(s+2)(u(s)+d(s)), which resembles the response of a relative degree one minimum phase plant. For example plant (28) may represent the approximated transfer function from the requested current to the obtained position for a DC electrical motor where the electrical time constant has been disregarded because the mechanical one is dominant. We assume that a controller (2) has been designed for

Conclusions

We proposed a novel anti-windup scheme for linear control systems over SISO plants with output magnitude saturation. The scheme, which has a hybrid formulation combines a switching logic detecting entry and exit of the plant output from suitably defined subsets of the unsaturated region, a finite-time/open-loop observer and a recovery filter driving the plant output back into the unsaturated region. For the first time we specified the anti-windup problem for output saturated plants in terms of

References (15)

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This work was partially supported by ANR project LimICoS contract number 12 BS03 005 01, by the iCODE institute, research project of the Idex Paris-Saclay, and by grant OptHySYS funded by the University of Trento.

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