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## A problem of robustness to uncertain metrology delay

Robustesse à des retards de métrologie incertains. Dans ce chapitre, nous présen-tons un exemple simple de système soumis à des retards de mesure variables et incertains. Ces retards n’ont pas de structure permettant de les compenser simplement dans le con-trôleur. Nous établissons un résultats de convergence robuste sous une condition de petit gain.

**Introduction**

In this chapter, we investigate the eﬀects of delay variability and uncertainty on the classic Internal Model Controller (IMC, see e.g [MZ89]) of a single-input single-output (SISO), static, nonlinear, sampled-data process with delayed measurements whose dating is uncertain. As is well-known, the uncertainty and the variability of delays lead to challenging control problems that may jeopardize closed-loop stability, see [Krs09], [HOS+16] and references therein. It is also known, see [WHQ+05], that metrology delays coupled with inaccurate process models could lead to closed-loop instability. As discussed earlier in this thesis, the general treatment of these issues is still an open problem.

The process under consideration and its controller constitute a sampled-data system (following the terminology employed in e.g. [CF95], [FSR04]) which can be reformulated using a classic discrete time representation. The specific case under consideration is actually also formally very similar to a scalar run-to-run controller, the robustness of which is not trivial. Run-to-run control is a popular and eﬃcient class of techniques, originally proposed in [SGHH91], specifically tailored for processes lacking in situ measurement for the quality of the production (see [WGD09]). Numerous examples of implementations have been reported in the semiconductor, and materials industry, in particular, see e.g. [WGD09, MdCH00] and references therein. Indeed, the field of run-to-run control encounters two of the practical problems addressed in this chapter: nonlinear model uncertainty and variable metrology delays. While these issues have often been reported (see, e.g. [WHQ+ 05], [GQ02], [FSB03], [WP15]), they have not received any definitive treatment from a theoretical viewpoint.

In the problem considered here, model uncertainty stems from the interactions between the input and the system states which can be rather complex, and, in turn, cause some non-negligible uncertainty on the quantitative eﬀects of the input. On the other hand, the measurements are available after a long time lag covering the various tasks of sample collection, receipt, preparation, analysis and transfer of data through an information technology (IT) system to the control system. Measurements are thus impacted by large delays, which can be varying to a large extent, and in some applications be state- or input-dependant. As detailed in the introduction of this thesis, this variability of the delay builds up with the intrinsic IT dating uncertainty, because, in numerous implementations, no reliable timestamp can be associated to the measurements. The delay variability cannot be easily represented by Gaussian models (e.g. additive noise on the measurement), nor can it be fully described as deterministic input or state dependant delay, nor known varying delays that could be exactly compensated for by predictor techniques (as done in e.g. [BPCP12, BPCP14, BLK13b, BLK13c, BLK13a]).

In the absence of measurement dating uncertainty, robust stability in the presence of model mismatch can be readily established, using the monotonicity of the system and model which is formulated here as an assumption. The study of measurement dating uncertainty eﬀects is more involved. Once expressed in the sampled time-scale, the control scheme exhibits a variable delay discrete-time dynamics. No straightforward eigenvalues or Nyquist criterion analysis (see [GQ02]) can be used to infer stability. Instead, a complete stability analysis in a space of suﬃciently large dimension, with a well chosen norm, yields a proof of robust stability under a small gain condition. Interestingly, this small-gain bound is reasonably sharp, so that it can serve as guideline for practical implementation. The novelty of the approach presented in this chapter lies in the proof technique. It does not treat the uncertainty of the delay using the Padé approximation approach considered in [ZCMC09], but directly uses an extended dimension of the discrete time dynamics. It is believed that these arguments of proof could be extended to address more general problems, in particular to higher dimensional forms (lifted forms) usually considered to recast general iterative learning control into run-to-run as is clearly explained in [WGD09].

The chapter has two objectives. After laying out a precise problem statement in Section 2.3, it establishes robust stability results with respect to model mismatch when measurements are delayed but exactly dated in Section 2.4.1. Then, it extends robust stability to small model errors when measurements are delayed and their dating is uncertain in Section 2.4.2. Those results are illustrated through simulations in Section 2.5.

### Plant under consideration (delay-free)

We note y the controlled variable (output) of the considered plant and u the control variable (input). It is assumed that there exists fp a strictly monotonous smooth function such that y = fp(u) (2.2)

Although fp is unknown, we can use a model of it, f, which is also smooth and monotonous1, such that fp(0) = f (0). Usually, f is a rough estimate of fp. Typical models are represented in Figure 2.1. For the simulations considered in this chapter, the model error can be as large as 20-40%, which is representative of industrial applications requirements.

The target value c for the controlled variable is assumed to be reachable by both the system and the model, i.e. there exists uc and u˜c verifying fp(uc) = c, f(˜uc) = c (2.3)

**Metrology delay**

A measurement system provides estimates of y with some time delay in a sampled manner. In many cases, this delay is time varying. Depending on the IT structure, measurements dating is usually done either using timestamping or an a priori estimation of the measurement delay. Either way, exact measurement dating is usually impractical, and some uncertainty on the measurement delay must be considered.

In the system considered in this chapter, the measurements available for feedback in a control loop thus have two specificities. They are delayed and the measurement delay 0 ≤ D itself is varying and uncertain. With 0 ≤ Dˆ the available estimation of D, we note Δ ˆ − the mismatch. Next, we formulate two modelling assumptions

Assumption 1. There exits Dmax such that D ≤ Dmax.

Assumption 2. There exits Δmax such that Δ ≤ Δmax. If Assumption 1 holds, it is clear from definition that −Dmax ≤ Δ.

#### Control problem

A closed-loop controller can be designed for system (2.2). Each time a measurement is received, the control is updated and the value of the control is kept constant until the next measurement is received, creating piece-wise constant control signals (with varying step-lengths). Repetitive application of this process generates a sequence of inputs and outputs. The delay results in shift of index in the measurement sequence.

Formally, the control design should aim at solving the following problem

Remark 2.5. In terms of controller design specifications, we can draw the following statements from our analysis :

• the sign of the estimated gain must be correct and its value cannot be too small compared with the reality. On the other hand, taking an estimated gain larger than the true one will slow the controller down but cannot jeopardize its stability

• variable delays cause no specific problem of convergence if we assume that exact measurement dating is available

• if there is dating uncertainty of the measurement, stability can still be retained, provided that the measurements filtering is strong enough (α small enough)

**Numerical illustration**

In this section, we consider a setting where each sample is analysed during a certain lapse of time during which no other sample is taken. New control actions are only implemented when a new measurement result is received. In that way, the time-sampled measurement delay is always zero (Dn = 0, Dmax = 0), i.e. the measurement we receive is always informative of the result of the last control action taken. This is a special case of Theorem 2.2 which is of practical importance in the implementation of many controllers. In classic run-to-run cases, nonzero Dmax could be considered, without loss of generality.

We will assume that the actual physical time required for the measurement to reach the controller, Tp, depends on the measured value according to the following relation Tp(y) = 8.5 − 0.75ymes + 10(ξ1 + ξ2) (2.36)

where ξ1,2 are the realizations of two independent random variables taking the values 0 or 1 with respective probabilities of 12 . In some simulations, we will assume that timestamps are not available and that the times at which the measurements are taken have to be estimated using an approximate model T of Tp. This estimation may be inexact, thus leading to dating uncertainty of the measurements, i.e. Δmax ≥ 0. In all simulations, the target will be c = −10.

#### Influence of pure dating uncertainty

In this subsection, the response of the system, fp, is assumed to be a simple linear function of the control u. We further assume that it is perfectly known, so that fp(u) = f(u) = −25u (2.37)

Furthermore, we consider a situation where no timestamp is available. As a consequence, Tisp must be estimated and we assume that the approximate model available to the controller T (y) = 17 − 1.5ymes (2.38)

This mismatch results into nonzero values of Δ which are plotted to provide a graphical view of its statistical distribution. We also corrupt the measurements with a zero-mean, uniform noise of small amplitude to excite the system.

We simulate the response of the system for various values of the gain α. The results show that while the closed-loop system remains stable for α small enough, as α increases, destabilization can arise in absence of any model error on the function fp, simply because of measurement dating uncertainty. This illustrates the fact that Theorem 2.2 does not hold after a critical value of α, which is conservatively estimated by conditions (2.34)

**Table of contents :**

**1 Introduction to varying delays and related control problems **

**2 A problem of robustness to uncertain metrology delay**

2.1 Introduction

2.2 Notations

2.3 Problem statement and proposed solution

2.4 Convergence analysis

2.5 Numerical illustration

2.6 Conclusions

**3 Application of direct simultaneous strategies for dynamic optimization of systems subject to hydraulic delays **

3.1 Introduction

3.2 Direct simultaneous methods for systems subject to delays

3.3 Example of application with hydraulic delay

3.4 Numerical implementation and results

3.5 Conclusions

**4 Calculus of variations for varying delays **

4.1 Introduction

4.2 Notations

4.3 Fixed time-varying delays

4.4 Input-dependant delays: a non-smooth problem

4.5 Conclusions

**5 Iterative resolution algorithm for the regularization of an input-dependant hydraulic input delay **

5.1 Introduction

5.2 Notations

5.3 Optimization algorithm

5.4 Numerical example

5.5 Conclusions

**6 Examples **

6.1 Introduction

6.2 Desirable features for a practical implementation of the algorithm

6.3 Relaxation to a non-regularized input for input-dependant input delays

6.4 Extension to input-dependant state delays

6.5 Conclusions

**7 Conclusion **

**Bibliography**

Appendices