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While it might appear that by using previously visited RTO points the gradients can be estimated ‘for free’, that is, without any additional experimental burden, in reality the steps taken by the RTO algorithm must be severely constrained to ensure good gradient estimates. ‘Dual MA’ algorithms attempt to guarantee accurate gradient estimates at

every iteration, by including constraints on the quality of the gradient estimates in the modified model-based optimization problem (Marchetti et al., 2010; Rodger and Chachuat, 2011; Marchetti, 2013)³. These Dual MA algorithms have two objectives:

a) optimize the process, b) ensure accurate gradient estimates. Unfortunately, as we will see, these are usually conflicting objectives. A new ‘dual’ constraint g_d,k−1(u) ≤ 0 is added to the model-based problem (2.3.4) in the MA algorithm. The aim of this constraint is to guarantee the accuracy of the gradient estimates. Two different dual constraint are discussed next, each one specific to the gradient-estimation method being used. In both cases, two distinct types of error are distinguished: the truncation error and the noise error. The truncation error occurs due to the curvature of the plant cost function, while all the gradient estimation equations in the preceding section assume it is locally linear. The noise error is due to the high-frequency noise affecting the measurements.

The first formulation was devised by Marchetti et al. (2010), and is specifically tailored to the case where the gradient is estimated using Equation (2.3.12). The authors proved that, at iteration k, the gradient estimation error is bounded as follows:

k∇φE,k− ∇φp(uk)k ≤ ε^t_k(u) + εⁿ_k(u), (2.3.14)

whereε^t_k(u) is the truncation error bound, andεⁿ_k(u) is the noise error bound.

The truncation error bound is given by:

ε^t_k(u) =σmax

2

°

°[(u − uk)^T(u − uk), . . . , (u − u_k−nu+1)^T(u − u_k−nu+1)]U⁻¹_k (u)°

°, (2.3.15) whereσmaxis an upper bound on the spectral radius of the Hessian of the plant cost function. Roughly speaking,ε^t_k(u) increases along with the maximum of the distances between all pairs of points in the set {u, uk, u_k−1, . . . , u_k−nu+1}. Hence, to keep the trun-cation error small, the past nu+ 1 points must be sufficiently close to each other. The noise error bound is given by:

εⁿ_k(u) = δ

l_min(u), (2.3.16)

where lmin(u) is the shortest distance between all possible pairs of complement affine subspaces that can be generated from S = [u,uk, u_k−1, . . . , u_k−nu+1] (Marchetti et al., 2010), andδ is the maximum noise value that can occur. Thus, in this case, it is assumed that the noise affecting the cost measurement in Equation (2.3.2) is interval bounded.

3This is in analogy to the concept of ‘dual control’ in the field of adaptive control, whereby there is a dichotomy between more excitation for better identification (exploration) and less excitation for better control (exploitation).

Roughly speaking, lmin(u) is the minimum of the orthogonal distances between each individual point in the set {u, uk, u_k−1, . . . , u_k−nu+1} and the hyperplane passing through the remaining points. Hence, in order to keep the noise error small, the past nu−1 input moves should be approximately orthogonal to each other and no two points should be too close to each other. The additional (scalar) ‘dual’ constraint that is added to the modified model-based optimization problem is:

gd,k(u) = ε^t_k(u) + εⁿ_k(u) − εmax, (2.3.17) which, as can be seen from Equation (2.3.14), should ensure that the maximum gradient error does not surpass the valueεmax. Note that this constraint can be non-convex and, for that reason, Marchetti et al. (2010) also proposes a simple convex relaxation. Also, note that while only the error affecting the cost gradient estimate was discussed here, this constraint also ensures the gradient estimation error remains bounded.

A second formulation of the dual constraint, proposed by Rodger and Chachuat (2011), is specifically tailored to the case when the gradient is estimated using Equation (2.3.13).

In this approach, two additional constraints are required. A first constraint aims to reduce the truncation error:

g_k^t(u) = 1 − (u − uk)^TΓ^TΓ(u − uk), (2.3.18) whereΓ is typically a diagonal matrix. This simple constraint ensures that each new point is sufficiently distant from the preceding operating point. A second constraints aims to reduce the noise error:

g_kⁿ(u) = minn³

whereαkis any non-zero vector orthogonal to the previous nu− 1 input moves, and Σ defines an ellipsoid around ukoutside which the next operating point must lie. Thus, reducing the noise error requires each input move to be both sufficiently large, and approximately orthogonal to the past nu−1 input moves. In this case, the dual constraint is a combination of the truncation-limiting constraint and the noise-limiting constraint:

g_d,k(u) =h

g_k^t(u) g_kⁿ(u)iT

. (2.3.20)

The feasible regions corresponding to both types of dual constraint for the nu= 2 case are shown in Figure 2.1, for typical values of the tuning parameters in these constraints.

Both constraints are similar: due to the trade off between truncation error and noise error, the next input move must neither be too large, nor too small. Also, the next input

Figure 2.1: Feasible regions corresponding to the dual constraint: a) given by the convex relaxation of Equation (2.3.17), b) given by Equation (2.3.20). u_k+1is constrained to the shaded region.

move must be approximately orthogonal to the past nu− 1 input moves. If past operat-ing points are used for gradient computation, then there is no doubt these constraints are necessary to ensure the gradient estimate does not become very inaccurate. How-ever, the dual constraints severely handicap the MA algorithm; by conflicting with the optimization objective, they negatively impact convergence towards the plant optimum.

In order to optimize the process, the MA algorithm should move in the one direction that most improves performance without violating constraints. However, the dual con-straints forbid moving in just one direction, instead requiring the exact opposite: that the MA algorithm should explore all directions of the input space. The larger nu, the more directions must be explored in order to ensure accurate gradient estimates, and hence the more the dual constraints interfere with the optimization objective. Thus, the larger the number of input variables, the slower the plant optimum is reached.