Computational burden

Let nAdenote the number of inequalities in the active set and nitthe number of (outer) active-set iterations. Then the computational burden is summarized in Table 5.1. Here we subdivide the work needed for solving the unconstrained problem and the work needed for the constrained solution, i.e. the iterations of the active-set method. Table 5.1. Overview of the computational burden subdivided into the unconstrained problem

(unc) and the constrained problem (con). The operations are: a Riccati recursion, a forward vector solve (fsolve), a partial forward vector solve (partial fsolve),a backward vector solve (bsolve) and solving a reduced QP (rQP).

Riccati fsolve partial fsolve bsolve rQP

unc 1 1 0 1 0

con 0 0 nA nit nit

total 1 1 nA nit+ 1 nit

5.4. Numerical examples

5.4.1. Waste water treatment process

Consider again the waste water treatment problem presented in Section 4.4.2. The state and disturbance estimates are identical (up to numerical accuracy) to those ob- tained with the interior-point MHE method and were shown and discussed in Chap- ter 4. Hence, we will only compare the performance of both algorithms and discuss the working of the Schur complement active-set method. The computation times are shown in Figure 5.4. It can be seen that the algorithm needs only 0.2 ms if an hori- zon of 10 is used. Hence, the algorithm is about a factor 2 faster than a comparable interior-point method, see Section 4.4.2.

Figure 5.5 shows that the algorithm typically needs only 2 or 3 active-set iterations. Once the number of active constraints stabilizes, also the number of constraints in the final working set stabilizes around the same number.

5.5. Conclusions

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x 10

time step

computation time

Figure 5.4. Computation times in seconds for the waste water treatment application and MHE

with horizon 10. The Schur complement active-set MHE method (dashed line) is about a factor 2 faster than the interior point method using ten iterations (solid line).

5.5. Conclusions

In this chapter a Schur complement active-set method was presented. It uses as the starting point the unconstrained MHE solution which can be computed efficiently us- ing a square-root Riccati based algorithm. By projecting onto the reduced space of active constraints a reduced non-negativity constrained QP is obtained. For this, a gradient-projection method using projected Newton steps and Cholesky factorizations is proposed. Cholesky updates are employed in the projected Newton iterations. Once a solution to the reduced QP is found, it is used to update primal and dual variables us- ing a Schur complement technique. The method allows multiple updates to the active set per iteration. Between (outer) active set iterations the reduced Hessian changes by adding some constraints to the working set. These changes are exploited by a Cholesky downdate. The performance of the algorithm was demonstrated by applica- tion of a C implementation to some numerical examples.

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active−set iterations

Figure 5.5. Schur complement active-set method applied to the waste water treatment process

for MHE with horizon 10. Top: Number of constraints in the final working set (solid line) and number of active constraints (dashed line). Bottom: Number of (outer) active-set iterations.

CHAPTER

6

Convex MHE formulations

6.1. Robust estimation using Huber penalty function

Traditionally, state estimators are based on a least-squares penalization of residuals. For linear unconstrained systems, this leads to the celebrated Kalman filter. For con- strained and/or nonlinear systems, moving horizon estimation (MHE) [44, 59, 89, 108, 134, 146, 155, 157, 202], has emerged as an attractive alternative. In MHE, a finite horizon optimization problem is solved in every time step. Past data outside the window is summarized in a so-called arrival cost. When a new measurement becomes available, the arrival cost is updated, the window is shifted and the process is repeated. The least-squares approach, however, is not always suitable when the process is char- acterized by structural defects in the model or by imperfect measurements. In such

6.1. Robust estimation using Huber penalty function

cases, robust methods, which are less sensitivive to large errors, are desirable. In robust statistics, estimators involving explicit or recursive optimization over (robust) penalty functions are referred to as M-estimators [103]. According to Zhang [205] a robust esimator should satisfy the following specifications: (1) have a bounded influ- ence function, i.e. derivative of the penalty function, and (2) return unique estimates, which implies that the norm function should be strictly convex. Theℓ1norm is such robust measure. However, in the least absolute deviation orℓ1approach, gross errors can still have a significant impact on the estimates as they are given equal weight as small residuals. A generalization of this is the least powers method orℓpapproach, us-

ing functions_|u|pwhich are convex for p_{≥ 1 [29]. The selection of an optimal value} of p for robust estimation has been ivestigated, and for p around 1.2 good estimates may be expected [154, 205].

Unfortunately, both the least absolute deviation and the least powers approach tend to produce more zero residuals than can be statistically explained in many cases. These drawbacks have motivated research into even more robust approaches.

Hybridℓ1-ℓ2combine robust treatment of large residuals with Gaussian treatment of small residuals. The Huber penalty function, introduced in 1973 by Peter Huber [102], is one such hybridℓ1 -ℓ2norm. It has been found very practical for robust estimation by several authors in certain areas.

For example, in geophysics, Guitton and Symes [29, 84, 85] have applied it to seismic data represented by a linear regression model, i.e. a robust inverse problem

min

x kAx − bkhuber (6.1)

instead of the standard least-squares problem min

x kAx − bk2 (6.2)

The authors do not employ the QP reformulation (see Section 6.1.2), but instead di- rectly apply standard nonlinear optimization to the Huber function. Since the Huber function is not twice differentiable, the convergence of any Newton method might be jeopardized. Nevertheless, the authors propose a quasi-Newton method using limited- memory BFGS updates and report satisfying results.

In the area of power engineering, Kyriakides et al [122] have applied the Huber penalty to estimate the parameters of a synchronous generator using a linear regres- sion model with structural defects, i.e. rank deficiency, in the process matrix. They present a statistical test and conclude that the Huber method outperforms the least- squares method especially when several parameters are unknown. Jabr [104] applied the Huber norm in the context of power system state estimation with output resid- ual penalization only. The author derives the quadratic reformulation and applies a

primal-dual interior-point method for offline state estimation with equality and in- equality constraints. The method is applied it to a network model of IEEE bus test systems. A posterior analysis of the performance and ability to detect outliers of the method is performed for two fixed values of tuning parameter for the Huber function. Robust model identification using theℓ1norm and the Huber function with application to type 1 diabetes modelling has recently been presented by Finan et al [58].

Wang et al [193] present a data dependent heuristic for determining the optimal tun- ing parameter for the Huber penalty and demonstrate their method on some robust regression examples.

Estimation problems using Huber penalty function have been approximated often throughout the literature by iteratively reweighted least-squares (IRLS), which avoids explicit optimization and thereby allowed its application to large offline problems or to online problems. With the advances in numerical optimization and increasing com- puting power, it has been applied using optimization methods in more recent years, although applications in state estimation are rare and no publications of Huber-based MHE are known to the author. The aim of the work presented in this chapter is twofold: first, we show that the use of Huber penalty functions in online estimation can yield an estimator with excellent robustness with regards to outliers and can be used to identify and reject otuliers, and second, we show that Huber penalty MHE can be solved efficiently using (square-root) Riccati based methods in combination with interior-point (Chapter 4) or active-set (Chapter 5) methods allowing computa- tion times comparable to standard MHE.