Figure 6-14: Zoom to the Figure6-12.
6.4 Appendix: proof of Lemma 2
For the sake of completeness, the proof of Lemma2is presented as in [Rao,2000].
Proof. Let x0denote the initial condition of the system (6-15) generating the output sequence{yk}, i.e., yk= CAkx0andε> 0. Using the state equation, for l ≤ N
Let ˆyk|t−1:= C ˆxk|t−1. By using the inverse triangle inequality, the following inequality is obtained
N−1
The above inequality can be rearranged to produce
Let define the observability matrix as O =h The following bound can be derived
N−1 k=0
∑
kCAjx0−CAjxˆ0k2≥ q
λmin(OTO)k(x0− ˆx0)k
The observability assumption guarantees that every eigenvalue of OTO is greater than zero, i.e., λmin(OTO) > 0 for N ≥ n. Hence, Substituting (6-21) and (6-20) into (6-19)
k ˆxt− Atx0k ≤ϕkAkN
Using the state equation, the following inequality is obtained for j≤ N:
kCAjxˆt−N|t−1− ˆyt−N+ jkt−1k ≤ kCk
j−1 l=0
∑
kAkj−1−lkGkk ˆwt−N+l|t − 1k (6-23) Substituting (6-23) into (6-22) the following bound on the estimation error is derived:
6.4 Appendix: proof of Lemma2 91
Without loss of generality, it is assumed that A is not vacuous to generate the expressions of the two inequalities above. Considerη = min
λmin(R−1),λmin(Q−1)
and chooseρ such that
ζ ≤η and the Lemma follows as claimed.
In this dissertation, a moving horizon approximation of the constrained estimation problem for uncertain, linear systems is investigated. Interesting elements were carefully set together to present an estimation strategy robust to unknown disturbances which address additional information in form of constraints in a moving horizon setting.
From the literature review, it was evidenced the lack of robust estimation strategies addressing system constraints. Although considerably work has been made over robust model predictive controllers, the results about robust moving horizon estimators are not as popular as it control counterpart. Moreover, available MHE-based schemes for uncertain, linear systems are mainly addressed by one research group. Their contributions are developed assuming that the uncertainty is mainly due to unknown parameters but the disturbing noises entering to the system. Therefore, there are no place for comparisons between their contributions with respect to ours.
The core of the contribution was presented in Chapters5and6. First in Chapter5both the H∞-FIE and the H∞-MHE were officially presented. For the sake of completeness, the H∞-FIE was pre-sented as a direct solution to the robust state estimation problem. The game-theoretical approach of the H∞filtering was used to provide robustness to the optimization-based estimator against un-known inputs. A different cost function was proposed in order to guarantee saddle-point properties of the solutions of the posed optimization problem. The main differences with the classic FIE were stated:
Classic FIE: the problem is to solve
φt∗= min
x0,{wk}φt where
φt = kx0− ¯x0k2P−1 0
+
t−1 k=0
∑
kwkk2Q−1+ kνkk2R−1
P0the initial condition of the Riccati equation associated with the estimation error covariance in the Kalman sense.
H∞-FIE: the problem is to solve
ψ¯t∗= min
x0
max{wk}ψ¯t
93
Π0the initial condition of the Riccati equation (5-44b).
To solve the computational infeasibility of the H∞-FIE, a moving horizon approximation was proposed, namely the H∞-MHE. To that end, a new arrival cost for the robust estimation problem was set by using a Riccati recursion of the estimation error weighting matrix obtained by means of the calculus of variations. It was shown by means of numerical examples the improvement of the estimates by the use of such a new technique when compared with estimation schemes for similar purposes. Again comparing with respect to the classic MHE:
Classic MHE: the problem is to solve
φˆt∗= min
Ptthe solution at time step t of the Riccati equation associated with the estimation error covariance in the Kalman sense. Πt the solution at time step t of the Riccati equation (5-44b).
In Chapter6, the constrained H∞-MHE was defined. A clever approximation was stated in order to solve at each sample time the associated minimax problem. Two quadratic programs must be solved instead of a complex minimax optimization scheme which reduces significantly the computational burden associated with the latter. It was shown by means of two numerical examples the feasibility of the approximation. Moreover the theoretical contribution was complemented with the stability proof of both the H∞-FIE and the H∞-MHE schemes by using a modified Lyapunov theory presented in previous contributions.
7.1 Future Work
There are still many unsolved issues around the main subject of the thesis. A brief list of some of them are described as follows:
• As it was stated in Chapter3, there also may exist uncertainty in the parameters of the model.
The question is, how well the proposed scheme behave against this type of uncertainty. If the proposed scheme is not able to overcome this type of uncertainty, what modifications are needed to account for?.
• An open issue even for classic schemes based on receding horizon schemes is about tuning.
There is not clear methodologies to perform an efficient tuning of this type of estimation strategies.
• Different less-conservative tools for guaranteeing robustness must be studied. As an exam-ple, the LMI tools allow a rather direct formulation. It is widely known that numerical tools for solving standard LMI’s are available.
• Nonlinear formulations of robust estimation schemes using the moving horizon philosophy seems to be more appealing that these based in linear models even with the well-known difficulties associated to nonlinear systems. The extension of the results presented in this dissertation into a nonlinear framework would be extremely useful mainly for systems with nonlinear and complex dynamics. Chemical, biochemical, and petrochemical processes are clearly the examples to account for.
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