Multi-parametric Moving Horizon Estimation

Moving horizon estimation is an estimation method based on optimization that considers a limited amount of past data. One of the main advantages of moving horizon estimation is the possibility to incorporate system knowledge as constraints in the estimation. In MHE the system states and disturbances are derived by solving the following optimization problem [230, 289, 339]: min ˆxT≠N/T, ˆWT - - ----ˆxT≠N/T ≠ xT≠N/T - - ----2 P_T≠1_{≠N/T ≠1}≠ - - ----Y_TT_≠N≠1 _{≠ Oˆx}T≠N/T ≠ cbUTT≠N≠2 - - ----2_P ≠1+ T_q≠1 k=T ≠N|| ˆwk||Q ≠1 k + ||ˆvk||R≠1k s.t. ˆxk+1 = Aˆxk+ Buk+ G ˆwk yk= Cˆxk+ ˆvk ˆxk œ X, ˆwk œ , ˆvkœ V. (2.3)

where T is the current time, Qk º 0, Rk º 0, PT≠N/T ≠1 º 0 are the covariances of wk,

vk, xT≠N assumed to be symmetric, N is the horizon length of the MHE, YTT≠N≠1 is a vector

containing the past N +1 measurements and UT≠1

T≠N is a vector containing the past N inputs.

x, v and w denote the variables of the system and ˆx, ˆv, and ˆwdenote the estimated variables

of the system and ˆxT /T≠N and ˆWT = WTT≠N≠1 denote the decision variable of the optimization

problem, the estimated state variable and the noise, respectively. In mp-MHE, the matrices

Qk, Rk and PT≠N/T ≠1 are constant. In particular PT≠N/T ≠1 = Pss which corresponds to the

steady-state Kalman covariance matrix.

To obtain an mp-MHE formulation, the problem in eq. (2.3) is reformulated as a multi- parametric programming problem:

min ˆxT≠N/T, ˆWTT≠N/T≠1 1 2 Ë ˆxT≠N/TT, ˆWTT≠N/T≠1 È H S UˆxT≠N/T ˆ WT≠1 T≠N/T T V+ ◊.f. S UˆxT≠N/T ˆ WT≠1 T≠N/T T V s.t. K. S UˆxT≠N/T ˆ W_TT_≠N/T≠1 T V_{Æ k.} (2.4)

The parameters of the multi-parametric programming problem (2.4) are the past and current measurements and inputs and the initial guess for the estimated states.

Current state

Unconstrained MHE: There are a few necessary steps that lead to incorporating the con-

strained MHE into robust MPC. The estimation error at the beginning of the horizon and at the current time have to be derived and the bounding set of the estimation error has to be obtained. The unconstrained moving horizon estimator is equivalent to the Kalman filter [114, 230], the estimation error and the bounding sets they generate should be equivalent so the Kalman filter can be used for comparison.

Constrained MHE: In order to formulate and solve the constrained moving horizon esti-

mator with multi-parametric programming, the optimization problem is reformulated into the standard multi-parametric quadratic form. Previous work has been performed on reformulating the MHE with the filtered arrival cost [83] and with the smoothed update of the arrival cost [349].

Recent developments

MHE with smoothed arrival cost: The formulation of the MHE with the smoothed ar-

into the standard multi-parametric quadratic form. The smoothed update of the ar- rival cost involves less parameters in the multi-parametric formulation of the MHE and hence it requires less computational effort to solve the mp-MHE than the equivalent estimation problem that utilizes the filtered arrival cost [350].

Simultaneous mp-MHE and mp-MPC: The implementation of explicit/multi-parametric

MPC, and in general, MPC, is based on the assumption that the state values are read- ily available from the system measurements and also that the measurable output is free of noise influence. However, in reality, the measured output may be noisy and the system measurements do not offer this information directly - instead the state infor- mation needs to be inferred from the available output measurements by using a state estimator which obtains an estimate ˆx of the real state x. The framework uses the con- strained MHE that gives improved estimation results compared to the unconstrained estimators. The estimation error remains inside the calculated error set and hence the MPC guarantees to satisfy the constraints [191, 351].

Simultaneous mp-MHE and mp-MPC for biomedical applications: Biomedical sys-

tems are complex systems with a high degree of nonlinearity. Estimation techniques play an important role in such processes since some of the parameters and the states of the systems cannot always be measured directly from the system outputs. In most of the biomedical applications, the optimal policies rely on patient-dependent data which might be unavailable or computationally impossible to retrieve in a reasonable time frame. This makes simultaneous mp-MHE and mp-MPC an important ongo- ing research area that can deal with some critical issues especially on topics such as: intravenous and volatile anaesthesia, type-1 diabetes and leukemia.

Future outlook

Simultaneous mp-MHE and mp-MPC for periodic systems: Simultaneous mp-MHE

and mp-MPC is an area that has been receiving more attention in the past years. The research work in this field has been acknowledged in several publications (give refer- ence). One area that represents an important research is the design of simultaneous control and estimation schemes for periodic systems. The periodic nature as well as the presence of multiple control objectives makes the control of such processes a challenging task.

mp-MHE for hybrid systems: The control of hybrid systems1 _{represents a demanding}

challenge on its own. So far the multi-parametric moving horizon estimation has not been addressed in the context of hybrid systems. Current research will be focusing on multi-parametric MIQP algorithm and a step-by-step procedure for the derivation of offline multi-parametric hybrid controllers [98, 248, 276] and an integrated soft- ware (PAROC) for the general design, operational optimization and control of process systems.