Fast moving horizon estimation for a two-dimensional distributed parameter system

Partial differential equations (PDEs) pose a challenge for control engineers, both in terms of theory and computational requirements. PDEs are usually approximated by ordinary differential equations or difference equations via the finite difference method, resulting in a high-dimensional state-space system. The obtained system matrix is oftentimes symmetric, which allows this high-dimensional system to be decomposed into a set of single-dimensional systems using its singular value decomposition. Any linear constraints in the original problem can also be simplified by replacing it with an ellipsoidal constraint. Based on this, speedup of the moving horizon estimation is achieved by employing an analytical solution obtained by augmenting the ellipsoidal constraint into the objective function as a penalty weighted by a decreasing scaling parameter. The approximated penalty method algorithm allows for efficient parallel computation for sub-problems. The proposed algorithm is demonstrated for a two-dimensional diffusion problem where the concentration field is estimated using distributed sensors. © 2014 Elsevier Ltd. All rights reserved. . Introduction Distributed parameter systems (DPS) are found in a variety of engineering applications in aerospace, materials, chemistry, and biology. he state and output responses of these systems are functions of spatial and temporal variables and thus described by a system of partial ifferential equations (PDEs). In comparison to ordinary differential equations (ODEs) or differential-algebraic equations (DAEs), PDEs escribe physical systems more accurately but are more challenging to handle both theoretically and computationally. For the purpose of controller design for DPS, infinite dimension PDEs are usually approximated by finite dimension ODEs or difference quations via the finite difference method (Morton & Mayers, 1995). The advantage of this approximation is that many control design ethods including model predictive control (MPC) are directly applicable to the approximate finite-dimensional system. However, to btain a reasonable numerical solution, the approximation often requires a very large number of variables, resulting in a system of ODEs f very high dimension. On the other hand, controller design of a DPS by directly using the PDE often requires mathematical knowledge such as infinite-dimensional operator theory (Curtain & Zwart, 1995) or non-harmonic Fourier series (Russell, 1967)) that is unfamiliar to ost control engineers. State estimation is an important element of modern control applications. The Kalman filter (KF) is the current standard for state stimation of a linear system (Kalman, 1960; Kalman & Bucy, 1961). However, the KF in its bare form cannot incorporate constraints, hich may come from nature of the physics or process knowledge. Quadratic programming (QP)-based moving horizon estimation (MHE) as been suggested as a practical way to incorporate inequality constraints into state estimation (Muske, Rawlings, & Lee, 1993; Rao, awlings, & Lee, 2001; Robertson, Lee, & Rawlings, 1996). A drawback of this optimization-based approach is the substantially higher omputational cost when compared to explicit estimators such as the KF. This drawback may limit the size of the problem to which the HE method can be applied. A preliminary version of the manuscript was published as H. Jang, K.-K. K. Kim, J. H. Lee, and R. D. Braatz, Fast moving horizon estimation for a distributed parameter ystem, In 12th International Conference on Control, Automation and Systems, pp. 533–538, 2012, Jeju Island, Korea. ∗ Corresponding author. Tel.: +82 42 350 3926; fax: +82 42 350 3910. E-mail addresses: [email protected], [email protected] (J.H. Lee). 098-1354/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. ttp://dx.doi.org/10.1016/j.compchemeng.2013.12.005 160 H. Jang et al. / Computers and Chemical Engineering 63 (2014) 159–172