Finite dimensional modeling and control of distributed parameter systems (original) (raw)
Related papers
Industrial Engineering Chemistry Research, 2002
In this work, a new system identification method that combines the characteristics of singular value decomposition (SVD) and the Karhunen-Loève (KL) expansion for distributed parameter systems is presented. This method is then demonstrated on two nonlinear reactor systems that can be described by systems of partial differential equations (PDEs). The results indicate that this new method provides satisfactory low-order models when compared to models developed using either the SVD approach or the KL expansion in a Galerkin method. In particular, it has the advantage of not requiring an exact PDE model, which is necessary for the KL solution and it captures the dynamics of the process in contrast to the SVD solution. This has important implications especially for applications such as control that require low-order models for implementable solutions.
IEEE Transactions on Automatic Control, 1999
Two reduced-order digital controllers for distributed parameter systems (DPS's) are described. Reduced-order models approximate the optimal finite past predictor and error covariance for the full system to minimize an approximation to the Kullback-Leibler information distance (KLID). An LQG controller based on a reduced-order system model is described. A reduced-order controller is found to minimize the KLID between the closed-loop system outputs with the full-and reduced-order controllers. Noncollocated control of a flexible beam is simulated.
Journal of Optimization Theory and Applications, 1985
tn order to implement feedback control for practical distributed-parameter systems (DPS), the resulting controllers must be finite-dimensional. The most natural approach to obtain such controllers is to make a finite-dimensional approximation, i.e., a reduced-order model, of the DPS and design the controller from this. In past work using perturbation theory, we have analyzed the stability of controllers synthesized this way, but used in the actual DPS; however, such techniques do not yield suboptimal performance results easily. In this paper, we present a modification of the above controller which allows us to more properly imbed the controller as part of the DPS. Using these modified controllers, we are able to show a bound on the suboptimality for an optimal quadratic DPS regulator implemented with a finitedimensional control, as well as stability bounds. The suboptimality result may be regarded as the distributed-parameter version of the 1968 results of Bongiorno and Youla.
SVD-Krylov based techniques for structure-preserving reduced order modelling of second-order systems
AIMS Mathematical Modelling and Control, 2021
We introduce an efficient structure-preserving model-order reduction technique for the large-scale second-order linear dynamical systems by imposing two-sided projection matrices. The projectors are formed based on the features of the singular value decomposition (SVD) and Krylov-based model-order reduction methods. The left projector is constructed by utilizing the concept of the observability Gramian of the systems and the right one is made by following the notion of the interpolation-based technique iterative rational Krylov algorithm (IRKA). It is well-known that the proficient model-order reduction technique IRKA cannot ensure system stability, and the Gramian based methods are computationally expensive. Another issue is preserving the second-order structure in the reduced-order mode. The structure-preserving model-order reduction provides a more exact approximation to the original model with maintaining some significant physical properties. In terms of these perspectives, the proposed method can perform better by preserving the second-order structure and stability of the system with minimized H2-norm. Several model examples are presented that illustrated the capability and accuracy of the introducing technique.
Computational methods for the control of distributed parameter systems
Proceedings of the IEEE Conference on Decision and Control
Finite dimensional approximation schemes that work well for distributed parameter systems are often not suitable for the analysis and implementation of feedback control systems. The relationship between approximation schemes for distributed parameter systems and their application to optimal control problems is discussed. A numerical example is given.
Two efficient SVD/Krylov algorithms for model order reduction of large scale systems
Electronic Transactions on Numerical Analysis, 2011
We present two efficient algorithms to produce a reduced order model of a time-invariant linear dynamical system by approximate balanced truncation. Attention is focused on the use of the structure and the iterative construction via Krylov subspaces of both controllability and observability matrices to compute low-rank approximations of the Gramians or the Hankel operator. This allows us to take advantage of any sparsity in the system matrices and indeed the cost of our two algorithms is only linear in the system dimension. Both algorithms efficiently produce good low-rank approximations (in the least square sense) of the Cholesky factor of each Gramian and the Hankel operator. The second algorithm works directly on the Hankel operator, and it has the advantage that it is independent of the chosen realization. Moreover it is also an approximate Hankel norm method. The two reduced order models produced by our methods are guaranteed to be stable and balanced. We study the convergence ...
Reduced order controllers for spatially distributed systems via proper orthogonal decomposition
SIAM J. Sci. Comput, 1999
A method for reducing controllers for systems described by partial di erential equations PDEs is presented. This approach di ers from an often used method of reducing the model and then designing the controller. The controller reduction is accomplished by projection of a large scale nite element approximation of the PDE controller onto low order bases that are computed using the proper orthogonal decomposition POD. Two methods for constructing input collections for POD, and hence low order bases, are discussed and computational results are included. The rst uses the method of snapshots found in POD literature. The second is a new idea that uses an integral representation of the feedback control law. Speci cally, the kernels, or functional gains, are used as data for POD. A low order controller derived by applying the POD process to functional gains avoids subjective criteria associated with implementing a time snapshot approach and performs favorably.
Control of Distributed Parameter Systems
Distributed parameter systems (DPS) is an established area of research in control which can trace its roots back to the sixties. While the general aims are the same as for lumped parameter systems, to adequately describe the distributed nature of the system one needs to use partial differential equation (PDE) models. The modelling issue is in itself nontrivial, especially when there is boundary control action and sensing on the boundary. Controllability and observability concepts are subtle and investigating these for a single PDE example leads to a sophisticated mathematical problem. The action of controlling the system introduces feedback into the PDE model which results in a more complicated mathematical model; the resulting closed-loop system may not be well-posed and this issue has only quite recently become well understood. At this stage, the mathematical machinery for formulating the basic control problems is available (although not so well known), and this has led to a wealth of new system theoretic results for DPS.