A method for algebraic analysis of the TLM algorithm

Abstract

SUMMARYAn analogy is described between the TLM algorithm and discrete state-space control theory. The analogy is used to derive the characteristic equations corresponding to parametrized model structures. Characteristic equations corresponding to several widely used node structures are derived by this means and found to be consistent with the observed behaviour of corresponding TLM models. An example is given of the use of the analogy in the predictive mode.

TLM nodal state estimator: An alternative method of initializing a two‐dimensional diffusion model

“…Using the TLM state-space analogy [3,8], analysis shows that the dynamics modelled by the node for a two-dimension parabolic model are…”

“…Thus, the use of an estimator algorithm is an attractive addition to the TLM model, especially for model initialization and input tracking. A direct analogy has been established between TLM and state-space control theory [8] and, thus, it is reasonable to make use of a form of estimator commonly used in state-space control. In the current application the estimator will be used to reconstruct the TLM pulse populations so that their values cause the output (potential) of the TLM model to approach that of the process.…”

“…A candidate approach for doing this is the state estimation algorithm and, having reviewed the operations of a TLM algorithm for solution of heat conduction problems, we now consider parameterization of the TLM network necessary to obtain a closed-form description of the network's dynamical behaviour. In doing this we make use of state-space analysis [8].…”

“…The associated characteristic equation for a parabolic stub loaded estimator derived using the TLM state-space analogy [8] is…”

Management of spatial and temporal thermal dynamics using TLM

“…Sc is the TLM scattering matrix in state-space form [5], B is the nodal input matrix, C is the nodal output matrix relating TLM pulses to nodal potentials and D is an optional matrix describing any direct coupling output error between input and output. QI is the time and space shift operator describing the movement of TLM pulses between neighbouring nodes.…”