Subspace Methods for System Identification

Compact thermal modeling for packaged microprocessor design with practical power maps

“…The subspace identification basically tries to first identify the system states (Kalman states), then the state matrices will be obtained by the least square based optimization method [29]. There are several implementations such as the Ho-Kalman' method, the MOSEP method and the N4SID (numerical algorithms for subspace system identification) method [28].…”

“…where P B ðA; CÞ represents an oblique projection of the row space of A onto the row space of C along row space of B [28][29][30]. By applying Singular Value Decomposition (SVD) on the left hand side of (9), there is…”

“…Such highly correlated power inputs, however, will lead to poor predictability when using the subspace identification method [30,29]. Fig.…”

“…In order to get a good system, the input signal should satisfy the so-called persistently exciting condition. That is, for an l-order system and l ok, rankðU 0j2kÀ1 Þ ¼ 2pk for a p-port system, assuming that the number of columns of the Hankel matrix N is sufficiently large in the N4SID method [29,30].…”

“…We consider practical measured power maps, which can be obtained from thermal lab based on the test thermal vehicle (testing thermal chips). We first observe that the subspace identification method may suffer from the lack of predictability problems in general [28,29], especially when the input power is given as series of 2-dimensional power distributions (called power maps) in which the input signal is highly spatially correlated. Power map-based thermal characterization is widely used in industry for thermal characterizations of package design as power maps can be easily obtained (measured or computed) practically.…”