Analysis Of Linear Systems Cheng Pdf
Su Strawderman
In this paper, we consider the relation between the switching dwell time and the stabilization of switched linear control systems. First of all, a concept of critical dwell time is given for switched linear systems without control inputs, and the critical dwell time is taken as an arbitrary given positive constant for a switched linear control systems with controllable switching models. Secondly, when a switched linear system has many stabilizable switching models, the problem of stabilization of the overall system is considered. An on-line feedback control is designed such that the overall system is asymptotically stabilizable under switching laws which depend only on those of uncontrollable subsystems of the switching models. Finally, when a switched system is partially controllable (While some switching models are probably unstabilizable), an on-line feedback control and a cyclic switching strategy are designed such that the overall system is asymptotically stabilizable if all switching models of this uncontrollable subsystems are asymptotically stable. In addition, algorithms for designing switching laws and controls are presented.
Lijun Zhang received the M.S. degree from Shaanxi Normal University in 1997, and the Ph.D. degree from Institute of Systems Science, Chinese Academy of Sciences, Beijing China in 2003. Since 2005, he is a Professor at the Harbin Engineering 470 University, China. His research interests include nonlinear system and control, switched systems control, etc. He is the author/co-author of 20 journal papers.
Chunwen Li graduated from Tsinghua University in 1982, also received the Ph.D. degree from Tsinghua University in 1989. Since 1996, he is a Professor working in Tsinghua University. His research interests include nonlinear systems and control, quantum control etc. He is the author/co-author of 50 papers.
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A fundamental assumption of nearly all functional magnetic resonance imaging (fMRI) analyses is that the relationship between local neuronal activity and the blood oxygenation level dependent (BOLD) signal can be described as following linear systems theory. With the advent of ultra-high field (7T and higher) MRI scanners, it has become possible to perform sub-millimeter resolution fMRI in humans. A novel and promising application of sub-millimeter fMRI is measuring responses across cortical depth, i.e. laminar imaging. However, the cortical vasculature and associated directional blood pooling towards the pial surface strongly influence the cortical depth-dependent BOLD signal, particularly for gradient-echo BOLD. This directional pooling may potentially affect BOLD linearity across cortical depth. Here we assess whether the amplitude scaling assumption for linear systems theory holds across cortical depth. For this, we use stimuli with different luminance contrasts to elicit different BOLD response amplitudes. We find that BOLD amplitude across cortical depth scales with luminance contrast, and that this scaling is identical across cortical depth. Although nonlinearities may be present for different stimulus configurations and acquisition protocols, our results suggest that the amplitude scaling assumption for linear systems theory across cortical depth holds for luminance contrast manipulations in sub-millimeter laminar BOLD fMRI.
The cerebral cortex consists of separate cortical regions that perform specialized computations. The first parcellation of the cortex into separate cortical regions was based on anatomical differences across cortical depth, i.e. cortical layers or laminae1,2,3,4,5,6, for reviews see7,8.
Magnetic resonance imaging (MRI) is one of the most popular techniques to study the human brain non-invasively. The recent development in static magnetic field strength to ultra-high fields of 7 Tesla and higher, has enabled researchers to investigate the human brain at a sub-millimeter (mesoscopic) scale. At this spatial resolution, it becomes possible to measure both anatomical and functional cortical depth-dependent signals that reflect contributions of different cortical layers. Sub-millimeter (laminar) functional MRI (fMRI) promises to complement anatomical measurements across cortical depth with functional properties that may indicate feed-forward and feedback processes9,10.
However, fMRI across cortical depth faces substantial challenges9,11,12. One dominant challenge relates to the cortical vascular organization. While fMRI detects hemodynamic consequences of neuronal activity13,14, the cortical vasculature has a specific organization across cortical layers: blood flows from pial and intracortical arteries and arterioles to the capillary bed that directly interfaces with neuronal tissue, and is then drained via venules and intracortical veins to larger pial veins at the cortical surface15,16. This likely explains a large part of the consistently reported finding that BOLD signal amplitude is larger at the cortical surface and systematically decreases toward deeper layers, see e.g.9,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31. The vascular architecture across cortical depth has also been modeled in detail e.g.32,33 and poses a fundamental challenge for the analysis of laminar fMRI as blood pooling effects might differently affect the blood oxygenation level dependent (BOLD) signal in deeper layers compared to more superficial ones34.
Furthermore, due to the directional blood pooling across layers, hemodynamic responses from deeper cortical layers influence signals at superficial layers. For example, if there is neuronal activity in the deeper cortical depth portions only and not in the more superficial ones, a BOLD signal change would be measured at all cortical depths, due to this directional blood pooling (see Fig. 1, lowest red line). Theoretically, this could occur in absence of feedforward signals due to clinical manifestations, see e.g.35, or in situations where neural activity is primarily driven by feedback signals -typically arriving in the supragranular and/or infragranular depth portions, as demonstrated by Kok and colleagues36 and Klein et al.37.
In summary, the specific cortical vascular organization might undermine fundamental assumptions in fMRI analysis at the sub-millimeter scale. Nearly all fMRI data-analysis techniques assume that fMRI responses are linearly proportional to a local average neuronal activity over a period of time following linear systems theory38. For this theory to hold, two assumptions must be met: scaling and temporal additivity. These assumptions have been tested for conventional supra-millimeter fMRI for which they largely hold, provided that the stimuli used are within a defined range of stimulus parameters that is commonly used in neuroimaging experiments, see e.g.13,38,39,40,41,42; for nonlinearities, mostly found in event-related designs, see e.g.43,44,45,46,47,48,49. Based on these results at conventional resolutions, the assumptions for a linear system are expected to largely hold for responses across cortical depth. However, the laminar BOLD signal amplitudes and directional blood pooling across layers potentially violate these assumptions, as different portions across cortical depth are not independent.
On a sub-millimeter level, however, the hemodynamic response at depth d does not only depend on the local neuronal response, but also on responses at deeper cortical depths because of the draining effects towards the cortical surface. Thus, the hemodynamic response at depth d can be expressed as:
where Hc(d) is the hemodynamic response at depth d; \(L[n(t)]_d\) is the hemodynamic transform of the local neuronal response over time at depth d and \(\mathop\sum \limits_i=d_0^d-1w_iH_c(i)\) reflects the draining from the sum of the hemodynamic response Hc(i) at all cortical depths below depth d, up to the gray-white matter surface d0, each weighted by a factor wi50,51. This weighting factor represents an estimation of the draining of altered deoxyhemoglobin content and increased blood pressure from lower layers51. The hemodynamic response at a given depth is thus a combination of the local neuronal activity and draining from deeper layers. Therefore, the hemodynamic response is not only dependent on the neural responses at that location (or depth) but also on the neural responses at other locations (depths). This dependence might violate the assumption that the hemodynamic response is proportional to the underlying local neural activity and could drive the system nonlinear. For example, if task demands drive deeper cortical depths more actively than more superficial depths due to feedback connections52,53, this would result in an increase in the hemodynamic response at all depths, while there is no additional local neuronal component at most depths. This is illustrated in Fig. 1, lower red curve. This violates the scaling assumption between cortical depths.
We used stimuli with different luminance contrasts (i.e. sinewave gratings with 5, 20, and 80% luminance contrast) to elicit gradually increasing neuronal responses. For linear amplitude scaling across cortical depth to hold, the BOLD response amplitude across cortical depth to a higher contrast (Fig. 1, green curve) should be equal to the response profile across cortical depth for another contrast, multiplied with a scaling factor k that is independent of cortical depth (Fig. 1, gray curve), therefore retaining the shape but changing in amplitude. Figure 1 also shows two changes in response amplitude that do not follow linear systems theory (Fig. 1, red lines). For example, an intercept shift due to e.g. a spatially restricted increase in neuronal activation does not satisfy linear systems theory requirements (Fig. 1, lower red curve).
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