This paper extends control barrier functions (CBFs) to high order control barrier functions (HOCBFs) that can be used for high relative degree constraints. The proposed HOCBFs are more general than recently proposed (exponential) HOCBFs. We introduce high order barrier functions (HOBF), and show that their satisfaction of Lyapunov-like conditions implies the forward invariance of the intersection of a series of sets. We then introduce HOCBF, and show that any control input that satisfies the HOCBF constraints renders the intersection of a series of sets forward invariant. We formulate optimal control problems with constraints given by HOCBF and control Lyapunov functions (CLF) and analyze the influence of the choice of the class K functions used in the definition of the HOCBF on the size of the feasible control region. We also provide a promising method to address the conflict between HOCBF constraints and control limitations by penalizing the class K functions. We illustrate the proposed method on an adaptive cruise control problem.

A growth trial was conducted to feed juvenile tilapia (initial weight,9.1 AE 0.1g), Oreochromis niloticus  O. aureus, isonitrogenous diets for 8 weeks. Six diets were formulated containing 29% crude protein from casein and gelatin, 10% crude fat from soybean oil and re¢ned soybean lecithin and varying levels of corn starch ranging from 6% to 46% at increments of 8%, with corresponding energy to protein (E/P) ratios of 35.6, 37.9, 40.2, 42.5, 44.8 and 47.1kJ g À1 . Weight gain (WG), speci¢c growth rate, feed e⁄ciency ratio and protein e⁄ciency ratio were signi¢cantly higher in ¢sh fed diets with starch ! 22% (or E/P ratio ! 40.2 kJ g À1 ) than in ¢sh fed diets with 6% or 14% starch (or E/P ratio of 35.6 or 37.9 kJ g À1 ).No further improvement was measured when dietary starch content increased beyond 22%. Body protein retention showed the same general pattern as WG, and was highest in ¢sh fed the 22% starch diet. Body composition was signi¢cantly a¡ected by dietary starch level. Fish fed diets with starch ! 30% had signi¢cantly higher lipid content than ¢sh fed diets with 6% or 14% starch. Ash content was negatively correlated with starch inclusion level, but moisture and protein contents did not show discernible trends among treatments. Results indicate that hybrid tilapia can utilize 46% dietary starch without growth retardation, while 22% starch in feed for juvenile tilapia containing 29% protein and 10% lipid, or an E/P ratio of 37.9 kJ g À1 is optimal. à Di¡erent superscripts within each column indicate signi¢cant (Po0.05) di¡erences between ¢sh fed di¡erent diets.Aquaculture Research, 2005 Dietary carbohydrate level in juvenile tilapia Y Wang et al.

To test the hypothesis that particle composition has a stronger influence on the community structure of particle‐attached than free‐living bacteria, elemental (C/N, δ13C, and δ15N) and chemical composition of particles and the size‐fractionated bacterial community composition were examined along the particle density gradient from the Pearl River to the open basin in the South China Sea. Microbial communities were collected at the three size fractions of 0.2–0.8, 0.8–3, and >3 µm, and the community composition was analyzed using high‐throughput sequencing of the 16S rRNA gene (V3–V4 regions). Multivariate analysis evaluating the similarities of bacterial community composition and chemical composition of particles revealed their general consistent spatial variations along the particle density gradient from freshwater to the sea basin. However, compositions of particulate organic matter were more strongly related to the free living than to the particle‐attached bacterial community composition, which was composed of relatively abundant anaerobic bacteria and those taxa preferring low‐oxygen conditions. This latter result might be caused by low‐oxygen microzones in particles. Community network models further revealed tighter interactions within the particle‐attached than in free‐living bacterial communities, suggesting that a relatively confined space in particles is more favorable for interactions within the community. These analyses indicated that particle microniche properties might be important in shaping particle‐attached community structure. In contrast, particulate organic matter compositions significantly influenced the free‐living bacterial community probably through the release of labile or semilabile organic matter from particles contributing to the bioavailability of dissolved organic carbon.

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