Ball on a beam: stabilization under saturated input control with large basin of attraction

This article is devoted to the stabilization of two underactuated planar systems, the well-known straight beam-and-ball system and an original circular beam-and-ball system. The feedback control for each system is designed, using the Jordan form of its model, linearized near the unstable equilibrium. The limits on the voltage, fed to the motor, are taken into account explicitly. The straight beam-and-ball system has one unstable mode in the motion near the equilibrium point. The proposed control law ensures that the basin of attraction coincides with the controllability domain. The circular beam-and-ball system has two unstable modes near the equilibrium point. Therefore, this device, never considered in the past, is much more difficult to control than the straight beam-and-ball system. The main contribution is to propose a simple new control law, which ensures by adjusting its gain parameters that the basin of attraction arbitrarily can approach the controllability domain for the linear case. For both nonlinear systems, simulation results are presented to illustrate the efficiency of the designed nonlinear control laws and to determine the basin of attraction

International society of sports nutrition position stand: caffeine and exercise performance

Following critical evaluation of the available literature to date, The International Society of Sports Nutrition (ISSN) position regarding caffeine intake is as follows: 1. Supplementation with caffeine has been shown to acutely enhance various aspects of exercise performance in many but not all studies. Small to moderate benefits of caffeine use include, but are not limited to: muscular endurance, movement velocity and muscular strength, sprinting, jumping, and throwing performance, as well as a wide range of aerobic and anaerobic sport-specific actions. 2. Aerobic endurance appears to be the form of exercise with the most consistent moderate-to-large benefits from caffeine use, although the magnitude of its effects differs between individuals. 3. Caffeine has consistently been shown to improve exercise performance when consumed in doses of 3–6 mg/ kg body mass. Minimal effective doses of caffeine currently remain unclear but they may be as low as 2 mg/kg body mass. Very high doses of caffeine (e.g. 9 mg/kg) are associated with a high incidence of side-effects and do not seem to be required to elicit an ergogenic effect. 4. The most commonly used timing of caffeine supplementation is 60 min pre-exercise. Optimal timing of caffeine ingestion likely depends on the source of caffeine. For example, as compared to caffeine capsules, caffeine chewing gums may require a shorter waiting time from consumption to the start of the exercise session. 5. Caffeine appears to improve physical performance in both trained and untrained individuals. 6. Inter-individual differences in sport and exercise performance as well as adverse effects on sleep or feelings of anxiety following caffeine ingestion may be attributed to genetic variation associated with caffeine metabolism, and physical and psychological response. Other factors such as habitual caffeine intake also may play a role in between-individual response variation. 7. Caffeine has been shown to be ergogenic for cognitive function, including attention and vigilance, in most individuals. 8. Caffeine may improve cognitive and physical performance in some individuals under conditions of sleep deprivation. 9. The use of caffeine in conjunction with endurance exercise in the heat and at altitude is well supported when dosages range from 3 to 6 mg/kg and 4–6 mg/kg, respectively. 10. Alternative sources of caffeine such as caffeinated chewing gum, mouth rinses, energy gels and chews have been shown to improve performance, primarily in aerobic exercise. 11. Energy drinks and pre-workout supplements containing caffeine have been demonstrated to enhance both anaerobic and aerobic performance