PI controllers for the general Saint-Venant equations
Copyright policy )PI controllers for the general Saint-Venant equations
We study the exponential stability in the H 2 norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single proportional-integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain of the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the input-to-state stability (ISS) of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.
Saint-Venant Equations, exponential stability, Lyapunov Stability, proportional integral control, Exponential stability, input-to-state stability, Mathematics - Analysis of PDEs, Shallow Water Equations, Input to state stability, partial differential equations, FOS: Mathematics, Stabilization of systems by feedback, Nonlinear systems in control theory, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Input-output approaches in control theory, Mathematics - Optimization and Control, 93D15, 35B35, 93D05, 93D09, 93D20, 93D25, Control/observation systems governed by partial differential equations, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Inhomogeneous, [SPI.AUTO] Engineering Sciences [physics]/Automatic, Optimization and Control (math.OC), Hyperbolic Equations, Saint-Venant equations, Boundary value problems for nonlinear first-order PDEs, nonlinear systems, Stability in context of PDEs, Analysis of PDEs (math.AP)
Saint-Venant Equations, exponential stability, Lyapunov Stability, proportional integral control, Exponential stability, input-to-state stability, Mathematics - Analysis of PDEs, Shallow Water Equations, Input to state stability, partial differential equations, FOS: Mathematics, Stabilization of systems by feedback, Nonlinear systems in control theory, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Input-output approaches in control theory, Mathematics - Optimization and Control, 93D15, 35B35, 93D05, 93D09, 93D20, 93D25, Control/observation systems governed by partial differential equations, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Inhomogeneous, [SPI.AUTO] Engineering Sciences [physics]/Automatic, Optimization and Control (math.OC), Hyperbolic Equations, Saint-Venant equations, Boundary value problems for nonlinear first-order PDEs, nonlinear systems, Stability in context of PDEs, Analysis of PDEs (math.AP)
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).6 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Top 10% influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Top 10%
Citations provided by BIP!Popularity provided by BIP!