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The method is based on the idea of a so-called fraction function . It is a scalar function, defined as the integral of a fluid's characteristic function in the control volume, namely the volume of a computational grid cell. The volume fraction of each fluid is tracked through every cell in the computational grid, while all fluids share a single ...
A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state. Typically, the function is defined to grow large when the system moves towards undesirable states. System stability is achieved by taking control actions that make the Lyapunov function drift in the negative direction towards zero.
The last condition is the key condition; in words it says that for each state x we can find a control u that will reduce the "energy" V. Intuitively, if in each state we can always find a way to reduce the energy, we should eventually be able to bring the energy asymptotically to zero, that is to bring the system to a stop.
In computer science, a control-flow graph (CFG) is a representation, using graph notation, of all paths that might be traversed through a program during its execution. The control-flow graph was conceived by Frances E. Allen, [1] who noted that Reese T. Prosser used boolean connectivity matrices for flow analysis before. [2]
Similarly, if the objective function of a minimization problem is a differentiable convex function, the necessary conditions are also sufficient for optimality. It was shown by Martin in 1985 that the broader class of functions in which KKT conditions guarantees global optimality are the so-called Type 1 invex functions. [12] [13]
The phrase H ∞ control comes from the name of the mathematical space over which the optimization takes place: H ∞ is the Hardy space of matrix-valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re(s) > 0; the H ∞ norm is the supremum singular value of the matrix over that space.
An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost function. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition also known as Pontryagin's minimum principle or simply Pontryagin's principle), [ 8 ] or by solving the Hamilton ...
Constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints, which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied.