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aircraft algorithm appearing approximation assumed asymptotic stability backward-time based on second block diagram boundary conditions calculus of variations Chap coefficients components condition for minimum conditional mean constant defined design problems dynamic programming dynamic-programming equation given error index error measure Euler-Lagrange equation expansion feedback gains first-order dynamic process fixed member Furthermore hamiltonian hence increment initial conditions input signal integral interval introduced iteration Lagrange multiplier linear optimum control linear system load disturbances mathematical matrix mean-square minimum value minimum-error function nonlinear normalized time-to-go notation obtained optimization theory optimum control equation optimum control signal optimum control system optimum impulse response optimum system optimum trajectory ordinary differential equations parameters partial derivatives partial differential equation quadratic rate of convergence satisfy second variations selection shown in Fig solution Specifically spectral density statistical signals steady-state Taylor series terminal-boundary condition time-varying transfer function two-point boundary-value problem variables weighting factors zero