In this paper, a procedure for construction of quadratic Lyapunov–Krasovskii functionals for line... more In this paper, a procedure for construction of quadratic Lyapunov–Krasovskii functionals for linear time-delay systems is proposed. It is shown that these functionals admit a quadratic low bound. The functionals are used to derive robust stability conditions.
This paper is concerned with spacecraft rendezvous with target spacecraft in an arbitrary ellipti... more This paper is concerned with spacecraft rendezvous with target spacecraft in an arbitrary elliptical orbit. With three independent control accelerations being the control of the resulting linearized Tschauner–Hempel equations, the spacecraft rendezvous problem can be reformulated as a regulation problem with controls of bounded magnitude and energy. A parametric Lyapunov differential equation approach is proposed in this paper to solve this constrained regulation problem. After establishing the fact that the Tschauner–Hempel equations are both null controllable with controls of bounded magnitude and energy, this paper proves that the proposed linear periodic controller semiglobally stabilizes the system. Equivalently, for any fixed initial conditions, the magnitude and energy of the control can be made as small as desired by tuning some free parameters in the feedback laws. In comparison with the existing quadratic-regulation-based approach, which requires solutions to nonlinear Riccati differential equations, the new approach requires only the solution of linear periodic Lyapunov differential equations, which are investigated in the paper by using the periodic generator approach. Numerical simulations of the nonlinear model of the spacecraft rendezvous instead of a linearized one show that both the magnitude and energy of the control can be reduced to an arbitrarily small level by reducing the values of some parameters in the controller and that the rendezvous mission can be accomplished satisfactorily. Nomenclature a = semimajor axis of target orbit A i , A i = state matrices for in-plane motion A o , A o = state matrices for out-of-plane motion a f = acceleration due to thrust forces on chaser B i , B i = control matrices for in-plane motion B o , B o = control matrices for out-of-plane motion C A = characteristic multiplier set for periodic matrix A E = eccentric anomaly e = eccentricity of target orbit h = orbital angular momentum of target orbit k = =h 3=2 K i , K o = feedback gains for in-plane and out-of-plane motions L i , L o = transformation matrices P i , P o = solutions to parametric Riccati differential equations R = vector from center of gravity to target spacecraft r = vector from target spacecraft to chaser spacecraft T = period of target orbit t = time tr A = trace of square matrix A T s = settling time u i , v i = control vectors of in-plane motion u o , v o = control vectors of out-of-plane motion kuk L 2 = L 2 norm of vector u t (i.e., the square root of R 1 t 0 ku tk 2 dt) kuk L 1 = L 1 norm of vector u t (i.e., sup t ku tk) W i , W o = solutions to Lyapunov differential equations x; y; z = rotating coordinate frame fixed at target spacecraft , = state vectors of Tschauner–Hempel equations i , i = state vectors of in-plane motion o , o = state vectors of out-of-plane motion = true anomaly A = eigenvalue set of matrix A = gravity constant = 1 e cos i ; # = state transition matrix of in-plane motion o ; # = state transition matrix of out-of-plane motion ! = orbital rate of target spacecraft
In this paper, a procedure for construction of quadratic Lyapunov–Krasovskii functionals for line... more In this paper, a procedure for construction of quadratic Lyapunov–Krasovskii functionals for linear time-delay systems is proposed. It is shown that these functionals admit a quadratic low bound. The functionals are used to derive robust stability conditions.
This paper is concerned with spacecraft rendezvous with target spacecraft in an arbitrary ellipti... more This paper is concerned with spacecraft rendezvous with target spacecraft in an arbitrary elliptical orbit. With three independent control accelerations being the control of the resulting linearized Tschauner–Hempel equations, the spacecraft rendezvous problem can be reformulated as a regulation problem with controls of bounded magnitude and energy. A parametric Lyapunov differential equation approach is proposed in this paper to solve this constrained regulation problem. After establishing the fact that the Tschauner–Hempel equations are both null controllable with controls of bounded magnitude and energy, this paper proves that the proposed linear periodic controller semiglobally stabilizes the system. Equivalently, for any fixed initial conditions, the magnitude and energy of the control can be made as small as desired by tuning some free parameters in the feedback laws. In comparison with the existing quadratic-regulation-based approach, which requires solutions to nonlinear Riccati differential equations, the new approach requires only the solution of linear periodic Lyapunov differential equations, which are investigated in the paper by using the periodic generator approach. Numerical simulations of the nonlinear model of the spacecraft rendezvous instead of a linearized one show that both the magnitude and energy of the control can be reduced to an arbitrarily small level by reducing the values of some parameters in the controller and that the rendezvous mission can be accomplished satisfactorily. Nomenclature a = semimajor axis of target orbit A i , A i = state matrices for in-plane motion A o , A o = state matrices for out-of-plane motion a f = acceleration due to thrust forces on chaser B i , B i = control matrices for in-plane motion B o , B o = control matrices for out-of-plane motion C A = characteristic multiplier set for periodic matrix A E = eccentric anomaly e = eccentricity of target orbit h = orbital angular momentum of target orbit k = =h 3=2 K i , K o = feedback gains for in-plane and out-of-plane motions L i , L o = transformation matrices P i , P o = solutions to parametric Riccati differential equations R = vector from center of gravity to target spacecraft r = vector from target spacecraft to chaser spacecraft T = period of target orbit t = time tr A = trace of square matrix A T s = settling time u i , v i = control vectors of in-plane motion u o , v o = control vectors of out-of-plane motion kuk L 2 = L 2 norm of vector u t (i.e., the square root of R 1 t 0 ku tk 2 dt) kuk L 1 = L 1 norm of vector u t (i.e., sup t ku tk) W i , W o = solutions to Lyapunov differential equations x; y; z = rotating coordinate frame fixed at target spacecraft , = state vectors of Tschauner–Hempel equations i , i = state vectors of in-plane motion o , o = state vectors of out-of-plane motion = true anomaly A = eigenvalue set of matrix A = gravity constant = 1 e cos i ; # = state transition matrix of in-plane motion o ; # = state transition matrix of out-of-plane motion ! = orbital rate of target spacecraft
Uploads
Papers by YM Patel