Andrea L'Afflitto

Finite-Time Stabilization and Optimal Feedback Control

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability have been developed in the literature using continuous Lyapunov functions. In this technical note, we develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite-time stability and finite-time stabilization. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a differential inequality involving fractional powers. This Lyapunov function can clearly be seen to be the solution to a partial differential equation that corresponds to a steady-state form of the Hamilton-Jacobi-Bellman equation, and hence, guaranteeing both finite-time stability and optimality.

Optimal singular control for nonlinear semistabilisation

ABSTRACT The singular optimal control problem for asymptotic stabilisation has been extensively studied in the literature. In this paper, the optimal singular control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Three approaches are presented to address the nonlinear semistable singular control problem. Namely, a singular perturbation method is presented to construct a state-feedback singular controller that guarantees closed-loop semistability for nonlinear systems. In this approach, we show that for a non-negative cost-to-go function the minimum cost of a nonlinear semistabilising singular controller is lower than the minimum cost of a singular controller that guarantees asymptotic stability of the closed-loop system. In the second approach, we solve the nonlinear semistable singular control problem by using the cost-to-go function to cancel the singularities in the corresponding Hamilton–Jacobi–Bellman equation. For this case, we show that the minimum value of the singular performance measure is zero. Finally, we provide a framework based on the concepts of state-feedback linearisation and feedback equivalence to solve the singular control problem for semistabilisation of nonlinear dynamical systems. For this approach, we also show that the minimum value of the singular performance measure is zero. Three numerical examples are presented to demonstrate the efficacy of the proposed singular semistabilisation frameworks.

Optimal control for linear and nonlinear semistabilization

The state feedback linear-quadratic optimal control problem for asymptotic stabilization has been extensively studied in the literature. In this paper, the optimal linear and nonlinear control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which involves convergent trajectories and Lyapunov stable equilibria and which is of paramount importance for consensus control of network dynamical systems. Specifically, we show that the optimal semistable state-feedback controller can be solved using a form of the Hamilton-Jacobi-Bellman conditions that does not require the cost-to-go function to be sign-definite. This result is then used to solve the optimal linear-quadratic regulator problem using a Riccati equation approach.

Abnormal Optimal Trajectory Planning of Multi-Body Systems in the Presence of Holonomic and Nonholonomic Constraints

In optimal control problems, the Hamiltonian function is given by the weighted sum of the integrand of the cost function and the dynamic equation. The coefficient multiplying the integrand of the cost function is either zero or one; and if this coefficient is zero, then the optimal control problem is known as abnormal; otherwise it is normal. This paper provides a characterization of the abnormal optimal control problem for multi-body mechanical systems, subject to external forces and moments, and holonomic and nonholonomic constraints. This study does not only account for first-order necessary conditions, such as Pontryagin’s principle, but also for higher-order conditions, which allow the analysis of singular optimal controls.

Differential games, asymptotic stabilization, and robust optimal control of nonlinear systems

In this paper, we develop a unified framework to solve the two-players zero-sum differential game problem over the infinite time horizon. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady-state form of the Hamilton-Jacobi-Isaacs equation, and hence, guaranteeing both asymptotic stability and the existence of a saddle point for the systems performance measure. The overall framework provides the foundation for extending optimal linear-quadratic controller synthesis to differential games involving nonlinear dynamical systems with nonlinear-nonquadratic performance measures. Connections to optimal linear and nonlinear regulation for linear and nonlinear dynamical systems with quadratic and nonlinear-nonquadratic cost functionals in the presence of exogenous disturbances are also provided.

Optimal singular control for nonlinear semistabilization

The singular optimal control problem for asymptotic stabilization has been extensively studied in the literature. In this paper, the optimal singular control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Two approaches are presented to address the nonlinear semistable singular control problem. Namely, we solve the nonlinear semistable singular control problem by using the cost-to-go function to cancel the singularities in the corresponding Hamilton-Jacobi-Bellman equation. For this case, we show that the minimum value of the singular performance measure is zero. In the second approach, we provide a framework based on the concepts of state-feedback linearization and feedback equivalence to solve the singular control problem for semistabilization of nonlinear dynamical systems. For this approach, we also show that the minimum value of the singular performance measure is zero. A numerical example is presented to demonstrate the efficacy of the proposed singular semistabilization frameworks.

Necessary Conditions for Control Effort Minimization of Euler-Lagrange Systems

The L1 norm of the control vector is a suitable measure for the effort needed to control a vehicle, since, in several cases of practical interest, it can provide accurate estimate of the fuel consumption. In this paper, we address the problem of minimizing the weighted sum of the L1 norms of the control vectors of N vehicles moving in formation. Specifically, modeling each agent as a six degrees-of-freedom rigid body subject to external forces and moments, and holonomic and nonholonomic constraints, we give necessary conditions for minimizing the formation’s control effort. In addition, we provide necessary conditions for the existence of singular controls for the abnormal and the normal optimal control problems. Two of our main results show that singular controls have order of singularity equal to one and are analytical in the junction between singular and non-singular arcs. In order to highlight the framework presented in this paper, we provide a numerical example concerning a formation of F-16 performing an Immelmann turn.

Partial-state stabilization and optimal feedback control

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization. Partial asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the Hamilton-Jacobi-Bellman equation, and hence, guaranteeing both partial stability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic controller synthesis to nonlinear-nonquadratic optimal partial-state stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying systems with quadratic and nonlinear nonquadratic cost functionals are also provided. An illustrative numerical example is presented to demonstrate the efficacy of the proposed linear and nonlinear partial stabilization framework.

Singular linear-quadratic control for semistabilization

Singular control has been extensively studied for the linear-quadratic regulator problem for achieving asymptotic stability of controlled linear systems with cheap control. In this paper, the singular control problem is extended to guarantee a weaker form of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Specifically, we show that the optimal state-feedback controller for the semistable linear-quadratic regulator problem can be solved using an algebraic Riccati equation. This result allows us to address the semistable singular control problem and obtain several expressions for the minimum cost of a semistabilizing singular controller. In particular, after establishing connections between Qui and Davisons formula for the minimum cost of a cheap controller and our results, we prove that the cost of a semistabilizing singular controller is zero if and only if the controlled system is minimum phase and right invertible.

Semistabilization, feedback dissipativation, system thermodynamics, and limits of performance in feedback control

In this paper, we develop a thermodynamic framework for semistabilization of linear and nonlinear dynamical systems. The proposed framework unifies system thermodynamic concepts with feedback dissipativity and control theory to provide a thermodynamic-based semistabilization framework for feedback control design. Specifically, we consider feedback passive and dissipative systems since these systems are not only widespread in systems and control, but also have clear connections to thermodynamics. In addition, we define the notion of entropy for a nonlinear feedback dissipative dynamical system. Then, we develop a state feedback control design framework that minimizes the time-averaged system entropy and show that, under certain conditions, this controller also minimizes the time-averaged system energy. The main result is cast as an optimal control problem characterized by an optimization problem involving two linear matrix inequalities.