Nonlinear Optimal Control: A Control Lyapunov Function and Receding Horizon Perspective

Abstract

Two well known approaches to nonlinear control involve the use of control Lyapunov functions (CLFs) and receding horizon control (RHC), also known as model predictive control (MPC). The on‐line Euler‐Lagrange computation of receding horizon control is naturally viewed in terms of optimal control, whereas researchers in CLF methods have emphasized such notions as inverse optimality. We focus on a CLF variation of Sontag's formula, which also results from a special choice of parameters in the so‐called pointwise minnorm formulation. Viewed this way, CLF methods have direct connections with the Hamilton‐Jacobi‐Bellman formulation of optimal control. A single example is used to illustrate the various limitations of each approach. Finally, we contrast the CLF and receding horizon points of view, arguing that their strengths are complementary and suggestive of new ideas and opportunities for control design. The presentation is tutorial, emphasizing concepts and connections over details and technicalities.

On the new notion of input-to-state safety

“…As shown in Proposition (1), a practical input-to-state safety can be shown if there exists B such that the inequalities (13) and (14) holds. When κ = 0 then (13) & (14) ⇒ the system (2) is input-to-state safe.…”

“…In [1] and [20], the authors proposed an optimization problem, in the form of a quadratic programming, where both control Lyapunov and control Barrier inequalities are formulated in the constraints. The proposed method generalizes the well-known pointwise minnorm control method for designing a control law using control Lyapunov functions via an optimization problem [13]. It has been successfully implemented in the cruise control of autonomous vehicle as reported in [10].…”

“…D where α(s,t) = Proof : Let x(t) be the solution of (2). Evaluating the time derivative of B(x(t)) along the trajectory of x, it follows from (13) and (14) thaṫ…”

“…2. An illustration of a practical ISSf barrier function which takes the form of a quadratic function, e.g., B(x) = −(x − x * ) T P(x − x * ) + c where P is a positive definite matrix, x * is the centroid of the unsafe set D and c is a constant that is chosen such that the zero level of B is equal to the boundary of D. The solid red-line is the plot of B and the dashed-line shows the possible lower and upper bound of B using the set distance function |x| D and a bias constant κ > 0 as used in Proposition (1), c.f., (13).…”

Receding Horizon Control

“…In Primbs et al [PND99], aspects of a stabilityguaranteeing, global control Lyapunov function (CLF) were used, via state and control constraints, to develop a stabilizing receding horizon scheme. Many of the nice characteristics of the CLF controller together with better cost performance were realized.…”