Abstract
This paper considers the synthesis of stabilizing controllers for nonlinear control-affine systems under multiple state constraints. A new control Lyapunov-barrier function approach is introduced for solving the considered problem. Assuming a classical control Lyapunov function, two possible methods for constructing new control Lyapunov-barrier functions are discussed. Sufficient conditions for the existence of new control Lyapunov-barrier functions are derived. With modifying the Sontag’s formula, an explicit state-constrained stabilizing feedback law is presented. Finally, two numerical examples are provided to illustrate the obtained theoretical results.
Highlights
The design of stabilizing controllers under state constraints is a critical research topic because the state trajectories of a practical control system are not allowed to enter certain unsafe regions
In [12]–[15], reference governors were applied for the satisfaction of state constraints
In [16], for nonlinear control-affine systems, a control barrier function (CBF) and a control Lyapunov function (CLF) were combined by weighted average to be a smooth control Lyapunov-barrier function (CLBF) and Sontag’s formula was applied for constructing continuous controllers to ensure both safety and stability
Summary
The design of stabilizing controllers under state constraints is a critical research topic because the state trajectories of a practical control system are not allowed to enter certain unsafe regions. In [16], for nonlinear control-affine systems, a CBF and a CLF were combined by weighted average to be a smooth control Lyapunov-barrier function (CLBF) and Sontag’s formula was applied for constructing continuous controllers to ensure both safety and stability. This paper introduces a new CLBF method for designing asymptotically stabilizing controllers for nonlinear controlaffine systems under multiple state constraints presented in terms of state functional inequalities. Different from the QP-based methods [5], [9], and [35]–[37], in our approach state-constrained controllers can be explicitly constructed by using a modified Sontag’s formula and, more importantly, asymptotical stability with the entire safe set being forward invariant and in the region of attraction can be guaranteed. Let BD ( ) := BD (0, ), B (S, ) := BRn (S, ), and B ( ) := BRn (0, )
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