Open-loop control laws

Differential models, such as any of those from Chapter 13, are usually referred to as control systems or just systems, a term that we have used already. These are divided into linear and nonlinear systems, as described in Sections 13.2.2 and 13.2.3, respectively. Formulation 14.1 can be considered in control terminology as the design of an open-loop control law for the system (subjected to nonconvex constraints on the state space). The open-loop part indicates that no feedback is used. Only the action trajectory needs to be specified over time (the feedback case is called closed-loop; recall Section 8.1). Once the initial state is given, the state trajectory can be inferred from the action trajectory. It may also be qualified as a feasible open-loop control law, to indicate that it satisfies all constraints but is not necessarily optimal. It is then interesting to consider designing an optimal open-loop control law. This is extremely challenging, even for problems that appear to be very simple. Elegant solutions exist for some restricted cases, including linear systems and some wheeled vehicle models, but in the absence of obstacles. These are covered in Chapter 15.