Event-based model predictive control with piecewise optimal state feedback laws

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Rapidly growing digital networks and ever growing computing power provide new possibilities for control and automation technology, but also give rise to new fundamental problems. In particular the central paradigm of permanent feedback must be questioned, if energy consumption and bandwidth requirements are to be minimized. Instead of permanently performing sensor and control computations, it is obviously desirable to limit computation and network traffic to instances when a controlled system actually needs attention. This is the central idea of event-based control.The proposed project addresses an event-based approach to model predictive control (MPC). In MPC, a mathematical optimization problem is solved periodically to determine the optimal future control sequence for the controlled system. The optimal control sequence needs to be updated periodically, because, among other reasons, disturbances need to be corrected for. Because the computational effort spent on solving underlying optimization problems is high, it is attractive to avoid them whenever possible. Consequently, event-based schemes are particularly attractive for MPC. Existing approaches exploit that not only a control signal but a control signal sequence is calculated in MPC. The approach proposed here is based on a different, new idea: Predictive control operates point-by-point, i.e. the optimization problem is solved for the current state of the system (a point in its state space). It has so far been overlooked that the point-wise solution implies the optimal feedback law on an entire region in the state space (a convex polytope) around the current state. In fact, the optimal feedback law is a state feedback law that has the same simple form as Kalman´s seminal optimal solution to the linear-quadratic regulator. (The proposed approach is not a variant of explicit MPC. No parametric optimizations need to be solved.)The optimal state feedback law and the region of its validity can be determined at practically no extra computational cost from the point-wise solution. The computationally expensive MPC problem only needs to be solved on demand, and the event causing the re-computation occurs when the controlled system leaves the region of validity of the current optimal state feedback law.The resulting event-based MPC method can be implemented by combining a lean (cheap, small, light, energy-efficient) local embedded system and a computationally powerful central node that is only called on demand. The local node only needs to evaluate optimal state feedback laws of the same simple form as Kalman´s solution to the linear-quadratic regulator. When its region of validity is left (a convex polytope), the local node requests a new state feedback law from the central node. The central node must therefore be capable of solving MPC problems. Because it is only called on demand, the central node can provide its service to multiple local nodes.