Optimal Control
Optimal control finds the input that minimizes a cost functional subject to the system dynamics, unifying many design methods under one principle.
Control as Optimization
Optimal control reframes design as a minimization: find the control input, as a function of time, that minimizes a cost functional while satisfying the system's differential-equation dynamics and any constraints. The cost encodes what good performance means, typically a blend of tracking accuracy, control effort, and terminal targets.
The ingredients
- Dynamics: the state equation x' = f(x, u, t) that the trajectory must obey.
- Cost functional: an integral of a running cost plus a terminal cost, to be minimized.
- Constraints: limits on inputs and states, and boundary conditions on the trajectory.
- The solution: an optimal input trajectory and the resulting optimal state trajectory.
Two solution routes
There are two principal ways to solve optimal control problems. Pontryagin's maximum principle gives necessary conditions through an adjoint (costate) system and characterizes the optimum along the trajectory. Dynamic programming, via the Hamilton-Jacobi-Bellman equation, gives a sufficient condition through a value function defined over the whole state space. The two are complementary views of the same problem.
The special linear-quadratic case
When the dynamics are linear and the cost is quadratic, optimal control has a clean closed-form answer: the linear-quadratic regulator, a constant state-feedback gain from a Riccati equation. This tractable case is the backbone of practical optimal design and the reason LQR is so widely used.
Why it unifies the field
Many control methods are optimal control in disguise. LQR minimizes a quadratic cost; the Kalman filter is optimal estimation, the dual problem; MPC solves a finite-horizon optimal control problem repeatedly online; H-infinity control minimizes a worst-case cost. Seeing them as instances of one optimization principle clarifies their relationships and trade-offs.
Optimal control's strength is that it makes objectives explicit: instead of tuning a controller until it looks acceptable, the engineer states what to minimize and derives the controller. Its challenge is that a good design still depends on choosing a cost that truly reflects the goal, which is itself a modeling art.