Minimising conservatism in infinite-horizon LQR control

A Microcontroller-Based Adaptive Model Predictive Control Platform for Process Control Applications

“…In theory, when an online optimal control features input constraints, a constrained optimization problem of infinite size must be solved at each time step. However, recent results (e.g., refer to [9] and the references therein) have shown that the constraints may be relaxed for some upper constraint bound N, under the assumption that the process enters a terminal state upon which the constraints are no longer active. Although this results in a finite dimension optimization problem, the bound N can be large enough to rule out feasible on-line optimization in many cases [9].…”

“…However, recent results (e.g., refer to [9] and the references therein) have shown that the constraints may be relaxed for some upper constraint bound N, under the assumption that the process enters a terminal state upon which the constraints are no longer active. Although this results in a finite dimension optimization problem, the bound N can be large enough to rule out feasible on-line optimization in many cases [9]. In the non-adaptive case, there are several ways to overcome this problem; for example, offline multi-parametric QP [1,2], dynamic programming [10], and partially precomputed active set methods [11] can produce control laws which essentially result in state look-up tables.…”

“…However, these methods require exponential time (normally O(3 N )) to determine the control law and are not suitable for on-line implementation in most adaptive situations or where the constraint horizon N is large. In other related works, however, it has been shown that the saturated linear quadratic regulator (LQR) control law is optimal in the presence of input constraints in many practical situations [9,12]. Importantly, saturated LQR is optimal for the class of first-order sampled data systems without delay [9].…”

“…In other related works, however, it has been shown that the saturated linear quadratic regulator (LQR) control law is optimal in the presence of input constraints in many practical situations [9,12]. Importantly, saturated LQR is optimal for the class of first-order sampled data systems without delay [9]. This result was leveraged by Short in [13] to create an efficient adaptive LQR controller for smart home applications; a drawback of the approach was that it was only applicable to process models of a first-order plus dead time (FOPDT) type.…”

Constrained linear time-varying quadratic regulation with guaranteed optimality

“…The special case of the above problem is linear time invariant quadratic regulation (LTIQR). LTIQR has been solved, the unconstrained case for several decades and the constrained case (CLTIQR) in recent years (see e.g., Sznaier and Damborg 1987, Choi et al 2000, Bemporad et al 2002, Marjanovic et al 2002. Unconstrained LTIQR is easily implemented through a constant linear state feedback that is derived by the solution of the algebraic Riccati equation.…”