State-space approach to interpolation in MPC

Alternative parameterisations for predictive control: How and why?

“…Here, by allowing H L to have a totally open structure, the specific structure of (8) is lost and with it the recursive feasibilty result given in Section III-C. A simple procedure does exist in the literature to recover this guarantee, but at the cost of introducing an extra degree of freedom [9]. In simple terms, one appends the d.o.f.…”

Interpolation based computationally efficient predictive control

“…Thus unlike the case of general interpolation which allows one to use (if desired) the previously computed predicted trajectory (time shifted by one instant), co-linearity requires a new decomposition at each time instant. Therefore one cannot (Mendez et al 2000) assert a monotonicity property for the cost (as in Theorem 2) or for 2 (as in Algorithm 2). Even worse it is not even possible to guarantee that feasibility at initial time ensures feasibility at all future instants.…”

“…Due to co-linearity one may not establish for either algorithm a monotonicity of cost proof, because it is not possible to use the shift of the previously computed predicted trajectory (Mendez et al 2000, Rossiter et al 2003. Hence the user is left needing an alternative approach.…”

“…Selecting one of these predicted trajectories to be the unconstrained linear quadratic (LQ) optimal ) affords the advantage of convergence to an optimal control law. Furthermore, the inclusion into the set of predicted trajectories of the 'tail', that is, the extension (shift) to current time of the trajectory computed at the previous time instant, enables the assertion of feasibility and closed loop stability (Mendez et al 2000). Despite their advantages, these approaches require the online solution of a QP (albeit in a small number of degrees of freedom) and do not provide a characterization of the regions of attraction.…”

“…For such systems conventional MPC strategies (Mayne et al 2000) are impracticably demanding unless one uses unreasonably short horizons thereby severely curtailing the size of the stabilizable set and leading to unacceptable degrees of suboptimality. Interpolation techniques provide an obvious and useful remedy, but amongst these: (i) the ones with guarantees of feasibility and stability either have no characterization of the stabilizable set (Mendez et al 2000) or require (Bacic et al 2003) the online implementation of an SDP which could still be computationally demanding; whereas (ii) the co-linear interpolation techniques (Rossiter et al 2003) are not amenable to the recursive feasibility analysis.…”