## Invited Session Tue.3.H 0112

#### Tuesday, 15:15 - 16:45 h, Room: H 0112

**Cluster 16: Nonlinear programming** [...]

### Real-time optimization III

**Chair: Victor M. Zavala and Sebastian Sager**

**Tuesday, 15:15 - 15:40 h, Room: H 0112, Talk 1**

**Markus Kögel**

On real-time optimization for model predictive control using multiplier methods and Nesterov’s gradient method

**Coauthor: Rolf Findeisen**

**Abstract:**

Model predictive control is an optimization based approach in automatic control to control systems. It allows taking constraints explicitly into account while optimizing the performance. Model predictive control requires solving in real-time optimization problem each time a new measurement becomes available.

We focus on the important special case of linear plants, quadratic cost criterions and convex constraints, in which the optimization problems are quadratic programs with a special structure. Although, multiple efficient algorithms exist by now, model predictive control is still challenging for fast, large systems or on embedded systems with limited computing power.

Therefore we present approaches using multiplier methods and Nesterov’s gradient method, which allow efficient real-time optimization. In particular, we outline how the solution can be parallelized or distributed. This enables the use of multiple processor cores or even multiples computers to decrease the solution time.

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We illustrate the proposed algorithms using application examples.

**Tuesday, 15:45 - 16:10 h, Room: H 0112, Talk 2**

**Gabriele Pannocchia**

On the convergence of numerical solutions to the continuous-time constrained LQR problem

**Coauthors: Mayne Q. David, Rawlings B. James**

**Abstract:**

A numerical procedure for computing the solution to the continuous-time infinite-horizon constrained linear quadratic regulator was presented in [1], which is based successive strictly convex QP problems where the decision variables are the control input value and slope at selected grid points. Each QP generates an upper bound to the optimal cost, and the accuracy is increased by using gradually refined grids computed offline to avoid any online integration. In this work we propose an adaptive method to gradually refine the grid where it is most needed, still without having to perform integration online, and we address the convergence properties of such algorithm as the number of grid points is increased. By means of suitable optimality functions, at each iteration given the current upper bound cost, we compute: (i) a lower bound approximation of the optimal cost which can be used to stop the algorithm within a guaranteed tolerance; (ii) for each grid interval, an estimate of the cost reduction that can obtained by bisecting it. Examples are presented.

- G. Pannocchia, J.B. Rawlings, D.Q. Mayne, W. Marquardt, IEEE Trans. Auto. Contr. 55 (2010), pp. 2192-2198.

**Tuesday, 16:15 - 16:40 h, Room: H 0112, Talk 3**

**Eric Kerrigan**

Breaking away from double-precision floating-point in interior point solvers

**Coauthors: George A. Constantinides, Juan L. Jerez, Stefano Longo**

**Abstract:**

We will show how one can modify interior point methods for solving constrained linear quadratic control problems in computing hardware with a fixed-point number representation or with significantly less bits than in single- or double-precision floating-point. This allows one to dramatically reduce the computational resources, such as time, silicon area and power, needed to compute the optimal input sequence at each sample instant. For fixed precision, we propose a simple pre-conditioner, which can be used with iterative linear solvers such as CG or MINRES, that allows one to compute tight bounds on the ranges of the variables in the Lanczos iteration, thereby allowing one to determine the best position of the radix point. To allow one to reduce the number of bits needed, we propose the use of the delta transform of Middleton and Goodwin in order to avoid numerical errors that would occur when using the usual shift transform to discretize the continuous-time optimal control problem. We also propose a Riccati method, tailored to the delta transform, for efficiently solving the resulting KKT systems that arise within an interior point solver.