Structure-Exploiting Numerical Algorithms for Optimal Control

Abstract

Cover illustration: An illustration of the structure of the Karush-Kuhn-Tucker matrix for the unconstrained finite-time optimal control problems considered in this thesis. Thank you Jonas Linder for helping me with the design.Linköping studies in science and technology. Dissertations.No. 1848 Tillägnas hela min underbara familj Structure-Exploiting Numerical Algorithms for Optimal Control AbstractNumerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (mpc), but can also be useful for advanced estimation techniques, such as moving horizon estimation (mhe). In mpc, the control input is computed by solving a constrained finite-time optimal control (cftoc) problem on-line, and in mhe the estimated states are obtained by solving an optimization problem that often can be formulated as a cftoc problem. Common types of optimization methods for solving cftoc problems are interior-point (ip) methods, sequential quadratic programming (sqp) methods and active-set (as) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (uftoc) problems. Hence, high-performing second-order methods for cftoc problems rely on efficient numerical algorithms for solving uftoc problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a cftoc problem is computed using an as type method, the aforementioned system of equations is only changed by a low-rank modification between two as iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the uftoc problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in ip, sqp and as methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the uftoc problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear mpc problems, an alternative approach to solving the cftoc problem on-line is to use multiparametric quadratic programming (mp-qp), where the corresponding cftoc problem can be solved explicitly off-line. This is referred to as explicit mpc. One of the main limitations with mp-qp is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-qp and explicit mpc more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the qp problem in the parametric solution in order to reduc...

Exploiting Chordality in Optimization Algorithms for Model Predictive Control

“…Also it is worth pointing out that the subproblem which is assigned to the root of clique tree can be seen as an LQ problem and hence we can use the procedure discussed above recursively. This is similar to what is presented in [24]. This is obtained in the example above by not connecting all of x 0 ,ū 0 , x 3 ,ū 1 and x 6 .…”

Direct Parallel Computations of Second-Order Search Directions for Model Predictive Control

“…In the general case, time-dependent matrices, linear and constant terms in the objective function, and an affine term in the dynamics equation 0000-0000/00$00.00 c year IEEE are present. However, these are omitted here for presentational brevity and the reader is instead referred to [22] for the details when they are present. The UFTOC problem is then given by min.…”

“…One way of solving the UFTOC problem (1) is to use the Riccati recursion, see for instance [1], [2], [5], [22]. It consists of a factorization of the KKT matrix (see, e.g., [23]) of the UFTOC problem (Algorithm 1) followed by a state recursion to compute the solution (Algorithm 2).…”

“…This approach avoids computing the orthonormal basis V which is required in Lemma 6. How to compute the Riccati factorization when G t+1 is singular due to Assumption 3 is shown in for example [5], [22]. To derive this alternative reduction strategy, Lemma 7 will be used.…”

“…Furthermore, in the algorithms presented here the solution is computed directly without involving any iterative updates of variables as is usually required in parallel first-order methods such as the alternating direction method of multipliers [21]. A more extensive and detailed presentation of the work presented here is available in the publication [22].…”