Vivek Dua

The explicit linear quadratic regulator for constrained systems

We present a technique to compute the explicit state-feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints. We show that this closed form solution is piecewise linear and continuous. As a practical consequence of the result, constrained linear quadratic regulation becomes attractive also for systems with high sampling rates, as on-line quadratic programming solvers are no more required for the implementation.

An Algorithm for the Solution of Multiparametric Mixed Integer Linear Programming Problems

In this paper, we present an algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems involving (i) 0-1 integer variables, and, (ii) more than one parameter, bounded between lower and upper bounds, present on the right hand side (RHS) of constraints. The solution is approached by decomposing the mp-MILP into two subproblems and then iterating between them. The first subproblem is obtained by fixing integer variables, resulting in a multiparametric linear programming (mp-LP) problem, whereas the second subproblem is formulated as a mixed integer linear programming (MILP) problem by relaxing the parameters as variables.

On-line optimization via off-line parametric optimization tools

In this paper, model predictive control (MPC) based optimization problems with a quadratic performance criterion and linear constraints are formulated as multi-parametric quadratic programs (mp-QP), where the input and state variables, corresponding to a plant model, are treated as optimization variables and parameters, respectively. The solution of such problems is given by (i) a complete set of profiles of all the optimal inputs to the plant as a function of state variables, and (ii) the regions in the space of state variables where these functions remain optimal. It is shown that these profiles are linear and the corresponding regions are described by linear inequalities. An algorithm for obtaining these profiles and corresponding regions of optimality is also presented. The key feature of the proposed approach is that the on-line optimization problem is solved off-line via parametric programming techniques. Hence (i) no optimization solver is called on-line, and (ii) only simple function evaluations are required, to obtain the optimal inputs to the plant for the current state of the plant.

The explicit solution of model predictive control via multiparametric quadratic programming

The control based on online optimization, popularly known as model predictive control (MPC), has long been recognized as the winning alternative for constrained systems. The main limitation of MPC is, however, its online computational complexity. For discrete-time linear time-invariant systems with constraints on inputs and states, we develop an algorithm to determine explicitly the state feedback control law associated with MPC, and show that it is piecewise linear and continuous. The controller inherits all the stability and performance properties of MPC, but the online computation is reduced to a simple linear function evaluation instead of the expensive quadratic program. The new technique is expected to enlarge the scope of applicability of MPC to small-size/fast-sampling applications which cannot be covered satisfactorily with anti-windup schemes.

A multiparametric programming approach for mixed-integer quadratic engineering problems

Abstract In this work we propose algorithms for the solution of multiparametric quadratic programming (mp-QP) problems and multiparametric mixed-integer quadratic programming (mp-MIQP) problems with a convex and quadratic objective function and linear constraints. For mp-QP problems it is shown that the optimal solution, i.e. the vector of continuous variables and Lagrange multipliers, is an affine function of parameters. The basic idea of the algorithm is to use this affine expression for the optimal solution to systematically characterize the space of parameters by a set of regions of optimality. The solution of the mp-MIQP problems is approached by decomposing it into two subproblems, which converge based upon an iterative methodology. The first subproblem, which is an mp-QP, is obtained by fixing the integer variables and its solution represents a parametric upper bound. The second subproblem is formulated as a mixed-integer non-linear programming (MINLP) problem and its solution provides a new integer vector, which can be fixed to obtain a parametric solution, which is better than the current upper bound. The algorithm terminates with an envelope of parametric profiles corresponding to different optimal integer solutions. Examples are presented to illustrate the basic ideas of the algorithms and their application in model predictive and hybrid control problems.

A bilevel programming framework for enterprise-wide process networks under uncertainty

Abstract Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where for example, one level may correspond to a local plant control/scheduling/planning problem and another level to a corresponding plant-wide planning/network problem. Such a multi-level decision network structure can be mathematically represented by using multi-level programming principles. In this paper, we specifically address bilevel decision-making problems under uncertainty in the context of enterprise-wide supply chain optimization with one level corresponding to a plant planning problem, while the other to a distribution network problem. We first describe how such problems can be modelled as bilevel programming problems and then we present an effective solution strategy based on parametric programming techniques. An attractive feature of the proposed strategy is the fact that it transforms the bilevel problem into a family of single parametric optimization problems, which can be solved to global optimality. A numerical example is presented to illustrate the proposed framework.

Design of robust model-based controllers via parametric programming

In this paper a method is presented for deriving the explicit robust model-based optimal control law for constrained linear dynamic systems. The controller is derived off-line via parametric programming before any actual process implementation takes place. The proposed control scheme guarantees feasible operation in the presence of bounded input uncertainties by (i) explicitly incorporating in the controller design stage a set of feasibility constraints and (ii) minimizing the nominal performance, or the expectation of the performance over the uncertainty space. An extension of the method to problems involving target point tracking in the presence of persistent disturbances is also discussed. The general concept is illustrated with two examples.

Global Optimization Issues in Multiparametric Continuous and Mixed-Integer Optimization Problems

In this paper, a number of theoretical and algorithmic issues concerning the solution of parametric nonconvex programs are presented. In particular, the need for defining a suitable overestimating subproblem is discussed in detail. The multiparametric case is also addressed, and a branch and bound (B&B) algorithm for the solution of parametric nonconvex programs is proposed.

Optimization techniques for process synthesis and material design under uncertainty

This paper describes developments in parametric mixed integer optimization for the solution of process synthesis and material design problems in the presence of uncertainty. For mixed integer linear models (MILP), such as the ones that arise in material design problems, a multiparametric MILP approach is described and illustrated with an example, whereas for mixed integer nonlinear convex models (MINLP), a parametric MINLP algorithm is presented and applied to a process synthesis example problem.

The Explicit Control Law for Hybrid Systems via Parametric Programming

In this paper an algorithm is presented for the derivation of the explicit optimal control policy for linear dynamical systems that also involve (i) logical decisions and (ii) constraints on process inputs and outputs. The control actions are usually computed by solving at regular time intervals an on-line optimization problem based on a set of measurements that specify the current process state. The approach presented in this paper derives the optimal control law off-line as a function of the state of the process, thus mating the repetitive solution of on-line optimization prob lems. Hence, the on-line implementation is reduced to a sequence of simple function evaluations. The key advantageous features of the algorithm are demonstrated via an illustrative example.