Multi-parametric Quadratic Programming Research Articles

Solution of the input-constrained LQR problem using dynamic programming

The input-constrained LQR problem is addressed in this paper; i.e., the problem of finding the optimal control law for a linear system such that a quadratic cost functional is minimised over a horizon of length N subject to the satisfaction of input constraints. A global solution (i.e., valid in the entire state space) for this problem, and for arbitrary horizon N, is derived analytically by using dynamic programming. The scalar input case is considered in this paper. Solutions to this problem (and to more general problems: state constraints, multiple inputs) have been reported recently in the literature, for example, approaches that use the geometric structure of the underlying quadratic programming problem and approaches that use multi-parametric quadratic programming techniques. The solution by dynamic programming proposed in the present paper coincides with the ones obtained by the aforementioned approaches. However, being derived using a different approach that exploits the dynamic nature of the constrained optimisation problem to obtain an analytical solution, the present result complements the previous methods and reveals additional insights into the intrinsic structure of the optimal solution.

FEASIBLE MODE ENUMERATION AND COST COMPARISON FOR EXPLICIT QUADRATIC MODEL PREDICTIVE CONTROL OF HYBRID SYSTEMS

For hybrid systems in piecewise affine (PWA) form, this paper presents a new methodology for computing the solution, defined over a set of (possibly overlapping) polyhedra, of the finite-time constrained optimal control problem based on quadratic costs. First, feasible mode sequences are determined via backward reachability analysis, and multiparametric quadratic programming is employed to determine candidate polyhedral regions of the solution and the corresponding value functions and optimal control gains. Then, the value functions associated with overlapping regions are compared in order to discard those regions whose associated control law is never optimal. The comparison problem is, in general, nonconvex and is tackled here as a DC (Difference of Convex functions) programming problem.

An algorithm for reducing complexity in parametric predictive control

Multi parametric quadratic programming solutions to predictive control can require large numbers of regions which would make practical implementation difficult. This paper demonstrates how an underexplored prediction class can be combined with parametric programming to give far simpler solutions and at little cost to optimality. The key proposal is to explore how one can give more flexibility in the shape of the predictions without increasing the required numbers of degrees of freedom; this flexibility allows terminal regions to be enlarged.

Spacecraft attitude control using explicit model predictive control

In this paper, an explicit model predictive controller for the attitude of a satellite is designed. Explicit solutions to constrained linear MPC problems can be computed by solving multi-parametric quadratic programs (mpQP), where the parameters are the components of the state vector. The solution to the mpQP is a piecewise affine (PWA) function, which can be evaluated at each sample to obtain the optimal control law. The on-line computation effort is restricted to a table-lookup, and the controller can be implemented on inexpensive hardware as fixed-point arithmetics can be used. This is useful for systems with limited power and CPU resources. An example of such systems is micro-satellites, which is the focus of this paper. In particular, the explicit MPC (eMPC) approach is applied to the SSETI/ESEO micro-satellite, initiated by the European Space Agency (esa). The theoretical results are supported by simulations.

Efficient Optimal Constrained Control Allocation via Multiparametric Programming

Constrained control allocation is studied, and it is shown how an explicit piecewise linear representation of the optimal solution can be computed numerically using multiparametric quadratic programming. Practical benefits of the approach include simple and efficient real-time implementation that permits software verifiability. Furthermore, it is shown how to handle control deficiency, reconfigurability, and flexibility to incorporate, for example, rate constraints. The algorithm is demonstrated on several overactuated aircraft control configurations, and the computational complexity is compared to other explicit approaches from the literature. The applicability of the method is further demonstrated using overactuated marine vessel dynamic position experiments on a scale model in a basin.

A MULTI PARAMETRIC QUADRATIC PROGRAMMING SOLUTION TO ROBUST PREDICTIVE CONTROL

Multi parametric quadratic programming is an alternative means of implementing conventional predictive control algorithms whereby one transfers much of the computational load to offline calculations. This paper demonstrates how one can formulate a robust MPC problem as a quadratic program and hence make it amenable to MPQP solutions. The paper then derives some MPQP solutions and discusses the efficacy of these.

MIN-MAX MODEL PREDICTIVE CONTROL AS A QUADRATIC PROGRAM

This paper deals with the implementation of min-max model predictive control for constrained linear systems with bounded additive uncertainties and quadratic cost functions. This type of controller has been shown to be a continuous piecewise affine function of the state vector by geometrical methods. However, no algorithm for computing the explicit solution has been given. In this paper, we show that the min-max optimization problem can be expressed as a multi-parametric quadratic program, and so, the explicit form of the controller may be determined by standard multi-parametric techniques.

Computation of the constrained infinite time linear quadratic regulator

This paper presents an efficient algorithm for computing the solution to the constrained infinite-time, linear quadratic regulator (CLQR) problem for discrete time systems. The algorithm combines multi-parametric quadratic programming with reachability analysis to obtain the optimal piecewise affine (PWA) feedback law. The algorithm reduces the time necessary to compute the PWA solution for the CLQR when compared to other approaches. It also determines the minimal finite horizon N ̄ S , such that the constrained finite horizon LQR problem equals the CLQR problem for a compact set of states S . The on-line computational effort for the implementation of the CLQR can be significantly reduced as well, either by evaluating the PWA solution or by solving the finite dimensional quadratic program associated with the CLQR for a horizon of N= N ̄ S .

Optimal constrained control allocation in marine surface vessels with rudders

A propeller with a rudder can produce a thrust vector within a range of directions and magnitudes in the horizontal plane for low-speed maneuvering and dynamic positioning. The set of attainable thrust vectors is non-convex because significant lift can be produced by the rudder only with forward thrust. We suggest to decompose the attainable thrust region into a finite union of convex polyhedral sets and derive a mixed-integer-like convex quadratic programming formulation of the optimal control allocation problem for marine vessels with any number of rudders as well as thrusters and other propulsion devices. Actuator rate and position constraints are explicitly taken into account. Using multi-parametric quadratic programming software, an explicit piecewise linear representation of the least-squares optimal control allocation law is pre-computed. It can be implemented with low computational complexity and high software reliability without the use of real-time optimization. The method is illustrated using a scale model ship in a basin.

Reduced explicit constrained linear quadratic regulators

In this note, it is studied how structural properties of certain linear systems can be exploited to derive reduced dimension multiparametric quadratic programs that lead to explicit piecewise linear feedback solutions to the state and input constrained linear quadratic regulation problem. The reduced dimensionality typically results in suboptimal controllers of lower complexity, with associated computational advantages in the online implementation. At the heart of the methods are state-space projections based on the singular value decomposition.

Suboptimal Explicit Receding Horizon Control via Approximate Multiparametric Quadratic Programming

Algorithms for solving multiparametric quadratic programming (MPQP) were recently proposed in Refs. 1–2 for computing explicit receding horizon control (RHC) laws for linear systems subject to linear constraints on input and state variables. The reason for this interest is that the solution to MPQP is a piecewise affine function of the state vector and thus it is easily implementable online. The main drawback of solving MPQP exactly is that, whenever the number of linear constraints involved in the optimization problem increases, the number of polyhedral cells in the piecewise affine partition of the parameter space may increase exponentially. In this paper, we address the problem of finding approximate solutions to MPQP, where the degree of approximation is arbitrary and allows to tradeoff between optimality and a smaller number of cells in the piecewise affine solution. We provide analytic formulas for bounding the errors on the optimal value and the optimizer, and for guaranteeing that the resulting suboptimal RHC law provides closed-loop stability and constraint fulfillment.

An algorithm for multi-parametric quadratic programming and explicit MPC solutions

Explicit solutions to constrained linear model predictive control problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the components of the state vector. We study the properties of the polyhedral partition of the state space induced by the multi-parametric piecewise affine solution and propose a new mp-QP solver. Compared to existing algorithms, our approach adopts a different exploration strategy for subdividing the parameter space, avoiding unnecessary partitioning and QP problem solving, with a significant improvement of efficiency.

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.

A multiparametric programming approach for mixed-integer quadratic engineering problems

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.

COMPLEXITY REDUCTION IN EXPLICIT LINEAR MODEL PREDICTIVE CONTROL

Explicit piecewise linear (PWL) state feedback laws solving constrained linear model predictive control (MPC) problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the elements of the state vector. This allows MPC to be implemented via a PWL function evaluation without real-time optimization. The main drawback of this approach is dramatic increase in the number of regions in the state space partition as the number of states, inputs and constraints increases. Here we study two approaches to complexity reduction. First, we consider input trajectory parameterization which significantly reduces the number of regions. Second, we develop a search tree that allows PWL function evaluation to be implemented in real time with low computational complexity.

On-line optimization via off-line parametric optimization tools

In this paper, on-line optimization problems with a quadratic performance criteria and linear constraints are formulated as multi-parametric quadratic programs, where the input and state variables, corresponding to a plant, 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, at each time interval (i) no optimization solver is called on-line, (ii) simple function evaluations are required for obtaining the optimal inputs to the plant for the current state of the plant.