Ye Pu

Fast Alternating Minimization Algorithm for Model Predictive Control

Abstract In this work, we apply the fast alternating minimization algorithm (FAMA) to model predictive control (MPC) problems with polytopic and second-order cone constraints. We present a splitting strategy, which speeds up FAMA by reducing each iteration to simple operations. We show that FAMA provides not only good performance for solving MPC problems when compared to other alternating direction methods, but also superior theoretical properties. Specifically, we derive complexity bounds on the number of iterations for both dual and primal variables, which are of particular relevance in the context of real-time MPC to bound the required online computation time. For MPC problems with polyhedral and ellipsoidal constraints, an off-line pre-conditioning method is presented to further improve the convergence speed of FAMA by decreasing the complexity upper-bounds and enlarging the step-size of the algorithm. Finally, we demonstrate the performance of FAMA compared to other alternating direction methods using a quadroter example.

Inexact fast alternating minimization algorithm for distributed model predictive control

This paper presents a new distributed optimization technique, the inexact fast alternating minimization algorithm (inexact FAMA), that allows for inexact local computation as well as for errors resulting from limited communication. We show that inexact FAMA is equivalent to the inexact accelerated proximal-gradient method applied to the dual problem and derive an upper-bound on the number of iterations for convergence for inexact FAMA. The second contribution of this work is that a weakened assumption for FAMA, as well as for its inexact version, is presented. The new assumption allows the strongly convex objective in the optimization problem to be subject to convex constraints, while still guaranteeing convergence of the algorithm, which facilitates its application to control problems. We apply inexact FAMA to distributed MPC problems and derive the convergence properties of the algorithm for this special case. By employing the upper-bound on the number of iterations, sufficient conditions on the errors are provided, which ensure converge of the algorithm. Finally, we demonstrate the performance of the algorithm and the theoretical findings using a randomly generated distributed MPC example.

Quantization Design for Distributed Optimization

We consider the problem of solving a distributed optimization problem using a distributed computing platform, where the communication in the network is limited: each node can only communicate with its neighbors and the channel has a limited data-rate. A common technique to address the latter limitation is to apply quantization to the exchanged information. We propose two distributed optimization algorithms with an iteratively refining quantization design based on the inexact proximal gradient method and its accelerated variant. We show that if the parameters of the quantizers, i.e., the number of bits and the initial quantization intervals, satisfy certain conditions, then the quantization error is bounded by a linearly decreasing function and the convergence of the distributed algorithms is guaranteed. Furthermore, we prove that after imposing the quantization scheme, the distributed algorithms still exhibit a linear convergence rate, and show complexity upper-bounds on the number of iterations to achieve a given accuracy. Finally, we demonstrate the performance of the proposed algorithms and the theoretical findings for solving a distributed optimal control problem.

Quantization design for unconstrained distributed optimization

We consider an unconstrained distributed optimization problem and assume that the bit rate of the communication in the network is limited. We propose a distributed optimization algorithm with an iteratively refining quantization design, which bounds the quantization errors and ensures convergence to the global optimum. We present conditions on the bit rate and the initial quantization intervals for convergence, and show that as the bit rate increases, the corresponding minimum initial quantization intervals decrease. We prove that after imposing the quantization scheme, the algorithm still provides a linear convergence rate, and furthermore derive an upper bound on the number of iterations to achieve a given accuracy. Finally, we demonstrate the performance of the proposed algorithm and the theoretical findings for solving a randomly generated example of a distributed least squares problem.

Complexity Certification of the Fast Alternating Minimization Algorithm for Linear MPC

In this technical note, the fast alternating minimization algorithm (FAMA) is proposed to solve model predictive control (MPC) problems with polytopic and second-order cone constraints. Two splitting strategies with efficient implementations for MPC problems are presented. We derive computational complexity certificates for both splitting strategies, by providing complexity upper-bounds on the number of iterations required to provide a certain accuracy of the dual function value and, most importantly, of the primal solution. This is of particular relevance in the context of real-time MPC in order to bound the required online computation time. We further address the computation of the complexity bounds, requiring the solution of a non-convex minimization problem. Finally, we demonstrate the performance of FAMA compared to other splitting methods using a quadrotor example.

Operator Splitting Methods in Control

The significant progress that has been made in recent years both in hardware implementations and in numerical computing has rendered real-time optimization-based control a viable option when it comes to advanced industrial applications. More recently, the need for control of a process in the presence of a limited amout of hardware resources has triggered research in the direction of embedded optimization-based control. At the same time, and standing at the other side of the spectrum, the field of big data has emerged, seeking for solutions to problems that classical optimization algorithms are incapable to provide. This triggered some interest to revisit the family of first order methods commonly known as decomposition schemes or operator splitting methods. Although it is established that splitting methods are quite beneficial when applied to large-scale problems, their potential in solving small to medium scale embedded optimization problems has not been studied so extensively. Our purpose is to study the behavior of such algorithms as solvers of control-related problems of that scale. Our effort focuses on identifying special characteristics of these problems and how they can be exploited by some popular splitting methods. G. Stathopoulos, H. Shukla, A. Szűcs, Y. Pu and C. N. Jones. Operator Splitting Methods in Control. Foundations and Trends

A consensus algorithm for networks with process noise and quantization error

In this paper we address the problem of quantized consensus where process noise or external inputs corrupt the state of each agent at each iteration. We propose a quantized consensus algorithm with progressive quantization, where the quantization interval changes in length at each iteration by a pre-specified value. We derive conditions on the design parameters of the algorithm to guarantee ultimate boundedness of the deviation from the average of each agent. Moreover, we determine explicitly the bounds of the consensus error under the assumption that the process disturbances are ultimately bounded within known bounds. A numerical example of cooperative path-following of a network of single integrators illustrates the performance of the proposed algorithm.

Splitting methods in control

The need for optimal control of processes under a restricted amount of resources renders first order optimization methods a viable option. Although computationally cheap, these methods typically suffer from slow convergence rates. In this work we discuss the family of first order methods known as decomposition schemes. We present three popular methods from this family, draw the connections between them and report all existing results that enable acceleration in terms of the convergence rate. The approach for splitting a problem into simpler ones so that the accelerated variants can be applied is also discussed and demonstrated via an example.

Quantization design for distributed optimization with time-varying parameters

We consider the problem of solving a sequence of distributed optimization problems with time-varying parameters and communication constraints, i.e. only neighbour-to-neighbour communication and a limited amount of information exchanged. By extending previous results and employing a warm-starting strategy, we propose an on-line algorithm for solving optimization problems under the given constraints and show that there exists a trade-off between the number of iterations for solving each problem in the sequence and the accuracy achieved by the algorithm. For a given accuracy ∈, we can find a number of iterations K, which guarantees that for the sequential realization of the parameter, the sub-optimal solution given by the algorithm satisfies the accuracy. We apply the method to solve a distributed model predictive control problem by considering the state measurement at each sampling time as the time-varying parameter and show that the simulation supports the theoretical results.

Design of a distributed quantized luenberger filter for bounded noise

This paper addresses the problem of distributed state estimation for linear systems with process and measurement noise, in the case of limited communication data rate, where the data exchanged between agents is quantized. We propose a linear distributed Luenberger observer and we derive a set of conditions on the design parameters of the quantizer to guarantee ultimate boundedness of the estimation error. The latter is shown to depend on the L2 norm of the disturbance signals and on the number of bits. A numerical example illustrates the performance of the proposed algorithm.