A Parametric Multi-Convex Splitting Technique with Application to Real-Time NMPC

A novel splitting scheme to solve parametric multiconvex programs is presented. It consists of a fixed number of proximal alternating minimisations and a dual update per time step, which makes it attractive in a real-time Nonlinear Model Predictive Control (NMPC) framework and for distributed computing environments. Assuming that the parametric program is semi-algebraic and that its KKT points are strongly regular, a contraction estimate is derived and it is proven that the sub-optimality error remains stable if two key parameters are tuned properly. Efficacy of the method is demonstrated by solving a bilinear NMPC problem to control a DC motor.Comment: To appear in Proceedings of the 53rd IEEE Conference on Decision and Control 201

High performance implementation of MPC schemes for fast systems

In recent years, the number of applications of model predictive control (MPC) is rapidly increasing due to the better control performance that it provides in comparison to traditional control methods. However, the main limitation of MPC is the computational e ort required for the online solution of an optimization problem. This shortcoming restricts the use of MPC for real-time control of dynamic systems with high sampling rates. This thesis aims to overcome this limitation by implementing high-performance MPC solvers for real-time control of fast systems. Hence, one of the objectives of this work is to take the advantage of the particular mathematical structures that MPC schemes exhibit and use parallel computing to improve the computational e ciency. Firstly, this thesis focuses on implementing e cient parallel solvers for linear MPC (LMPC) problems, which are described by block-structured quadratic programming (QP) problems. Speci cally, three parallel solvers are implemented: a primal-dual interior-point method with Schur-complement decomposition, a quasi-Newton method for solving the dual problem, and the operator splitting method based on the alternating direction method of multipliers (ADMM). The implementation of all these solvers is based on C++. The software package Eigen is used to implement the linear algebra operations. The Open Message Passing Interface (Open MPI) library is used for the communication between processors. Four case-studies are presented to demonstrate the potential of the implementation. Hence, the implemented solvers have shown high performance for tackling large-scale LMPC problems by providing the solutions in computation times below milliseconds. Secondly, the thesis addresses the solution of nonlinear MPC (NMPC) problems, which are described by general optimal control problems (OCPs). More precisely, implementations are done for the combined multiple-shooting and collocation (CMSC) method using a parallelization scheme. The CMSC method transforms the OCP into a nonlinear optimization problem (NLP) and de nes a set of underlying sub-problems for computing the sensitivities and discretized state values within the NLP solver. These underlying sub-problems are decoupled on the variables and thus, are solved in parallel. For the implementation, the software package IPOPT is used to solve the resulting NLP problems. The parallel solution of the sub-problems is performed based on MPI and Eigen. The computational performance of the parallel CMSC solver is tested using case studies for both OCPs and NMPC showing very promising results. Finally, applications to autonomous navigation for the SUMMIT robot are presented. Specially, reference tracking and obstacle avoidance problems are addressed using an NMPC approach. Both simulation and experimental results are presented and compared to a previous work on the SUMMIT, showing a much better computational e ciency and control performance.Tesi

A parametric augmented Lagrangian algorithm for real-time economic NMPC

In this paper, a novel optimality-tracking algorithm for solving Economic Nonlinear Model Predictive Control (ENMPC) problems in real-time is presented. Developing online schemes for ENMPC is challenging, since it is unclear how convexity of the Quadratic Programming (QP) problem, which is obtained by linearisation of the NMPC program around the current iterate, can be enforced efficiently. Therefore, we propose addressing the problem by means of an augmented Lagrangian formulation. Our tracking scheme consists of a fixed number of inexact Newton steps computed on an augmented Lagrangian subproblem followed by a dual update per time step. Under mild assumptions on the number of iterations and the penalty parameter, it can be proven that the sub-optimality error provided by the parametric algorithm remains bounded over time. This result extends the authors' previous works from a theoretical and a computational perspective. Efficacy of the approach is demonstrated on an ENMPC example consisting of a bioreactor

Algorithms and Applications for Nonlinear Model Predictive Control with Long Prediction Horizon

Fast implementations of NMPC are important when addressing real-time control of systems exhibiting features like fast dynamics, large dimension, and long prediction horizon, as in such situations the computational burden of the NMPC may limit the achievable control bandwidth. For that purpose, this thesis addresses both algorithms and applications. First, fast NMPC algorithms for controlling continuous-time dynamic systems using a long prediction horizon have been developed. A bridge between linear and nonlinear MPC is built using partial linearizations or sensitivity update. In order to update the sensitivities only when necessary, a Curvature-like measure of nonlinearity (CMoN) for dynamic systems has been introduced and applied to existing NMPC algorithms. Based on CMoN, intuitive and advanced updating logic have been developed for different numerical and control performance. Thus, the CMoN, together with the updating logic, formulates a partial sensitivity updating scheme for fast NMPC, named CMoN-RTI. Simulation examples are used to demonstrate the effectiveness and efficiency of CMoN-RTI. In addition, a rigorous analysis on the optimality and local convergence of CMoN-RTI is given and illustrated using numerical examples. Partial condensing algorithms have been developed when using the proposed partial sensitivity update scheme. The computational complexity has been reduced since part of the condensing information are exploited from previous sampling instants. A sensitivity updating logic together with partial condensing is proposed with a complexity linear in prediction length, leading to a speed up by a factor of ten. Partial matrix factorization algorithms are also proposed to exploit partial sensitivity update. By applying splitting methods to multi-stage problems, only part of the resulting KKT system need to be updated, which is computationally dominant in on-line optimization. Significant improvement has been proved by giving floating point operations (flops). Second, efficient implementations of NMPC have been achieved by developing a Matlab based package named MATMPC. MATMPC has two working modes: the one completely relies on Matlab and the other employs the MATLAB C language API. The advantages of MATMPC are that algorithms are easy to develop and debug thanks to Matlab, and libraries and toolboxes from Matlab can be directly used. When working in the second mode, the computational efficiency of MATMPC is comparable with those software using optimized code generation. Real-time implementations are achieved for a nine degree of freedom dynamic driving simulator and for multi-sensory motion cueing with active seat

Numerical Methods for PDE Constrained Optimization with Uncertain Data

Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization. The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods

Nonlinear Dynamic Analysis and Control of Chemical Processes Using Dynamic Operability

Nonlinear dynamic analysis serves an increasingly important role in process systems engineering research. Understanding the nonlinear dynamics from the mathematical model of a process helps to find the boundaries of all achievable process conditions and identify the system instabilities. The information on such boundaries is beneficial for optimizing the design and formulating a control structure. However, a systematic approach to analyzing nonlinear dynamics of chemical processes considering such boundaries in a quantifiable and adaptable way is yet to exist in the literature. The primary aim of this work is to formulate theoretical concepts for dynamic operability, as well as develop the practical implementation methods for the analysis of dynamic performance in chemical processes. Process operability is a powerful tool for analyzing the relationships between the input variables, the output variables, and the disturbances via the geometric computation of variable sets. The operability sets are described by unions of polyhedra, which can be translated to sets of inequality constraints, so the results of the operability analysis can be used for process optimization and advanced process control. Nonetheless, existing process operability approaches in the literature are currently limited for steady-state processes and a generalized definition of dynamic operability that retains the core principles of steady-state operability as a controllability measure. A unified dynamic operability concept is proposed in this dissertation with two different adaptations to represent the complex relationships between the design, control structure, and control law of a given process. The existing operability mapping methods discretize the input space by partitioning the ranges of each input variable evenly, and all possible input combinations are simulated to achieve the output sets. The procedure is repeated for each value in the expected disturbance set to find the output regions that are guaranteed to be achieved regardless of the disturbance scenario. However, for dynamic systems, the same set of manipulated inputs can take different values at different time intervals, so the number of possible input combinations, which is also the number of simulations required, increases exponentially with the number of time intervals. This tractability challenge motivates the development of novel dynamic operability mapping approaches. A linear time-invariant dynamic system is first considered to tackle the dynamic mapping of achievable output sets. For a linear system, the achievable output set (AOS) at a fixed predicted time is the smallest convex hull that contains all the images of the extreme points of the available input set (AIS) when propagated through the dynamic model. Given a collection of AOS’s at all predicted times, referred to as the achievable funnel, a set of output constraints is infeasible if its intersection with the achievable funnel is empty. Under the influence of a stochastic disturbance, the achievable funnel is shifted according to the definition of the expected disturbance set (EDS). If the EDS is bounded, the intersection of all achievable funnels at each disturbance realization is the tightest set of transient output constraints that is operable. Additionally, given a fixed setpoint, an AOS is referred to as a feasible AOS if a series of inputs from the AIS always brings any output to the setpoint regardless of the realization of the disturbance within the EDS. Thus, novel developed theories and algorithms to update the dynamic operability mapping according to the current state variables and the disturbance propagations are proposed to reduce the online computational time of the constraint calculation task. Dynamic operability mapping for nonlinear processes is an expansion of the above linear mapping. A novel state-space projection mapping is proposed by taking advantage of the discrete-time state-space structure of the dynamic model to reduce the number of input mapping combinations. This method augments the AIS at the current step to include the AOS of the state variables from the previous time step. The nonlinear dynamic operability mapping framework consists of three components: the AOS inspector, the AIS divider, and the merger of the AOS from the previous time with the AIS. Specifically, the AOS inspector evaluates if the current input-output combinations are approximately accurate to the real AOS when all input combinations are mapped to the output space. If the AOS inspector gauges that the current AOS is not sufficiently precise, the AIS divider systematically generates more input-output combinations based on the current AOS. This feedback process is repeated until an accuracy tolerance is reached. Finally, a novel grey-box model identification algorithm for process control is developed by integrating dynamic operability mapping and Bayesian calibration. The proposed dynamic discrepancy reduced-order model-based approach calibrates the rates of changes of the grey-box model to match the plant instead of compensating for the time-varying output differences. The model reduction framework is divided into three steps: formulating the dynamic discrepancy terms, calibrating the hyperparameters, and selecting the least complex model that is neither underfitted nor overfitted. To demonstrate the effectiveness of the reduced-order model, the developed approach is implemented into a model predictive controller for a high-fidelity model as the simulated plant

A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

In this chapter an algorithm for nonlinear explicit model predictive control is presented. A low complexity receding horizon control law is obtained by approximating the optimal control law using multiscale basis function approximation. Simultaneously, feasibility and stability of the approximate control law is ensured through the computation of a capture basin (region of attraction) for the closed-loop system. In a previous work, interval methods were used to construct the capture basin (feasible region), yet this approach suffered due to slow computation times and high grid complexity. In this chapter, we suggest an alternative to interval analysis based on zonotopes. The suggested method significantly reduces the complexity of the combined function approximation and verification procedure through the use of DC (difference of convex) programming, and recursive splitting. The result is a multiscale function approximation method with improved computational efficiency for fast nonlinear explicit model predictive control with guaranteed stability and constraint satisfaction