Robust nonlinear MPC for tracking piece-wise constant reference signals

This book provides a comprehensive study of nonlinear adaptive robust model predictive control (MPC). Chapters 2–5 present a framework for the analysis and synthesis of nonlinear robust MPC. This framework includes the treatment of robustness, computation methods, and performance improvement. Chapters 6–7 show how to develop the basic ideas for the design and analysis of the nonlinear adaptive robust MPC. One of the key techniques is the set-based approach, in which the internal model identifier allows the MPC to compensate for future changes in the parameter estimates and uncertainty associated with the unknown model parameters. Chapters 8–12 illustrate how to implement the synthesis approaches for nonlinear adaptive robust MPC, and a robust adaptive economic MPC is also proposed. This text also gives a finite-time identification method, which can be used to estimate the unknown parameters in finite time, provided a persistence of excitation (PE) condition is satisfied. This identification method is particularly effective in the online implementation of MPC. The early chapters study continuous-time systems, and Chapters 13–14 extend the set-based estimation and robust adaptive MPC to discrete-time problems. While adaptive robust MPC is an improvement on robust MPC, this book shows that feedback MPC can be used to improve the open-loop MPC. At each sampling instant, a sequence of parameter estimates can be performed/invoked to improve the control performance. Economic MPC is also incorporated so as to improve the control performance in a broader way. This book is intended for someone learning functions of a complex variable and who enjoys using Matlab. It will enhance the experience of learning complex-variable theory and will strengthen the knowledge of someone already trained in this branch of advanced calculus. Supplying students with a bridge between the functions of complex-variable theory and Matlab, this supplemental text enables instructors to easily add a Matlab component to their complex-variables courses. The book shows students how Matlab can be a powerful learning aid in such staples of complex-variable theory as conformal mapping, infinite series, contour integration, and Laplace and Fourier transforms. In addition to Matlab programming problems, the text includes many examples in each chapter along with Matlab code.

Removing noise from piecewise constant (PWC) signals is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need to be separated into stratigraphic zones, and in biophysics, jumps between molecular dwell states have to be extracted from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited. This paper (part I, the first of two) shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following and coordinate descent. In the second paper, part II, we introduce novel PWC denoising methods, and comparisons between these methods performed on synthetic and real signals, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.

A Markov random field model for change points detection

This paper presents a model predictive control (MPC) based reference governor approach for control of constrained linear systems. A nominal closed-loop system is first designed to guarantee that, in the unconstrained case, asymptotic zero-error regulation for (piecewise) constant reference signals is achieved. Then, a couple of exogenous signals are added to the reference signal and to the control variable and their value is determined by formulating a MPC problem in order to guarantee that (i) when the state and control constraints are not active, the nominal closed-loop system is recovered, (ii) in transient conditions the constraints are always satisfied and the difference of the performances between the real and the nominal closed-loop systems is minimised, and (iii) when the reference signal is infeasible, the output is brought to the nearest feasible value. A simulation example is reported to witness the potentialities of the approach.

A Distributed Predictive Control (DPC) algorithm for tracking (piecewise) constant output reference signals is presented in this paper. The overall system is assumed to be composed by a number of discrete-time linear subsystems interconnected through the states and/or the inputs. The algorithm is non-cooperative, based on neighbor-to-neighbor communication, and does not require an iterative exchange of information among neighbors. Unfeasible reference signals, that cannot be reached due to state and control constraints, can also be considered by computing the nearest feasible reference value. A simulation example is reported.

We consider model predictive control (MPC) of disturbed linear time-invariant (LTI) multiple-input multiple-output (MIMO) systems without constraints. For the considered system class, the MPC problem has an analytical solution which allows for prediction horizons with arbitrary length. In addition, we introduce integral error feedback which assures steady-state accuracy for (piece-wise) constant disturbances and reference signals. The proposed MPC scheme is applied to grid-side current control of grid-connected power converters with LCL-filter. Simulation results illustrate and compare the control performance of the proposed MPC scheme with and without integral error feedback under (symmetric) grid perturbations.

Self-triggered MPC with performance guarantee for tracking piecewise constant reference signals

Model predictive control (MPC) has been a very successful advanced process control technique for many applications especially in process industries because of its ability to handle hard constraints and multiple inputs and outputs. However, the presence of uncertainty deteriorates the performance of nominal MPC, and a robust model predictive control becomes necessary. The existing development of robust (nonlinear) MPC has yet to be widely applied in industry, mainly due to conservatism of the algorithm and also the impracticality of implementation; thereby robust NMPC has been mostly conceptual until a scenariobasedrobust NMPC recently emerged. A scenario tree is generated to represent the evolution of states with respect to uncertain parameters, and a multistage stochastic programming formulation has been employed to continuously solve for the optimal control actionin a moving horizon fashion. This is a good place to start, however, many challenges still remain and need to be addressed. This thesis develops easily implementable robust NMPC strategies which provide performance guarantees and computational efficiency. One of the major issues of multistage NMPC approaches is computational complexity. Due to the construction of the scenario tree, multistage NMPC models are inevitably larger than their nominal counterparts, and their size grows exponentially with respect to the number of uncertain parameters and the length of robust horizons. To solve this issue, we present an efficient parallelizable advanced-step multistage NMPC (as-msNMPC) approach, which explicitly deals with two types of uncertainty: model parameters and unmeasured noise. The first type is attended to by incorporating multistage scenario trees and the second by applying nonlinear programming (NLP) sensitivity. The framework of as-msNMPC has been demonstrated on two examples with robust performance andsignificantly faster online computation compared to benchmark methods. We also conduct a stability analysis for the newly constructed robust NMPC schemes. Based on Lyapunov stability theory, we show that the origin is asymptotically stable under standard NMPC if the model is perfect. And when additive disturbance is present, thesystem is robustly stable to a neighborhood around the origin. When the system allows both types of uncertainty, we first show that recursive feasibility can be ensured for fully expanded multistage NMPC under mild assumptions. With both advanced-step and ideal multistage NMPC, we then show that robust stability can be achieved with input-to-statepractical stability (ISpS). Last but not least, a sensitivity-assisted multistage NMPC (samNMPC) algorithm hasbeen proposed to deal with the size of the multistage formulation on a linear algebra level. A block-bordered-diagonal structure of the KKT matrix naturally arises with themultistage NLP problem, and Schur complement decomposition can be performed to decouple scenarios. In this case, we can approximate many scenarios with the solution to the nominal scenario. In addition, we apply a scenario generation technique to determine which scenario is most likely to violate constraints. We then formulate an approximate multistage formulation that has a much smaller problem size. The tracking performance of samNMPC and exact multistage NMPC have been compared and similar performance has been reached with only a fraction of the computational effort of the exact multistage formulation.

This paper presents a Distributed Predictive Control (DPC) method for tracking piecewise constant reference signals. The system under control is assumed to be composed by a number of non-overlapping subsystems interconnected through states and inputs. The algorithm is non cooperative, based on neighbor-to-neighbor communication, does not require an iterative exchange of information among neighbors and relies on the robustness properties of the “tube-based” approach developed of the design of robust model predictive controllers. Convergence results are stated and a simulation example is reported to illustrate the performance of DPC.

Robust tracking of piece‐wise constant reference signals for constrained systems with parametric plant uncertainty and additive disturbances is addressed in this paper. The parametric uncertainty is decreased online by set‐membership estimation and a nominal model is updated for improving set‐point tracking. The online estimated parametric uncertainty is used for an online‐determined terminal set which enlarges the set of reachable references close to the system constraints when compared to an offline worst‐case consideration. An artificial target state is introduced which can deviate from the nominal target state. This new target state is used to ensure recursive feasibility for unreachable references and changes in the reference signal. Moreover, a novel “recovery mode” is specified which is deployed in case the new nominal model yields an infeasible control problem. Control algorithms are developed for time‐invariant systems and systems with arbitrarily fast changing plants but known relative bounds. Constraint satisfaction and ‐stability are guaranteed for the proposed algorithms. Controlling the engine load of a self‐propelled work machine is used as a practical example.

A hierarchical Nonlinear Model Predictive Control (NMPC) scheme with guaranteed Input-to-State-practical-Stability (ISpS) is proposed. The controller is formed by an Integral Sliding Mode (ISM) controller and a NMPC one. The ISM, relying on the knowledge of the nominal continuous-time model of the system and of the piecewise constant control signal generated by the NMPC produces a control action aimed at reducing the difference between the dynamics of the nominal closed-loop system and the actual evolution of the state. The NMPC in this way can be designed based on a system with reduced uncertainty. In order to prove the stability of the overall control scheme, some general Regional ISpS results for continuous-time systems are proven.

In the present work, a novel denoising technique for piecewise constant or linear signals is presented termed as signal split. The proposed method separates the sharp edges or transitions from the noise elements by splitting the into different parts. Unlike many noise removal techniques, the method works only in the nonorthogonal domain. The new method utilizes Stein unbiased risk estimate (SURE) to split the signal, Lipschitz exponents to identify noise elements, and a polynomial fitting approach for the sub reconstruction. At the final stage, merging of all parts yield in the fully denoised at a very low computational cost. Statistical results are quite promising and performs better than the conventional shrinkage methods in the case of different types of noise, i.e., speckle, Poisson, and white Gaussian noise. The method has been compared with the state of the art SURE-linear expansion of thresholds denoising technique as well and performs equally well. The method has been extended to the multisplitting approach to identify small edges which are difficult to identify due to the mutual influence of their adjacent strong edges.

Optimal piecewise constant reference command for approximate output synthesis of scalar systems: an interpolation approach

A robust nonlinear tracking MPC using qLPV embedding and zonotopic uncertainty propagation

SummaryThis work deals with the problem of trajectory tracking for a nonlinear system with unknown but bounded model parameter uncertainties. First, this work focuses on the design of a robust nonlinear model predictive control (RNMPC) law subject to model parameter uncertainties implying solving a min‐max optimization problem. Secondly, a new approach is proposed, consisting in relating the min‐max problem to a more tractable optimization problem based on the use of linearization techniques to ensure a good trade‐off between tracking accuracy and computation time. The developed strategy is applied in simulation to a simplified macroscopic continuous photobioreactor model and is compared to the RNMPC and nonlinear model predictive controllers. Its efficiency and its robustness against parameter uncertainties and/or perturbations are illustrated through numerical results.