Multi-Level Iteration Schemes with Adaptive Level Choice for Nonlinear Model Predictive Control

In this thesis we develop the Multi-Level Iteration schemes (MLI), a numerical method for Nonlinear Model Predictive Control (NMPC) where the dynamical models are described by ordinary differential equations. The method is based on Direct Multiple Shooting for the discretization of the optimal control problems to be solved in each sample. The arising parametric nonlinear problems are solved approximately by setting up a generalized tangential predictor in a preparation phase. This generalized tangential predictor is given by a quadratic program (QP), which implicitly defines a piecewise affine linear feedback law. The feedback law is then evaluated in a feedback phase by solving the QP for the current state estimate as soon as it becomes known to the controller. The method developed in this thesis yields significant computational savings by updating the matrix and vector data of the tangential predictor in a hierarchy of four levels. The lowest level performs no updates and just calculates the feedback for a new initial state estimate. The second level updates the QP constraint functions and approximates the QP gradient. The third level updates the QP constraint functions and calculates the exact QP gradient. The fourth level evaluates all matrix and vector data of the QP. Feedback schemes are then assembled by choosing a level for each sample. This yields a successive update of the piecewise affine linear feedback law that is implicitly defined by the generalized tangential predictor. We present and discuss four strategies for data communication between the levels in a scheme and we describe how schemes with fixed level choices can be assembled in practice. We give local convergence theory for each level type holding its own set of primal-dual variables for fixed initial values, and discuss existing convergence theory for the case of a closed-loop process. We outline a modification of the levels that yields additional computational savings. For the adaptive choice of the levels at runtime, we develop two contraction-based criteria to decide whether the currently used linearization remains valid and use them in an algorithm to decide which level to employ for the next sample. Furthermore, we propose a criterion applicable to online estimation. The criterion provides additional information for the level decision for the next sample. Focusing on the second lowest level, we propose an efficient algorithm for suboptimal NMPC. For the presented algorithmic approaches, we describe structure exploitation in the form of tailored condensing, outline the Online Active Set Strategy as an efficient way to solve the quadratic subproblems and extend the method to linear least-squares problems. We develop iterative matrix-free methods for one contraction-based criterion, which estimates the spectral radius of the iteration matrix. We describe three application fields where MLI provides significant computational savings compared to state-of-the-art numerical methods for NMPC. For both fixed and adaptive MLI schemes, we carry out extensive numerical testings for challenging nonlinear test problems and compare the performance of MLI to a state-of-the-art numerical method for NMPC. The schemes obtained by adaptive MLI are computationally much cheaper while showing comparable performance. By construction, the adaptive MLI allows giving feedback with a much higher frequency, which significantly improves controller performance for the considered test problems. To perform the numerical experiments, we have implemented the proposed method within a MATLAB(R) based software called MLI, which makes use of a software package for the automatic derivative generation of first and higher order for the solution of the dynamic model as well as objective and constraint functions, which performs structure exploitation by condensing, and which efficiently solves the parametric quadratic subproblems by using a software package that provides an implementation of the Online Active Set Strategy

Nonlinear model predictive control using automatic differentiation

Although nonlinear model predictive control (NMPC) might be the best choice for a nonlinear plant, it is still not widely used. This is mainly due to the computational burden associated with solving online a set of nonlinear differential equations and a nonlinear dynamic optimization problem in real time. This thesis is concerned with strategies aimed at reducing the computational burden involved in different stages of the NMPC such as optimization problem, state estimation, and nonlinear model identification. A major part of the computational burden comes from function and derivative evaluations required in different parts of the NMPC algorithm. In this work, the problem is tackled using a recently introduced efficient tool, the automatic differentiation (AD). Using the AD tool, a function is evaluated together with all its partial derivative from the code defining the function with machine accuracy. A new NMPC algorithm based on nonlinear least square optimization is proposed. In a first–order method, the sensitivity equations are integrated using a linear formula while the AD tool is applied to get their values accurately. For higher order approximations, more terms of the Taylor expansion are used in the integration for which the AD is effectively used. As a result, the gradient of the cost function against control moves is accurately obtained so that the online nonlinear optimization can be efficiently solved. In many real control cases, the states are not measured and have to be estimated for each instance when a solution of the model equations is needed. A nonlinear extended version of the Kalman filter (EKF) is added to the NMPC algorithm for this purpose. The AD tool is used to calculate the required derivatives in the local linearization step of the filter automatically and accurately. Offset is another problem faced in NMPC. A new nonlinear integration is devised for this case to eliminate the offset from the output response. In this method, an integrated disturbance model is added to the process model input or output to correct the plant/model mismatch. The time response of the controller is also improved as a by–product. The proposed NMPC algorithm has been applied to an evaporation process and a two continuous stirred tank reactor (two–CSTR) process with satisfactory results to cope with large setpoint changes, unmeasured severe disturbances, and process/model mismatches. When the process equations are not known (black–box) or when these are too complicated to be used in the controller, modelling is needed to create an internal model for the controller. In this thesis, a continuous time recurrent neural network (CTRNN) in a state–space form is developed to be used in NMPC context. An efficient training algorithm for the proposed network is developed using AD tool. By automatically generating Taylor coefficients, the algorithm not only solves the differentiation equations of the network but also produces the sensitivity for the training problem. The same approach is also used to solve online the optimization problem of the NMPC. The proposed CTRNN and the predictive controller were tested on an evaporator and two–CSTR case studies. A comparison with other approaches shows that the new algorithm can considerably reduce network training time and improve solution accuracy. For a third case study, the ALSTOM gasifier, a NMPC via linearization algorithm is implemented to control the system. In this work a nonlinear state–space class Wiener model is used to identify the black–box model of the gasifier. A linear model of the plant at zero–load is adopted as a base model for prediction. Then, a feedforward neural network is created as the static gain for a particular output channel, fuel gas pressure, to compensate its strong nonlinear behavior observed in open–loop simulations. By linearizing the neural network at each sampling time, the static nonlinear gain provides certain adaptation to the linear base model. The AD tool is used here to linearize the neural network efficiently. Noticeable performance improvement is observed when compared with pure linear MPC. The controller was able to pass all tests specified in the benchmark problem at all load conditions.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Novel Methodologies in State Estimation for Constrained Nonlinear Systems under Non-Gaussian Measurement Noise & Process Uncertainty

Chemical processes often involve scheduled/unscheduled changes in the operating conditions that may lead to non-zero mean non-Gaussian (e.g., uniform, multimodal) process uncertainties and measurement noises. Moreover, the distribution of the variables of a system subjected to process constraints may not often follow Gaussian distributions. It is essential that the state estimation schemes can properly capture the non-Gaussianity in the system to successfully monitor and control chemical plants. Kalman Filter (KF) and its extension, i.e., Extended Kalman Filter (EKF), are well-known model-driven state estimation schemes for unconstrained applications. The present thesis initially performed state estimation using this approach for an unconstrained large-scale gasifier that supports the efficiency and accuracy offered by KF. However, the underlying assumption considered in KF/EKF is that all state variables, input variables, process uncertainties, and measurement noises follow Gaussian distributions. The existing EKF-based approaches that consider constraints on the states and/or non-Gaussian uncertainties and noises require significantly larger computational costs than those observed in EKF applications. The current research aims to introduce an efficient EKF-based scheme, referred to as constrained Abridged Gaussian Sum Extended Kalman Filter (constrained AGS EKF), that can generalize EKF to perform state estimation for constrained nonlinear applications featuring non-zero mean non-Gaussian distributions. Constrained AGS-EFK uses Gaussian mixture models to approximate the non-Gaussian distributions of the constrained states, process uncertainties, and measurement noises. In the present abridged Gaussian sum framework, the main characteristics of the overall Gaussian mixture models are used to represent the distributions of the corresponding non-Gaussian variable. Constrained AGS-EKF includes new modifications in both prior and posterior estimation steps of the standard EKF to capture the non-zero mean distribution of the process uncertainties and measurement noises, respectively. These modified prior and posterior steps require the same computational costs as in EKF. Moreover, an intermediate step is considered in the constrained AGS-EKF framework that explicitly applies the constraints on the priori estimation of the distributions of the states. The additional computational costs to perform this intermediate step is relatively small when compared to the conventional approaches such as Gaussian Sum Filter (GSF). Note that the constrained AGS-EKF performs the modified EKF (consists of modified prior, intermediate, and posterior estimation steps) only once and thus, avoids additional computational costs and biased estimations often observed in GSFs. Moving Horizon Estimation (MHE) is an optimization-based state estimation approach that provides the optimal estimations of the states. Although MHE increases the required computation costs when compared to EKF, MHE is best known for the constrained applications as it can take into account all the process constraints. This PhD thesis initially provided an error analysis that shows that EKF can provide accurate estimates if it is constantly initialized by a constrained estimation scheme such as MHE (even though EKF is unconstrained state estimator). Despite the benefits provided by MHE for constrained applications, this framework assumes that the distributions the process uncertainties and measurement noises are zero-mean Gaussian, known a priori, and remain unchanged throughout the operation, i.e., known time-independent distributions, which may not be accurate set of assumptions for the real-world applications. Performing a set of MHEs (one MHE per each Gaussian component in the mixture model) more likely become computationally taxing and hence, is discouraged. Instead, the abridged Gaussian sum approach introduced in this thesis for AGS-EKF framework can be used to improve the MHE performance for the applications involving non-Gaussian random noises and uncertainties. Thus, a new extended version of MHE, i.e., referred to as Extended Moving Horizon Estimation (EMHE), is presented that makes use of the Gaussian mixture models to capture the known time-dependent non-Gaussian distributions of the process uncertainties and measurement noises use of the abridged Gaussian sum approach. This framework updates the Gaussian mixture models to represent the new characteristics of the known time-dependent distribution of noises/uncertainties upon scheduled changes in the process operation. These updates require a relatively small additional CPU time; thus making it an attractive estimation scheme for online applications in chemical engineering. Similar to the standard MHE and despite the accuracy and efficiency offered by the EMHE scheme, the application of EMHE is limited to the scenarios where the changes in the distribution of noises and uncertainties are known a priori. However, the knowledge of the distributions of measurement noises or process uncertainties may not be available a priori if any unscheduled operating changes occur during the plant operation. Motivated by this aspect, a novel robust version of MHE, referred to as Robust Moving Horizon Estimation (RMHE), is introduced that improves the robustness and accuracy of the estimation by modelling online the unknown distributions of the measurement noises or process uncertainties. The RMHE problem involves additional constraints and decision variables than the standard MHE and EMHE problems to provide optimal Gaussian mixture models that represent the unknown distributions of the random noises or uncertainties along with the optimal estimated states. The additional constraints in the RMHE problem do not considerably increase the required computational costs than that needed in the standard MHE and consequently, both the present RMHE and the standard MHE require somewhat similar CPU time on average to provide the point estimates. The methodologies developed through this PhD thesis offers efficient MHE-based and EKF-based frameworks that significantly improve the performance of these state estimation schemes for practical chemical engineering applications

Expanding the Horizons of Manufacturing: Towards Wide Integration, Smart Systems and Tools

This research topic aims at enterprise-wide modeling and optimization (EWMO) through the development and application of integrated modeling, simulation and optimization methodologies, and computer-aided tools for reliable and sustainable improvement opportunities within the entire manufacturing network (raw materials, production plants, distribution, retailers, and customers) and its components. This integrated approach incorporates information from the local primary control and supervisory modules into the scheduling/planning formulation. That makes it possible to dynamically react to incidents that occur in the network components at the appropriate decision-making level, requiring fewer resources, emitting less waste, and allowing for better responsiveness in changing market requirements and operational variations, reducing cost, waste, energy consumption and environmental impact, and increasing the benefits. More recently, the exploitation of new technology integration, such as through semantic models in formal knowledge models, allows for the capture and utilization of domain knowledge, human knowledge, and expert knowledge toward comprehensive intelligent management. Otherwise, the development of advanced technologies and tools, such as cyber-physical systems, the Internet of Things, the Industrial Internet of Things, Artificial Intelligence, Big Data, Cloud Computing, Blockchain, etc., have captured the attention of manufacturing enterprises toward intelligent manufacturing systems

Robust Model Predictive Control for Linear Parameter Varying Systems along with Exploration of its Application in Medical Mobile Robots

This thesis seeks to develop a robust model predictive controller (MPC) for Linear Parameter Varying (LPV) systems. LPV models based on input-output display are employed. We aim to improve robust MPC methods for LPV systems with an input-output display. This improvement will be examined from two perspectives. First, the system must be stable in conditions of uncertainty (in signal scheduling or due to disturbance) and perform well in both tracking and regulation problems. Secondly, the proposed method should be practical, i.e., it should have a reasonable computational load and not be conservative. Firstly, an interpolation approach is utilized to minimize the conservativeness of the MPC. The controller is calculated as a linear combination of a set of offline predefined control laws. The coefficients of these offline controllers are derived from a real-time optimization problem. The control gains are determined to ensure stability and increase the terminal set. Secondly, in order to test the system's robustness to external disturbances, a free control move was added to the control law. Also, a Recurrent Neural Network (RNN) algorithm is applied for online optimization, showing that this optimization method has better speed and accuracy than traditional algorithms. The proposed controller was compared with two methods (robust MPC and MPC with LPV model based on input-output) in reference tracking and disturbance rejection scenarios. It was shown that the proposed method works well in both parts. However, two other methods could not deal with the disturbance. Thirdly, a support vector machine was introduced to identify the input-output LPV model to estimate the output. The estimated model was compared with the actual nonlinear system outputs, and the identification was shown to be effective. As a consequence, the controller can accurately follow the reference. Finally, an interpolation-based MPC with free control moves is implemented for a wheeled mobile robot in a hospital setting, where an RNN solves the online optimization problem. The controller was compared with a robust MPC and MPC-LPV in reference tracking, disturbance rejection, online computational load, and region of attraction. The results indicate that our proposed method surpasses and can navigate quickly and reliably while avoiding obstacles

New approaches for the real-time optimization of process systems under uncertainty

In the process industry, the economical operation of systems is of utmost importance for stakeholders to remain competitive. Moreover, economic incentives can be used to drive the development of sustainable processes, which must be deployed to ensure continued human and ecological welfare. In the process systems engineering paradigm, model predictive control (MPC) and real-time optimization (RTO) are methods used to achieve operational optimality; however, both methods are subject to uncertainty, which can adversely affect their performance. Along with the challenges of uncertainty, formulations of economic optimization problems are largely problem-specific as process utilities and products vary significantly by application; thus, many nascent processes have not received a tailored economic optimization treatment. In this thesis, the focus is on avenues of economic optimization under uncertainty, namely, the two-step RTO method, which updates process models via parameters; and the modifier adaptation (MA) method, which updates process models via error and gradient correction. In the case of parametric model uncertainty, the two-step RTO method is used. The parameter estimation (PE) step that accompanies RTO requires plant measurements that are often noisy, which can cause the propagation of noise to the parameter estimates and result in poor RTO performance. In the present work, a noise-abatement scheme is proposed such that high-fidelity parameter estimates are used to update a process model for economic optimization. This is achieved through parameter estimate bootstrapping to compute bounds and determine the measurement-set that results in the lowest parameter variation; thus, the scheme is dubbed low-variance parameter estimation (lv-PE). This method is shown to result in improved process economics through truer set points and reduced dynamic behaviour. In the case of structural model mismatch (i.e., unmodelled phenomena), the MA approach is used, whereby gradient modifier (i.e., correction) terms must be recursively estimated until convergence. These modifier terms require plant perturbations to be performed, which incite time-consuming plant dynamics that delay operating point updates. In cases with frequent disturbances, MA may have poor performance well as there is limited time to refine the modifiers. Herein, a partial modifier adaptation (pMA) method is proposed, which selects a subset of modifications to be made, thus reducing the number of necessary perturbations. Through this reduced experimental burden, the operating point refinement process is accelerated resulting in quicker convergence to advantageous operating points. Additionally, constraint satisfaction during this refinement process can also result in poor performance via wasted below-specification products. Accordingly, the pMA method also includes an adjustment step that can drive the system to constraint-satisfying regions at each iteration. The pMA method is shown to economically outperform both the standard MA method as well as a related directional MA method in cases with frequent periodic disturbances. The economic optimization methods described above are implemented in novel processes to improve their economics, which can incite further technological uptake. Post-combustion carbon capture (PCC) is the most advanced carbon capture technology as it has been investigated extensively. PCC takes industrial flue gases and separates the carbon dioxide for later repurposing or storage. Most PCC operating schemes make decisions using simplified models since a mechanistic PCC model is large and difficult to solve. To this end, this thesis provides the first robust MPC that can address uncertainty in PCC with a mechanistic model. The advantage of the mechanistic model in robust optimal control is that it allows for a precise treatment of uncertainties in phenomenological parameters. Using the multi-scenario approach, discrete realizations of the uncertain parameters inside a given uncertainty region can be incorporated into the controller to produce control actions that result in a robust operation in closed-loop. In the case of jointly uncertainty activity coefficients and flue gas flowrates, the proposed robust MPC is shown to lead to improved performance with respect to a nominal controller (i.e., one that does not hedge against uncertainty) under various operational scenarios. In addition to the PCC robust control problem, the mechanistic model is used for economic optimization and state estimation via RTO and moving horizon estimation (MHE) layers respectively. While the former computes economical set points, the latter uses few measurements to compute the full system state, which is necessary for the controller that uses a mechanistic model. These layers are integrated to operate the system economically via a new economic function that accounts for the most significant economic aspects of PCC, including the carbon economy, energy, chemical, and utility costs. A new proposed MPC layer is novel in its ability to enable flexible control of the plant by manipulating fresh material streams to impact CO2 capture and the MHE layer is the first to provide accurate system estimates to the controller with realistically accessible measurements. A joint MPC-MHE-RTO scheme is deployed for PCC, which is shown to lead to more economical steady-state operation compared to constant set point counterfactuals under cofiring, diurnal operation, and price variation scenarios. The lv-PE scheme is also deployed for the PCC system where it is found to improve set point economics with respect to traditional PE methods. The improvements are observed to occur through reduced emissions and more efficient energy used, thus having environmental co-benefits. Moreover, the lv-PE algorithm is used for uncertainty quantification to develop a robust RTO that leads to more conservative set points (i.e., less economic improvement) but lower set point variation (i.e., less control burden). The methodologies developed in this PhD thesis provide improvements in efficacy as well as applicability of online economic optimization in engineering applications, where uncertainty is often present. These can be deployed by both academic as well as industrial practitioners that wish to improve the economic performance on their processes

Robust Empirical Model-Based Algorithms for Nonlinear Processes

This research work proposes two robust empirical model-based predictive control algorithms for nonlinear processes. Chemical process are generally highly nonlinear thus predictive control algorithms that explicitly account for the nonlinearity of the process are expected to provide better closed-loop performance as compared to algorithms based on linear models. Two types of models can be considered for control: first-principles and empirical. Empirical models were chosen for the proposed algorithms for the following reasons: (i) they are less complex for on-line optimization, (ii) they are easy to identify from input-output data and (iii) their structure is suitable for the formulation of robustness tests. One of the key problems of every model that is used for prediction within a control strategy is that some model parameters cannot be known accurately due to measurement noise and/or error in the structure of the assumed model. In the robust control approach it is assumed that processes can be represented by models with parameters' values that are assumed to lie between a lower and upper bound or equivalently, that these parameters can be represented by a nominal value plus uncertainty. When this uncertainty in control parameters is not considered by the controller the control actions might be insufficient to effectively control the process and in some extreme cases the closed-loop may become unstable. Accordingly, the two robust control algorithms proposed in the current work explicitly account for the effect of uncertainty on stability and closed-loop performance. The first proposed controller is a robust gain-scheduling model predictive controller (MPC). In this case the process is represented within each operating region by a state-affine model obtained from input-output data. The state-affine model matrices are used to obtain a state-space based MPC for every operating region. By combining the state-affine, disturbance and controller equations a closed-loop representation was obtained. Then, the resulting mathematical representation was tested for robustness with linear matrix inequalities (LMI's) based on a test where the vertices of the parameter box were obtained by an iterative procedure. The result of the LMI's test gives a measure of performance referred to as γ that relates the effect of the disturbances on the process outputs. Finally, for the gain-scheduling part of the algorithm a set of rules was proposed to switch between the available controllers according to the current process conditions. Since every combination of the controller tuning parameters results in a different value of γ, an optimization problem was proposed to minimize γ with respect to the tuning parameters. Accordingly, for the proposed controller it was ensured that the effect of the disturbances on the output variables was kept to its minimum. A bioreactor case study was presented to show the benefits of the proposed algorithm. For comparison purposes a non-robust linear MPC was also designed. The results show that the proposed algorithm has a clear advantage in terms of performance as compared to non-robust linear MPC techniques. The second controller proposed in this work is a robust nonlinear model predictive controller (NMPC) based on an empirical Volterra series model. The benefit of using a Volterra series model for this case is that its structure can be split in two sections that account for the nominal and uncertain parameter values. Similar to the previously proposed gain-scheduled controller the model parameters were obtained from input-output data. After identifying the Volterra model, an interconnection matrix and its corresponding uncertainty description were found. The interconnection matrix relates the process inputs and outputs and is built according to the type of cost function that the controller uses. Based on the interconnection representing the system a robustness test was proposed based on a structured singular value norm calculation (SSV). The test is based on a min-max formulation where the worst possible closed-loop error is minimized with respect to the manipulated variables. Additional factors that were considered in the cost function were: manipulated variables weighting, manipulated variables restrictions and a terminal condition. To show the benefits of this controller two case studies were considered, a single-input-single-output (SISO) and a multiple-input-multiple-output (MIMO) process. Both case studies show that the proposed controller is able to control the process. The results showed that the controller could efficiently track set-points in the presence of disturbances while complying with the saturation limits imposed on the manipulated variables. This controller was also compared against a non-robust linear MPC, non-robust NMPC and non-robust first-principles NMPC. These comparisons were performed for different levels of uncertainty and for different values of the suppression or control actions weights. It was shown through these comparisons that a tradeoff exists between nominal performance and robustness to model error. Thus, for larger weights the controller is less aggressive resulting in more sluggish performance but less sensitivity to model error thus resulting in smaller differences between the robust and non-robust schemes. On the other hand when these weights are smaller the controller is more aggressive resulting in better performance at the nominal operating conditions but also leading to larger sensitivity to model error when the system is operated away from nominal conditions. In this case, as a result of this increased sensitivity to model error, the robust controller is found to be significantly better than the non-robust one

Integration of Process Design, Scheduling, and Control Via Model Based Multiparametric Programming

The conventional approach to assess the multiscale operational activities sequentially often leads to suboptimal solutions and even interruptions in the manufacturing process due to the inherent differences in the objectives of the individual constituent problems. In this work, integration of the traditionally isolated process design, scheduling, and control problems is investigated by introducing a multiparametric programming-based framework, where all decision layers are based on a single high fidelity model. The overall problem is dissected into two constituent parts, namely (i) design and control, and (ii) scheduling and control problems. The proposed framework was first assessed on these constituent subproblems, followed by the implementation on the overall problem. The fundamental steps of the framework consists of (i) developing design dependent offline control and scheduling strategies, and (ii) exact implementation of these offline rolling horizon strategies in a mixed-integer dynamic optimization problem for the optimal design. The design dependence of the offline operational strategies allows for the integrated problem to consider the design, scheduling, and control problems simultaneously. The proposed framework is showcased on (i) a binary distillation column for the separation of toluene and benzene, (ii) a system of two continuous stirred tank reactor, (iii) a small residential heat and power network, and (iv) two batch reactor systems. Furthermore, a novel algorithm for large scale multiparametric programming problems is proposed to solve the classes of problems frequently encountered as a result of the integration of rolling horizon strategies