Teaching Mathematics to Control Engineers [Focus on Education

This paper proposes a solution method for the discrete-time long-term dynamic portfolio optimization problem with state and asset allocation constraints. We use the ideas of Model Predictive Control (MPC) to solve the constrained stochastic control problem. MPC is a solution technique which was developed to solve constrained optimal control problems for deterministic control applications. MPC solves the optimal control problem with a receding horizon where a series of consecutive open-loop optimal control problems is solved. The aim of this paper is to develop an MPC approach to the problem of long-term portfolio optimization when the expected returns of the risky assets are modeled using a factor model based on stochastic Gaussian processes. We prove that MPC is a suboptimal control strategy for stochastic systems which uses the new information advantageously and thus is better than the pure optimal open-loop control. For the open-loop optimal control optimization, we derive the conditional portfolio distribution and the corresponding conditional portfolio mean and variance. The mean and the variance depend on future decision about the asset allocation. For the dynamic portfolio optimization problem, we consider constraints on the asset allocation as well as probabilistic constraints on the attainable values of the portfolio wealth. We discuss two different objectives, a classical mean–variance objective and the objective to maximize the probability of exceeding a predetermined value of the portfolio. The dynamic portfolio optimization problem is stated, and the solution via MPC is explained in detail. The results are then illustrated in a case study.

SummaryThe chapter deals with neural networks and learning machines for engine control applications, particularly in modeling for control. In the first section, basic features of engine control in a layered engine management architecture are reviewed. The use of neural networks for engine modeling, control and diagnosis is then briefly described. The need for descriptive models for model-based control and the link between physical models and black box models are emphasized by the grey box approach discussed in this chapter. The second section introduces the neural models frequently used in engine control, namely, MultiLayer Perceptrons (MLP) and Radial Basis Function (RBF) networks. A more recent approach, known as Support Vector Regression (SVR), to build models in kernel expansion form is also presented. The third section is devoted to examples of application of these models in the context of turbocharged Spark Ignition (SI) engines with Variable Camshaft Timing (VCT). This specific context is representative of modern engine control problems. In the first example, the airpath control is studied, where open loop neural estimators are combined with a dynamical polytopic observer. The second example considers modeling the in-cylinder residual gas fraction by Linear Programming SVR (LP-SVR) based on a limited amount of experimental data and a simulator built from prior knowledge. Each example demonstrates that models based on first principles and neural models must be joined together in a grey box approach to obtain effective and acceptable results.

Early control textbooks in Norway

Chapter Eleven Modelling and Monitoring Environmental Outcomes in Adaptive Management

The need for cost-effective fuel economy improvements has driven the introduction of automatic transmissions with an increasing number of gear ratios. Incorporation of interlocking dog clutches in these transmissions decreases package space and increases efficiency, as compared to conventional dry or wet clutches. Unlike friction based clutches, interlocking dog clutches require very precise rotational speed matching prior to engagement. Precise engine speed control is therefore critical to maintaining high shift quality. This research focuses on controlling the engine speed during a gearshift period by manipulating throttle position and combustion phasing. Model predictive control (MPC) is advantageous in this application since the speed profile of a future prediction horizon is known with relatively high confidence. The MPC can find the optimal control actions to achieve the designated speed target without invoking unnecessary actuator manipulation and violating hardware and combustion constraints. This research utilizes linear parameter varying (LPV) MPC to control the engine speed during the gearshift period. Combustion stability constraints are considered with a control oriented covariance of indicated mean effective pressure model (COV of IMEP). The proposed MPC engine speed controller is validated with a high-fidelity 0-dimensional engine model with crank angle resolution. Four case studies, based on simulation, investigate the impact of different MPC design parameters. They also demonstrate that the proposed MPC engine controller successfully achieves the speed reference tracking objective while considering combustion variation constraints.

This book is designed for senior undergraduate and graduate students pursuing courses in mathematics, physics, engineering and biology. The text begins with a study of ordinary differential equations. The concepts of first- and second-order equations are covered initially. It moves further to linear systems, series solutions, regular Sturm–Liouville theory, boundary value problems and qualitative theory. Thereafter, partial differential equations are explored. Topics such as first-order partial differential equations, classification of partial differential equations and Laplace and Poisson equations are also discussed in detail. The book concludes with heat equation, one-dimensional wave equation and wave equation in higher dimensions. It highlights the importance of analysis, linear algebra and geometry in the study of differential equations. It provides sufficient theoretical material at the beginning of each chapter, which will enable students to better understand the concepts and begin solving problems straightaway.

Model Predictive Control (MPC) is a solution towards more energy-efficient waste treatment without compromising on treatment quality. A key component is the process model describing how the inputs and outputs correlate. MPC uses this model to predict future outputs over a finite horizon to decide on step changes to make at the input. These step changes are made so that the output reaches and maintains at a user specified set point. For MPC to be effective, the process model needs to accurately describe the process behaviour. This is a difficult challenge in waste treatment processes due to a combination of slow response, process complexity, and large disturbances. This research project investigated two research avenues towards developing better modelling techniques. This would result in more accurate models or achieve a sufficiently accurate model with fewer experiments. The first avenue is Constrained Model Identification (CMI). Model identification is an optimisation problem to estimate the model parameters. In CMI, process knowledge from first principles and operator experience is translated into optimisation constraints to aid data-driven model identification. The second avenue is Sequential Optimal Experiment Design (SOED). This uses the concept of measuring a value representing information content of a dataset. Like MPC, SOED uses the model to make output predictions. The expected output response to a sequence of input steps form a dataset, and SOED is an optimisation problem to maximise the information content of that expected dataset, by changing the input step sequence. Once optimised, this step sequence is applied in the next experiment. The third part of this work focused on farm-fed anaerobic digestion. It is a renewable energy technology fuelled by agricultural waste. They rely on government incentives to be profitable, but these incentives have steadily been decreased. This project investigated methods to help farmers in the day-to day operation of the unit, including biogas production estimation, automated fault identification and partial diagnosis.

This thesis addresses the design of optimization-based control laws for the case where convergence to a desired set-point, minimization of an arbitrary performance index, or a combination of the two, is the desired objective. The results are developed within the sample-data Model Predictive Control (MPC) framework considering constrained nonlinear continuous-time time-varying dynamical systems. For a given time sampling, a sample-data MPC control strategy consists of i) choosing among all future finite horizon predictions of state and input trajectories of the system the one that minimizes the given performance index, ii) applying the optimal input trajectory to the system until a new time sample is reached, and iii) iterating this process. The performance index is chosen to describe the specific control problem under consideration. In a classic Tracking-MPC framework, where the main goal is to steer the state of the system to a desired steady-state, the performance index is properly chosen to penalize the distance from the current state to a desired one. In order to capture more complex control objectives, in recent years a growing attention has been dedicated to a new class of controllers that goes under the name of Economic-MPC. Here, the term economic is used to stress the fact that the performance index is a general index of interest that we wish to minimize, e.g., economic, which does not denote the distance to a desired set point. This setting makes full use of the potentialities of optimization-based control strategies. Although, it comes with disadvantages. In fact, by choosing an arbitrary performance index, it is difficult to predict, and therefore certify, the evolution of the closed-loop system, which could potentially manifest undersirable behaviors. This thesis provides analysis and certification of a variety of closed-loop behaviors stemming from the use Economic-MPC controllers. A set of tools for design of provably correct MPC controllers is provided for the case where the performance index is of the Tracking-MPC type, purely economic, or a combination of the two. The results focus the certification of both closed-loop economic performance and closed-loop state evolution. The proposed strategies are applied to a range of motion control problems for underactuated vehicles. An MPC controller for Trajectory-Tracking and Path-Following with convergence guarantees is first proposed and then extended, using the results presented on Economic-MPC, to address the control problems of distributed formation keeping, energy efficient trajectory-tracking, and target-following through highly observable trajectories.

3.1 Introduction Recently, the application of reduced-order models to optimal control problems for partial differential equations (PDEs) has received an increasing amount of attention. The reduced-order approach is based on projecting the dynamical system onto subspaces consisting of basis elements that contain characteristics of the expected solution. This is in contrast to, e.g., finite element techniques, where the elements of the subspaces are uncorrelated to the physical properties of the system that they approximate. One reduced-basis method is proper orthogonal decomposition (POD). It has been successfully used in a variety of fields; see, for instance, [9, 22] for detailed reference lists. Output feedback controller synthesis for linear control systems that meet desired performance and/or robustness specifications is an attractive model-based control design tool and has been an active research area of the control community for several decades (see, e.g., [7, 30]). It is not always possible to have full access to the state of the control system, and a controller based on the measurements has to be used. Output feedback synthesis without additional complexity constraints yields in general a controller order equal to nx, the dimension of the dynamical system. The computation of the controller action becomes more expensive with increasing controller dimension. This is one reason why full-order synthesis control has not been widely used in industry. Recently, linear matrix inequalities (LMIs) have attained much attention in control engineering [4], since many control problems can be formulated in terms of LMIs and thus solved via convex programming approaches.

Safe operation of Gas Turbine Engines (GTEs) is one of the paramount missions in the GTE industry. Ensuring reliable operation and control performance is a challenging task that involves optimizing engine performance across various operating points. The Model Predictive Control (MPC) technique is a powerful approach that provides an optimization framework to meet the management and control requirements of GTEs during their operation. This work presents a systematic control framework to efficiently control a GTE. The proposed control algorithm employs an adaptive MPC methodology that ensures both safe and efficient engine operation. The MPC structure is adapted across varying operating points, and its parameters are optimized to provide control inputs that ensure best engine performance during both transient and steadystate engine operation. To enhance control reliability, the MPC is systematically coupled with min-max limiters to maintain critical variables within allowable bounds, ultimately to prevent engine surge, flame-out, and exceeding unsafe variable limits. Through this approach, the aim is to achieve a superior level of safety and operability for GTE control and management. Simulation results on a generic commercial turbofan engine with a 40,000-pound thrust show that the proposed algorithm outperforms a gain-scheduled PI controller with min-max limiters, and demonstrate enhanced performance in engine management and control.

This paper presents a survey of the most recent literature developments in the field of computational guidance and control for space applications. In recent years, the space sector has changed significantly, shifting towards solutions requiring increased performance and autonomy. This change has also been accelerated by the technical and economic needs of the new space approach. However, traditional guidance and control techniques cannot always meet the increasing requirements of autonomy and performance. For this reason, computational guidance and control has become of increasing interest to the aerospace sector. In contrast to traditional guidance and control, computational guidance and control generates commands in real time by using on-board numerical computation. It represents, therefore, a fundamental step on the way to system autonomy and autonomous operations, with the potential to increase system capabilities and reduce operational costs. On-board computation introduces, however, a series of challenges, including the need to develop algorithms which are computationally efficient, reliable and robust. The literature has classified computational guidance and control techniques into two major categories, namely model-based and data-based. The model-based category encompasses all techniques in which the guidance and control laws are explicitly related to the system dynamic and kinematic models. On the contrary, data-based techniques infer guidance and control laws from a set of data, derived from the models, but do not explicitly use such models in the algorithm itself. Model-based guidance and control includes optimal control, model predictive control and convex optimisation. For the specific case of space attitude re-orientation manoeuvres, additional techniques are path planning and artificial potential functions. The general concept behind the Model Predictive Control (MPC) framework is to solve an optimal control problem repeatedly, for example whenever new measurements are available. The MPC implements only the first element of the returned solution, and the process is then repeated. By doing so, the system is better able to cope autonomously with model uncertainties, failures, or errors. For spacecraft with limited on-board capability, having to repeatedly solve an optimisation problem on-board could represent a challenge. In this case, explicit implementations of the MPC have been proposed, in which the guidance and control laws are precomputed offline and implemented on-board using look-up tables or simple functions. If the optimal control problem of the MPC is convex, or if it can be convexified, convex optimisation can be used directly. If the problem cannot be convexified, the solution can be found by sequentially solving a convex approximation of the original non-convex problem. Convex optimisation can be used either repeatedly in the context of an MPC framework, or as a single function call to generate guidance and control laws, e.g. at the beginning of an attitude re-orientation manoeuvre. Convex optimisation has received increased attention in recent years, also due to its suitability to on-board implementation. In fact, if the optimal control problem is convex, it can be solved reliably and efficiently on-board. Convex optimisation has been successfully applied to the space sector with the notable examples of the SpaceX rocket booster landing and of the Mars pinpoint landing demonstration aboard the Masten Xombie sounding rocket. Model-based guidance techniques, not based on optimisation, have been increasingly explored in recent years, as guidance strategies employed in other engineering fields, such as robotics, are being applied to spacecraft. Among these strategies, of particular importance are path planning and artificial potential field methods, due to their applicability to spacecraft autonomous attitude manoeuvres. Path planning algorithms have been applied for decades to ground vehicles and robotic manipulators. These algorithms consist of techniques to find the best route between two points of a graph representing the search space of a problem. For spacecraft re-orientation applications, the literature has proposed different attitude representation conventions and various cost functions associated to the graph search algorithm. Each approach comes with advantages and disadvantages, which will be analysed in the paper. Along with path planning, Artificial Potential Function (APF) techniques have been developed in the field of robotics, and are now also considered for space applications, to track or maintain a specific reference under a set of constraints. The main idea behind these methods is to synthetise the controller using a potential function. This function is specifically built to grant asymptotic stability to the system, according to Lyapunov’s second stability theorem. This formulation grants flexibility to accommodate system constraints, in a computationally light-weight and efficient manner. Data-based guidance and control laws are mainly based on Artificial Intelligence (AI) technologies, and generate guidance commands after training an algorithm with a large set of data. The interest of the space sector towards AI has accelerated in the last years due to the greater maturity of the technology. Some of the most powerful examples lie in the area of machine learning, like reinforcement learning, supervised learning and deep learning. These algorithms are used to extract patterns from large amounts of data derived from complex nonlinear dynamical and kinematical models. Examples of such data can be numerical solutions of an optimal control problem. The patterns extracted from the data allow the system to identify, in a short computational time, the critical parameters needed to relate the state and control variables. Creating a sufficiently large dataset for training the algorithm requires a considerable computational effort, though, and for certain methods optimality cannot be guaranteed due to the lack of theoretical support derived from the optimality conditions. Advancements made in the last years span several research directions and open a wide variety of opportunities and challenges, which need to be examined to guide further exploration. In this paper, both model-based and data-based techniques and applications are presented. The considered techniques include model predictive control, convex optimisation and machine learning. In addition, for the implementation of autonomous attitude manoeuvres, path planning methods and artificial potential function methods are analysed. These techniques have been applied, in the literature, to a range of different applications. The applications discussed in this paper include, in addition to attitude manoeuvres, powered descent landing guidance, low-thrust orbital transfers, proximity operations and station keeping.

We consider the speed control of a spark ignition engine with an open torque converter during vehicle deceleration. In these conditions, the torque transmitted from the engine to the wheels through the torque converter, and the corresponding torque load on the engine, is related to the torque converter speed ratio, the ratio between the turbine speed and the engine/impeller speed. The engine speed controller needs to effectively coordinate airflow and spark timing and to enforce constraints including engine stall avoidance, misfire avoidance, and desired actuator operating ranges. We formulate a model of the relevant powertrain dynamics, and propose a parametrization that removes a multiplicative nonlinearity, hence enabling the application of linear quadratic model predictive control (MPC). The MPC controller is synthesized as a piecewise affine state feedback and evaluated in simulations and in an experimental vehicle showing good performance in reference tracking and settling time, which translates in smooth and more consistent deceleration profiles and potential for fuel economy improvements.

The topic for the current Mini-Workshop organized by Karl Kunisch (Univ. Graz), Angela Kunoth (Univ. Bonn) and Rolf Rannacher (Univ. Heidelberg) emerged from the Oberwolfach Workshop “Numerical Techniques for Optimization Problems with PDE Constraints” which was held February 16–22, 2003. It was realized that numerically solving control problems which are constrained by time-dependent nonlinear PDEs (Partial Differential Equations) are particularly challenging with respect to the complexity of the problem. Mathematically, one has to minimize a functional under PDE constraints and possibly additional constraints on the state and the control. Standard discretizations on uniform grids in space and time will only yield solutions where the inherent structures of the problem (nonlinearity, constraints) are not sufficiently captured. Certain optimization problems for large coupled systems of partial differential equations are currently not yet numerically treatable or do not satisfy the time constraints required in practice. Overcoming this barrier can only be achieved by designing new mathematically founded algorithmic approaches. The road towards this goal leads to many interesting problems in optimization, linear algebra, numerics, analysis, and approximation theory. The conference had 21 participants which represented continuous optimization, numerical analysis and scientific computing. 18 talks were given. The 10 longer, overview-style talks were on optimization with PDEs, focussing on These talks were intended to bridge the gap between the different research fields Optimization and Numerics. They were complemented by 8 shorter talks on more specialized research topics, ranging from efficiency indices for optimization over iterative methods for the coupled systems and multigrid acceleration to modelling issues in optimization, and data compression by means of proper orthogonal decompositions and central Voronoi tesselations. Different modern approaches to overcome the complexity issues in numerical simulations for PDE-constrained optimization have been presented and discussed. One of the approaches is to employ fast iterative solvers like multigrid on uniform grids. The methodology which conceptually provides the largest potential is to introduce adaptivity. This drastically reduces complexity but depending on the context may require solving an additional problem. Wavelet approaches particularly allow to resolve each of the variables separately and in addition provide a built-in preconditioning. Yet another approach uses compressed information in order to efficiently solve the primal-dual system. All these issues are currently very active research areas. The extensive discussions held during this workshop have produced a number of new ideas and connections. It was agreed upon that a mere black-box-style matching of efficient PDE codes with optimization tool boxes would on one hand remain much below its potential and on the other hand not help overcoming complexity barriers. Some concepts from automatic differentiation seem to carry over to adaptive methods. Even combining adaptivity with proper orthogonal decompositions may be a very promising direction. The many new ideas discussed during this workshop will have to be further elaborated in future.

Wiener model identification and predictive control of a pH neutralisation process is presented. Input-output data from a nonlinear, first principles simulation model of the pH neutralisation process are used for subspace-based identification of a black-box Wiener-type model. The proposed nonlinear subspace identification method has the advantage of delivering a Wiener model in a format which is suitable for its use in a standard linear-model-based predictive control scheme. The identified Wiener model is used as the internal model in a model predictive controller (MPC) which is used to control the nonlinear white-box simulation model. To account for the unmeasurable disturbance, a nonlinear observer is proposed. The performance of the Wiener model predictive control (WMPC) is compared with that of a linear MPC, and with a more traditional feedback control, namely a PID control. Simulation results show that the WMPC outperforms the linear MPC and the PID controllers.

Unlike steady-state optimization and linear model predictive control, dynamic optimization and nonlinear model predictive control is underway to industry application. This paper discusses the implementation issues of two-level control strategy integrated dynamic real-time optimization and nonlinear model predictive control of a heat-integrated distillation column system. To realize this scheme, four nonlinear programming problems: economic operation optimization, data reconciliation and state estimation, parameter estimation and nonlinear model predictive control, are considered based on system rigorous dynamic model. The economic operation optimization is to minimize the cost of the plant. The unmeasured states are estimated by measurements of temperatures, pressures and flow rates. Time-variant parameters such as tray efficiencies and feed composition are updated on-line. The sequential quadratic programming and orthogonal collocation are used to solve the dynamic nonlinear optimization problems.