This dissertation aims to realize the goal of developing robust control solutions that can enable legged robots to navigate complex unknown environments. The idea of creating articulated-legged machines that can mimic animal locomotion has fueled the imagination of many researchers. These legged robots are designed to assist humans in their day-to-day tasks and challenging scenarios such as monitoring remote, inhospitable environments, disaster response, and other dangerous environments. Despite several decades of research, legged robots have yet to reach the dexterity or dynamic stability needed for real-world deployments. A fundamental gap exists in the understanding and development of reliable and scalable algorithms required for the real-time planning and control of legged robots. The overarching goal of this thesis is to formally develop computationally tractable, robust controllers based on nonlinear hybrid systems theory, model predictive control, and optimization for the real-time planning and control of agile locomotion in quadrupedal robots. Toward this objective, this thesis first investigates layered control architectures. In particular, we propose a two-level hierarchical control architecture in which the higher level is based on a reduced-order model predictive control (MPC), and the lower level is based on a full-order quadratic programming (QP) based virtual constraints controller. Specifically, two MPC architectures are explored: 1) An event-based MPC scheme that generates the optimal center of mass (COM) trajectories using a reduced-order linear inverted pendulum (LIP) model, and 2) A time-based MPC scheme that computes the optimal COM and ground reaction forces (GRF) using the reduced-order single rigid body (SRB) dynamics model. The optimal COM trajectories in the event-based MPC and the optimal COM trajectories, along with the ground reaction forces in the time-based MPC, are then tracked by the low-level virtual constraints controller. The event-based MPC scheme is numerically validated on the Vision 60 platform in a physics-based simulation environment. It has significantly reduced the computational burden associated with real-time planning-based MPC schemes. However, owing to the quasi-static nature of the optimal trajectories generated by the LIP model, we explored a time-based MPC scheme using Single Rigid Body Dynamics. This time-based MPC scheme is also numerically validated using the mathematical model of the A1 quadrupedal robot. Most MPC schemes use a reduced-order model to generate optimal trajectories. However, the abstraction and unmodeled dynamics in template models significantly increase the gap between reduced- and full-order models, limiting the robot's full scope and potential. In the second part of the thesis, we aim to develop a computationally tractable robust model predictive control (RMPC) scheme based on convex QPs to bridge this gap. The RMPC framework considers the single rigid body model subject to a set of unmodeled dynamics and plans for the optimal reduced-order trajectory and GRFs. The generated optimal GRFs of the high-level RMPC are then mapped to the full-order model using a low-level nonlinear controller based on virtual constraints and QP. The key innovation of the proposed RMPC framework is that it allows the integration of the hierarchical controller with Reinforcement Learning (RL) techniques to train a neural network to compute the vertices of the uncertainty set numerically. The proposed hierarchical control algorithm is validated numerically and experimentally for robust and blind locomotion of the A1 quadrupedal robot on different indoor and outdoor terrains and at different speeds. The numerical analysis of the RMPC suggests significant improvement in the performance of the rough terrain locomotion compared to the nominal MPC. In particular, the proposed RMPC algorithm outperforms the nominal MPC by over 60% during rough terrain locomotion over 550 uneven terrains. Our experimental studies also indicate a significant reduction in the gap between the reduced full-order models by comparing the desired and actual GRFs. Finally, the last part of the thesis presents a formal approach for synthesizing robust $mathcal{H}_2$- and $mathcal{H}_infty$-optimal MPCs to stabilize the periodic locomotion of legged robots. The proposed algorithm builds on the existing optimization-based control stack. We outline the set of conditions under which the closed-loop nonlinear dynamics around a periodic orbit can be transformed into a linear time-invariant (LTI) system using Floquet theory. We then outline an approach to systematically generate parameterized $mathcal{H}_2$- and $mathcal{H}_infty$- robust controllers using linear matrix inequalities (LMIs). We subsequently established a set of conditions guaranteeing the existence of such robust optimal controllers. The proposed $mathcal{H}_2$- and $mathcal{H}_infty$-optimal MPCs are extensively validated both numerically and experimentally for the robust locomotion of the A1 quadrupedal robot subject to various external disturbances and uneven terrains. Our numerical analysis suggests a significant improvement in the performance of robust locomotion compared to the nominal MPC. / Doctor of Philosophy / Legged robots have always been envisioned to work alongside humans, assisting them in mundane day-to-day tasks to challenging scenarios such as monitoring remote locations, planetary exploration, and supporting relief programs in disaster situations. Furthermore, research into legged locomotion can aid in designing and developing powered prosthetic limbs and exoskeletons. With these advantages in mind, several researchers have created sophisticated-legged robots and even more complicated algorithms to control them. Despite this, a significant gap exists between the agility, mobility, and dynamic stability shown by the existing legged robots and their biological counterparts. To work alongside humans, legged robots have to interact with complex environments and deal with uncertainties in the form of unplanned contacts and unknown terrains. Developing robust control solutions to accommodate disturbances explicitly marks the first step towards safe and reliable real-world deployment of legged robots. Toward this objective, this thesis aims to establish a formal foundation to develop computationally tractable robust controllers for the real-time planning and control of legged robots. Initial investigations in this thesis report on the use of layered control architectures, specifically event-based and time-based Model Predictive Control(MPC) schemes. These layered control architectures consist of an MPC scheme built around a reduced-order model at the high level and a virtual constraints-based nonlinear controller at the low level. Using these layered control architectures, this thesis proposed two robust control solutions to improve the rough terrain locomotion of legged robots. The first proposed robust control solution aims to mitigate one of the issues of layered control architecture. In particular, layered control architectures rely on a reduced order model at the high level to remain computationally tractable. However, the approximation of fullorder models with reduced-order models limits the full scope and potential of the robot. The proposed algorithm aims to bridge the gap between reduced- and full-order models with the integration of model-free Reinforcement Learning (RL) techniques. The second algorithm proposes a formal approach to generate robust optimal control solutions that can explicitly accommodate the disturbances and stabilize periodic legged locomotion. Under some mild conditions, the MPC control solution is analyzed, and an auxiliary feedback control solution that can handle disturbances explicitly is proposed. The thesis also theoretically establishes the sufficient conditions for the existence of such controllers. Both the proposed control solutions are extensively validated using numerical simulations and experiments using an A1 quadrupedal robot as a representative example.

The objective of this thesis is to understand the interactions between the computational mechanisms, described by algorithms and software, and the physical world, described by differential equations, in the context of networked systems. Such systems can be denoted as cyber-physical nodes connected over a network. In this work, the power grid is used as a guiding example and a rich source of problems which can be generalized to networked cyber-physical systems. We address specific problems that arise in cyber-physical networks due to the presence of a computational network and a physical network as well as provide directions for future research.

Designing high performance control of rotorcraft unmanned aerial vehicle (UAV) requires a mathematical model that describes the dynamics of the vehicle. The model is derived from first principle modelling, such as rigid-body dynamics, actuator dynamics and etc. It is found that simplified decoupled model of RUAV has slightly better data fitting compared with the complex model for helicopter attitude dynamics in hover or near hover flight condition. In addition, the simplified modelling approach has made the analysis of system dynamics easy. System identification method is applied to identify the unknown intrinsic parameters in the nominal model, where manual piloted flight experiment is carried out and input-output data about a nominal operating region is recorded for parameters identification process. Integral-based parameter identification algorithm is then used to identify model parameters that give the best matching between the simulation and measured output response. The results obtained show that the dominant dynamics is captured. The advantages of using integral-based method include the fast computation time, insensitive to initial parameter value and fast convergence rate in comparison with other contemporary system identification methods such as prediction error method (PEM), maximum likelihood method, equation error method and output error method. Besides, the integral-based parameter identification method can be readily extended to tackle slow time-varying model parameters and fast varying disturbances. The model prediction is found to be improved significantly when the iterative integral-based parameter identification is employed and thus further validates the minimal modelling approach. From the literature review, many control schemes have been designed and validated in simulation. However, few of them has really been implemented in real flight as well as under windy and severe conditions, where unpredictable large system parameters variations and unexpected disturbances are present. Therefore, the emphasis on this part will be on the control design that would have satisfactory reference sequence tracking or regulation capability in the presence of unmodelled dynamics and external disturbances. Generalised Predictive Controller (GPC) is particularly considered as the helicopter attitude dynamics control due to its insensitivity with respect to model mismatch and its capability to address the control problem of nominal model with deadtime. The robustness analysis shows that the robustness of the basic GPC is significantly improved using the Smith Predictor (SP) in place of optimal predictor in basic GPC. The effectiveness of the proposed robust GPC was well proven with the control of helicopter heading on the test rig in terms of the reference sequence tracking performance and the input disturbance rejection capability. The second motivation is the investigation of adaptive GPC from the perspective of performance improvements for the robust GPC. The promising experimental results prove the feasibility of the adaptive GPC controller, and especially evident when the underlying robust GPC is tuned with low robustness and legitimates the use of simplified model. Another approach of robust model predictive control is considered where disturbance is identified in real‐time using an iterative integral‐based method.

A typical petro-chemical or oil-refining plant is known to operate with hundreds if not thousands of control loops. All critical loops are primarily required to operate at their respective optimal levels in order for the plant to run efficiently. With such a large number of vital loops, it is difficult for engineers to monitor and maintain these loops with the intention that they are operating under optimum conditions at all times. Parts of processes are interactive, more so nowadays with increasing integration, requiring the use of a more advanced protocol of control systems. The most widely applied advanced process control system is the Model Predictive Controller (MPC). The success of these controllers is noted in the large number of applications worldwide. These controllers rely on a process model in order to predict future plant responses. Naturally, the performance of model-based controllers is intimately linked to the quality of the process models. Industrial project experience has shown that the most difficult and time-consuming work in an MPC project is modeling and identification. With time, the performance of these controllers degrades due to changes in feed, working regime as well as plant configuration. One of the causes of controller degradation is this degradation of process models. If a discrepancy between the controller’s plant model and the plant itself exists, controller performance may be adversely affected. It is important to detect these changes and re-identify the plant model to maintain control performance over time. In order to avoid the time-consuming process of complete model identification, a model validation tool is developed which provides a model quality indication based on real-time plant data. The focus has been on developing a method that is simple to implement but still robust. The techniques and algorithms presented are developed as far as possible to resemble an on-line software environment and are capable of running parallel to the process in real time. These techniques are based on parametric (regression) and nonparametric (correlation) analyses which complement each other in identifying problems -iiwithin on-line models. These methods pinpoint the precise location of a mismatch. This implies that only a few inputs have to be perturbed in the re-identification process and only the degraded portion of the model is to be updated. This work is carried out for the benefit of SASOL, exclusively focused on the Secunda plant which has a large number of model predictive controllers that are required to be maintained for optimal economic benefit. The efficacy of the methodology developed is illustrated in several simulation studies with the key intention to mirror occurrences present in industrial processes. The methods were also tested on an industrial application. The key results and shortfalls of the methodology are documented. / Thesis (M.Sc.Eng.)-University of KwaZulu-Natal, Durban, 2010.

The work in this thesis focuses on the development of a Stochastic Model Predictive Control (SMPC) algorithm for linear systems with additive and multiplicative stochastic uncertainty subjected to linear input/state constraints. Constraints can be in the form of hard constraints, which must be satisfied at all times, or soft constraints, which can be violated up to a pre-defined limit on the frequency of violation or the expected number of violations in a given period. When constraints are included in the SMPC algorithm, the difficulty arising from stochastic model parameters manifests itself in the online optimization in two ways. Namely, the difficulty lies in predicting the probability distribution of future states and imposing constraints on closed loop responses through constraints on predictions. This problem is overcome through the introduction of layered tubes around a centre trajectory. These tubes are optimized online in order to produce a systematic and less conservative approach of handling constraints. The layered tubes centered around a nominal trajectory achieve soft constraint satisfaction through the imposition of constraints on the probabilities of one-step-ahead transition of the predicted state between the layered tubes and constraints on the probability of one-step-ahead constraint violations. An application in the field of Sustainable Development policy is used as an example. With some adaptation, the algorithm is extended the case where the uncertainty is not identically and independently distributed. Also, by including linearization errors, it is extended to non-linear systems with additive uncertainty.

A novel continuous low-thrust fuel-efficient model predictive control strategy for multi-satellite formations flying in low earth orbit is presented. State prediction relies on a full nonlinear relative motion model, based on quasi-nonsingular relative orbital elements, including earth oblateness effects and, through state augmentation, differential drag. The optimal control problem is specically designed to incorporate latest theoretical results concerning maneuver optimality in the state-space, yielding to a sensible total delta-V reduction, while assuring feasibility and stability though imposition of a Lyapunov constraint. The controller is particularly suitable for networked architectures since it exploits the predictive strategy and the dynamics knowledge to provide robustness against feedback losses and delays. The Networked MPC is validated through real missions simulation scenarios using a high-fidelity orbital propagator which accounts for high-order geopotential, solar radiation pressure, atmospheric drag and third-body effects.

A thesis submitted to the Faculty of Engineering, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy Johannesburg, October 1997 / This thesis develops and investigates signal processing models that are useful both for interpreting and implementing certain types of Model Predictive Control. Two types of Model Predictive Control are investigated, namely, techniques based on Internal Model Control and Long Range Predictive Control. [Abbreviated abstract. Open document to view full version].

O presente texto tem por objetivo avaliar diferentes aplicações do algoritmo PLS-PH (Partial Least Squares Prediction Horizon), desenvolvido por (LAURI et al., 2010) para a identificação de sistemas, com o objetivo de desenvolvimento de controladores MPC. Desta maneira, é avaliada a capacidade do algoritmo gerar modelos lineares para realizar predições múltiplos passos à frente, para sistemas SISO e MIMO, com dados coletados em malha fechada. É também avaliada a capacidade do algoritmo de identificar modelos não-lineares baseados na estrutura NARX polinomial. / The objective of this work consists in evaluating different applications of the PLS-PH (Partial Least Squares Prediction Horizon) algorithm, developed by (LAURI et al., 2010), in order to identify models for MPC controllers. The algorithms capacity of producing linear models capable of performing multiple steps-ahead prediction for both SISO and MIMO systems, with data collected in closed-loop. The algorithms capability of identifying non-linear models with the NARX polynomial structure is also evaluated.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / This work proposes a study and application of advanced controller to trajectory tracking of wheeled mobile robots. This kind of problem is a challenger for controllers once its models has two inputs and three outputs and is a non-linear model. In the literature there are various solutions to wheeled mobile robots trajectory tracking, among them the Model Predictive Control (MPC) with linearization model and a non linear control which in this work will be nominated as Klancar Controller. The Predictive Controllers can be applied efficiently in plants which has multiple inputs an multiple outputs, in situation that a future reference trajectory is known and systems with input and output constraints . However, the main disadvantage of MPC is the high computational effort which limits its practical application. Thus, this specific controller uses the plants linearization model. On the other hand, the Klancar Controller may be more efficient than the ones based on linear models, once the model is non linear. However, its solution, by definition, does not match the optimized criteria which can be a disadvantage mainly in systems that has constrains and a known future reference. Furthermore, this work proposes the application of the Predictive Control Extended Prediction Self Adaptive Control (EPSAC) to wheeled mobile robot trajectory tracking. This control strategy uses explicitly the non linear robot model, future reference, constraints on the variables and has a optimized solution. And, to the matter of this work, it has not been found reports of the EPSAC applied in mobile robotics, and is thus an unprecedented application. Simulation results are presented comparing the controllers studied using performance indices. Else, the controllers were applied in a mobile robot. / Este trabalho propÃe o estudo e aplicaÃÃo de controladores avanÃados ao seguimento de trajetÃrias de robÃs mÃveis com rodas. Este tipo de problema à bastante desafiador do ponto de vista de controle uma vez que o modelo tem duas entradas e trÃs saÃdas, alÃm disso, trata-se de um modelo nÃo linear. Na literatura existem diversas soluÃÃes para o controle de trajetÃria de robÃs mÃveis, dentre eles tem-se o Controle Preditivo Baseado em Modelo (MPC) por meio de modelos linearizados e um controlador nÃo linear denominado neste trabalho de controlador de Klancar. Os controladores preditivos podem ser aplicados de forma eficiente em plantas com modelos multivariÃveis, em situaÃÃes na qual a trajetÃria futura de referÃncia à conhecida e em sistemas com restriÃÃes nas vaiÃveis de entrada e de saÃda. PorÃm, a principal desvantagem do MPC linearizado à o alto custo computacional o que limita as aplicaÃÃes prÃticas. AlÃm disso, esse controlador especÃfico utiliza um modelo linearizado da planta. Por outro lado, o controlador de Klancar pode ser mais eficiente que os baseados em modelos lineares, devido Ãs nÃo linearidades inerentes do modelo. No entanto, a sua soluÃÃo, por definiÃÃo, nÃo corresponde a critÃrios Ãtimos o que pode representar uma desvantagem principalmente em sistemas com restriÃÃes e referÃncia futura conhecida. AlÃm disso, neste trabalho à proposta a aplicaÃÃo do controle preditivo EPSAC (Extended Prediction Self Adaptive Control) para o controle de seguimento de trajetÃrias. Esta estratÃgia utiliza de forma explÃcita o modelo nÃo linear do robÃ, a referÃncia futura, as restriÃÃes nas variÃveis do robà e soluÃÃo corresponde a um critÃrio Ãtimo. Atà onde foi pesquisado pelo autor deste trabalho, nÃo existem relatos da utilizaÃÃo do EPSAC na robÃtica mÃvel, sendo desta forma uma aplicaÃÃo inÃdita. Resultados de simulaÃÃo sÃo apresentados comparando os controladores estudados, utilizando Ãndices de desempenhos. AlÃm disso, os mesmo foram implementados em um robà mÃvel.

O atual ambiente de alta competitividade do Mercado tem levado os produtores a operar com margens de lucro cada vez mais restritas. Nesse sentido, é imperativa a racionalização dos custos de produção, bem como a otimização dos processos produtivos. Diante de tal cenário, o controle preditivo baseado em modelos tem sido apresentado como uma poderosa alternativa para a obtenção dos objetivos acima mencionados. Sendo assim, é essencial que seja estabelecida uma metodologia de análise, baseada em critérios claros, acompanháveis e mensuráveis. Atualmente, encontram-se disponíveis no mercado distintas metodologias e suas respectivas ferramentas de suporte, as quais auxiliam na realização de tais análises. Quando se observa o número de soluções de controle avançado disponíveis, juntamente com as metodologias e ferramentas de análise de desempenho disponíveis, nota-se que existe um amplo espectro de possíveis combinações a ser avaliado. O objetivo desse trabalho é estudar algumas dessas combinações. São aqui avaliados o desempenho de controladores preditivos, à luz de algumas das consagradas técnicas de avaliação, bem como a própria efetividade e aplicabilidade de tais técnicas. São utilizados e analisados os seguintes controles avançados: Controlador Preditivo Generalizado (GPC); Controlador Multivariável Robusto; e Controlador ESSMPC. Para a avaliação de desempenho, serão utilizadas e estudadas as seguintes técnicas: Controller Performance Index (CPI); Cp e Cpk; e Índice de Yu e Qin. Os resultados mostraram que o Controlador Robusto Multivariável apresentou desempenho similar ao ESSMPC e ambos apresentaram desempenho melhor que o GPC. Todos os algoritmos apresentaram maior sensibilidade às mudanças nos pesos das variáveis controladas e menor nos pesos das manipuladas. No caso da inserção de erros, os algoritmos apresentam sensibilidade maior até 35% de erro. Após tal valor, a diferença de desempenho não é tão significativa. Além disso, o Cp, Cpk e Índice de Yu e Qin se comportaram de forma similar, mas diferentes do CPI. / The current environment of high competitiveness of the market has led producers to operate with profit margins increasingly restricted. Therefore, it is imperative to streamlining production costs, as well as the optimization of production processes. Faced with this scenario, the model predictive control has been presented as a powerful alternative for obtaining the objectives stated above. Therefore, it is essential to establish a methodology of analysis, based on clear and measurable criteria. Currently, there are different methods and support tools available in the Market which help in such analysis. These methodologies and tools may evaluate the problem only quantitatively (increased production of a particular unit, for example) or qualitatively (how close to the predictions of advanced control solution is the actual behavior of the plant, for example). When one observes the number of advanced control solutions available, along with methodologies and tools available for performance analysis, we note that there is a wide spectrum of possible combinations to be evaluated. The aim of this work is to study some of these combinations. It will be observed the performance of advanced control solutions, through some of the most famous evaluation techniques, as well as their own effectiveness and applicability of such techniques. For the execution of the work will be used and analyzed the following advanced control solutions: Generalized Predictive Controller (GPC); Robust Multivariable Controller; and Controller ESSMPC. For the performance assessment, it will be used and studied the following techniques: Controller Performance Index; Cp and Cpk; and Qin and Yu Index. The results showed that the Robust Multivariable Controller performance was similar to ESSMPC and both performed better than the GPC. All algorithms showed greater sensitivity to changes in the weights of the controlled variables than on weights for manipulated variables. In the case of error insertion, the algorithms exhibit greater sensitivity up to 35% of mismatch. After this value, the performance difference was not very significant. Moreover, the Cp index Cpk and Qin and Yu behaved similar but different than CPI.