Design of multi-parametric NCO-tracking controllers for linear continuous-time systems

Process optimization for industrial applications aims to achieve performance enhancements while satisfying system constraints. A major challenge for any such method lies in the problem of uncertainty stemming from model mismatch and process disturbances. Classical approaches such as model predictive control usually handle the uncertainty by repeatedly solving the optimization problem on-line, which may prove a rather computationally demanding task nonetheless and cause serious delays for fast dynamic systems. Existing approaches for mitigating the on-line computational burden via off-line optimization include multi-parametric programming and NCO-tracking. Multi-parametric programming aims to generate a mapping of control strategies as a function of given parameters; whereas NCO-tracking involves tracking the necessary conditions of optimality (NCOs) based on a precomputed control switching structure, which enables a dynamic real-time optimization problem to be transferred into an on-line tracking problem using a feedback controller. A methodology, called multi-parametric (mp-)NCO-tracking is developed in this thesis, whereby multi-parametric dynamic optimization and NCO-tracking methods are combined into a unified framework. An algorithm for the design of mp-NCO-tracking controllers for continuous-time, linear-quadratic optimal control problems is presented in Chapter 2. The off-line step defines the multi-parametric control structure mapped to given uncertain (measurable) parameters in terms of so-called critical regions and feedback laws. Specifically, each critical region corresponds to a unique control switching structure in terms of the sequence of active constraints. The on-line step involves determining the current critical region once the parameter value has been revealed, and then applying the corresponding feedback control laws in a receding horizon manner. The mp-NCO-tracking approach provides a means for relaxing the invariant switching structure assumption in NCO-tracking by constructing critical regions for various switching structures. Moreover, addressing the problem directly in continuous-time can potentially reduce the number of critical regions compared with standard multi-parametric programming based on a time discretization and a control vector parameterization. The methodology and its benefits are illustrated for a number of simple case studies. To obtain the mathematical representation of the generally nonlinear critical regions, Chapter 3 investigates a machine learning model as a classifier, based on deep neural network. This feed-forward network is selected for its representational power as a universal approximator for arbitrary continuous functions. Here, the classifier takes the unknown parameter as input and maps the corresponding critical regions in terms of their switching structures. An algorithm for training the classifier is presented, which involves generating the training data set, setting up a neural network architecture, and applying optimization based training. By using a Softmax classifier in the output layer of the network, a normalized probability distribution is obtained, which consist of a vector with as many elements as the total number of critical regions, and each element representing the likelihood for a region to be the correct one. The classifier is conveniently embedded into the multi-parametric NCO-tracking controller for choosing the real-time switching structure in on-line control. Lastly, a robustification of the mp-NCO-tracking methodology is developed in Chapter 4, where constraints are guaranteed to be satisfied under all possible uncertainty scenarios, which leads to a min-max formulation. A robust counterpart formulation of the multi-parametric dynamic optimization problem is presented, which considers both additive or multiplicative time-varying disturbances. The approach involves backing-off the path and terminal constraints of the linear-quadratic optimal control problem based on a worst-case uncertainty propagation computed using either interval or ellipsoidal reachability tubes. The uncertain system state is decomposed into a nominal reference and a perturbed component, and a convex enclosure of the reachable set for the perturbed component is precomputed via some auxiliary differential equations. Conservative constraint back-offs are obtained from the precomputed reachability tubes, which enables the controller design procedure in the nominal case to be directly applied for the robust control problem, and to retain the same computational effort as in the nominal case. These developments are demonstrated by numerical case studies, and ways of extending this approach to more general, nonlinear optimal control problems are discussed in Chapter 5.Open Acces

Advanced multiparametric optimization and control studies for anaesthesia

Anaesthesia is a reversible pharmacological state of the patient where hypnosis, analgesia and muscle relaxation are guaranteed and maintained throughout the surgery. Analgesics block the sensation of pain; hypnotics produce unconsciousness, while muscle relaxants prevent unwanted movement of muscle tone. Controlling the depth of anaesthesia is a very challenging task, as one has to deal with nonlinearity, inter- and intra-patient variability, multivariable characteristics, variable time delays, dynamics dependent on the hypnotic agent, model analysis variability, agent and stability issues. The modelling and automatic control of anaesthesia is believed to (i) benefit the safety of the patient undergoing surgery as side-effects may be reduced by optimizing the drug infusion rates, and (ii) support anaesthetists during critical situations by automating the drug delivery systems. In this work we have developed several advanced explicit/multi-parametric model predictive (mp-MPC) control strategies for the control of depth of anaesthesia. State estimation techniques are developed and used simultaneously with mp-MPC strategies to estimate the state of each individual patient, in an attempt to overcome the challenges of inter- and intra- patient variability, and deal with possible unmeasurable noisy outputs. Strategies to deal with the nonlinearity have been also developed including local linearization, exact linearization as well as a piece-wise linearization of the Hill curve leading to a hybrid formulation of the patient model and thereby the development of multiparametric hybrid model predictive control methodology. To deal with the inter- and intra- patient variability, as well as the noise on the process output, several robust techniques and a multiparametric moving horizon estimation technique have been design and implemented. All the studies described in the thesis are performed on clinical data for a set of 12 patients who underwent general anaesthesia.Open Acces

Addressing Stability Robustness, Period Uncertainties, and Startup of Multiple-Period Repetitive Control for Spacecraft Jitter Mitigation

Repetitive Control (RC) is a relatively new form of control that seeks to converge to zero tracking error when executing a periodic command, or when executing a constant command in the presence of a periodic disturbance. The design makes use of knowledge of the period of the disturbance or command, and makes use of the error observed in the previous period to update the command in the present period. The usual RC approaches address one period, and this means that potentially they can simultaneously address DC or constant error, the fundamental frequency for that period, and all harmonics up to Nyquist frequency. Spacecraft often have multiple sources of periodic excitation. Slight imbalance in reaction wheels used for attitude control creates three disturbance periods. A special RC structure was developed to allow one to address multiple unrelated periods which is referred to as Multiple-Period Repetitive Control (MPRC). MPRC in practice faces three main challenges for hardware implementation. One is instability due to model errors or parasitic high frequency modes, the second is degradation of the final error level due to period uncertainties or fluctuations, and the third is bad transients due to issues in startup. Regarding these three challenges, the thesis develops a series of methods to enhance the performance of MPRC or to assist in analyzing its performance for mitigating optical jitter induced by mechanical vibration within the structure of a spacecraft testbed. Experimental analysis of MPRC shows contrasting advantages over existing adaptive control algorithms, such as Filtered-X LMS, Adaptive Model Predictive Control, and Adaptive Basis Method, for mitigating jitter within the transmitting beam of Laser Communication (LaserCom) satellites

Stochastic and Optimal Distributed Control for Energy Optimization and Spatially Invariant Systems

Improving energy efficiency and grid responsiveness of buildings requires sensing, computing and communication to enable stochastic decision-making and distributed operations. Optimal control synthesis plays a significant role in dealing with the complexity and uncertainty associated with the energy systems. The dissertation studies general area of complex networked systems that consist of interconnected components and usually operate in uncertain environments. Specifically, the contents of this dissertation include tools using stochastic and optimal distributed control to overcome these challenges and improve the sustainability of electric energy systems. The first tool is developed as a unifying stochastic control approach for improving energy efficiency while meeting probabilistic constraints. This algorithm is applied to demonstrate energy efficiency improvement in buildings and improving operational efficiency of virtualized web servers, respectively. Although all the optimization in this technique is in the form of convex optimization, it heavily relies on semidefinite programming (SP). A generic SP solver can handle only up to hundreds of variables. This being said, for a large scale system, the existing off-the-shelf algorithms may not be an appropriate tool for optimal control. Therefore, in the sequel I will exploit optimization in a distributed way. The second tool is itself a concrete study which is optimal distributed control for spatially invariant systems. Spatially invariance means the dynamics of the system do not vary as we translate along some spatial axis. The optimal H2 [H-2] decentralized control problem is solved by computing an orthogonal projection on a class of Youla parameters with a decentralized structure. Optimal H∞ [H-infinity] performance is posed as a distance minimization in a general L∞ [L-infinity] space from a vector function to a subspace with a mixed L∞ and H∞ space structure. In this framework, the dual and pre-dual formulations lead to finite dimensional convex optimizations which approximate the optimal solution within desired accuracy. Furthermore, a mixed L2 [L-2] /H∞ synthesis problem for spatially invariant systems as trade-offs between transient performance and robustness. Finally, we pursue to deal with a more general networked system, i.e. the Non-Markovian decentralized stochastic control problem, using stochastic maximum principle via Malliavin Calculus