Robust nonlinear receding horizon control with constraint tightening: off line approximation and application to networked control system

2007/2008Nonlinear Receding Horizon (RH) control, also known as moving horizon control or nonlinear Model Predictive Control (MPC), refers to a class of algorithms that make explicit use of a nonlinear process model to optimize the plant behavior, by computing a sequence of future ma- nipulated variable adjustments. Usually the optimal control sequence is obtained by minimizing a multi-stage cost functional on the basis of open-loop predictions. The presence of uncertainty in the model used for the optimization raises the question of robustness, i.e., the maintenance of certain properties such as stability and performance in the presence of uncertainty. The need for guaranteeing the closed-loop stability in presence of uncertainties motivates the conception of robust nonlinear MPC, in which the perturbations are explicitly taken in account in the design of the controller. When the nature of the uncertainty is know, and it is assumed to be bounded in some compact set, the robust RH control can be determined, in a natural way, by solving a min–max optimal control problem, that is, the performance objective is optimized for the worst-case scenario. However, the use of min-max techniques is limited by the high computational burden required to solve the optimization problem. In the case of constrained system, a possibility to ensure the robust constraint satisfaction and the closed-loop stability without resorting to min-max optimization consists in imposing restricted (tightened) constraints on the the predicted trajectories during the optimization. In this framework, an MPC scheme with constraint tightening for discrete-time nonlinear systems affected by state-dependent and norm bounded uncertainties is proposed and discussed. A novel method to tighten the constraints relying on the nominal state prediction is described, leading to less conservative set contractions than in the existing approaches. Moreover, by imposing a stabilizing state constraint at the end of the control horizon (in place of the usual terminal one placed at the end of the prediction horizon), less stringent assumptions can be posed on the terminal region, while improving the robust stability properties of the MPC closed-loop system. The robust nonlinear MPC formulation with tightened constraints is then used to design off- line approximate feedback laws able to guarantee the practical stability of the closed-loop system. By using off-line approximations, the computational burden due to the on-line optimization is removed, thus allowing for the application of the MPC to systems with fast dynamics. In this framework, we will also address the problem of approximating possibly discontinuous feedback functions, thus overcoming the limitation of existent approximation scheme which assume the continuity of the RH control law (whereas this condition is not always verified in practice, due to both nonlinearities and constraints). Finally, the problem of stabilizing constrained systems with networked unreliable (and de- layed) feedback and command channels is also considered. In order to satisfy the control ob- jectives for this class of systems, also referenced to as Networked Control Systems (NCS’s), a control scheme based on the combined use of constraint tightening MPC with a delay compen- sation strategy will be proposed and analyzed. The stability properties of all the aforementioned MPC schemes are characterized by using the regional Input-to-State Stability (ISS) tool. The ISS approach allows to analyze the depen- dence of state trajectories of nonlinear systems on the magnitude of inputs, which can represent control variables or disturbances. Typically, in MPC the ISS property is characterized in terms of Lyapunov functions, both for historical and practical reasons, since the optimal finite horizon cost of the optimization problem can be easily used for this task. Note that, in order to study the ISS property of MPC closed-loop systems, global results are in general not useful because, due to the presence of state and input constraints, it is impossible to establish global bounds for the multi-stage cost used as Lyapunov function. On the other hand local results do not allow to analyze the properties of the predictive control law in terms of its region of attraction. There- fore, regional ISS results have to employed for MPC controlled systems. Moreover, in the case of NCS, the resulting control strategy yields to a time-varying closed-loop system, whose stability properties can be analyzed using a novel regional ISS characterization in terms of time-varying Lyapunov functions.XXI Ciclo198

Advanced control designs for output tracking of hydrostatic transmissions

The work addresses simple but efficient model descriptions in a combination with advanced control and estimation approaches to achieve an accurate tracking of the desired trajectories. The proposed control designs are capable of fully exploiting the wide operation range of HSTs within the system configuration limits. A new trajectory planning scheme for the output tracking that uses both the primary and secondary control inputs was developed. Simple models or even purely data-driven models are envisaged and deployed to develop several advanced control approaches for HST systems

Systems and control : 21th Benelux meeting, 2002, March 19-21, Veldhoven, The Netherlands

Book of abstract

Stabilizing nonlinear model predictive control in presence of disturbances and off - line approximations of the control law

2009/2010One of the more recent and promising approaches to control is the Receding Horizon one. Due to its intrinsic characteristics, this methodology, also know as Model Predictive Control, allows to easily face disturbances and model uncertainties: indeed at each sampling instant the control action is recalculated on the basis of the reached state (closed loop). More in detail, the procedure consists in the minimization of an adequate cost function with respect to a control input sequence; then the first element of the optimal sequence is applied. The whole procedure is then continuously reiterated. In this thesis, we will focus in particular on robust control of constrained systems. This is motivated by the fact that, in practice, every real system is subjected to uncertainties, disturbances and constraints, in particular on state and input (for instance, plants can work without being damaged only in a limited set of configurations and, on the other side, control actions must be compatible with actuators' physical limits). With regard to the first aspect, maintaining the closed loop stability even in presence of disturbances or model mismatches can result in an essential strategy: moreover it can be exploited in order to design an approximate stabilizing controller, as it will be shown. The control input values are obtained recurring to a Nearest Neighbour technique or, in more favourable cases, to a Neural Network based approach to the exact RH law, which can be then calculated off line: this implies a strong improvement related to the applicability of MPC policy in particular in terms of on line computational burden. The proposed scheme is capable to guarantee stability even for systems that are not stabilizable by means of a continuous feedback control law. Another interesting framework in which the study of the influence of uncertainties on stability can lead to significant contributions is the networked MPC one. In this case, due to the absence of physical interconnections between the controller and the systems to be controlled, stability can be obtained only taking into account of the presence of disturbances, delays and data losses: indeed this kind of uncertainties are anything but infrequent in a communication network. The analysis carried out in this thesis regards interconnected systems and leads to two distinct procedures, respectively stabilizing the linear systems with TCP protocol and nonlinear systems with non-acknowledged protocol. The core of both the schemes resides in the online solution of an adequate reduced horizon optimal control problem.Una delle strategie di controllo emerse più recentemente, più promettenti e di conseguenza più studiate negli ultimi anni è quella basata sull'approccio Receding Horizon. Grazie alle caratteristiche che contraddistinguono questa tecnica, cui si fa spesso riferimento anche col nome di Model Predictive Control, risulta piuttosto agevole trattare eventuali disturbi e incertezze di modellazione; tale metodo prevede infatti il calcolo di un nuovo ingresso di controllo per ciascun istante di campionamento, in seguito alla minimizzazione ad ogni passo di un'opportuna funzione di costo rispetto ad una sequenza di possibili futuri ingressi, inizializzata sulla base del valore dello stato del sistema all'istante considerato. Il controllo è dato dal primo elemento di tale sequenza ottima; tutto questo viene continuamente ripetuto, il che comporta un aggiornamento costante del segnale di controllo. Gli inconvenienti di questa tecnica risiedono nelle elevate risorse computazionali e nei tempi di calcolo richiesti, così da ridurne drasticamente l'applicabilità specie nel caso di sistemi con elevata dinamica. In questa tesi ci si concentrerà sulle caratteristiche di robustezza del controllore: l'importanza di quest'analisi risiede nel fatto che ogni sistema reale è soggetto a incertezze e disturbi di varia origine cui bisogna far fronte durante le normali condizioni di funzionamento. Inoltre, la capacità di gestire errori di modellazione, come si vedrà, può essere sfruttata per ottenere un notevole incremento delle prestazioni nella stima del valore da fornire in ingresso all'impianto: si tratta di ripartire l'errore complessivo in modo da garantirsi dei margini che consentano di lavorare con un'approssimazione della legge di controllo, come specificato più avanti. In tutto il lavoro si considereranno sistemi vincolati: l'interesse per questa caratteristica dipende dal fatto che nella pratica vanno sempre tenuti in considerazione eventuali vincoli su stato e ingressi: basti pensare al fatto che ogni impianto è progettato per lavorare solo all'interno un determinato insieme di configurazioni, determinato ad esempio da vincoli fisici su attuatori, sensori e così via: non riporre sufficiente attenzione in tali restrizioni può risultare nel danneggiamento del sistema di controllo o dell'impianto stesso. Le caratteristiche di stabilità di un sistema controllato mediante MPC dipendono in modo determinante dalla scelta dei parametri e degli attributi della funzione di costo da minimizzare; nel seguito, con riferimento al caso dei sistemi non lineari, saranno forniti suggerimenti e strumenti utili in tal senso, al fine di ottenere la stabilità anche in presenza di disturbi (che si assumeranno opportunamente limitati). Successivamente tale robustezza verrà sfruttata per la progettazione di controllori stabilizzanti approssimati: si dimostrerà infatti che, una volta progettato adeguatamente il sistema di controllo “esatto” basato su approccio RH e conseguentemente calcolati off-line i valori ottimi degli ingressi su una griglia opportunamente costruita sul dominio dello stato, il ricorso a una conveniente approssimazione di tali valori non compromette le proprietà di stabilità del sistema complessivo, che continua per di più a mantenere una certa robustezza. Da notare che ciò vale anche per sistemi non stabilizzabili mediante legge di controllo feedback continua: la funzione approssimante può essere ottenuta in questo caso con tecniche di tipo Nearest Neighbour; qualora invece la legge di controllo sia sufficientemente regolare si potrà far ricorso ad approssimatori smooth, quali ad esempio le reti neurali. Tutto ciò comporta un notevole miglioramento delle prestazioni del controllore RH sia dal punto di vista del tempo di calcolo richiesto che (nel secondo caso) della memoria necessaria ad immagazzinare i parametri del controllore, risultando nell'applicabilità dell'approccio basato su MPC anche al caso di sistemi con elevata dinamica. Un altro ambito in cui lo studio dell'influenza delle incertezze e dei disturbi sulla stabilità richiede una notevole attenzione è quello dei sistemi networked; anche in questo caso il ricorso all'MPC può portare a ottimi risultati di stabilità robusta, a patto di individuare un' opportuna struttura per il sistema complessivo ed effettuare scelte adeguate per il problema di ottimizzazione. In particolare, si considererà il caso di trasmissione di dati tra un controllore centralizzato e le varie parti dell'impianto in assenza di collegamento fisico diretto. Lo studio della stabilità dovrà allora tenere in considerazione la presenza di perdite di pacchetti o ritardi di trasmissione, condizioni tutt'altro che infrequenti per le reti. Saranno quindi proposte due distinte procedure, che si dimostreranno essere in grado di garantire robustezza a sistemi rispettivamente lineari comunicanti con protocolli di tipo TCP e non lineari in presenza di protocolli UDP. Questo secondo caso è senz'altro il più complesso ma allo stesso tempo il più concreto tra i due. Il nucleo del controllo è ancora basato su una tecnica MPC, ma stavolta il controllore è chiamato a risolvere il problema di ottimizzazione su un orizzonte “ridotto”, che consente la gestione dei ritardi e di eventuali perdite di pacchetto su determinati canali. La lunghezza dell'orizzonte dipenderà dalla presenza o meno dei segnali di ricezione del pacchetto (acknowledgement).XXIII Ciclo197

Advanced Path Planning and Collision Avoidance Algorithms for UAVs

The thesis aims to investigate and develop innovative tools to provide autonomous flight capability to a fixed-wing unmanned aircraft. Particularly it contributes to research on path optimization, tra jectory tracking and collision avoidance with two algorithms designed respectively for path planning and navigation. The complete system generates the shortest path from start to target avoiding known obstacles represented on a map, then drives the aircraft to track the optimum path avoiding unpredicted ob jects sensed in flight. The path planning algorithm, named Kinematic A*, is developed on the basis of graph search algorithms like A* or Theta* and is meant to bridge the gap between path-search logics of these methods and aircraft kinematic constraints. On the other hand the navigation algorithm faces concurring tasks of tra jectory tracking and collision avoidance with Nonlinear Model Predictive Control. When A* is applied to path planning of unmanned aircrafts any aircraft kinematics is taken into account, then practicability of the path is not guaranteed. Kinematic A* (KA*) generates feasible paths through graph-search logics and basic vehicle characteristics. It includes a simple aircraft kinematic-model to evaluate moving cost between nodes of tridimensional graphs. Movements are constrained with minimum turning radius and maximum rate of climb. Furtermore, separation from obstacles is imposed, defining a volume around the path free from obstacles (tube-type boundaries). Navigation is safe when the tracking error does not exceed this volume. The path-tracking task aims to link kinematic information related to desired aircraft positions with dynamic behaviors to generate commands that minimize the error between reference and real tra jectory. On the other hand avoid obstacles in flight is one of the most challenging tasks for autonomous aircrafts and many elements must be taken into account in order to implement an effective collision avoidance maneuver. Second part of the thesis describes a Nonlinear Model Predictive Control (NMPC) application to cope with collision avoidance and path tracking tasks. First contribution is the development of a navigation system able to match concurring problems: track the optimal path provided with KA* and avoid unpredicted obstacles detected with sensors. Second Contribution is the Sense & Avoid (S&A) technique exploiting spherical camera and visual servoing control logics

Real-Time Optimization for Estimation and Control: Application to Waste Heat Recovery for Heavy Duty Trucks

This thesis aims at the investigation and development of the control of waste heat recovery systems (WHR) for heavy duty trucks based on the organic Rankine cycle. It is desired to control these systems in real time so that they recover as much energy as possible, but this is no trivial task since their highly nonlinear dynamics are strongly affected by external inputs (disturbances). Additionally, nonlinear operational constraints must be satisfied. To deal with this problem, in this thesis a dynamic model of a WHR that is based on first principles and empirical relationships from thermodynamics and heat transfer is formulated. This model corresponds to a DAE of index 1. In view of the requirements of the employed numerical methods, it includes a spline-based evaluation method for the thermophysical properties needed to evaluate the model. Therewith, the continuous differentiability of the state trajectories with respect to controls and states on its domain of evaluation is achieved. Next, an optimal control problem (OCP) for a fixed time horizon is formulated. From the OCP, a nonlinear model-predictive control (NMPC) scheme is formulated as well. Since NMPC corresponds to a state feedback strategy, a state estimator is also formulated in the form of a moving horizon estimation (MHE) scheme. In this thesis, we make use of efficient numerical methods based on the direct multiple shooting (DMS) method for optimal control, backward differentiation formulae for the solution of initial value problems for DAE, and the corresponding versions of the real-time iteration (RTI) scheme in order to approximately solve the OCP and implement the MHE and NMPC schemes. The simultaneous implementation of NMPC and MHE schemes based on RTI has been already proven to be stable in the control literature. Several numerical instances of the DMS method for the proposed OCP, NMPC and MHE schemes are tested assuming a given real-world operation scenario consisting of truck exhaust gas data recorded during a real trip. These data have been kindly provided by our industry cooperation partner Daimler AG. Additionally, the PI and LQGI control strategies, of wide-spread use in the literature of control of WHR, are also considered for comparison with the proposed scheme. An important result of this thesis is that, considering the highest energy recovery obtained from both strategies as a reference for the given operation scenario, the proposed NMPC scheme is able to reach an additional energy generation of around 3% when the full state vector is assumed to be known, and its computational speed allows it to update the control function in times shorter than the considered sampling time of 100 [ms], which makes it a suitable candidate for real-time implementation. In a more realistic scenario in which the state has to be estimated from noisy measurements, a combination of both aforementioned NMPC and MHE schemes yields an additional energy generation of around 2%. Concretely, this thesis presents novel results and advances in the following areas: • A first principles DAE model of the WHR is presented. The model is derived from the energy and mass conservation considerations and empirical heat transfer relationships; and features a tailored evaluation method of thermophysical properties with which it possesses the property of being at least continuously differentiable with respect to its controls and states on its whole domain of evaluation. • A new real-time optimization control strategy for the WHR is developed. It consists of an NMPC strategy based on efficient simulation, optimization and control tools developed in previous works. The scheme is able to explicitly handle nonlinear constraints on controls and states. In contrast to other NMPC instances for the WHR found in the literature, our scheme's efficient numerical treatment make it real-time feasible even if the full nonlinear WHR dynamics are considered. • To the author's knowledge, this is the first implementation that considers both the NMPC and the MHE approaches used simultaneously in the control of the WHR. The combination of NMPC and MHE produces a closed-loop, model-based implementation that can treat realistic measurements as inputs and calculates the corresponding control functions as outputs

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

Fault tolerant flight control system design for unmanned aerial vehicles

Safety and reliability of air vehicles is of the utmost importance. This is particularly true for large civil transport aircraft where a large number of human lives depend on safety critical design. With the increase in the use of unmanned aerial vehicles (UAVs) in our airspace it is essential that UAV safety is also given attention to prevent devastating failures which could ultimately lead to loss of human lives. While civil aircraft have human operators, the pilot, to counteract any unforeseen faults, autonomous UAVs are only as good as the on board flight computer. Large civil aircraft also have the luxury of weight hence redundant actuators (control surfaces) can be installed and in the event of a faulty set of actuators the redundant actuators can be brought into action to negate the effects of any faults. Again weight is a luxury that UAVs do not have. The main objective of this research is to study the design of a fault tolerant flight controller that can exploit the mathematical redundancies in the flight dynamic equations as opposed to adding hardware redundancies that would result in significant weight increase. This thesis presents new research into fault tolerant control for flight vehicles. Upon examining the flight dynamic equations it can be seen, for example, that an aileron, which is primarily used to perform a roll manoeuvre, can be used to execute a limited pitch moment. Hence a control method is required that moves away from the traditional fixed structure model where control surface roles are clearly defined. For this reason, in this thesis, I have chosen to study the application of model predictive control (MPC) to fault tolerant control systems. MPC is a model based method where a model of the plant forms an integral part of the controller. An optimisation is performed based on model estimations of the plant and the inputs are chosen via an optimisation process. One of the main contributions of this thesis is the development of a nonlinear model predictive controller for fault tolerant flight control. An aircraft is a highly nonlinear system hence if a nonlinear model can be integrated into the control process the cross-coupling effects of the control surface contributions can be easily exploited. An active fault tolerant control system comprises not only of the fault tolerant controller but also a fault detection and isolation subsystem. A common fault detection method is based on parameter estimation using filtering techniques. The solution proposed in this thesis uses an unscented Kalman filter (UKF) for parameter estimation and controller updates. In summary the main contribution of this thesis is the development of a new active fault tolerant flight control system. This new innovative controller exploits the idea of analytical redundancy as opposed to hardware redundancy. It comprises of a nonlinear model predictive based controller using pseudospectral discretisation to solve the nonlinear optimal control problem. Furthermore a UKF is incorporated into the design of the active fault tolerant flight control system

Reverse Engineering Biological Control Systems for Applications in Process Control.

The main emphasis of this dissertation is the development of nonlinear control strategies based on biological control systems. Commonly utilized biological control schemes have been studied in order to reverse engineer the important concepts for applications in process control. This approach has led to the development of a nonlinear habituating control strategy and nonlinear model reference adaptive control schemes. Habituating control is a controller design strategy for nonlinear systems with more manipulated inputs than controlled outputs. Nonlinear control laws that provide input-output linearization while simultaneously minimizing the cost of affecting control are derived. Local stability analysis shows the controller can provide a simple solution to singularity and non-minimum phase problems. A direct adaptive control strategy for a class of single-input, single-output non-linear systems is presented. The major advantage is that a detailed dynamic non-linear model is not required for controller design. Unknown controller functions in the associated input-output linearizing control law are approximated using locally supported radial basis functions. Lyapunov stability analysis is used to derive parameter update laws which ensure the state vector remains bounded and the plant output asymptotically tracks the output of a linear reference model. A nonlinear model reference adaptive control strategy in which a linear model (or multiple linear models) is embedded within the nonlinear controller is presented. The nonlinear control law is constructed by embedding linear controller gains derived from models obtained using standard linear system identification techniques within the associated input-output linearizing control law. Higher-order controller functions are approximated with radial basis functions. Lyapunov stability analysis is used to derive stable parameter update laws. The major disadvantage of the previous techniques is computational expense. Two modifications have been developed. First, the effective dimension is reduced by applying nonlinear principal component analysis to the state variable data obtained from open-loop tests. This allows basis functions to be placed in a lower dimensional space than the original state space. Second, the total number of basis functions is fixed a priori and an algorithm which adds/prunes basis function centers to surround the current operating point on-line is utilized