Explicit Reference Governor for Continuous Time Nonlinear Systems Subject to Convex Constraints

This paper introduces a novel closed-form strategy that dynamically modifies the reference of a pre-compensated nonlinear system to ensure the satisfaction of a set of convex constraints. The main idea consists of translating constraints in the state space into constraints on the Lyapunov function and then modulating the reference velocity so as to limit the value of the Lyapunov function. The theory is introduced for general nonlinear systems subject to convex constraints. In the case of polyhedric constraints, an explicit solution is provided for the large and highly relevant class of nonlinear systems whose Lyapunov function is lower-bounded by a quadratic form. In view of improving performances, further specializations are provided for the relevant cases of linear systems and robotic manipulators.Comment: Submitted to: IEEE Transactions on Automatic Contro

Nonlinear constrained and saturated control of power electronics and electromechanical systems

Power electronic converters are extensively adopted for the solution of timely issues, such as power quality improvement in industrial plants, energy management in hybrid electrical systems, and control of electrical generators for renewables. Beside nonlinearity, this systems are typically characterized by hard constraints on the control inputs, and sometimes the state variables. In this respect, control laws able to handle input saturation are crucial to formally characterize the systems stability and performance properties. From a practical viewpoint, a proper saturation management allows to extend the systems transient and steady-state operating ranges, improving their reliability and availability. The main topic of this thesis concern saturated control methodologies, based on modern approaches, applied to power electronics and electromechanical systems. The pursued objective is to provide formal results under any saturation scenario, overcoming the drawbacks of the classic solution commonly applied to cope with saturation of power converters, and enhancing performance. For this purpose two main approaches are exploited and extended to deal with power electronic applications: modern anti-windup strategies, providing formal results and systematic design rules for the anti-windup compensator, devoted to handle control saturation, and “one step” saturated feedback design techniques, relying on a suitable characterization of the saturation nonlinearity and less conservative extensions of standard absolute stability theory results. The first part of the thesis is devoted to present and develop a novel general anti-windup scheme, which is then specifically applied to a class of power converters adopted for power quality enhancement in industrial plants. In the second part a polytopic differential inclusion representation of saturation nonlinearity is presented and extended to deal with a class of multiple input power converters, used to manage hybrid electrical energy sources. The third part regards adaptive observers design for robust estimation of the parameters required for high performance control of power systems

Advances in Constrained Spacecraft Relative Motion Planning

This dissertation considers Spacecraft Relative Motion Planning (SRMP), where maneuvers are planned for one or more spacecraft to execute in close proximity to obstacles or to each other. The need for this type of maneuver planning has grown in recent years as the space environment becomes more cluttered, and the focus on space situational awareness increases. In SRMP, maneuvers must accommodate non-linear and non-convex constraints, be robust to disturbances, and be implementable on-board spacecraft with limited computational capabilities. Consequently, many standard optimization or path planning techniques cannot be directly applied to SRMP. In this dissertation, three novel SRMP techniques are developed and simulations are presented to illustrate the implementation of each method. Firstly, an invariance-based SRMP technique is proposed. Maneuvers are planned to transition a spacecraft between specified natural motion trajectories, which require no control to follow, while avoiding obstacles and accommodating minimum and maximum actuation limits. The method is based on a graph search applied to a ``virtual net'' with nodes corresponding to natural motion trajectories. Adjacency rules in the virtual net are based on safe positively invariant tubes built around each natural motion trajectory. These rules guarantee safe transitions between adjacent natural motion trajectories, even when set-bounded disturbances are present. Procedures to construct the safe positively invariant tubes and the virtual net are developed. Methods to reduce calculations are proposed and shown to significantly reduce computation time, with tradeoffs related to maneuver planning flexibility. Secondly, a SRMP technique is developed for the specific problem of satellite inspection. In this setting, an inspector spacecraft maneuvers to gather information about a target spacecraft. An information collection model is developed and used to construct a rapidly computable analytical control law based on the local gradient of the information rate. This control law drives the inspector spacecraft on a path along which the rate of information collection is strictly increasing. To ensure constraint satisfaction, the local gradient control law is combined with a state feedback control law, and rules are developed to govern switches between the two controllers. The method is shown to be effective in generating trajectories to gather information about a specified target point while accommodating disturbances. Finally, a control strategy is proposed to generate a formation containing an arbitrary number of vehicles. This strategy is based on an add-on predictive control mechanism known as a parameter governor. Parameter governors work by modifying parameters, such as gains or offsets, in a nominal closed-loop system to enforce constraints and improve performance. The parameter governor is first developed in a general setting, using generic non-linear system dynamics and an arbitrary formation design. Required calculations are minimized, and non-convex constraints are accommodated through use of a parameter update strategy based on graph colorability theory, and by limiting parameter values to a discrete set. A convergence analysis is presented, proving that under reasonable assumptions, the parameter governor is guaranteed to generate the desired formation. Two specific parameter governors, referred to as the Scale Shift Governor and Time Shift Governor, are proposed and applied to generate formations of spacecraft. These parameter governors enforce constraints by modifying either scale- or time-shifts applied to the target trajectory provided to each spacecraft in formation. Simulation case studies show the effectiveness of each method and demonstrate robustness to disturbances.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145795/1/gfrey_1.pd

Reference governors: Theoretical Extensions and Practical Applications.

As systems become downsized and operate at the limits of performance, control systems must be designed to ensure that system state and control constraints are satisfied; however, conventional control schemes are often designed without taking constraints into account. Reference governors and the related, more flexible, extended command governors are add-on, constraint enforcement schemes that modify reference signals to conventionally designed, closed-loop systems for the purpose of enforcing output constraints. The focus of this dissertation is on theoretical and methodological extensions of reference and extended command governors, and on their practical applications. Various theoretical results are presented. The first is the development of reduced-order reference and extended command governors, which enables constraint enforcement schemes using simplified models. The second, related development is that of reference governors for decentralized systems that may or may not communicate over a network. The third considers command governors with penalty functions that are used to enforce prioritized sets of constraints, as well as reference governors that are applied to a sequence of prioritized references. The fourth considers the often overlooked case of applying reference governors to linear systems subject to nonlinear constraints; various formulations of constraints are considered, including quadratic constraints and mixed logical-dynamic constraints. The final theoretical development considers using contractive sets to design reference governors for systems with time-varying reference inputs or subject to time-dependent constraints. Numerical simulations are used throughout to illustrate the theoretical advances. The design of reference governor schemes for three systems arising in practical applications is also presented. The first scheme enforces compressor surge constraints for turbocharged gasoline engines, ensuring that the compressor does not surge. The second scheme is designed for an airborne wind energy system that is subject to various flight constraints including constraints on altitude and angle of attack. The third and final scheme is designed for the constrained control of spacecraft attitude, whose discrete-time dynamics evolve on the configuration space SO(3). In the case of the first application, experimental vehicle results are reported that show successful avoidance of surge. For the other two applications, nonlinear model simulation results are reported that show enforcement of system constraints.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113518/1/kalabic_1.pd

Reference Governors: From Theory to Practice

Control systems that are subject to constraints due to physical limitations, hardware protection, or safety considerations have led to challenging control problems that have piqued the interest of control practitioners and theoreticians for many decades. In general, the design of constraint management schemes must meet several stringent requirements, for example: low computational burden, performance, recovery mechanisms from infeasibility conditions, robustness, and formulation simplicity. These requirements have been particularly difficult to meet for the following three classes of systems: stochastic systems, linear systems driven by unmodeled disturbances, and nonlinear systems. Hence, in this work, we develop three constraint management schemes, based on Reference Governor (RG), for these classes of systems. The first scheme, which is referred to as Stochastic RG, leverages the ideas of chance constraints to construct a Stochastic Robustly Invariant Maximal Output Admissible set (SR-MAS) in order to enforce constraints on stochastic systems. The second scheme, which is called Recovery RG (RRG), addresses the problem of recovery from infeasibility conditions by implementing a disturbance observer to update the MAS, and hence recover from constraint violations due to unmodeled disturbances. The third method addresses the problem of constraint satisfaction on nonlinear systems by decomposing the design of the constraint management strategy into two parts: enforcement at steady-state, and during transient. The former is achieved by using the forward and inverse steady-state characterization of the nonlinear system. The latter is achieved by implementing an RG-based approach, which employs a novel Robust Output Admissible Set (ROAS) that is computed using data obtained from the nonlinear system. Added to this, this dissertation includes a detailed literature review of existing constraint management schemes to compare and highlight advantages and disadvantages between them. Finally, all this study is supported by a systematic analysis, as well as numerical and experimental validation of the closed-loop systems performance on vehicle roll-over avoidance, turbocharged engine control, and inverted pendulum control problems

Low Complexity Model Predictive Control of a Diesel Engine Airpath.

The diesel air path (DAP) system has been traditionally challenging to control due to its highly coupled nonlinear behavior and the need for constraints to be considered for driveability and emissions. An advanced control technology, model predictive control (MPC), has been viewed as a way to handle these challenges, however, current MPC strategies for the DAP are still limited due to the very limited computational resources in engine control units (ECU). A low complexity MPC controller for the DAP system is developed in this dissertation where, by "low complexity," it is meant that the MPC controller achieves tracking and constraint enforcement objectives and can be executed on a modern ECU within 200 microseconds, a computation budget set by Toyota Motor Corporation. First, an explicit MPC design is developed for the DAP. Compared to previous explicit MPC examples for the DAP, a significant reduction in computational complexity is achieved. This complexity reduction is accomplished through, first, a novel strategy of intermittent constraint enforcement. Then, through a novel strategy of gain scheduling explicit MPC, the memory usage of the controller is further reduced and closed-loop tracking performance is improved. Finally, a robust version of the MPC design is developed which is able to enforce constraints in the presence of disturbances without a significant increase in computational complexity compared to non-robust MPC. The ability of the controller to track set-points and enforce constraints is demonstrated in both simulations and experiments. A number of theoretical results pertaining to the gain scheduling strategy is also developed. Second, a nonlinear MPC (NMPC) strategy for the DAP is developed. Through various innovations, a NMPC controller for the DAP is constructed that is not necessarily any more computationally complex than linear explicit MPC and is characterized by a very streamlined process for implementation and calibration. A significant reduction in computational complexity is achieved through the novel combination of Kantorovich's method and constrained NMPC. Zero-offset steady state tracking is achieved through a novel NMPC problem formulation, rate-based NMPC. A comparison of various NMPC strategies and developments is presented illustrating how a low complexity NMPC strategy can be achieved.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120832/1/huxuli_1.pd

Improving Tracking in Optimal Model Predictive Control

The thesis deals with the improvement in the tracking in model predictive control(MPC). The main motivation is to explore high embedding performance controllers with constraint handling capabilities in a simple fashion. There are several techniques available for effectively using an infinite horizon rather than a finite horizon. First, there has been relatively little discussion so far on how to make effective use of advance information on target changes in the predictive control literature. While earlier work has indicated that the default solutions from conventional algorithms are often poor, very few alternatives have been proposed. This thesis demonstrates the impact of future information about target changes on performance, and proposes a pragmatic method for identifying the amount of future information on the target that can be utilised effectively in infinite horizon algorithms. Numerical illustrations in MATLAB demonstrate that the proposal is both systematic and beneficial. This thesis introduces several important issues related to model predictive control (MPC)tracking that have been hitherto neglected in the literature, by first deriving a control law for future information about target changes within optimal predictive control (OMPC) for both nominal and constraints cases. This thesis proposes a pragmatic design for scenarios in which the target is unreachable. In order to deal with an unreachable target, the proposed design allows an artificial target into the MPC optimisation problem. Numerical illustrations in MATLAB provide evidence of the efficacy of the proposals. This thesis extends efficient, robust model predictive control (MPC) approaches for Linear Parameter-Varying (LPV) systems to tracking scenarios. A dual-mode approach is used and future information about target changes is included in the optimisation tracking problem. The controller guarantees recursive feasibility by adding an artificial target as an extra degree of freedom. Convergence to admissible targets is ensured by constructing a robustly invariant set to track any admissible target. The efficacy of the proposed algorithm is demonstrated by MATLAB simulation. The thesis considers the tractability of parametric solvers for predictive control based optimisations, when future target information is incorporated. It is shown that the inclusion of future target information can significantly increase the implied parametric dimension to an extent that is undesirable and likely to lead to intractable problems. The thesis then proposes some alternative methods for incorporating the desired target information, while minimising the implied growth in the parametric dimensions, at some possibly small cost to optimality. Feasibility is an important issue in predictive control, but the influence of many important parameters such as the desired steady-state, the target and the current value of the input is rarely discussed in the literature. At this point, the thesis makes two contributions. First, it gives visibility to the issue that including the core parameters, such as the target and the current input, vastly increases the dimension of the parametric space, with possible consequences on the complexity of any parametric solutions. Secondly, it is shown that a simple re-parametrization of the degrees of freedom to take advantage of allowing steady-state offset can lead to large increases in the feasible volumes, with no increases in the dimension of the required optimisation variables. Simulation with MAT LAB 2017a provides the evidence of the efficacy of all proposals

Advanced Computational-Effective Control and Observation Schemes for Constrained Nonlinear Systems

Constraints are one of the most common challenges that must be faced in control systems design. The sources of constraints in engineering applications are several, ranging from actuator saturations to safety restrictions, from imposed operating conditions to trajectory limitations. Their presence cannot be avoided, and their importance grows even more in high performance or hazardous applications. As a consequence, a common strategy to mitigate their negative effect is to oversize the components. This conservative choice could be largely avoided if the controller was designed taking all limitations into account. Similarly, neglecting the constraints in system estimation often leads to suboptimal solutions, which in turn may negatively affect the control effectiveness. Therefore, with the idea of taking a step further towards reliable and sustainable engineering solutions, based on more conscious use of the plants' dynamics, we decide to address in this thesis two fundamental challenges related to constrained control and observation. In the first part of this work, we consider the control of uncertain nonlinear systems with input and state constraints, for which a general approach remains elusive. In this context, we propose a novel closed-form solution based on Explicit Reference Governors and Barrier Lyapunov Functions. Notably, it is shown that adaptive strategies can be embedded in the constrained controller design, thus handling parametric uncertainties that often hinder the resulting performance of constraint-aware techniques. The second part of the thesis deals with the global observation of dynamical systems subject to topological constraints, such as those evolving on Lie groups or homogeneous spaces. Here, general observability analysis tools are overviewed, and the problem of sensorless control of permanent magnets electrical machines is presented as a case of study. Through simulation and experimental results, we demonstrate that the proposed formalism leads to high control performance and simple implementation in embedded digital controllers

Reference Governors for MIMO Systems and Preview Control: Theory, Algorithms, and Practical Applications

The Reference Governor (RG) is a methodology based on predictive control for constraint management of pre-stablized closed-loop systems. This problem is motivated by the fact that control systems are usually subject to physical restrictions, hardware protection, and safety and efficiency considerations. The goal of RG is to optimize the tracking performance while ensuring that the constraints are satisfied. Due to structural limitations of RG, however, these requirements are difficult to meet for Multi-Input Multi-Output (MIMO) systems or systems with preview information. Hence, in this dissertation, three extensions of RG for constraint management of these classes of systems are developed. The first approach aims to solve constraint management problem for linear MIMO systems based on decoupling the input-output dynamics, followed by the deployment of a bank of RGs for each decoupled channel, namely Decoupled Reference Governor (DRG). This idea was originally developed in my previous work based on transfer function decoupling, namely DRG-tf. This dissertation improves the design of DRG-tf, analyzes the transient performance of DRG-tf, and extends the DRG formula to state space representations. The second scheme, which is called Preview Reference Governor, extends the applicability of RG to systems incorporated with the preview information of the reference and disturbance signals. The third subject focuses on enforcing constraints on nonlinear MIMO systems. To achieve this goal, three different methods are established. In the first approach, which is referred to as the Nonlinear Decoupled Reference Governor (NL-DRG), instead of employing the Maximal Admissible set and using the decoupling methods as the DRG does, numerical simulations are used to compute the constraint-admissible setpoints. Given the extensive numerical simulations required to implement NL-DRG, the second approach, namely Modified RG (M-RG), is proposed to reduce the computational burden of NL-DRG. This solution consists of the sequential application of different RGs based on linear prediction models, each robustified to account for the worst-case linearization error as well as coupling behavior. Due to this robustification, however, M-RG may lead to a conservative response. To lower the computation time of NL-DRG while improving the performance of M-RG, the third approach, which is referred to as Neural Network DRG (NN-DRG), is proposed. The main idea behinds NN-DRG is to approximate the input-output mapping of NL-DRG with a well-trained NN model. Afterwards, a Quadratic Program is solved to augment the results of NN such that the constraints are satisfied at the next timestep. Additionally, motivated by the broad utilization of quadcopter drones and the necessity to impose constraints on the angles and angle rates of drones, the simulation and experimental results of the proposed nonlinear RG-based methods on a real quadcopter are demonstrated