Provably-Correct Task Planning for Autonomous Outdoor Robots

Autonomous outdoor robots should be able to accomplish complex tasks safely and reliably while considering constraints that arise from both the environment and the physical platform. Such tasks extend basic navigation capabilities to specify a sequence of events over time. For example, an autonomous aerial vehicle can be given a surveillance task with contingency plans while complying with rules in regulated airspace, or an autonomous ground robot may need to guarantee a given probability of success while searching for the quickest way to complete the mission. A promising approach for the automatic synthesis of trusted controllers for complex tasks is to employ techniques from formal methods. In formal methods, tasks are formally specified symbolically with temporal logic. The robot then synthesises a controller automatically to execute trusted behaviour that guarantees the satisfaction of specified tasks and regulations. However, a difficulty arises from the lack of expressivity, which means the constraints affecting outdoor robots cannot be specified naturally with temporal logic. The goal of this thesis is to extend the capabilities of formal methods to express the constraints that arise from outdoor applications and synthesise provably-correct controllers with trusted behaviours over time. This thesis focuses on two important types of constraints, resource and safety constraints, and presents three novel algorithms that express tasks with these constraints and synthesise controllers that satisfy the specification. Firstly, this thesis proposes an extension to probabilistic computation tree logic (PCTL) called resource threshold PCTL (RT-PCTL) that naturally defines the mission specification with continuous resource threshold constraints; furthermore, it synthesises an optimal control policy with respect to the probability of success. With RT-PCTL, a state with accumulated resource out of the specified bound is considered to be failed or saturated depending on the specification. The requirements on resource bounds are naturally encoded in the symbolic specification, followed by the automatic synthesis of an optimal controller with respect to the probability of success. Secondly, the thesis proposes an online algorithm called greedy Buchi algorithm (GBA) that reduces the synthesis problem size to avoid the scalability problem. A framework is then presented with realistic control dynamics and physical assumptions in the environment such as wind estimation and fuel constraints. The time and space complexity for the framework is polynomial in the size of the system state, which is efficient for online synthesis. Lastly, the thesis proposes a synthesis algorithm for an optimal controller with respect to completion time given the minimum safety constraints. The algorithm naturally balances between completion time and safety. This work proves an analytical relationship between the probability of success and the conditional completion time given the mission specification. The theoretical contributions in this thesis are validated through realistic simulation examples. This thesis identifies and solves two core problems that contribute to the overall vision of developing a theoretical basis for trusted behaviour in outdoor robots. These contributions serve as a foundation for further research in multi-constrained task planning where a number of different constraints are considered simultaneously within a single framework

Correct-By-Construction Control Synthesis for Systems with Disturbance and Uncertainty

This dissertation focuses on correct-by-construction control synthesis for Cyber-Physical Systems (CPS) under model uncertainty and disturbance. CPSs are systems that interact with the physical world and perform complicated dynamic tasks where safety is often the overriding factor. Correct-by-construction control synthesis is a concept that provides formal performance guarantees to closed-loop systems by rigorous mathematic reasoning. Since CPSs interact with the environment, disturbance and modeling uncertainty are critical to the success of the control synthesis. Disturbance and uncertainty may come from a variety of sources, such as exogenous disturbance, the disturbance caused by co-existing controllers and modeling uncertainty. To better accommodate the different types of disturbance and uncertainty, the verification and control synthesis methods must be chosen accordingly. Four approaches are included in this dissertation. First, to deal with exogenous disturbance, a polar algorithm is developed to compute an avoidable set for obstacle avoidance. Second, a supervised learning based method is proposed to design a good student controller that has safety built-in and rarely triggers the intervention of the supervisory controller, thus targeting the design of the student controller. Third, to deal with the disturbance caused by co-existing controllers, a Lyapunov verification method is proposed to formally verify the safety of coexisting controllers while respecting the confidentiality requirement. Finally, a data-driven approach is proposed to deal with model uncertainty. A minimal robust control invariant set is computed for an uncertain dynamic system without a given model by first identifying the set of admissible models and then simultaneously computing the invariant set while selecting the optimal model. The proposed methods are applicable to many real-world applications and reflect the notion of using the structure of the system to achieve performance guarantees without being overly conservative.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145933/1/chenyx_1.pd

Design of of model-based controllers via parametric programming

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Inductive Certificate Synthesis for Control Design

The focus of this thesis is developing a framework for designing correct-by-construction controllers using control certificates. We use nonlinear dynamical systems to model the physical environment (plants). The goal is to synthesize controllers for these plants while guaranteeing formal correctness w.r.t. given specifications. We consider different fundamental specifications including stability, safety, and reach-while-stay. Stability specification states that the execution traces of the system remain close to an equilibrium state and approach it asymptotically. Safety specification requires the execution traces to stay in a safe region. Finally, for reach-while-stay specification, safety is needed until a target set is reached.The design task consists of two phases. In the first phase, the control design problem is reduced to the question of finding a control certificate. More precisely, the goal of the first phase is to define a class of control certificates with a specific structure. This definition should guarantee the following: ``Having a control certificate, one can systematically design a controller and prove its correctness at the same time."The goal in the second phase is to find such a control certificate. We define a potential control certificate space (hypothesis space) using parameterized functions. Next, we provide an inductive search framework to find proper parameters, which yield a control certificate. Finally, we evaluate our framework. We show that discovering control certificates is practically feasible and demonstrate the effectiveness of the automatically designed controllers through simulations and real physical systems experiments

Design and Certification of Industrial Predictive Controllers

Three decades have passed since milestone publications by several industrial and academic researchers spawned a flurry of research and commercial, industrial activities on model predictive control (MPC). The improvement in efficiency of the on-line optimization part of MPC led to its adoption in mechanical and mechatronic systems from process control and petrochemical applications. However, the massive strides made by the academic community in guaranteeing stability through state-space MPC have not always been directly applicable in an industrial setting. This thesis is concerned with design and a posteriori certification of feasibility/stability of input-output MPC controllers for industrial applications without terminal conditions (i.e. terminal penalty, terminal constraint, terminal control). MPC controllers which differ in their modelling and prediction method are categorized into three major groups, and a general equivalence between these forms is established. Then an overview on robust set invariance is given as it plays a fundamental role in our analysis of the constrained control systems. These tools are used to give new tuning guidelines as well as a posteriori tests for guaranteeing feasibility of the suboptimal or optimal predictive control law without terminal conditions, which is fundamental towards stability of the closed loop. Next, penalty adaptation is used as a systematic procedure to derive asymptotic stability without any terminal conditions and without using set invariance or Lyapunov arguments. This analysis however is restricted to repetitive systems with input constraints. Then, predictive control without terminal conditions is considered for nonlinear and distributed systems. The invariance tools are extended to switching nonlinear systems, a proof of convergence is given for the iterative nonlinear MPC (NMPC), and a guarantee on overall cost decrease is developed for distributed NMPC, all without terminal conditions. Reference generation and parameter adaptation are shown to be effective mechanisms for NMPC and distributed NMPC (DNMPC) under changing environmental conditions. This is demonstrated on two benchmark test-cases i.e. the wet-clutch and hydrostatic drivetrain, respectively. Terminal conditions in essence are difficult to compute, may compromise performance and are not used in the industry. The main contribution of the thesis is a systematic development and analysis of MPC without terminal conditions for linear, nonlinear and distributed systems.This work was supported within the framework of the LeCoPro project (grant nr. 80032) of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen)

Non-Convex Optimization and Applications to Bilinear Programming and Super-Resolution Imaging

Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering applications, economics, and operations research, and have a natural encoding to quadratic programs. They appear in the study of Lyapunov functions used to deduce the stability of solutions to differential equations describing dynamical systems. For multivariate dynamical systems, the problem formulation for computing an appropriate Lyapunov function is a BLP. In electric power systems engineering, one of the most practically important and well-researched subfields of constrained nonlinear optimization is Optimal Power Flow wherein one attempts to optimize an electric power system subject to physical constraints imposed by electrical laws and engineering limits, which can be naturally formulated as a quadratic program. In a recent publication, we studied the relationship between data flow constraints for numerical domains such as polyhedra and bilinear constraints. The problem of recovering an image from its Fourier modulus, or intensity, measurements emerges in many physical and engineering applications. The problem is known as Fourier phase retrieval wherein one attempts to recover the phase information of a signal in order to accurately reconstruct it from estimated intensity measurements by applying the inverse Fourier transform. The problem of recovering phase information from a set of measurements can be formulated as a quadratic program. This problem is well-studied but still presents many challenges. The resolution of an optical device is defined as the smallest distance between two objects such that the two objects can still be recognized as separate entities. Due to the physics of diffraction, and the way that light bends around an obstacle, the resolving power of an optical system is limited. This limit, known as the diffraction limit, was first introduced by Ernst Abbe in 1873. Obtaining the complete phase information would enable one to perfectly reconstruct an image; however, the problem is severely ill-posed and the leads to a specialized type of quadratic program, known as super-resolution imaging, wherein one attempts to learn phase information beyond the limits of diffraction and the limitations imposed by the imaging device

Advances in combined architecture, plant, and control design

The advancement of many engineering systems relies on novel design methodologies, design formulations, design representations, and other advancements. In this dissertation, we consider three broad design domains: architecture, plant, and control. These domains cover most of the potential design decision elements in an actively-controlled engineering system. In this dissertation, strategic aspects of this combined problem are addressed. The task of representing and generating candidate architectures is addressed with methods developed based on colored graphs built by enumerating all perfect matchings of a specified catalog of components. The proposed approach captures all architectures under specific assumptions. General combined plant and control design (or co-design) problems are examined. Previous work in co-design theory imposed restrictions on the type of problems that could be posed. Here many of those restrictions are lifted. The problem formulations and optimality conditions for both the simultaneous and nested solution strategies are given along with a detailed discussion of the two methods. Direct transcription is also discussed as it enables the solution of general co-design problems by approximating the problem. Motivated primarily by the need for efficient methods to solve certain control problems that emerge using the nested co-design method, an automated problem generation procedure is developed to support easy specification of linear-quadratic dynamic optimization problems using direct transcription and quadratic programming. Pseudospectral and single-step methods (including the zero-order hold) are all implemented in this unified framework and comparisons are made. Three detailed engineering design case studies are presented. The results from the enumeration and evaluation of all passive analog circuits with up to a certain number of components are used to synthesize low-pass filters and circuits that match a certain magnitude response. Advantages and limitations of enumerative approaches are highlighted in this case study, along with comparisons to circuits synthesized via evolutionary computation; many similarities are found in the topologies. The second case study tackles a complex co-design problem with the design of strain-actuated solar arrays for spacecraft precision pointing and jitter reduction. Nested co-design is utilized along with a linear-quadratic inner loop problem to obtain solutions efficiently. A simpler, scaled problem is analyzed to gain general insights into these results. This is accomplished with a unified theory of scaling in dynamic optimization. The final case study involves the design of active vehicle suspensions. All three design domains are considered in this problem. A class of architecture, plant, and control design problems which utilize linear physical elements is discussed. This problem class can be solved using the methods in this dissertation

Wind power forecasting and its applications to the power system

textThe goal of research in this dissertation is to bring more wind resources into the power grid by mitigating the uncertainty of the current wind power, by developing a new algorithm to respond to the fluctuation of the future wind power, and by building additional transmission lines to bring more wind resources from a remote area to the load center. First, in order to overcome the wind power uncertainty, the probabilistic and ensemble wind power forecasting is proposed to increase the forecasting accuracy and to deliver the probability density function of the uncertainty. Accurate wind power forecasting reduces the amounts and cost of ancillary services (AS). As the mismatch between the bid and actual amount of delivered energy decreases, the imbalance between supply and demand also decreases. If the forecasting ahead is increased up to 24 hours, accurate wind power forecasting can also help wind farm owners bid the exact amount of wind power in the day ahead (DA) market. Furthermore, wind power owners can use the parametric probabilistic density of error distributions for hedging the price risk and building a better offer curve. Second, a novel algorithm to generate many wind power scenarios as a function of installed capacity of wind power is proposed based on an analysis of the power spectral density of wind power. Scenarios can be used to simulate the power system to estimate the required amount of AS to respond to the fluctuation of future wind power as the installed capacity of wind power increases. Scenarios have statistical characteristics of the future wind power that are regressed as a function of the installed capacity of wind power from the statistical characteristics of the current wind power. This algorithm can generate many possible scenarios to simulate the power system in many different situations. Third, optimal transmission expansion by simulating the power system with the multiple load and wind power scenarios in different locations is planned to prepare the preliminary result to bring more wind resources in remote areas to the load center in Texas. In this process, the geographical smoothing effects of wind power and the stochastic correlation structure between the load and wind power are considered. Furthermore, the generalized dynamic factor model (GDFM) is used to synthesize load and wind power scenarios to keep their correlation structure. The premise of the GDFM is that a few factors can drive the correlated movements of load and wind power simultaneously, so the scenario generation process is parsimonious.Electrical and Computer Engineerin

Polynomial Approximation of Value Functions and Nonlinear Controller Design with Performance Bounds

For any suitable Optimal Control Problem (OCP) which satisfies the Principle of Optimality, there exists a value function, defined as the unique viscosity solution to a HJB PDE, and which can be used to design an optimal feedback controller for the given OCP. Solving the HJB analytically is rarely possible, and existing numerical approximation schemes largely rely on discretization - implying that the resulting approximate value functions may not have the useful property of being uniformly less than or equal to the true value function (ie be sub-value functions). Furthermore, controllers obtained from such schemes currently have no associated bound on performance. To address these issues, for a given OCP, we propose a sequence of Sum-Of-Squares (SOS) problems, each of which yields a polynomial sub-solution to the HJB PDE, and show that the resulting sequence of polynomial sub-solutions converges to the value function of the OCP in the $L^1$ norm. Furthermore, for each polynomial sub-solution in this sequence we define an associated sublevel set, and show that the resulting sequence of sublevel sets converges to the sub-level set of the value function of the OCP in the volume metric. Next, for any approximate value function, obtained from an SOS program or any other method (e.g. discretization), we construct an associated feedback controller, and show that sub-optimality of this controller as applied to the OCP is bounded by the distance between the approximate and true value function of the OCP in the $W^{1,\infty}$ (Sobolev) norm. This result implies approximation of value functions in the $W^{1,\infty}$ norm results in feedback controllers with performance that can be made arbitrarily close to optimality. Finally, we demonstrate numerically that by solving our proposed SOS problem we are able to accurately approximate value functions, design controllers and estimate reachable sets