We present a nonlinear model predictive control (MPC) scheme for tracking of dynamic target signals. The scheme combines stabilization and dynamic trajectory planning in one layer, thus ensuring constraint satisfaction irrespective of changes in the dynamic target signal. For periodic target signals we ensure exponential stability of the optimal reachable periodic trajectory using suitable terminal ingredients and a convexity condition for the underlying periodic optimal control problem. Furthermore, we introduce an online optimization of the terminal set size to automate the trade-off between fast convergence and operation close to the constraints. In addition, we show how stabilization and dynamic trajectory planning can be formulated as partially decoupled optimization problems, which reduces the computational demand while ensuring recursive feasibility and convergence. The main tool to enable the proposed design is a novel reference generic offline computation that provides suitable terminal ingredients for tracking of dynamic reference trajectories. The practicality of this approach is demonstrated on benchmark examples, which demonstrates superior performance compared to state of the art approaches.
In this paper, we investigate system theoretic properties of transient average constrained economic model predictive control (MPC) without terminal constraints. We show that the optimal open-loop solution passes by the optimal steady-state for consecutive time instants. Using this turnpike property and suitable controllability conditions, we provide closed-loop performance bounds. Furthermore, stability is proved by combining the rotated value function with an input-to-state (ISS) Lyapunov function of an extended state related to the transient average constraints. The results are illustrated with a numerical example.
Solving nonlinear model predictive control problems in real time is still an important challenge despite of recent advances in computing hardware, optimization algorithms and tailored implementations. This challenge is even greater when uncertainty is present due to disturbances, unknown parameters or measurement and estimation errors. To enable the application of advanced control schemes to fast systems and on low-cost embedded hardware, we propose to approximate a robust nonlinear model controller using deep learning and to verify its quality using probabilistic validation techniques. We propose a probabilistic validation technique based on finite families, combined with the idea of generalized maximum and constraint backoff to enable statistically valid conclusions related to general performance indicators. The potential of the proposed approach is demonstrated with simulation results of an uncertain nonlinear system.
We propose a data-driven tracking model predictive control (MPC) scheme to control unknown discrete-time linear time-invariant systems. The scheme uses a purely data-driven system parametrization to predict future trajectories based on behavioral systems theory. The control objective is tracking of a given input-output setpoint. We prove that this setpoint is exponentially stable for the closed loop of the proposed MPC, if it is reachable by the system dynamics and constraints. For an unreachable setpoint, our scheme guarantees closed-loop exponential stability of the optimal reachable equilibrium. Moreover, in case the system dynamics are known, the presented results extend the existing results for model-based setpoint tracking to the case where the stage cost is only positive semidefinite in the state. The effectiveness of the proposed approach is illustrated by means of a practical example.
This paper presents a heterogeneously parameterized tube-based model predictive control (MPC) design applicable to linear parameter-varying (LPV) systems. In a heterogeneous tube, the parameterizations of the tube cross sections and the associated control laws are allowed to vary along the prediction horizon. Two extreme cases that can be described in this framework are scenario MPC (high complexity, larger domain of attraction) and homothetic tube MPC with a simple time-invariant control parameterization (low complexity, smaller domain of attraction). In the proposed framework, these extreme parameterizations, as well as other parameterizations of intermediate complexity, can be combined within a single tube. By allowing for more flexibility in the parameterization design, one can influence the trade-off between computational cost and the size of the domain of attraction. Sufficient conditions on the parameterization structure are developed under which recursive feasibility and closed-loop stability are guaranteed. A specific parameterization that combines the principles of scenario and homothetic tube MPC is proposed and it is shown to satisfy the required conditions. The properties of the approach, including its capability of achieving improved complexity/performance trade-offs, are demonstrated using two numerical examples.
Reachability analysis, in general, is a fundamental method that supports formally-correct synthesis, robust model predictive control, set-based observers, fault detection, invariant computation, and conformance checking, to name but a few. In many of these applications, one requires to compute a reachable set starting within a previously computed reachable set. While it was previously required to re-compute the entire reachable set, we demonstrate that one can leverage the dependencies of states within the previously computed set. As a result, we almost instantly obtain an over-approximative subset of a previously computed reachable set by evaluating analytical maps. The advantages of our novel method are demonstrated for falsification of systems, optimization over reachable sets, and synthesizing safe maneuver automata. In all of these applications, the computation time is reduced significantly.
Buildings sector is one of the major consumers of energy in the United States. The buildings HVAC (Heating, Ventilation, and Air Conditioning) systems, whose functionality is to maintain thermal comfort and indoor air quality (IAQ), account for almost half of the energy consumed by the buildings. Thus, intelligent scheduling of the building HVAC system has the potential for tremendous energy and cost savings while ensuring that the control objectives (thermal comfort, air quality) are satisfied. Recently, several works have focused on model-free deep reinforcement learning based techniques such as Deep Q-Network (DQN). Such methods require extensive interactions with the environment. Thus, they are impractical to implement in real systems due to low sample efficiency. Safety-aware exploration is another challenge in real systems since certain actions at particular states may result in catastrophic outcomes. To address these issues and challenges, we propose a model-based reinforcement learning approach that learns the system dynamics using a neural network. Then, we adopt Model Predictive Control (MPC) using the learned system dynamics to perform control with random-sampling shooting method. To ensure safe exploration, we limit the actions within safe range and the maximum absolute change of actions according to prior knowledge. We evaluate our ideas through simulation using widely adopted EnergyPlus tool on a case study consisting of a two zone data-center. Experiments show that the average deviation of the trajectories sampled from the learned dynamics and the ground truth is below $20\%$. Compared with baseline approaches, we reduce the total energy consumption by $17.1\% \sim 21.8\%$. Compared with model-free reinforcement learning approach, we reduce the required number of training steps to converge by 10x.
A gradual takeover strategy is proposed, in which the dynamic driving privilege assignment in real-time and the driving privilege gradual handover are realized. Firstly, the driving privilege assignment based on the risk level is achieved. The naturalistic driving data is applied to study the driver behavior during danger. TTC (time to collision) is defined as an obvious risk measure, whereas the time before the host vehicle has to brake assuming that the target vehicle is braking is defined as the potential risk measure, i.e. the time margin (TM). A risk assessment algorithm is proposed based on the obvious risk and potential risk. Secondly, the driving privilege gradual handover is realized. The non-cooperative MPC (model predictive control) is employed to resolve the conflicts between the driver and active safety system. The naturalistic driving data are applied to verify the effectiveness of the risk assessment algorithm, and the risk assessment algorithm performs better than TTC in the ROC (receiver operating characteristic). It is identified that the Nash equilibrium of the non-cooperative MPC can be achieved by using a non-iterative method. The driving privilege gradual handover is realized by using the confidence matrixes updating. The simulation verification shows that the gradual takeover strategy can achieve the driving privilege gradual handover between the driver and active safety system.
This paper analyzes the stability-related properties of Long Short-Term Memory (LSTM) networks and investigates their use as the model of the plant in the design of Model Predictive Controllers (MPC). First, sufficient conditions guaranteeing the Input-to-State stability (ISS) and Incremental Input-to-State stability (dISS) of LSTM are derived. These properties are then exploited to design an observer with guaranteed convergence of the state estimate to the true one. Such observer is then embedded in a MPC scheme solving the tracking problem. The resulting closed-loop scheme is proved to be asymptotically stable. The training algorithm and control scheme are tested numerically on the simulator of a pH reactor, and the reported results confirm the effectiveness of the proposed approach.
This paper deals with the problem of formulating an adaptive Model Predictive Control strategy for constrained uncertain systems. We consider a linear system, in presence of bounded time varying additive uncertainty. The uncertainty is decoupled as the sum of a process noise with known bounds, and a time varying offset that we wish to identify. The time varying offset uncertainty is assumed unknown point-wise in time. Its domain, called the Feasible Parameter Set, and its maximum rate of change are known to the control designer. As new data becomes available, we refine the Feasible Parameter Set with a Set Membership Method based approach, using the known bounds on process noise. We consider two separate cases of robust and probabilistic constraints on system states, with hard constraints on actuator inputs. In both cases, we robustly satisfy the imposed constraints for all possible values of the offset uncertainty in the Feasible Parameter Set. By imposing adequate terminal conditions, we prove recursive feasibility and stability of the proposed algorithms. The efficacy of the proposed robust and stochastic Adaptive MPC algorithms is illustrated with detailed numerical examples.
The electrical grid is evolving from a network consisting of mostly synchronous machines to a mixture of synchronous machines and inverter-based resources such as wind, solar, and energy storage. This transformation has led to a decrease in mechanical inertia, which necessitate a need for the new resources to provide frequency responses through their inverter interfaces. In this paper we proposed a new strategy based on model predictive control to determine the optimal active-power set-point for inverters in the event of a disturbance in the system. Our framework explicitly takes the hard constraints in power and energy into account, and we show that it is robust to measurement noise, limited communications and delay by using an observer to estimate the model mismatches in real-time. We demonstrate the proposed controller significantly outperforms an optimally tuned virtual synchronous machine on a standard 39-bus system under a number of scenarios. In turn, this implies optimized inverter-based resources can provide better frequency responses compared to conventional synchronous machines.
Lithium-ion battery packs are usually composed of hundreds of cells arranged in series and parallel connections. The proper functioning of these complex devices requires suitable Battery Management Systems (BMSs). Advanced BMSs rely on mathematical models to assure safety and high performance. While many approaches have been proposed for the management of single cells, the control of multiple cells has been less investigated and usually relies on simplified models such as equivalent circuit models. This paper addresses the management of a battery pack in which each cell is explicitly modelled as the Single Particle Model with electrolyte and thermal dynamics. A nonlinear Model Predictive Control (MPC) is presented for optimally charging the battery pack while taking voltage and temperature limits on each cell into account. Since the computational cost of nonlinear MPC grows significantly with the complexity of the underlying model, a sensitivity-based MPC (sMPC) is proposed, in which the model adopted is obtained by linearizing the dynamics along a nominal trajectory that is updated over time. The resulting sMPC optimizations are quadratic programs which can be solved in real-time even for large battery packs (e.g. fully electric motorbike with 156 cells) while achieving the same performance of the nonlinear MPC.
We study operations of a battery energy storage system under a baseline-based demand response (DR) program with an uncertain schedule of DR events. Baseline-based DR programs may provide undesired incentives to inflate baseline consumption in non-event days, in order to increase "apparent" DR reduction in event days and secure higher DR payments. Our goal is to identify and quantify such incentives. To understand customer decisions, we formulate the problem of determining hourly battery charging and discharge schedules to minimize expected net costs, defined as energy purchase costs minus energy export rebates and DR payments, over a sufficiently long time horizon (e.g., a year). The complexity of this stochastic optimization problem grows exponentially with the time horizon considered. To obtain computationally tractable solutions, we propose using multistage model predictive control with scenario sampling. Numerical results indicate that our solutions are near optimal (e.g., within 3% from the optimum in the test cases). Finally, we apply our solutions to study an example residential customer with solar photovoltaic and battery systems participating in a typical existing baseline-based DR program. Results reveal that over 66% of the average apparent load reduction during DR events could result from inflation of baseline consumption during non-event days.
In this paper, we present a robust adaptive model predictive control (MPC) scheme for linear systems subject to parametric uncertainty and additive disturbances. The proposed approach provides a computationally efficient formulation with theoretical guarantees (constraint satisfaction and stability), while allowing for reduced conservatism and improved performance due to online parameter adaptation. A moving window parameter set identification is used to compute a fixed complexity parameter set based on past data. Robust constraint satisfaction is achieved by using a computationally efficient tube based robust MPC method. The predicted cost function is based on a least mean squares point estimate, which ensures finite-gain $\mathcal{L}_2$ stability of the closed loop. The overall algorithm has a fixed (user specified) computational complexity. We illustrate the applicability of the approach and the trade-off between conservatism and computational complexity using a numerical example. This paper is an extended version of~[1], and contains additional details regarding the theoretical proof of Theorem~1, the numerical example, and the offline computations in Appendix~A--B.
In this work, the control of snake robot locomotion via economic model predictive control (MPC) is studied. Only very few examples of applications of MPC to snake robots exist and rigorous proofs for recursive feasibility and convergence are missing. We propose an economic MPC algorithm that maximizes the robot's forward velocity and integrates the choice of the gait pattern into the closed loop. We show recursive feasibility of the MPC optimization problem, where some of the developed techniques are also applicable for the analysis of a more general class of system. Besides, we provide performance results and illustrate the achieved performance by numerical simulations. We thereby show that the economic MPC algorithm outperforms a standard lateral undulation controller and achieves constraint satisfaction. Surprisingly, a gait pattern different to lateral undulation results from the optimization.
We propose a novel two-layer multi-agent architecture aimed at efficient real-time control of large-scale and complex-dynamics systems. The proposed architecture integrates intelligent control approaches (which have a low computation time and fit real-time applications) with model-predictive control (which takes care of the optimality requirements of control). The bottom control layer (intelligent-control module) includes several distributed intelligent-control agents, the design parameters of which are tuned by the top layer (model-predictive control module). The model-predictive control module fulfills two significant roles: looking ahead to the effects of the control decisions, and coordinating the intelligent-control agents of the lower control layer. The resulting multi-agent control system has a very low computation time, and provides adaptivity, control coordination, and aims at excellent performance. Additionally, we give a general treatment of type-2 fuzzy membership functions, and introduce two categories for them: probabilistic-fuzzy (which is a novel concept introduced in this paper) and fuzzy-fuzzy (which is a new treatment of the existing type-2 fuzzy membership functions). The performance of the proposed modeling and control approaches are assessed via a case study involving a simple urban traffic network: the results show that the novel concept of probabilistic-fuzzy membership function outperforms the type-1 and type-2 membership functions that have already been introduced in the literature. Furthermore, the proposed two-layer integrated multi-agent control architecture significantly outperforms a multi-agent decentralized fuzzy control system (without coordination among the agents), while requiring a comparable computation time.
This paper proposes a model predictive control (MPC)-based energy management system to address communication failures in an islanded microgrid (MG). The energy management system relies on a communication network to monitor and control power generation in the units. A communication failure to a unit inhibits the ability of the MPC to change the power of the unit. However, this unit is electrically connected to the network and ignoring this aspect could have adverse effect in the operation of the microgrid. Therefore, this paper considers an MPC design that includes the electrical connectivity of the unit during communication failures. This paper also highlights the benefit of including the lower layer control behaviour of the MG to withstand communication failures. The proposed approaches are validated with a case study.
This paper addresses the design of an optimization-based cooperative path-following control law for multiple robotic vehicles that optimally balances the transient trade-off between coordination and path-following errors. To this end, we formulate a more general multi-agent framework where each agent is associated with (i) a continuous-time dynamical model, which governs the evolution of its state, and (ii) an output equation that is a function of both the state of the agent and a coordination vector. According to a given network topology, each agent can access its state and coordination vector, as well as the coordination vectors of the neighboring agents. In this setup, the goal is to design a distributed control law that steers the output signals to the origin, while simultaneously driving the coordination vectors of the agents of the network to consensus. To solve this, we propose a model predictive control scheme that builds on a pre-existing auxiliary consensus control law to design a performance index that combines the output regulation objective with the consensus objective. Convergence guarantees under which one can solve this coordinated output regulation problem are provided. Numerical simulations display the effectiveness of the proposed scheme applied to a cooperative path following control problem of a network of 3D nonholonomic robotic vehicles.
Modeling error or external disturbances can severely degrade the performance of Model Predictive Control (MPC) in real-world scenarios. Robust MPC (RMPC) addresses this limitation by optimizing over feedback policies but at the expense of increased computational complexity. Tube MPC is an approximate solution strategy in which a robust controller, designed offline, keeps the system in an invariant tube around a desired nominal trajectory, generated online. Naturally, this decomposition is suboptimal, especially for systems with changing objectives or operating conditions. In addition, many tube MPC approaches are unable to capture state-dependent uncertainty due to the complexity of calculating invariant tubes, resulting in overly-conservative approximations. This work presents the Dynamic Tube MPC (DTMPC) framework for nonlinear systems where both the tube geometry and open-loop trajectory are optimized simultaneously. By using boundary layer sliding control, the tube geometry can be expressed as a simple relation between control parameters and uncertainty bound; enabling the tube geometry dynamics to be added to the nominal MPC optimization with minimal increase in computational complexity. In addition, DTMPC is able to leverage state-dependent uncertainty to reduce conservativeness and improve optimization feasibility. DTMPC is demonstrated to robustly perform obstacle avoidance and modify the tube geometry in response to obstacle proximity.
In pedestrian-dense traffic scenarios, an autonomous vehicle may have to safely drive through a crowd of pedestrians while the vehicle tries to keep the desired speed as much as possible. This requires a model that can predict the motion of crowd pedestrians and a method for the vehicle to predictively adjust its speed. In this study, the model-based predictive control (MPC) was combined with a social-force based vehicle-crowd interaction (VCI) model to regulate the longitudinal speed of the autonomous vehicle. The predictive feature of the VCI model can be precisely utilized by the MPC. A criterion for simultaneously guaranteeing pedestrian safety and keeping the desired speed was designed, and consequently, the MPC was formulated as a standard quadratic programming (QP) problem, which can be easily solved by standard QP toolbox. The proposed approach was compared with the traditional proportional-integral-derivative (PID) control approach for regulating longitudinal speed. Scenarios of different pedestrian density were evaluated in simulation. The results demonstrated the merits of the proposed method to address this type of problem. It also shows the potential of extending the method to address more complex vehicle-pedestrian interaction situations.