This paper introduces a framework for analyzing a general class of uncertain nonlinear discrete-time systems with given state-, control-, and disturbance constraints. In particular, we propose a set-theoretic generalization of the concept of dissipativity of systems that are affected by external disturbances. The corresponding theoretical developments build upon set based analysis methods and lay a general theoretical foundation for a rigorous stability analysis of economic tube model predictive controllers. Besides, we discuss practical prodecures for verifying set-dissipativity of constrained linear control systems with convex stage costs.
We consider nonlinear model predictive control (MPC) with multiple competing cost functions. In each step of the scheme, a multiobjective optimal control problem with a nonlinear system and terminal conditions is solved. We propose an algorithm and give performance guarantees for the resulting MPC closed loop system. Thereby, we significantly simplify the assumptions made in the literature so far by assuming strict dissipativity and the existence of a compatible terminal cost for one of the competing objective functions only. We give conditions which ensure asymptotic stability of the closed l…
Economic Model Predictive Control (MPC) dissipativity theory is central to discussing the stability of policies resulting from minimizing economic stage costs. In its current form, the dissipativity theory for economic MPC applies to problems based on deterministic dynamics or to very specific classes of stochastic problems, and does not readily extend to generic Markov Decision Processes. In this paper, we clarify the core reason for this difficulty, and propose a generalization of the economic MPC dissipativity theory that circumvents it. This generalization focuses on undiscounted infinite-…
We consider nonlinear model predictive control (MPC) with multiple competing cost functions. This leads to the formulation of multiobjective optimal control problems (MO OCPs). Since the design of MPC algorithms for directly solving multiobjective problems is rather complicated, particularly if terminal conditions shall be avoided, we use an indirect approach via a weighted sum formulation for solving these MO OCPs. This way, for each set of weights we obtain an optimal control problem with a single objective. In economic MPC it is known that strict dissipativity is the key assumption for conc…
Economic Model Predictive Control has recently gained popularity due to its ability to directly optimize a given performance criterion, while enforcing constraint satisfaction for nonlinear systems. Recent research has developed both numerical algorithms and stability analysis for the undiscounted case. The introduction of a discount factor in the cost, however, can be desirable in some cases of interest, e.g., economics, stochastically terminating processes, Markov decision processes, etc. Unfortunately, the stability theory in this case is still not fully developed. In this paper we propose …
The close link between dissipativity and optimal control is already apparent in Jan C. Willems' first papers on the subject. In recent years, research on this link has been revived with a particular focus on nonlinear problems and applications in model predictive control (MPC). This paper surveys these recent developments and some of Willems' and other authors' earlier results.
In the Economic Nonlinear Model Predictive (ENMPC) context, closed-loop stability relates to the existence of a storage function satisfying a dissipation inequality. Finding the storage function is in general -- for nonlinear dynamics and cost -- challenging, and has attracted attentions recently. Q-Learning is a well-known Reinforcement Learning (RL) techniques that attempts to capture action-value functions based on the state-input transitions and stage cost of the system. In this paper, we present the use of the Q-Learning approach to obtain the storage function and verify the dissipativity…
This paper is about robust Model Predictive Control (MPC) for linear systems with additive and multiplicative uncertainty. A novel class of configuration-constrained polytopic robust forward invariant tubes is introduced, which admit a joint parameterization of their facets and vertices. They are the foundation for the development of novel Configuration-Constrained Tube MPC (CCTMPC) controllers that freely optimize the shape of their polytopic tube, subject to conic vertex configuration constraints, as well as associated vertex control laws by solving convex optimization problems online. It is…
This paper is about a class of distributionally robust model predictive controllers (MPC) for nonlinear stochastic processes that evaluate risk and control performance measures by propagating ambiguity sets in the space of state probability measures. A framework for formulating such ambiguity tube MPC controllers is presented, which is based on modern measure-theoretic methods from the field of optimal transport theory. Moreover, a supermartingale based analysis technique is proposed, leading to stochastic stability results for a large class of distributionally robust controllers for linear an…
Tube-based model predictive control (MPC) methods leverage tubes to bound deviations from a nominal trajectory due to uncertainties in order to ensure constraint satisfaction. This paper presents a novel tube-based MPC formulation based on system level disturbance reachable sets (SL-DRS), which leverage the affine system level parameterization (SLP). We show that imposing a finite impulse response (FIR) constraint on the affine SLP guarantees containment of all future deviations in a finite sequence of SL-DRS. This allows us to formulate a system level tube-MPC (SLTMPC) method using the SL-DRS…
This paper is concerned with tube-based model predictive control (MPC) for both linear and nonlinear, input-affine continuous-time dynamic systems that are affected by time-varying disturbances. We derive a min-max differential inequality describing the support function of positive robust forward invariant tubes, which can be used to construct a variety of tube-based model predictive controllers. These constructions are conservative, but computationally tractable and their complexity scales linearly with the length of the prediction horizon. In contrast to many existing tube-based MPC implemen…
Robust tube-based model predictive control (MPC) methods address constraint satisfaction by leveraging an a priori determined tube controller in the prediction to tighten the constraints. This paper presents a system level tube-MPC (SLTMPC) method derived from the system level parameterization (SLP), which allows optimization over the tube controller online when solving the MPC problem, which can significantly reduce conservativeness. We derive the SLTMPC method by establishing an equivalence relation between a class of robust MPC methods and the SLP. Finally, we show that the SLTMPC formulati…
In this paper, we provide non-averaged and transient performance guarantees for recently developed, tube-based robust economic model predictive control (MPC) schemes. In particular, we consider both tube-based MPC schemes with and without terminal conditions. We show that the closed-loop performance obtained by applying such MPC schemes is approximately optimal when evaluated both on finite and infinite time horizons. These performance bounds are similar to those derived previously for nominal economic MPC. The theoretical results are discussed in a numerical example.
Georgia Institute of Technology（佐治亚理工学院）；NASA Jet Propulsion Laboratory, California Institute of Technology（美国国家航空航天局喷气推进实验室，加州理工学院）
Learning-based control aims to construct models of a system to use for planning or trajectory optimization, e.g. in model-based reinforcement learning. In order to obtain guarantees of safety in this context, uncertainty must be accurately quantified. This uncertainty may come from errors in learning (due to a lack of data, for example), or may be inherent to the system. Propagating uncertainty forward in learned dynamics models is a difficult problem. In this work we use deep learning to obtain expressive and flexible models of how distributions of trajectories behave, which we then use for n…
This paper is about a parallel algorithm for tube-based model predictive control. The proposed control algorithm solves robust model predictive control problems suboptimally, while exploiting their structure. This is achieved by implementing a real-time algorithm that iterates between the evaluation of piecewise affine functions, corresponding to the parametric solution of small-scale robust MPC problems, and the online solution of structured equality constrained QPs. The performance of the associated real-time robust MPC controllers is illustrated by a numerical case study.
Adaptive model predictive control (MPC) methods using set-membership identification to reduce parameter uncertainty are considered in this work. Strong duality is used to reformulate the set-membership equations exactly within the MPC optimization. A predicted worst-case cost is then used to enable performance-oriented exploration. The proposed approach guarantees robust constraint satisfaction and recursive feasibility. It is shown that method can be implemented using homothetic tube and flexible tube parameterizations of state tubes, and a simulation study demonstrates performance improvemen…
Over the last years, the development of Autonomous Underwater Vehicles (AUV) with attached robotic manipulators, the so-called Underwater Vehicle Manipulator System (UVMS), has gained significant research attention, due to the ability of interaction with underwater environments. In such applications, force/torque controllers which guarantee that the end-effector of the UVMS applies desired forces/torques towards the environment, should be designed in a way that state and input constraints are taken into consideration. Furthermore, due to their complicated structure, unmodeled dynamics as well …
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 …
This chapter compares different formulations for Economic nonlinear Model Predictive Control (EMPC) which are all based on an established dissipativity assumption on the underlying Optimal Control Problem (OCP). This includes schemes with and without stabilizing terminal constraints, respectively, or with stabilizing terminal costs. We recall that a recently proposed approach based on gradient correcting terminal penalties implies a terminal constraint on the adjoints of the OCP. We analyze the feasibility implications of these dual/adjoint terminal constraints and we compare our findings to a…
This work proposes a novel robust model predictive control (MPC) algorithm for linear systems affected by dynamic model uncertainty and exogenous disturbances. The uncertainty is modeled using a linear fractional perturbation structure with a time-varying perturbation matrix, enabling the algorithm to be applied to a large model class. The MPC controller constructs a state tube as a sequence of parameterized ellipsoidal sets to bound the state trajectories of the system. The proposed approach results in a semidefinite program to be solved online, whose size scales linearly with the order of th…
Model predictive control (MPC) for uncertain systems in the presence of hard constraints on state and input is a non-trivial problem, and the challenge is increased manyfold in the absence of state measurements. In this paper, we propose an adaptive output feedback MPC technique, based on a novel combination of an adaptive observer and robust MPC, for single-input single-output discrete-time linear time-invariant systems. At each time instant, the adaptive observer provides estimates of the states and the system parameters that are then leveraged in the MPC optimization routine while robustly …
In this paper, a novel tube-based economic Model Predictive Control (MPC) scheme for uncertain systems that uses neither terminal costs nor terminal constraints is investigated. We show that the results from the undisturbed case can be extended to systems with bounded disturbances by using similar turnpike arguments and a properly modified stage cost. We prove robust guarantees on the closed-loop performance, convergence, and stability under suitable dissipativity and controllability conditions and discuss them in a numerical example.
This paper presents a distributionally robust stochastic model predictive control (SMPC) approach for linear discrete-time systems subject to unbounded and correlated additive disturbances. We consider hard input constraints and state chance constraints, which are approximated as distributionally robust (DR) Conditional Value-at-Risk (CVaR) constraints over a Wasserstein ambiguity set. The computational complexity is reduced by resorting to a tube-based MPC scheme with indirect feedback, such that the error scenarios can be sampled offline. Recursive feasibility is guaranteed by softening the …
In this work, we propose a tube-based MPC scheme for state- and input-constrained linear systems subject to dynamic uncertainties characterized by dynamic integral quadratic constraints (IQCs). In particular, we extend the framework of $ρ$-hard IQCs for exponential stability analysis to external inputs. This result yields that the error between the true uncertain system and the nominal prediction model is bounded by an exponentially stable scalar system. In the proposed tube-based MPC scheme, the state of this error bounding system is predicted along with the nominal model and used as a scalin…
Model predictive control (MPC) is an effective approach to control multivariable dynamic systems with constraints. Most real dynamic models are however affected by plant-model mismatch and process uncertainties, which can lead to closed-loop performance deterioration and constraint violations. Methods such as stochastic MPC (SMPC) have been proposed to alleviate these problems; however, the resulting closed-loop state trajectory might still significantly violate the prescribed constraints if the real system deviates from the assumed disturbance distributions made during the controller design. …
This paper investigates adaptive model predictive control (MPC) for a class of constrained linear systems with unknown model parameters. This is also posed as the dual control problem consisting of system identification and regulation. We first propose an online strategy for simultaneous unknown parameter identification and uncertainty set estimation based on the recursive least square technique. The designed strategy provides a contractive sequence of uncertain parameter sets, and the convergence of parameter estimates is achieved under certain conditions. Second, by integrating tube MPC with…
National University of Defense Technology（国防科技大学）；Jilin University（吉林大学）
Robust model predictive control (MPC) is a well-known control technique for model-based control with constraints and uncertainties. In classic robust tube-based MPC approaches, an open-loop control sequence is computed via periodically solving an online nominal MPC problem, which requires prior model information and frequent access to onboard computational resources. In this paper, we propose an efficient robust MPC solution based on receding horizon reinforcement learning, called r-LPC, for unknown nonlinear systems with state constraints and disturbances. The proposed r-LPC utilizes a Koopma…
A multirate nonlinear model predictive control (NMPC) strategy is proposed for systems with dynamics and control inputs evolving on different timescales. The proposed multirate formulation of the system model and receding horizon optimal control problem allows larger time steps in the prediction horizon compared to single-rate schemes, providing computational savings while ensuring recursive feasibility. A multirate variational model is used with a tube-based successive linearization NMPC strategy. This allows either Jacobian linearization or linearization using quadratic and linear Taylor ser…
Learning-Based Shrinking Disturbance-Invariant Tubes for State- and Input-Dependent Uncertainty
Electrical and Computer Engineering (ECE), Smith Engineering and with Ingenuity Labs Research Institute, Queen’s University（电气与计算机工程系（ECE）、Smith工程学院及Ingenuity Labs研究研究所、皇后大学）；School of Computing and with Ingenuity Labs Research Institute, Queen’s University（计算机学院及Ingenuity Labs研究研究所、皇后大学）
本文提出了一种基于学习的框架，用于在 state 和 input 依赖的不确定性下构建缩小的扰动不变管，并通过固定点映射验证安全性，同时在数据丰富区域保持不变性。
In the current context of high integration of renewable energies, maximizing infrastructures capabilities for electricity transmission is a general need for Transmission System Operators (TSO). The French TSO, RTE, is developing levers to control power flows in real-time: renewable production curtailment is already employed and large battery storage systems are planned to be installed for congestion management in early 2020. The combination of these levers with the use of Dynamic Line Rating (DLR) helps exploiting the lines at the closest of their limit by managing their temperature in real-ti…
We present a framework to design nonlinear robust output feedback model predictive control (MPC) schemes that ensure constraint satisfaction under noisy output measurements and disturbances. We provide novel estimation methods to bound the magnitude of the estimation error based on: stability properties of the observer; detectability; set-membership estimation; moving horizon estimation (MHE). Robust constraint satisfaction is guaranteed by suitably incorporating these online validated bounds on the estimation error in a homothetic tube based MPC formulation. In addition, we show how the perfo…
