The stability of stochastic Model Predictive Control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that ensure closed-loop performance bounds and boundedness of the state, but tight ultimate bounds for the state and non-conservative performance bounds are typically not determined. In this work we use an input-to-state stability property to find conditions that imply convergence with probability 1 of a disturbed nonlinear system to a minimal robust positively invariant set. We discuss implications for the convergence of the state and control laws of stochastic MPC formulations, and we prove convergence results for several existing stochastic MPC formulations for linear and nonlinear systems.
The local stability and convergence for Model Predictive Control (MPC) of unconstrained nonlinear dynamics based on a linear time-invariant plant model is studied. Based on the long-time behavior of the solution of the Riccati Differential Equation (RDE), explicit error estimates are derived that clearly demonstrate the influence of the two critical parameters in MPC: the prediction horizon $T$ and the control horizon $τ$. In particular, if the MPC-controller has access to an exact (linear) plant model, the MPC-controls and the corresponding optimal state trajectories converge exponentially to…
We propose a robust adaptive Model Predictive Control (MPC) strategy with online set-based estimation for constrained linear systems with unknown parameters and bounded disturbances. A sample-based test applied to predicted trajectories is used to ensure convergence of parameter estimates by enforcing a persistence of excitation condition on the closed loop system. The control law robustly satisfies constraints and has guarantees of feasibility and input-to-state stability. Convergence of parameter set estimates to the actual system parameter vector is guaranteed under conditions on reachabili…
This paper introduces the proximally stabilized Fischer-Burmeister method (FBstab); a new algorithm for convex quadratic programming that synergistically combines the proximal point algorithm with a primal-dual semismooth Newton-type method. FBstab is numerically robust, easy to warmstart, handles degenerate primal-dual solutions, detects infeasibility/unboundedness and requires only that the Hessian matrix be positive semidefinite. We outline the algorithm, provide convergence and convergence rate proofs, report some numerical results from model predictive control benchmarks, and also include…
We present a predictive feedback control method for a class of quasilinear hyperbolic systems with one boundary control input. Assuming exact model knowledge, convergence to the origin, or tracking at the uncontrolled boundary, are achieved in finite time. A robustness certificate is provided, showing that at least under more restrictive assumptions on the system coefficients, the control method has inherent robustness properties with respect to small errors in the model, measurements and control input. Rigorous, although conservative conditions on the time derivative of the initial condition …
For systems with uncertain linear models, bounded additive disturbances and state and control constraints, a robust model predictive control algorithm incorporating online model adaptation is proposed. Sets of model parameters are identified online and employed in a robust tube MPC strategy with a nominal cost. The algorithm is shown to be recursively feasible and input-to-state stable. Computational tractability is ensured by using polytopic sets of fixed complexity to bound parameter sets and predicted states. Convex conditions for persistence of excitation are derived and are related to pro…
This paper develops a distributed model predictive control (DMPC) strategy for a class of discrete-time linear systems with consideration of globally coupled constraints. The DMPC under study is based on the dual problem concerning all subsystems, which is solved by means of the primal-dual gradient optimization in a distributed manner using Laplacian consensus. To reduce the computational burden, the constraint tightening method is utilized to provide a capability of premature termination with guaranteeing the convergence of the DMPC optimization. The contraction theory is first adopted in th…
For constrained linear systems with bounded disturbances and parametric uncertainty, we propose a robust adaptive model predictive control strategy with online parameter estimation. Constraints enforcing persistently exciting closed loop control actions are introduced for a set-membership parameter identification scheme. The algorithm requires the online solution of a convex program, satisfies constraints robustly, and ensures recursive feasibility and input-to-state stability. Almost sure convergence to the actual system parameters is demonstrated under assumptions on stabilizability, reachab…
What has become known as Stahl's Theorem in power engineering circles has been used to justify a convergence guarantee of the Holormorphic Embedding Method (HEM) as it applies to the power flow (PF) problem. In this two-part paper, we examine in more detail the implications of Stahl's theorems to both theoretcial and numerical convergence for a wider range of problems to which these theorems are now being applied. In Pt. 1, we introduce the theorem using the necessary mathematical parlance and then translate the language to show its implications to convergence of nonlinear problems in general …
Stochastic gradient methods are dominant in nonconvex optimization especially for deep models but have low asymptotical convergence due to the fixed smoothness. To address this problem, we propose a simple yet effective method for improving stochastic gradient methods named predictive local smoothness (PLS). First, we create a convergence condition to build a learning rate which varies adaptively with local smoothness. Second, the local smoothness can be predicted by the latest gradients. Third, we use the adaptive learning rate to update the stochastic gradients for exploring linear convergen…
We prove that the fixed point iteration of arbitrary positive concave mappings with nonempty fixed point set converges geometrically for any starting point. We also show that positivity is crucial for this result to hold, and the concept of (nonlinear) spectral radius of asymptotic mappings provides us with information about the convergence factor. As a practical implication of the results shown here, we rigorously explain why some power control and load estimation algorithms in wireless networks, which are particular instances of the fixed point iteration, have shown geometric convergence in …
Multi-forecast model predictive control (MF-MPC) is a control policy that creates a plan of actions over a horizon for each of a given set of forecasted scenarios or contingencies, with the constraint that the first action in all plans be the same. In this note we show how these coupled plans can be found by solving a sequence of single plans, using an incremental proximal method. We refer to this policy as incremental proximal model predictive control (IP-MPC). We have observed that even when the iterations in IP-MPC are not carried out to convergence, we obtain a policy that achieves much of…
In this technical note we analyse the performance improvement and optimality properties of the Learning Model Predictive Control (LMPC) strategy for linear deterministic systems. The LMPC framework is a policy iteration scheme where closed-loop trajectories are used to update the control policy for the next execution of the control task. We show that, when a Linear Independence Constraint Qualification (LICQ) condition holds, the LMPC scheme guarantees strict iterative performance improvement and optimality, meaning that the closed-loop cost evaluated over the entire task converges asymptotica…
The paper presents a distributed model predictive control (DMPC) scheme for continuous-time nonlinear systems based on the alternating direction method of multipliers (ADMM). A stopping criterion in the ADMM algorithm limits the iterations and therefore the required communication effort during the distributed MPC solution at the expense of a suboptimal solution. Stability results are presented for the suboptimal DMPC scheme under two different ADMM convergence assumptions. In particular, it is shown that the required iterations in each ADMM step are bounded, which is also confirmed in simulati…
In this paper, we propose a novel model predictive control (MPC) framework for output tracking that deals with partially unknown constraints. The MPC scheme optimizes over a learning and a backup trajectory. The learning trajectory aims to explore unknown and potentially unsafe areas, if and only if this might lead to a potential performance improvement. On the contrary, the backup trajectory lies in the known space, and is intended to ensure safety and convergence. The cost function for the learning trajectory is divided into a tracking and an offset cost, while the cost function for the back…
In this paper, we develop a robust economic model predictive controller for the containment of stochastic Susceptible-Exposed-Infected-Vigilant (SEIV) epidemic processes which drives the process to extinction quickly, while minimizing the rate at which control resources are used. The work we present here is significant in that it addresses the problem of efficiently controlling general stochastic epidemic systems without relying on mean-field approximation, which is an important issue in the theory of stochastic epidemic processes. This enables us to provide rigorous convergence guarantees on …
In this letter, an accelerated quadratic programming (QP) algorithm is proposed based on the proximal gradient method. The algorithm can achieve convergence rate $O(1/p^α)$, where $p$ is the iteration number and $α$ is the given positive integer. The proposed algorithm improves the convergence rate of existing algorithms that achieve $O(1/p^{2})$. The key idea is that iterative parameters are selected from a group of specific high order polynomial equations. The performance of the proposed algorithm is assessed on the randomly generated model predictive control (MPC) optimization problems. The…
This paper considers solving convex quadratic programs (QPs) in a real-time setting using a regularized and smoothed Fischer-Burmeister method (FBRS). The Fischer-Burmeister function is used to map the optimality conditions of the quadratic program to a nonlinear system of equations which is solved using Newton's method. Regularization and smoothing are applied to improve the practical performance of the algorithm and a merit function is used to globalize convergence. FBRS is simple to code, easy to warmstart, robust to early termination, and has attractive theoretical properties, making it ap…
The stability of stochastic Model Predictive Control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that guarantee closed-loop performance bounds and boundedness of the state, but convergence to a terminal control law is typically not shown. In this work we use results on general state space Markov chains to find conditions that guarantee convergence of disturbed nonlinear systems to terminal modes of operation, so that they converge in probability to a priori known terminal linear feedback laws and achieve time-average…
In this paper, we propose a chance constrained stochastic model predictive control scheme for reference tracking of distributed linear time-invariant systems with additive stochastic uncertainty. The chance constraints are reformulated analytically based on mean-variance information, where we design suitable Probabilistic Reachable Sets for constraint tightening. Furthermore, the chance constraints are proven to be satisfied in closed-loop operation. The design of an invariant set for tracking complements the controller and ensures convergence to arbitrary admissible reference points, while a …
Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. A Continuation/GMRES Method for NMPC, suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation $Ax=b$ of the Continuation NMPC (CNMPC) equations on every time step. The coefficient matrix $A$ of the linear system is often ill-conditioned, resulting in poor GMRES convergence, slowing down the on-line computation of the control by CNMPC, and reducing control quality. We…
Tight Stability, Convergence, and Robustness Bounds for Predictive Coding Networks
University of South Florida（南佛罗里达大学）；VERSES AI Research Lab（VERSES AI 研究院）；Vienna University of Technology（维也纳技术大学）；Rochester Institute of Technology（罗切斯特理工学院）
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 real-time optimization method for nonlinear model predictive control (NMPC) of systems governed by partial differential equations (PDEs). The NMPC problem to be solved is formulated by discretizing the PDE system in space and time by using the finite difference method. The proposed method is called the double-layer Jacobi method, which exploits both the spatial and temporal sparsities of the PDE-constrained NMPC problem. In the upper layer, the NMPC problem is solved by ignoring the temporal couplings of either the state or costate (Lagrange multiplier corresponding to th…
We develop a one-Newton-step-per-horizon, online, lag-$L$, model predictive control (MPC) algorithm for solving discrete-time, equality-constrained, nonlinear dynamic programs. Based on recent sensitivity analysis results for the target problems class, we prove that the approach exhibits a behavior that we call superconvergence; that is, the tracking error with respect to the full horizon solution is not only stable for successive horizon shifts, but also decreases with increasing shift order to a minimum value that decays exponentially in the length of the receding horizon. The key analytical…
In this paper we present a Learning Model Predictive Control (LMPC) strategy for linear and nonlinear time optimal control problems. Our work builds on existing LMPC methodologies and it guarantees finite time convergence properties for the closed-loop system. We show how to construct a time varying safe set and terminal cost function using closed-loop data. The resulting LMPC policy is time varying and it guarantees recursive constraint satisfaction and non-decreasing performance. Computational efficiency is obtained by convexifing the safe set and terminal cost function. We demonstrate that,…
Many systems exhibit a mixture of continuous and discrete dynamics. We consider a family of mixed-integer non-convex non-linear optimisation problems obtained in discretisations of optimal control of such systems. For this family, a branch-and-bound algorithm solves the discretised problem to global optimality. As an example, we consider water systems, where variations in flow and variations in water levels are continuous, while decisions related to fixed-speed pumps and whether gates that may be opened and closed are discrete. We show that the related optimal-control problems come from the fa…
This paper presents a new concept of controlled dissipativity as an extension of the standard dissipativity property to systems with parameter-varying storage functions under the framework of economic model predictive control (EMPC). Based on this concept, two EMPC controllers, integrated with the dissipation inequality constraints rendering the storage function parameters as decision variables, are formulated and the associated recursive feasibility is ensured. Then, the asymptotic convergence to an optimal equilibrium in closed-loop, without requiring the standard dissipativity assumption, i…
Vehicle platooning has been shown to be quite fruitful in the transportation industry to enhance fuel economy, road throughput, and driving comfort. Model Predictive Control (MPC) is widely used in literature for platoon control to achieve certain objectives, such as safely reducing the distance among consecutive vehicles while following the leader vehicle. In this paper, we propose a Distributed Nonlinear MPC (DNMPC), based upon an existing approach, to control a heterogeneous dynamic platoon with unidirectional topologies, handling possible cut-in/cut-out maneuvers. The introduced method add…
