In this paper, we present a novel stochastic output-feedback MPC scheme for distributed systems with additive process and measurement noise. The chance constraints are treated with the concept of probabilistic reachable sets, which, under an unimodality assumption on the disturbance distributions are guaranteed to be satisfied in closed-loop. By conditioning the initial state of the optimization problem on feasibility, the fundamental property of recursive feasibility is ensured. Closed-loop chance constraint satisfaction, recursive feasibility and convergence to an asymptotic average cost bound are proven. The paper closes with a numerical example of three interconnected subsystems, highlighting the chance constraint satisfaction and average cost compared to a centralized setting.
In model predictive control (MPC), an optimal control problem (OCP) is solved for the current state and the first input of the solution, the optimal feedback law, is applied to the system. This procedure requires to solve the OCP in every time step. Recently, a new approach was suggested for linear MPC. The parametric solution of a linear quadratic OCP is a piecewise-affine feedback law. The solution at a point in state space provides an optimal feedback law and a domain on which this law is the optimal solution. As long as the system remains in the domain, the law can be reused and the calcul…
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 …
Recently, a regional MPC approach has been proposed that exploits the piecewise affine structure of the optimal solution (without computing the entire explicit solution before). Here, regional refers to the idea of using the affine feedback law that is optimal in a vicinity of the current state of operation, and therefore provides the optimal input signal without requiring to solve a QP. In the present paper, we apply the idea of regional MPC to min-max MPC problems. We show that the new robust approach can significantly reduce the number of QPs to be solved within min-max MPC resulting in a r…
In this paper we propose a parallel coordinate descent algorithm for solving smooth convex optimization problems with separable constraints that may arise e.g. in distributed model predictive control (MPC) for linear network systems. Our algorithm is based on block coordinate descent updates in parallel and has a very simple iteration. We prove (sub)linear rate of convergence for the new algorithm under standard assumptions for smooth convex optimization. Further, our algorithm uses local information and thus is suitable for distributed implementations. Moreover, it has low iteration complexit…
In this paper, we propose the inexact alternating minimization algorithm (inexact AMA), which allows inexact iterations in the algorithm, and its accelerated variant, called the inexact fast alternating minimization algorithm (inexact FAMA). We show that inexact AMA and inexact FAMA are equivalent to the inexact proximal-gradient method and its accelerated variant applied to the dual problem. Based on this equivalence, we derive complexity upper-bounds on the number of iterations for the inexact algorithms. We apply inexact AMA and inexact FAMA to distributed optimization problems, with an emp…
This paper presents a Distributed Stochastic Model Predictive Control algorithm for networks of linear systems with multiplicative uncertainties and local chance constraints on the states and control inputs. The chance constraints are approximated via Cantelli's inequality by means of expected value and covariance. The cooperative control algorithm is based on the distributed Alternating Direction Method of Multipliers, which renders the controller fully distributedly implementable, recursively feasible and ensures point-wise convergence of the states. The aforementioned properties are guarant…
This paper consists of a tutorial on the Augmented Lagrangian based Alternating Direction Inexact Newton method (ALADIN) and its application to distributed model predictive control (MPC). The focus is - for simplicity of presentation - on convex quadratic programming (QP) formulations of MPC. It is explained how ALADIN can be used to synthesize sparse QP solvers for large-scale linear-quadratic optimal control by combining ideas from augmented Lagrangian methods, sequential quadratic programming, as well as barrier or interior point methods. The highlight of this tutorial is a real-time ALADIN…
The paper considers constrained linear systems with stochastic additive disturbances and noisy measurements transmitted over a lossy communication channel. We propose a model predictive control (MPC) law that minimizes a discounted cost subject to a discounted expectation constraint. Sensor data is assumed to be lost with known probability, and data losses are accounted for by expressing the predicted control policy as an affine function of future observations, which results in a convex optimal control problem. An online constraint-tightening technique ensures recursive feasibility of the onli…
Model predictive control (MPC) samples a generally unknown and complicated feedback law point by point. The solution for the current state $x$ contains, however, more information than only the optimal signal $u$ for this particular state. In fact, it provides an optimal affine feedback law $x\rightarrow u(x)$ on a polytope $Π\subset \mathbb{R}^n$, i.e., on a full-dimensional state space set. It is an obvious idea to reuse this affine feedback law as long as possible. Reusing it on its polytope $Π$ is too conservative, however, because any $Π$ is a state space set with a common affine law $x\ri…
Time-distributed Optimization (TDO) is an approach for reducing the computational burden of Model Predictive Control (MPC). When using TDO, optimization iterations are distributed over time by maintaining a running solution estimate and updating it at each sampling instant. In this paper, TDO applied to input constrained linear MPC is studied in detail, and analytic expressions for the system gains and a bound on the number of optimization iterations per sampling instant required to guarantee closed-loop stability is derived. Further, it is shown that the closed-loop stability of TDO-based MPC…
We combine conditional state density construction with an extension of the Scenario Approach for stochastic Model Predictive Control to nonlinear systems to yield a novel particle-based formulation of stochastic nonlinear output-feedback Model Predictive Control. Conditional densities given noisy measurement data are propagated via the Particle Filter as an approximate implementation of the Bayesian Filter. This enables a particle-based representation of the conditional state density, or information state, which naturally merges with scenario generation from the current system state. This appr…
In dual decomposition, the dual to an optimization problem with a specific structure is solved in distributed fashion using (sub)gradient and recently also fast gradient methods. The traditional dual decomposition suffers from two main short-comings. The first is that the convergence is often slow, although fast gradient methods have significantly improved the situation. The second is that computation of the optimal step-size requires centralized computations, which hinders a fully distributed implementation of the algorithm. In this paper, the first issue is addressed by providing a tighter c…
Recent studies have shown that multi-step optimization based on Model Predictive Control (MPC) can effectively coordinate the increasing number of distributed renewable energy and storage resources in the power system. However, the computation complexity of MPC is usually high which limits its use in practical implementation. To improve the efficiency of MPC, in this paper, we apply a distributed optimization method to MPC. The approach consists of a partitioning technique based on spectral clustering that determines the best system partition and an improved Optimality Condition Decomposition …
This paper considers linear discrete-time systems with additive disturbances, and designs a Model Predictive Control (MPC) law incorporating a dynamic feedback gain to minimise a quadratic cost function subject to a single chance constraint. The feedback gain is selected online and we provide two selection methods based on minimising upper bounds on predicted costs. The chance constraint is defined as a discounted sum of violation probabilities on an infinite horizon. By penalising violation probabilities close to the initial time and assigning violation probabilities in the far future with va…
In this paper we propose and analyze two dual methods based on inexact gradient information and averaging that generate approximate primal solutions for smooth convex optimization problems. The complicating constraints are moved into the cost using the Lagrange multipliers. The dual problem is solved by inexact first order methods based on approximate gradients and we prove sublinear rate of convergence for these methods. In particular, we provide, for the first time, estimates on the primal feasibility violation and primal and dual suboptimality of the generated approximate primal and dual so…
CAV platooning technology has received considerable attention in the past few years, driven by the next generation smart transportation systems. Unlike most of the existing platooning methods that focus on linear vehicle dynamics of CAVs, this paper considers nonlinear vehicle dynamics and develops fully distributed optimization based CAV platooning control schemes via the model predictive control (MPC) approach for a possibly heterogeneous CAV platoon. The nonlinear vehicle dynamics leads to several major difficulties in distributed algorithm development and control analysis and design. Speci…
Yerevan State University（埃里温国立大学）；Center For Scientific Innovation and Education（科学与教育创新中心）；Institute of Mathematics of National Academy of Sciences of Republic of Armenia（亚美尼亚共和国国家科学院数学研究所）；Department of Mechanical Science & Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校机械科学与工程系）
In this paper, we consider the problem of distributed optimal control of linear dynamical systems with a quadratic cost criterion. We study the case of output feedback control for two interconnected dynamical systems, and show that the linear optimal solution can be obtained from a combination of two uncoupled Riccati equations and two coupled Riccati equations.
We exploit an adaptive control technique, namely funnel control, in order to establish both initial and recursive feasibility in Model Predictive Control (MPC) for output-constrained nonlinear systems. Moreover, we show that the resulting feedback controller outperforms the funnel controller both w.r.t. the required sampling rate for a zero-order-hold implementation and required control action. We further propose a combination of funnel control and MPC, exploiting the performance guarantees of the model-free funnel controller during a learning phase and the advantages of the model-based MPC sc…
Virginia Tech（弗吉尼亚理工大学）；California Institute of Technology（加州理工学院）
This paper presents a layered control approach for real-time trajectory planning and control of robust cooperative locomotion by two holonomically constrained quadrupedal robots. A novel interconnected network of reduced-order models, based on the single rigid body (SRB) dynamics, is developed for trajectory planning purposes. At the higher level of the control architecture, two different model predictive control (MPC) algorithms are proposed to address the optimal control problem of the interconnected SRB dynamics: centralized and distributed MPCs. The distributed MPC assumes two local quadra…
In this paper we investigate data-driven predictive control of discrete-time linear descriptor systems. Specifically, we give a tailored variant of Willems' fundamental lemma, which shows that for descriptor systems the non-parametric modelling via a Hankel matrix requires less data compared to linear time-invariant systems without algebraic constraints. Moreover, we use this description to propose a data-driven framework for optimal control and predictive control of discrete-time linear descriptor systems. For the latter, we provide a sufficient stability condition for receding-horizon contro…
A standard way of finding a feedback law that stabilizes a control system to an operating point is to recast the problem as an infinite horizon optimal control problem. If the optimal cost and the optmal feedback can be found on a large domain around the operating point then a Lyapunov argument can be used to verify the asymptotic stability of the closed loop dynamics. The problem with this approach is that is usually very difficult to find the optimal cost and the optmal feedback on a large domain for nonlinear problems with or without constraints. Hence the increasing interest in Model Predi…
The increasing presence of large-scale distributed systems highlights the need for scalable control strategies where only local communication is required. Moreover, in safety-critical systems it is imperative that such control strategies handle constraints in the presence of disturbances. In response to this need, we present the Distributed and Localized Model Predictive Control (DLMPC) algorithm for large-scale linear systems. DLMPC is a distributed closed-loop model predictive control (MPC) scheme wherein only local state and model information needs to be exchanged between subsystems for the…
Robust design of autonomous systems under uncertainty is an important yet challenging problem. This work proposes a robust controller that consists of a state estimator and a tube based predictive control law. The class of linear systems under ellipsoidal uncertainty is considered. In contrast to existing approaches based on polytopic sets, the constraint tightening is directly computed from the ellipsoidal sets of disturbances without over-approximation, thus leading to less conservative bounds. Conditions to guarantee robust constraint satisfaction and robust stability are presented. Further…
A robust model predictive control (MPC) method is presented for linear, time-invariant systems affected by bounded additive disturbances. The main contribution is the offline design of a disturbance-affine feedback gain whereby the resulting constraint tightening is minimized. This is achieved by formulating the constraint tightening problem as a convex optimization problem with the feedback term as a variable. The resulting MPC controller has the computational complexity of nominal MPC, and guarantees recursive feasibility, stability and constraint satisfaction. The advantages of the proposed…
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…
In this paper we consider a network with $P$ nodes, where each node has exclusive access to a local cost function. Our contribution is a communication-efficient distributed algorithm that finds a vector $x^\star$ minimizing the sum of all the functions. We make the additional assumption that the functions have intersecting local domains, i.e., each function depends only on some components of the variable. Consequently, each node is interested in knowing only some components of $x^\star$, not the entire vector. This allows for improvement in communication-efficiency. We apply our algorithm to m…
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…
A typical bottleneck of model predictive control algorithms is the computational burden in order to compute the receding horizon feedback law which is predominantly determined by the length of the prediction horizon. Based on a relaxed Lyapunov inequality we present techniques which allow us to show stability and suboptimality estimates for a reduced prediction horizon. In particular, the known structural properties of suboptimality estimates based on a controllability condition are used to cut the gap between theoretic stability results and numerical observations.
In distributed model predictive control (MPC), the control input at each sampling time is computed by solving a large-scale optimal control problem (OCP) over a finite horizon using distributed algorithms. Typically, such algorithms require several (virtually, infinite) communication rounds between the subsystems to converge, which is a major drawback both computationally and from an energetic perspective (for wireless systems). Motivated by these challenges, we propose a suboptimal distributed MPC scheme in which the total communication burden is distributed also in time, by maintaining a run…
This paper designs a model predictive control (MPC) law for constrained linear systems with stochastic additive disturbances and noisy measurements, minimising a discounted cost subject to a discounted expectation constraint. It is assumed that sensor data is lost with a known probability. Taking into account the data losses modelled by a Bernoulli process, we parameterise the predicted control policy as an affine function of future observations and obtain a convex linear-quadratic optimal control problem. Constraint satisfaction and a discounted cost bound are ensured without imposing bounds …
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 …
This paper presents a distributed optimization algorithm tailored to solve optimization problems arising in smart grids. In detail, we propose a variant of the Augmented Lagrangian based Alternating Direction Inexact Newton (ALADIN) method, which comes along with global convergence guarantees for the considered class of linear-quadratic optimization problems. We establish local quadratic convergence of the proposed scheme and elaborate its advantages compared to the Alternating Direction Method of Multipliers (ADMM). In particular, we show that, at the cost of more communication, ALADIN requir…
Anytime Plug-and-Play Control with Contract-Based Distributed MPC
Institute for Dynamic Systems and Control, ETH Zurich（动态系统与控制研究所，苏黎世联邦理工学院）；Institute of Mechanical Engineering, École Polytechnique Fédérale de Lausanne（机械工程研究所，洛桑联邦理工学院）
