Model predictive control with robust feasibility

This paper presents a new model predictive control (MPC) method that provides robust feasibility with tractable, real-time computation. The method optimizes the closed-loop system dynamics, which involves models of the process (with parametric uncertainty) and controller at each step in the prediction horizon. Such problems are often formulated as a multi-stage stochastic program that suffers from the curse of dimensionality. This paper presents an alternative formulation that yields a bilevel stochastic optimization problem that is transformed by a series of reformulation steps into a tractable problem such that it can be solved through a limited number of second order cone programming sub-problems. The method addresses robust feasibility, manipulated saturation, state and output soft constraints, exogenous and endogenous uncertainty, and uncertainty in the state estimation in an integrated manner. Case study results demonstrate the advantages of the proposed robust MPC over nominal MPC and several other robust MPC formulations.     

Stochastic model predictive control with joint chance constraints

ABSTRACT

This article investigates model predictive control (MPC) of linear systems subject to arbitrary (possibly unbounded) stochastic disturbances. An MPC approach is presented to account for hard input constraints and joint state chance constraints in the presence of unbounded additive disturbances. The Cantelli–Chebyshev inequality is used in combination with risk allocation to obtain computationally tractable but accurate surrogates for the joint state chance constraints when only the mean and variance of the arbitrary disturbance distributions are known. An algorithm is presented for determining the optimal feedback gain and optimal risk allocation by iteratively solving a series of convex programs. The proposed stochastic MPC approach is demonstrated on a continuous acetone–butanol–ethanol fermentation process, which is used in the production of biofuels.     

Steady-state target optimization designs for integrating real-time optimization and model predictive control

In industrial practice, the optimal steady-state operation of continuous-time processes is typically addressed by a control hierarchy involving various layers. Therein, the real-time optimization (RTO) layer computes the optimal operating point based on a nonlinear steady-state model of the plant. The optimal point is implemented by means of the model predictive control (MPC) layer, which typically uses a linear dynamical model of the plant. The MPC layer usually includes two stages: a steady-state target optimization (SSTO) followed by the MPC dynamic regulator. In this work, we consider the integration of RTO with MPC in the presence of plant-model mismatch and constraints, by focusing on the design of the SSTO problem. Three different quadratic program (QP) designs are considered: (i) the standard design that finds steady-state targets that are as close as possible to the RTO setpoints; (ii) a novel optimizing control design that tracks the active constraints and the optimal inputs for the remaining degrees of freedom; and (iii) an improved QP approximation design were the SSTO problem approximates the RTO problem. The main advantage of the strategies (ii) and (iii) is in the improved optimality of the stationary operating points reached by the SSTO-MPC control system. The performance of the different SSTO designs is illustrated in simulation for several case studies.     

Least-restrictive robust periodic model predictive control applied to room temperature regulation

State-feedback model predictive control (MPC) of constrained discrete-time periodic affine systems is considered. The periodic systems’ states and inputs are subject to periodically time-dependent, hard, polyhedral constraints. Disturbances are additive, bounded and subject to periodically time-dependent bounds. The objective is to design MPC laws that robustly enforce constraint satisfaction in a manner that is least-restrictive, i.e., have the largest possible domain. The proposed design method is demonstrated on a building climate control example. The proposed method is directly applicable to time-invariant MPC.     

Robust model predictive control under redundant channel transmission with applications in networked DC motor systems

In networked systems, intermittent failures in data transmission are usually inevitable due to the limited bandwidth of the communication channel, and an effective countermeasure is to add redundance so as to improve the reliability of the communication service. This paper is concerned with the model predictive control (MPC) problem by using static output feedback for a class of polytopic uncertain systems with redundant channels under both input and output constraints. By utilizing the min–max control approach combined with stochastic analysis, sufficient conditions are established to guarantee the feasibility of the designed MPC scheme that ensures the robust stability of the closed‐loop system. In terms of the solution to an auxiliary optimization problem, an easy‐to‐implement MPC algorithm is proposed to obtain the desired sub‐optimal control sequence as well as the upper bound of the quadratic cost function. Finally, to illustrate its effectiveness, the proposed design method is applied to control a networked direct current motor system. Copyright © 2016 John Wiley & Sons, Ltd.     

Chance-constrained model predictive control for spacecraft rendezvous with disturbance estimation

A robust Model Predictive Controller (MPC) is used to solve the problem of spacecraft rendezvous, using the Hill-Clohessy-Wiltshire model with additive disturbances and line-of-sight constraints. Since a standard (non-robust) MPC is not able to cope with disturbances, a robust MPC is designed using a chance-constrained approach for robust satisfaction of constraints in a probabilistic sense. Disturbances are modeled as Gaussian allowing for an explicit transformation of the probabilistic constraints into simple algebraic constraints. To estimate the distribution parameters a predictor of disturbances is proposed. Both robust and non-robust MPC control laws are compared using the Monte Carlo method, which shows the superiority of the robust MPC.     

Affine and predictive control policies for a class of nonlinear systems

The input-state linear horizon (ISLH) for a nonlinear discrete-time system is defined as the smallest number of time steps it takes the system input to appear nonlinearly in the state variable. In this paper, we employ the latter concept and show that the class of constraint admissible N-step affine state-feedback policies is equivalent to the associated class of constraint admissible disturbance-feedback policies, provided that N is less than the system’s ISLH. The result generalizes a recent result in [Goulart, P. J., Kerrigan, E. C., Maciejowski, J. M. (2006). Optimization over state feedback policies for robust control with constraints. Automatica, 42(4), 523-533] and is significant because it enables one: (i) to determine a constraint admissible state-feedback policy by employing well-known convex optimization techniques; and (ii) to guarantee robust recursive feasibility of a class of model predictive control (MPC) policies by imposing a suitable terminal constraint. In particular, we propose an input-to-state stabilizing MPC policy for a class of nonlinear systems with bounded disturbance inputs and mixed polytopic constraints on the state and the control input. At each time step, the proposed MPC policy requires the solution of a single convex quadratic programme parameterized by the current system state.     

Stochastic model predictive control for probabilistically constrained Markovian jump linear systems with additive disturbance

Model predictive control (MPC) for Markovian jump linear systems with probabilistic constraints has received much attention in recent years. However, in existing results, the disturbance is usually assumed with infinite support, which is not considered reasonable in real applications. Thus, by considering random additive disturbance with finite support, this paper is devoted to a systematic approach to stochastic MPC for Markovian jump linear systems with probabilistic constraints. The adopted MPC law is parameterized by a mode‐dependent feedback control law superimposed with a perturbation generated by a dynamic controller. Probabilistic constraints can be guaranteed by confining the augmented system state to a maximal admissible set. Then, the MPC algorithm is given in the form of linearly constrained quadratic programming problems by optimizing the infinite sum of derivation of the stage cost from its steady‐state value. The proposed algorithm is proved to be recursively feasible and to guarantee constraints satisfaction, and the closed‐loop long‐run average cost is not more than that of the unconstrained closed‐loop system with static feedback. Finally, when adopting the optimal feedback gains in the predictive control law, the resulting MPC algorithm has been proved to converge in the mean square sense to the optimal control. A numerical example is given to verify the efficiency of the proposed results.     

A control matching model predictive control approach to string stable vehicle platooning

A predictive control strategy for vehicle platoons is presented in this paper, accommodating both string stability and constraints (e.g., physical and safety) satisfaction. In the proposed design procedure, the two objectives are achieved by matching a model predictive controller (MPC), enforcing constraints satisfaction, with a linear controller designed to guarantee string stability. The proposed approach neatly combines the straightforward design of a string stable controller in the frequency domain, where a considerable number of approaches have been proposed in literature, with the capability of an MPC-based controller enforcing state and input constraints.A controller obtained with the proposed design procedure is validated both in simulations and in the field test, showing how string stability and constraints satisfaction can be simultaneously achieved with a single controller. The operating region that the MPC controller is string stable is characterized by the interior of feasible set of the MPC controller.     

小型无人直升机的模型预测控制算法研究

针对无人直升机在阵风干扰环境中的姿态控制精度低的问题.本文将非线性刚体动力学模型在悬停点应用小扰动理论得到了线性化数学模型.考虑系统输入输出和控制量约束,采用模型预测控制将控制器的设计问题转化为每个采样时刻求解一个带不等式和等式约束的凸二次规划问题.通过设计终端状态约束解决了有限时域模型预测控制(model predictive control, MPC)算法的稳定性问题,并通过引入松弛变量使得约束优化问题更容易求解.随机和常值阵风干扰下无人机悬停仿真验证了本文MPC预测控制器具有幅度不超过0.25 m/s的良好干扰抑制能力,性能明显优于线性二次型调节器(linear-quadratic regulator, LQR).     

Output‐feedback model predictive control for stochastic systems with multiplicative and additive uncertainty

This paper studies the output‐feedback model predictive control (MPC) design problem for linear systems with multiplicative and additive random uncertainty. We first present an off‐line optimization algorithm to optimize feedback gains of the observer and the dual‐mode control policy. After that, by defining a cuboid tube whose center and boundary are both time‐varying variables, we develop a set sequence with increased freedom to contain stochastic system trajectories. A quadratic performance function with analytic upper and lower bounds is minimized such that it decreases exponentially to a finite range under the expectation. The resulting MPC algorithms are proved to guarantee practically stochastic input‐to‐state stability. A numerical example of the wind turbine model illustrates the properties of the MPC algorithms.     

Economic MPC of nonlinear systems with nonmonotonic Lyapunov functions and its application to HVAC control

This paper proposes a Lyapunov‐based economic model predictive control (MPC) scheme for nonlinear systems with nonmonotonic Lyapunov functions. Relaxed Lyapunov‐based constraints are used in the MPC formulation to improve the economic performance. These constraints will enforce a Lyapunov decrease after every few steps. Recursive feasibility and asymptotical convergence to the steady state can be achieved using Lyapunov‐like stability analysis. The proposed economic MPC can be applied to minimize energy consumption in heating ventilation and air conditioning control of commercial buildings. The Lyapunov‐based constraints in the online MPC problem enable the tracking of the desired set‐point temperature. The performance is demonstrated by a virtual building composed of 2 adjacent zones.     

Adaptive model predictive control for a class of constrained linear systems based on the comparison model

This paper proposes an adaptive model predictive control (MPC) algorithm for a class of constrained linear systems, which estimates system parameters on-line and produces the control input satisfying input/state constraints for possible parameter estimation errors. The key idea is to combine the robust MPC method based on the comparison model with an adaptive parameter estimation method suitable for MPC. To this end, first, a new parameter update method based on the moving horizon estimation is proposed, which allows to predict an estimation error bound over the prediction horizon. Second, an adaptive MPC algorithm is developed by combining the on-line parameter estimation with an MPC method based on the comparison model, suitably modified to cope with the time-varying case. This method guarantees feasibility and stability of the closed-loop system in the presence of state/input constraints. A numerical example is given to demonstrate its effectiveness.     

Robust distributed model predictive control of linear systems with structured time-varying uncertainties

In this work, synthesis of robust distributed model predictive control (MPC) is presented for a class of linear systems subject to structured time-varying uncertainties. By decomposing a global system into smaller dimensional subsystems, a set of distributed MPC controllers, instead of a centralised controller, are designed. To ensure the robust stability of the closed-loop system with respect to model uncertainties, distributed state feedback laws are obtained by solving a min–max optimisation problem. The design of robust distributed MPC is then transformed into solving a minimisation optimisation problem with linear matrix inequality constraints. An iterative online algorithm with adjustable maximum iteration is proposed to coordinate the distributed controllers to achieve a global performance. The simulation results show the effectiveness of the proposed robust distributed MPC algorithm.     

Coherent quantum LQG control

Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as “coherent feedback control”. It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints, our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as an initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones.     

Output feedback model predictive control for LPV systems based on quasi-min–max algorithm

This paper proposes a robust output feedback model predictive control (MPC) scheme for linear parameter varying (LPV) systems based on a quasi-min–max algorithm. This approach involves an off-line design of a robust state observer for LPV systems using linear matrix inequality (LMI) and an on-line robust output feedback MPC algorithm using the estimated state. The proposed MPC method for LPV systems is applicable for a variety of systems with constraints and guarantees the robust stability of the output feedback systems. A numerical example for an LPV system subject to input constraints is given to demonstrate its effectiveness.     

Predictive metamorphic control

In spite of its easy implementation, ability to handle constraints and nonlinearities, etc., model predictive control (MPC) does have drawbacks including tuning difficulties. In this paper, we propose a refinement to the basic MPC strategy by incorporating a tuning parameter such that one can move smoothly from an existing controller to a new MPC strategy. Each change of this tuning parameter leads to a new stabilising control law, therefore, allowing one to gradually move from an existing control law to a new and better one. For the infinite horizon case without constraints and for the general case with state and input constraints, stability results are established. We also examine the practical applicability of the proposed approach by employing it in the nominal prediction model of the tube-based output feedback robust MPC method. The merits of the proposed method are illustrated by examples.     

Switched model predictive control for performance enhancement

This article presents a switched model predictive control (MPC) algorithm for non-linear discrete time systems where the weights on the state and control variables in the cost function to be minimised depend on the current value of the state. In so doing, with a reduced computational burden one can easily include, in the problem formulation, a number of control specifications, such as the requirement to avoid critical regions in the state space or to reduce as much as possible the use of some actuators in other zones of the state space. The proposed MPC method has stability properties and is applied for control of a thermal system. The reported simulation results witness its favourable characteristics with respect to a standard MPC implementation.     

Robust output feedback model predictive control using off-line linear matrix inequalities

A fundamental question about model predictive control (MPC) is its robustness to model uncertainty. In this paper, we present a robust constrained output feedback MPC algorithm that can stabilize plants with both polytopic uncertainty and norm-bound uncertainty. The design procedure involves off-line design of a robust constrained state feedback MPC law and a state estimator using linear matrix inequalities (LMIs). Since we employ an off-line approach for the controller design which gives a sequence of explicit control laws, we are able to analyze the robust stabilizability of the combined control laws and estimator, and by adjusting the design parameters, guarantee robust stability of the closed-loop system in the presence of constraints. The algorithm is illustrated with two examples.     

Real‐time model predictive control based on dual gradient projection: Theory and fixed‐point FPGA implementation

This paper proposes a method to design robust model predictive control (MPC) laws for discrete‐time linear systems with hard mixed constraints on states and inputs, in case of only an inexact solution of the associated quadratic program is available, because of real‐time requirements. By using a recently proposed dual gradient‐projection algorithm, it is proved that the discrepancy of the optimal control law as compared with the obtained one is bounded even if the solver is implemented in fixed‐point arithmetic. By defining an alternative MPC problem with tightened constraints, a feasible solution is obtained for the original MPC problem, which guarantees recursive feasibility and asymptotic stability of the closed‐loop system with respect to a set including the origin, also considering the presence of external disturbances. The proposed MPC law is implemented on a field‐programmable gate array in order to show the practical applicability of the method. Copyright © 2016 John Wiley & Sons, Ltd.