MPC for stable linear systems with model uncertainty

In this paper, we developed a model predictive controller, which is robust to model uncertainty. Systems with stable dynamics are treated. The paper is mainly focused on the output-tracking problem of a system with unknown steady state. The controller is based on a state-space model in which the output is represented as a continuous function of time. Taking advantage of this particular model form, the cost functions is defined in terms of the integral of the output error along an infinite prediction horizon. The model states are assumed perfectly known at each sampling instant (state feedback). The controller is robust for two classes of model uncertainty: the multi-model plant and polytopic input matrix. Simulations examples demonstrate that the approach can be useful for practical application.     

Resource-aware MPC for constrained nonlinear systems: A self-triggered control approach

In systems with resource constraints, such as actuation limitations or limited communication bandwidth, it is desired to obtain control signals that are either sparse or sporadically changing in time to reduce resource utilization. In this paper, we propose a resource-aware self-triggered MPC strategy for discrete-time nonlinear systems subject to state and input constraints that has three important features: Firstly, significant reductions in resource utilization can be realized without modifying the cost function by input regularization or explicitly penalizing resource usage. Secondly, the control laws and triggering mechanisms are synthesized so that a priori chosen performance levels (in terms of the original cost function) are guaranteed by design next to asymptotic stability and constraint satisfaction. Thirdly, we address the co-design problem of jointly designing the feedback law and the triggering condition. By means of numerical examples, we show the effectiveness of this novel strategy.     

A robust gradient-based MPC for integrating real time optimizer (RTO) with control

A gradient-based model predictive control (MPC) strategy was recently proposed to reduce the computational burden derived from the explicit inclusion of an economic real time optimization (RTO). The main idea is to compute a suboptimal solution, which is the convex combination of a feasible solution and a solution of an approximated (linearized) problem. The main benefits of this strategy are that convergence is still guaranteed and good economic performances are obtained, according to several simulation scenarios. The formulation, however, is developed only for the nominal case, which significantly reduces its applicability. In this work, an extension of the gradient-based MPC to explicitly account for disturbances is made. The resulting robust formulation considers a nominal prediction model, but restricted constraints (in order to account for the effect of additive disturbances). The nominal economic performance is preserved and robust stability is ensured. An illustrative example shows the benefits of the proposal.     

An optimal hierarchical control scheme for smart generation units: An application to combined steam and electricity generation

Optimal management of thermal and energy grids with fluctuating demand and prices requires to orchestrate the generation units (GU) among all their operating modes. A hierarchical approach is proposed to control coupled energy nonlinear systems. The high level hybrid optimization defines the unit commitment, with the optimal transition strategy, and best production profiles. The low level dynamic model predictive control (MPC), receiving the set-points from the upper layer, safely governs the systems considering process constraints. To enhance the overall efficiency of the system, a method to optimal start-up the GU is here presented: a linear parameter-varying MPC computes the optimal trajectory in closed-loop by iteratively linearizing the system along the previous optimal solution. The introduction of an intermediate equilibrium state as additional decision variable permits the reduction of the optimization horizon, while a terminal cost term steers the system to the target set-point. Simulation results show the effectiveness of the proposed approach.     

Robust MPC for systems with output feedback and input saturation

In this work, it is proposed an MPC control algorithm with proved robust stability for systems with model uncertainty and output feedback. It is assumed that the operating strategy is such that system inputs may become saturated at transient or steady state. The developed strategy aims at the case in which the controller performs in the output-tracking scheme following an optimal set point that is provided by an upper optimization layer of the plant control structure. In this case, the optimal operating point usually lies at the boundary of the region where the input is defined. Assuming that the system remains stabilizable in the presence of input saturation, the design of the robust controller is performed off-line and an on-line implementation strategy is proposed. At each sampling step, a sub optimal control law is obtained by combining control configurations that correspond to particular subsets of available manipulated inputs. Stability of the closed-loop system is forced by considering in the off-line step of the controller design, a state contracting restriction for the closed-loop system. To produce an offset free controller and to attend the case of unknown steady state, the method is developed for a state-space model in the incremental form. The method is illustrated with simulation examples extracted from the process industry.     

Distributed predictive control of continuous-time systems

In the last years focus has been put in the development of distributed Model Predictive Control (MPC) algorithms. With a few exceptions, they have been mostly developed in the discrete-time framework. However, discretization of large-scale systems may destroy the sparsity of the original continuous-time models, making distributed control design and implementation more difficult. Also, more in general, discrete-time control of continuous-time systems does not allow to consider the process inter-sampling behavior. In this paper we present a novel non-cooperative distributed predictive control algorithm for continuous-time systems based on robust MPC concepts. The convergence properties of the proposed control scheme are stated, and its realizability is tested through a simulation case study.     

Min-max MPC using a tractable QP problem

Min-max model predictive controllers (MMMPC) suffer from a great computational burden that is often circumvented by using approximate solutions or upper bounds of the worst possible case of a performance index. This paper proposes a computationally efficient MMMPC control strategy in which a close approximation of the solution of the min-max problem is computed using a quadratic programming problem. The overall computational burden is much lower than that of the min-max problem and the resulting control is shown to have a guaranteed stability. A simulation example is given in the paper.     

Robust distributed MPC for load frequency control of uncertain power systems

Reliable Load frequency control (LFC) is crucial to the operation and design of modern electric power systems. However, the power systems are always subject to uncertainties and external disturbances. Considering the LFC problem of a multi-area interconnected power system, this paper presents a robust distributed model predictive control (RDMPC) based on linear matrix inequalities. The proposed algorithm solves a series of local convex optimization problems to minimize an attractive range for a robust performance objective by using a time-varying state-feedback controller for each control area. The scheme incorporates the two critical nonlinear constraints, e.g., the generation rate constraint (GRC) and the valve limit, into convex optimization problems. Furthermore, the algorithm explores the use of an expanded group of adjustable parameters in LMI to transform an upper bound into an attractive range for reducing conservativeness. Good performance and robustness are obtained in the presence of power system dynamic uncertainties.     

Output feedback MPC with guaranteed robust stability

This paper focuses on the issues of robust stability of model predictive control (MPC). The control problem is formulated as linear matrix inequalities (LMI) optimization problem. A suboptimal solution for the output feedback control problem is proposed. The size of the resulting MP controller is reduced by using a suitable state-space representation of the process. Guaranteed stability conditions for the output feedback MPC are enforced via a Lyapunov type constraint. An iterative algorithm is developed resulting in a pair of coupled LMI optimization problems which provide a robustly stable output feedback gain. Model uncertainties are considered via a polytopic set of process models. The methodology is illustrated with the simulation of the control problem of two chemical processes. The results show that the proposed strategy eliminates the need to detune the MP controller improving the performance for most of the cases considered.     

Uniting bounded control and MPC for stabilization of constrained linear systems

In this work, a hybrid control scheme, uniting bounded control with model predictive control (MPC), is proposed for the stabilization of linear time-invariant systems with input constraints. The scheme is predicated upon the idea of switching between a model predictive controller, that minimizes a given performance objective subject to constraints, and a bounded controller, for which the region of constrained closed-loop stability is explicitly characterized. Switching laws, implemented by a logic-based supervisor that constantly monitors the plant, are derived to orchestrate the transition between the two controllers in a way that safeguards against any possible instability or infeasibility under MPC, reconciles the stability and optimality properties of both controllers, and guarantees asymptotic closed-loop stability for all initial conditions within the stability region of the bounded controller. The hybrid control scheme is shown to provide, irrespective of the chosen MPC formulation, a safety net for the practical implementation of MPC, for open-loop unstable plants, by providing a priori knowledge, through off-line computations, of a large set of initial conditions for which closed-loop stability is guaranteed. The implementation of the proposed approach is illustrated, through numerical simulations, for an exponentially unstable linear system.     

A receding horizon control approach to sampled-data implementation of continuous-time controllers

We propose a novel way for sampled-data implementation (with the zero order hold assumption) of continuous-time controllers for general nonlinear systems. We assume that a continuous-time controller has been designed so that the continuous-time closed-loop satisfies all performance requirements. Then, we use this control law indirectly to compute numerically a sampled-data controller. Our approach exploits a model predictive control (MPC) strategy that minimizes the mismatch between the solutions of the sampled-data model and the continuous-time closed-loop model. We propose a control law and present conditions under which stability and sub-optimality of the closed loop can be proved. We only consider the case of unconstrained MPC. We show that the recent results in [G. Grimm, M.J. Messina, A.R. Teel, S. Tuna, Model predictive control: for want of a local control Lyapunov function, all is not lost, IEEE Trans. Automat. Control 2004, to appear] can be directly used for analysis of stability of our closed-loop system.     

A delta transform approach to loop gain-phase shaping design of robust digital control systems

This paper addresses the existence of loop gain-phase shaping (LGPS) solutions for the design of robust digital control systems for SISO, minimum-phase, continuous-time processes with parametric uncertainty. We develop the frequency response properties of LGPS for discrete-time systems using the Δ-transform, a transform method that applies to both continuous-time and discrete-time systems. A theorem is presented which demonstrates that for reasonable specifications there always exists a sampling period such that the robust digital control problem has a solution. Finally, we offer a procedure for estimating the maximum feasible sampling period for LGPS solutions to robust digital control problems.     

Resource-aware MPC for constrained linear systems: Two rollout approaches

In systems with resource constraints, such as actuation limitations in sparse control applications or limited bandwidth in networked control systems, it is desirable to use control signals that are either sparse or sporadically changing in time. Motivated by these applications, in this paper we propose two resource-aware MPC schemes for discrete-time linear systems subject to state and input constraints. The two MPC schemes exploit ideas from rollout strategies to determine simultaneously the new (continuous) control inputs and the (discrete) time instants at which the control actions are updated. The first scheme provides performance guarantees by design, in the sense that it allows the user to select a desired suboptimal level of performance, where the degree of suboptimality provides a trade-off between the guaranteed closed-loop control performance on the one hand and the utilization of (communication/actuation) resources on the other hand. The second scheme provides a guaranteed (average) resource utilization, while cleverly allocating these resources in order to maximize the control performance. By means of numerical examples, we demonstrate the effectiveness of the proposed strategies.     

Applying variable dissolved oxygen set point in a two level hierarchical control structure to a wastewater treatment process

This paper presents a two-level hierarchical control structure for biological wastewater treatment plants, with the goal of improving effluent quality and reducing operational costs. The Benchmark Simulation Model No. 1 is used as working scenario. The hierarchical structure allows to adjust the dissolved oxygen in the fifth tank (S_O,5) according with the working conditions, instead of keeping it in a fixed value. Model Predictive Control (MPC) with inlet flow rate feedforward control (MPC + FF) is proposed for the lower level to control nitrate nitrogen concentration of the second tank and S_O,5. MPC, Affine Function and fuzzy controller are tested for the higher level to adjust the S_O,5 set point of the lower level based on the ammonium and ammonia nitrogen concentration in the fifth tank. Modifying the tuning parameters of the higher level, a tuning region is determined, in which the effluent quality and operational costs are simultaneously improved.     

An improved robust model predictive control approach to systems with linear fractional transformation perturbations

In this paper, a robust model predictive control approach is proposed for a class of uncertain systems with time-varying, linear fractional transformation perturbations. By adopting a sequence of feedback control laws instead of a single one, the control performance can be improved and the region of attraction can be enlarged compared with the existing model predictive control (MPC) approaches. Moreover, a synthesis approach of MPC is developed to achieve high performance with lower on-line computational burden. The effectiveness of the proposed approach is verified by simulation examples.     

Modified reference tracking MPC for constrained linear systems

This article presents a model predictive control for tracking piecewise constant references with a new steady-state parametrisation. The modified algorithm is based on the artificial reference idea, but the number of decision variables is equal to the standard MPC for regulation. The proposed strategy is able to track admissible constant references with an admissible evolution. If the reference is not admissible, the system is steered to the closest admissible stationary point. A modified initialisation algorithm is proposed to recover the enlarged domain of attraction provided by related artificial reference-based strategies. Simulation examples are presented to illustrate the benefits of the proposed strategy.     

A switching robust model predictive control approach for nonlinear systems

This work considers enhancing the stability and improving the economic performance of nonlinear model predictive control in the presence of disturbances or model uncertainties. First, a robust control Lyapunov function (RCLF)-based predictive control strategy is proposed. Second, the approximate dynamic programming (ADP) is employed to further improve regulation performance. Finally, the ADP and RCLF-MPC are combined to provide a switching control scheme, which is illustrated on a CSTR example to show its effectiveness.     

A robust MPC approach for the rebalancing of mobility on demand systems

A control-oriented model for mobility-on-demand systems is here proposed. The system is first described through dynamical stochastic state-space equations, and then suitably simplified in order to obtain a control-oriented model, on which two control strategies based on Model Predictive Control are designed. The first strategy aims at keeping the expected value of the number of vehicles parked in stations within prescribed bounds; the second strategy specifically accounts for stochastic fluctuations around the expected value. The model includes the possibility of weighting the control effort, leading to control solutions that may trade off efficiency and cost. The models and control strategies are validated over a dataset of logged trips of ToBike, the bike-sharing systems in the city of Turin, Italy.     

An H-minimax approach to the design of robust control systems

A realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices. A solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone.