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.     

Optimal MPC for tracking of constrained linear systems

Model predictive control (MPC) is one of the few techniques which is able to handle constraints on both state and input of the plant. The admissible evolution and asymptotic convergence of the closed-loop system is ensured by means of suitable choice of the terminal cost and terminal constraint. However, most of the existing results on MPC are designed for a regulation problem. If the desired steady-state changes, the MPC controller must be redesigned to guarantee the feasibility of the optimisation problem, the admissible evolution as well as the asymptotic stability. Recently, a novel MPC has been proposed to ensure the feasibility of the optimisation problem, constraints satisfaction and asymptotic evolution of the system to any admissible target steady-state. A drawback of this controller is the loss of a desirable property of the MPC controllers: the local optimality property. In this article, a novel formulation of the MPC for tracking is proposed aimed to recover the optimality property maintaining all the properties of the original formulation.     

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.     

Model predictive control for constrained uncertain piecewise linear systems

For constrained piecewise linear (PWL) systems, the possible existing model uncertainty will bring the difficulties to the design approaches of model predictive control (MPC) based on mixed integer programming (MIP). This paper combines the robust method and hybrid method to design the MPC for PWL systems with structured uncertainty. For the proposed approach, as the system model is known at current time, a free control move is optimized to be the current control input. Meanwhile, the MPC controller uses a sequence of feedback control laws as the future control actions, where each feedback control law in the sequence corresponds to each partitions and the arbitrary switching technique is adopted to tackle all the possible switching. Furthermore, to reduce the online computational burden of MPC, the segmented design procedure is suggested by utilizing the characteristics of the proposed approach. Then, an offline design algorithm is proposed, and the reserved degree of freedom can be online used to optimize the control input with lower computational burden.     

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.     

H∞ model predictive control for constrained discrete‐time piecewise affine systems

This paper investigates stability analysis for piecewise affine (PWA) systems and specifically contributes a new robust model predictive control strategy for PWA systems in the presence of constraints on the states and inputs and with l₂ or norm‐bounded disturbances. The proposed controller is based on piecewise quadratic Lyapunov functions. The problem of minimization of the cost function for model predictive control design is changed to minimization of the worst case of the cost function. Then, this objective is reduced to minimization of a supremum of the cost function subject to a terminal inequality by considering the induced l₂‐norm. Finally, the predictive controller design problem is turned into a linear matrix inequality feasibility exercise with constraints on the input signal and state variables. It is shown that the closed‐loop system is asymptotically stable with guaranteed robust performance. The validity of the proposed method is verified through 3 well‐known examples of PWA systems. Simulation results are provided to show good convergence properties along with capability of the proposed controller to reject disturbances.     

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.     

State tracking model reference adaptive control for switched nonlinear systems with linear uncertain parameters

Model reference adaptive control(MRAC)is considered for a class of switched nonlinear systems in which the unknown parameters appear linearly.The linear uncertain parameters in each subsystem can be expressed as a vector and the uncertain vectors in different subsystems are estimated individually by different vector variables.Update laws are designed such that the parameter estimation will ’freeze’ until its corresponding subsystem is active.Controllers for subsystems are given to ensure asymptotic states tracking under arbitrary switchings.Two examples are presented to validate the proposed method.     

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.     

Stabilizing low complexity feedback control of constrained piecewise affine systems

Piecewise affine (PWA) systems are powerful models for describing both non-linear and hybrid systems. One of the key problems in controlling these systems is the inherent computational complexity of controller synthesis and analysis, especially if constraints on states and inputs are present. In addition, few results are available which address the issue of computing stabilizing controllers for PWA systems without placing constraints on the location of the origin.This paper first introduces a method to obtain stability guarantees for receding horizon control of discrete-time PWA systems. Based on this result, two algorithms which provide low complexity state feedback controllers are introduced. Specifically, we demonstrate how multi-parametric programming can be used to obtain minimum-time controllers, i.e., controllers which drive the state into a pre-specified target set in minimum time. In a second segment, we show how controllers of even lower complexity can be obtained by separately dealing with constraint satisfaction and stability properties. To this end, we introduce a method to compute PWA Lyapunov functions for discrete-time PWA systems via linear programming. Finally, we report results of an extensive case study which justify our claims of complexity reduction.     

MPC for tracking with optimal closed-loop performance

In the recent paper [Limon, D., Alvarado, I., Alamo, T., & Camacho, E.F. (2008). MPC for tracking of piece-wise constant references for constrained linear systems. Automatica, 44, 2382-2387], a novel predictive control technique for tracking changing target operating points has been proposed. Asymptotic stability of any admissible equilibrium point is achieved by adding an artificial steady state and input as decision variables, specializing the terminal conditions and adding an offset cost function to the functional.In this paper, the closed-loop performance of this controller is studied and it is demonstrated that the offset cost function plays an important role in the performance of the model predictive control (MPC) for tracking. Firstly, the controller formulation has been enhanced by considering a convex, positive definite and subdifferential function as the offset cost function. Then it is demonstrated that this formulation ensures convergence to an equilibrium point which minimizes the offset cost function. Thus, in case of target operation points which are not reachable steady states or inputs for the constrained system, the proposed control law steers the system to an admissible steady state (different to the target) which is optimal with relation to the offset cost function. Therefore, the offset cost function plays the role of a steady-state target optimizer which is built into the controller. On the other hand, optimal performance of the MPC for tracking is studied and it is demonstrated that under some conditions on both the offset and the terminal cost functions optimal closed-loop performance is locally achieved.     

Robust output tracking of constrained perturbed linear systems via model predictive sliding mode control

A robustifying strategy for constrained linear multivariable systems is proposed. A combination of tracking model predictive control with output integral sliding mode techniques is used to completely reject bounded matched perturbations. It can be guaranteed that all constraints on inputs, states, and outputs are satisfied although only output information is used. Finally, real‐world experiments with an unstable plant are presented in order to demonstrate the validity and the effectiveness of the proposed approach.     

Input-to-state stable finite horizon MPC for neutrally stable linear discrete-time systems with input constraints

MPC or model predictive control is representative of control methods which are able to handle inequality constraints. Closed-loop stability can therefore be ensured only locally in the presence of constraints of this type. However, if the system is neutrally stable, and if the constraints are imposed only on the input, global asymptotic stability can be obtained; until recently, use of infinite horizons was thought to be inevitable in this case. A globally stabilizing finite-horizon MPC has lately been suggested for neutrally stable continuous-time systems using a non-quadratic terminal cost which consists of cubic as well as quadratic functions of the state. The idea originates from the so-called small gain control, where the global stability is proven using a non-quadratic Lyapunov function. The newly developed finite-horizon MPC employs the same form of Lyapunov function as the terminal cost, thereby leading to global asymptotic stability. A discrete-time version of this finite-horizon MPC is presented here. Furthermore, it is proved that the closed-loop system resulting from the proposed MPC is ISS (Input-to-State Stable), provided that the external disturbance is sufficiently small. The proposed MPC algorithm is also coded using an SQP (Sequential Quadratic Programming) algorithm, and simulation results are given to show the effectiveness of the method.     

Minimum-time trajectory generation for constrained linear systems using flatness and B-splines

In this article, we discuss minimum-time trajectory generation for input-and-state constrained continuous-time LTI systems in the light of the notion of flatness and B-splines parametrisation. Flat systems have the useful property that the input and the state trajectories can be completely characterised by the (so-called) flat output. We propose a splines parametrisation for the flat output, and the corresponding parametrisations for the performance output, the states and the inputs. Using this parametrisation the problem of minimum-time constrained trajectory planning is cast into a feasibility-search problem in the splines control-point space, in which the constraint region is characterised by a polytope. A close approximation of the minimum-time trajectory is obtained by systematically searching the end-time that makes the constraint polytope to be minimally feasible.     

Model predictive control for tracking randomly varying references

This paper proposes a model predictive control scheme for tracking a-priori unknown references varying in a wide range and analyses its performance. It is usual to assume that the reference eventually converges to a constant in which case convergence to zero of the tracking error can be established. In this note we remove this simplifying assumption and characterise the set to which the tracking error converges and the associated region of convergence.     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

Stable MPC for tracking with maximal domain of attraction

In this work, a stable MPC that maximizes the domain of attraction of the closed-loop system is proposed. The proposed approach is suitable to real applications in the sense that it accounts for the case of output tracking, it is offset free if the output target is reachable and minimizes the offset if some of the constraints are active at steady state. The new approach is based on the definition of a Minkowski functional related to the input and terminal constraints of the stable infinite horizon MPC. It is also shown that the domain of attraction is defined by the system model and the constraints, and it does not depend on the controller tuning parameters. The proposed controller is illustrated with small order examples of the control literature.     

Convergence properties of constrained linear system under MPC control law using affine disturbance feedback

This paper shows new convergence properties of constrained linear discrete time system with bounded disturbances under Model Predictive Control (MPC) law. The MPC control law is obtained using an affine disturbance feedback parametrization with an additional linear state feedback term. This parametrization has the same representative ability as some recent disturbance feedback parametrization, but its choice together with an appropriate cost function results in a different closed-loop convergence property. More exactly, the state of the closed-loop system converges to a minimal invariant set with probability one. Deterministic convergence to the same minimal invariant set is also possible if a less intuitive cost function is used. Numerical experiments are provided that validate the results.     

Networked predictive control of constrained linear systems with input quantisation

This article investigates the problem of stabilising predictive control for constrained systems, wherein communication from the controller to the plant input is through a digital channel subject to quantisation and delay. A novel model with structured norm-bounded uncertainties is proposed to describe control system with input quantisation. Under a multirate scheme, a delay compensation strategy is presented. The networked predictive control synthesis approach is developed by solving a finite receding horizon optimisation problem with free control moves. It is shown that the proposed predictive controller not only efficiently reduces the negative effects of the quantisation and communication delays but also guarantees the closed-loop stability and satisfies constraints. Finally, a simulation example illustrates the effectiveness of the derived method.