ROBUST MODEL PREDICTIVE CONTROL FOR INPUT SATURATED AND SOFTENED STATE CONSTRAINTS

This paper starts with a brief review of robust model predictive control (RMPC) algorithsms for uncertain systems using linear matrix inequalities (LMIs) subject to input and/or output saturated constraints. However when RMPC has both input and state constraints, a difficulty will arise due to the inability of the optimizer to satisfy the state constraints due to the constraints on inputs. Therefore, a novel RMPC scheme is presented that softens the state constraints as penalty terms are added to its objective function. These terms maintain state violation at low values until a constrained solution is returned. The state violation can be regulated by changing the value of the weighting factor. A novel robust predictive controller for input saturated and softened state constraints for linear time varying (LTV) systems with polytopic model uncertainties is presented.     

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.     

On Stability of Constrained Receding Horizon Control with Finite Terminal Weighting Matrix

In this paper, a new stabilizing receding horizon control, based on a finite input and state horizon cost with a finite terminal weighting matrix, is proposed for time-varying discrete linear systems with constraints. We propose matrix inequality conditions on the terminal weighting matrix under which closed-loop stability is guaranteed for both cases of unconstrained and constrained systems with input and state constraints. We show that such a terminal weighting matrix can be obtained by solving a linear matrix inequality (LMI). In the case of constrained time-invariant systems, an artificial invariant ellipsoid constraint is introduced in order to relax the conventional terminal equality constraint and to handle constraints. Using the invariant ellipsoid constraints, a feasibility condition of the optimization problem is presented and a region of attraction is characterized for constrained systems with the proposed receding horizon control.     

Robust nonlinear MPC without terminal constraint for tracking changing setpoints in the presence of non-additive unknown disturbance

This paper develops a novel robust tracking model predictive control (MPC) without terminal constraint for discrete-time nonlinear systems capable to deal with changing setpoints and unknown non-additive bounded disturbances. The MPC scheme without terminal constraint avoids difficult computations for the terminal region and is thus simpler to design and implement. However, the existence of disturbances and/or sudden changes in a setpoint may lead to feasibility and stability issues in this method. In contrast to previous works that considered changing setpoints and/or additive slowly varying disturbance, the proposed method is able to deal with changing setpoints and non-additive non-slowly varying disturbance. The key idea is the addition of tightened input and state (tracking error) constraints as new constraints to the tracking MPC scheme without terminal constraints based on artificial references. In the proposed method, the optimal tracking error converges asymptotically to the invariant set for tracking, and the perturbed system tracking error remains in a variable size tube around the optimal tracking error. Closed-loop input-to-state stability and recursive feasibility of the optimization problem for any piece-wise constant setpoint and non-additive disturbance are guaranteed by tightening input and state constraints as well as weighting the terminal cost function by an appropriate stabilizing weighting factor. The simulation results of the satellite attitude control system are provided to demonstrate the efficiency of the proposed predictive controller.     

Use of model predictive controller in dual-loop control of three-phase PWM AC/DC converter

In the classical dual-loop voltage control scheme for an AC/DC converter, this paper proposes a simple stabilizing inner-loop model predictive controller (MPC) to regulate the output current and q-frame current to their references. The proposed MPC minimizes a cost function of the tracking error without any use of numerical methods using the specific property of the input matrix of the converter. It is shown that this MPC globally stabilizes the converter in the presence of input constraints. As the same manner of the classical dual-loop control scheme, PI controllers are adopted in the outerloop to regulate the output voltage while maintaining the maximum power factor. The simulation results show that the proposed inner-loop MPC considerably enhances the closed-loop performance despite the load changes.     

Novel MPC‐Based Fault Tolerant Tracking Control Against Sensor Faults

The problem of active fault‐tolerant tracking control with control input and system output constraints is studied for a class of discrete‐time systems subject to sensor faults. A time‐varying fault‐tolerant observer is first developed to estimate the real system state from the faulty sensor output and control input signals. Then by using the estimated state at each time step, a model predictive control (MPC)‐based fault‐tolerant tracking control scheme is presented to guarantee the desired tracking performance and the given input and output constraints on the faulty system. In comparison with many existing fault‐tolerant MPC methods, its main contribution is that the proposed state estimator is designed by the simple and online numerical computation to tolerate the possible sensor faults, so that the regular MPC algorithm without fault information can be adopted for the online calculation of fault‐tolerant control signal. The potential recursive infeasibility and computational complexity due to the faults are avoided in the scheme. Additionally, the closed‐loop stability of the post‐fault system is discussed. Simulative results of an electric throttle control system verify the effectiveness of the proposed method.     

LMI-based robust model predictive control and its application to an industrial CSTR problem

In this paper, robust model predictive control (MPC) is studied for a class of uncertain linear systems with structured time-varying uncertainties. This general class of uncertain systems is useful for nonlinear plant modeling in many circumstances. The controller design is characterizing as an optimization problem of the “worst-case” objective function over infinite moving horizon, subject to input and output constraints. A sufficient state-feedback synthesis condition is provided in the form of linear matrix inequality (LMI) optimizations, and will be solved on-line. The stability of such a control scheme is determined by the feasibility of the optimization problem. To demonstrate its usefulness, this robust MPC technique is applied to an industrial continuous stirred tank reactor (CSTR) problem with explicit input and output constraints. Its relative merits to conventional MPC approaches are also discussed.     

Synthesis of Mixed Objective Output Feedback Robust Model Predictive Control

Aiming at the constrained polytopic uncertain system with energy‐bounded disturbance and unmeasurable states, a novel synthesis scheme to design the output feedback robust model predictive control(MPC)is put forward by using mixed H₂/H_∞ design approach. The proposed scheme involves an offline design of a robust state observer using linear matrix inequalities(LMIs)and an online output feedback robust MPC algorithm using the estimated states in which the desired mixed objective robust output feedback controllers are cast into efficiently tractable LMI‐based convex optimization problems. In addition, the closed‐loop stability and the recursive feasibility of the proposed robust MPC are guaranteed through an appropriate reformulation of the estimation error bound (EEB). A numerical example subject to input constraints illustrates the effectiveness of the proposed controller.     

Constrained MPC algorithm for uncertain time-varying systems with state-delay

In this note, we present a model predictive control (MPC) algorithm for uncertain time-varying systems with input constraints and state-delay. Uncertainty is assumed to be polytopic, and delay is assumed to be unknown but with a known upper bound. For a memoryless state-feedback MPC law, we define an optimization problem that minimizes a cost function and relaxes it to two other optimization problems by finding an upper bound of the cost function. One is solvable and the other is not. We prove equivalence and feasibilities of the two optimization problems under a certain assumption on the weighting matrix. Based on these properties and optimality, we show that feasible MPC from the optimization problems stabilizes the closed-loop system. Then, we present an improved MPC algorithm that includes relaxation procedures of the assumption on the weighting matrix and stabilizes the closed-loop system. Finally, a numerical example illustrates the performance of the proposed algorithm.     

Adaptive receding horizon predictive control for constrained discrete-time linear systems with parameter uncertainties

This paper proposes a discrete-time model predictive control (MPC) scheme combined with an adaptive mechanism. To this end, first, an adaptive parameter estimation algorithm suitable for MPC is proposed, which uses the available input and output signals to estimate the unknown system parameters. It enables the prediction of a monotonically decreasing worst-case estimation error bound over the prediction horizon of MPC. These distinctive features allow for future model improvement to be explicitly considered in MPC. Thus, a less conservative adaptive-type MPC controller can be developed based on the proposed estimation method. Second, we show how the discrete-time adaptive-type state-feedback MPC controller is constructed by combining the on-line parameter estimation scheme with a modified robust MPC method based on the comparison model. The developed MPC controller guarantees feasibility and stability of the closed-loop system theoretically in the presence of input and state constraints. A numerical example is given to demonstrate its effectiveness.     

The use of a cell filter for state estimation in closed-loop NMPC of low dimensional systems

Combining variants of the Kalman filter and moving horizon estimation (MHE) with nonlinear MPC has been studied before. The MHE is appealing due to its ability to impose constraints and demonstrated superiority over extended Kalman filter. However, nonlinear MPC based on MHE requires solutions to two back to back nonlinear programs. In this paper we propose to use the cell filter (CF) to provide state feedback to the MPC regulator. The cell filter is a piecewise constant approximation of the conditional probability density of the states, whose temporal evolution is modeled by an aggregate Markov chain. Since the CF is based on discretized state, input and output spaces, the curse of dimensionality limits its application to low dimensional and constrained systems. In this paper we present simulation examples of closed-loop MPC for a nonlinear reactor and agricultural pest control based on state feedback from both CF and MHE.     

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.     

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.     

An improved structure for model predictive control using non-minimal state space realisation

This paper describes a new method for the design of model predictive control (MPC) using non-minimal state space models, in which the state variables are chosen as the set of measured input and output variables and their past values. It shows that the proposed design approach avoids the use of an observer to access the state information and, as a result, the disturbance rejection, particularly the system input disturbance rejection, is significantly improved when constraints become activated. In addition, when there is no model/plant mismatch, the paper shows that the system output constraints can be realised in the proposed approach. Furthermore, closed-form transfer function representation of the model predictive control system enables the application of frequency response analysis tools to the nominal performance of the system.     

Inherent robustness properties of quasi-infinite horizon nonlinear model predictive control

We consider inherent robustness properties of model predictive control (MPC) for continuous-time nonlinear systems with input constraints and terminal constraints. We show that MPC with a nominal prediction model and persistent but bounded disturbances has some degree of inherent robustness when the terminal control law and the terminal penalty matrix are chosen as the linear quadratic control law and the related Lyapunov matrix, respectively. We emphasize that the input constraint sets can be any compact set rather than convex sets, and our results do not depend on the continuity of the optimal cost function or of the control law in the interior of the feasible region.     

Exponential stability of constrained receding horizon control withterminal ellipsoid constraints

In this paper, state- and output-feedback receding horizon controllers are proposed for linear discrete time systems with input and state constraints. The proposed receding horizon controllers are obtained from the finite horizon optimization problem with the finite terminal weighting matrix and the artificial invariant ellipsoid constraint, which is less restrictive than the conventional terminal equality constraint. Both hard constraints and mixed constraints are considered in the state-feedback case, and mixed constraints are considered in the output-feedback case. It is shown that all proposed state- and output-feedback receding horizon controllers guarantee the exponential stability of closed-loop systems for all feasible initial sets using the Lyapunov approach     

Receding horizon guidance laws for constrained missiles with autopilot lags

In this paper, a new stabilizing receding horizon control (RHC) scheme is proposed for linear continuous time-invariant systems with input and state constraints. The control scheme is based on the minimization of the finite horizon cost with a finite terminal weighting matrix subject to constraints. A new algorithm is suggested to implement the RHC scheme for a constrained receding horizon guidance law (CRHG). The proposed CRHG, which does not use information on the time-to-go, is obtained by using linear matrix inequality (LMI) optimization. Through simulation examples, the performance of the proposed CRHG is illustrated.     

Output feedback robust MPC for LPV system with polytopic model parametric uncertainty and bounded disturbance

The output feedback robust model predictive control (MPC), for the linear parameter varying (LPV) system with norm-bounded disturbance, is addressed, where the model parametric matrices are only known to be bounded within a polytope. The previous techniques of norm-bounding technique, quadratic boundedness (QB), dynamic output feedback, and ellipsoid (true-state bound; TSB) refreshment formula for guaranteeing recursive feasibility, are fused into the newly proposed approaches. In the notion of QB, the full Lyapunov matrix is applied for the first time in this context. The single-step dynamic output feedback robust MPC, where the infinite-horizon control moves are parameterised as a dynamic output feedback law, is the main topic of this paper, while the multi-step method is also suggested. In order to strictly guarantee the physical constraints, the outer bound of the true state replaces the true state itself, so tightness of this bound has a major effect on the control performance. In order to tighten the TSB, a procedure for refreshing the real-time ellipsoid based on that of the last sampling instant is given. This paper is conclusive for the past results and far-reaching for the future researches. Two benchmark examples are given to show the effectiveness of the novel results.     

Constrained predictive pole-placement control with linear models

Predictive pole-placement (PPP) control is a continuous-time MPC using a particular set of basis functions leading to pole-placement behaviour in the unconstrained case. This paper presents two modified versions of the PPP controller which are each shown to have desirable stability properties when controlling systems with input, output and state constraints.     

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.