Ignacio Alvarado

Brief paper: MPC for tracking piecewise constant references for constrained linear systems

In this paper, a novel model predictive control (MPC) for constrained (non-square) linear systems to track piecewise constant references is presented. This controller ensures constraint satisfaction and asymptotic evolution of the system to any target which is an admissible steady-state. Therefore, any sequence of piecewise admissible setpoints can be tracked without error. If the target steady state is not admissible, the controller steers the system to the closest admissible steady state. These objectives are achieved by: (i) adding an artificial steady state and input as decision variables, (ii) using a modified cost function to penalize the distance from the artificial to the target steady state (iii) considering an extended terminal constraint based on the notion of invariant set for tracking. The control law is derived from the solution of a single quadratic programming problem which is feasible for any target. Furthermore, the proposed controller provides a larger domain of attraction (for a given control horizon) than the standard MPC and can be explicitly computed by means of multiparametric programming tools. On the other hand, the extra degrees of freedom added to the MPC may cause a loss of optimality that can be arbitrarily reduced by an appropriate weighting of the offset cost term.

MPC FOR TRACKING OF PIECE-WISE CONSTANT REFERENCES FOR CONSTRAINED LINEAR SYSTEMS

Abstract Model predictive control (MPC) is one of the few techniques which is able to handle with constraints on both state and input of the plant. The admissible evolution and asymptotically convergence of the closed loop system is ensured by means of a suitable choice of the terminal cost and terminal contraint. 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 optimization problem, the admissible evolution as well as the asymptotic stability. In this paper a novel formulation of the MPC is proposed to track varying references. This controller ensures the feasibility of the optimization problem, constraint satisfaction and asymptotic evolution of the system to any admissible steady-state. Hence, the proposed MPC controller ensures the offset free tracking of any sequence of piece-wise constant admissible set points. Moreover this controller requires the solution of a single QP at each sample time, it is not a switching controller and improves the performance of the closed loop system.

Technical communique: 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.

Cooperative distributed MPC for tracking

This paper proposes a cooperative distributed linear model predictive control (MPC) strategy for tracking changing setpoints, applicable to any finite number of subsystems. The proposed controller is able to drive the whole system to any admissible setpoint in an admissible way, ensuring feasibility under any change of setpoint. It also provides a larger domain of attraction than standard distributed MPC for regulation, due to the particular terminal constraint. Moreover, the controller ensures convergence to the centralized optimum, even in the case of coupled constraints. This is possible thanks to the warm start used to initialize the optimization Algorithm, and to the design of the cost function, which integrates a Steady-State Target Optimizer (SSTO). The controller is applied to a real four-tank plant.

MPC for tracking with optimal closed-loop performance

In this paper, a novel model predictive control (MPC) formulation has been proposed to solve tracking problems, considering a generalized offset cost function. Sufficient conditions on this function are given to ensure the local optimality property. This novel formulation allows to consider as target operation points, states which may be not equilibrium points of the linear systems. In this case, it is proved in this paper that 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 proposed controller for tracking achieves an optimal closed-loop performance during the transient as well as an optimal steady state in case of not admissible target. These properties are illustrated in an example.

Applying Risk Management to Combined Heat and Power Plants

This paper shows how risk management can be applied to schedule the operation of combined heat and power plants in order to consider process uncertainties. The main innovative point is the consideration of mitigation actions to reduce exposure to the identified risks. Model predictive control is used to select the strategic plan of mitigation actions.

Output feedback Robust tube based MPC for tracking of piece-wise constant references

This paper presents a novel formulation of a robust output feedback model predictive controller to track piecewise constant references (output feedback RMPCT). The real plant is assumed to be modelled as a linear system with additive bounded uncertainties on the states. Under mild assumptions, the proposed MPC can steer the uncertain system in an admissible evolution to any admissible steady state, that is, under any change of the set point. The proposed output feedback controller consists of a stable state estimator and a recently developed robustly stabilizing, tube based, model predictive control for tracking. Feasibility of the proposed controller for any admissible setpoint is achieved by adding an artificial steady state as decision variable. Robust constraint satisfaction is guaranteed by tube-based approach and considering nominal predictions. Robust stability and convergence to (a neighborhood of) the desired steady state is ensured by considering a modified cost function and an extended terminal constraint. The cost function penalizes the tracking error with the artificial reference and the deviation between the artificial and desired steady state; the terminal constraint restricts the terminal state and the artificial steady state. The optimization problem to be solved is a quadratic programming problem, which allows explicit implementations. Robust stability is guaranteed under mild conditions which allows one to consider tracking specifications and disturbance rejection conditions. Moreover, a simple method to achieve offset-free control, compensating the steady tracking error of the outputs, is presented.

Robust control of the distributed solar collector field ACUREX using MPC for tracking.

Abstract This paper presents the application of a robust model predictive control for tracking of piece-wise constant references (RMPCT) to a distributed collector field, ACUREX, at the solar power plant of PSA (Solar Plant of Almeria). The main characteristic of a solar power plant is that the primary energy source, solar radiation, cannot be manipulated. Solar radiation varies throughout the day, causing changes in plant dynamics and strong disturbances in the process. The real plant is assumed to be modeled as a linear system with additive bounded uncertainties on the states. Under mild assumptions, the proposed RMPCT can steer the uncertain system in an admissible evolution to any admissible steady state, that is, under any change of the set point. This allows us to reject constant disturbances compensating the effect of then changing the setpoint.

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.

MPC for tracking of constrained nonlinear systems

This paper deals with the tracking problem for constrained nonlinear systems using a model predictive control (MPC) law. MPC provides a control law suitable for regulating constrained linear and nonlinear systems to a given target steady state. However, when the target operating point changes, the feasibility of the controller may be lost and the controller fails to track the reference. In this paper, a novel MPC for tracking changing constant references is presented. The main characteristics of this controller are: (i) considering an artificial steady state as a decision variable, (ii) minimizing a cost that penalizes the error with the artificial steady state, (iii) adding to the cost function an additional term that penalizes the deviation between the artificial steady state and the target steady state (the so-called offset cost function) and (iv) considering an invariant set for tracking as extended terminal constraint. The calculation of the stabilizing parameters of the proposed controller is studied and some methods are proposed. The properties of this controller has been tested on a constrained CSTR simulation model.