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MPC for tracking of constrained nonlinear systems
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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.
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... En trabajos como [16], [26], [8], [27], [28], [2] se propone la utilización de variables de decisión auxiliares con el objetivo de permitir que el sistema pueda ser llevado a cualquier punto de funcionamiento respetando las restricciones dadas sin necesidad de ampliar el horizonte de control. Una característica significativa de estos controladores es que si la referencia a seguir es infactible, pueden estabilizar la planta en el punto factible más cercano a la referencia evitando relajar las restricciones propias del sistema. ...
... Por ello, en [16], [27], [28], [2] se propone un diseño de MPC para el seguimiento de una secuencia admisible de referencias utilizando un modelo LTI. En el mismo se considera una región terminal ampliada de manera que tome en cuenta los posibles cambios de referencia sin necesidad de formular nuevamente el controlador. ...
... , u a − u s (r)) asegura que la variable artificial converja a la referencia establecida. Además, se debe tener en cuenta que una adecuada penalización del estado terminal x(N ) puede conducir a la estabilidad asintótica con buenos rendimientos, como se evidencia en [27]. Luego, el problema de optimización P N (x) para el MPC analizado se define como: mín ...
Modern industrial systems, based on providing better quality and uniformity of their products while making better use of available resources and favoring care for the environment, incorporate increasingly complex control systems. In particular, the chemical process industry has a significant development, accompanied by advances in computing, control and optimization problems. Among these advances, advanced control techniques have been established to improve the performance and ensure the stability of the controlled system. Consequently, optimization-based controllers are implemented in a wide range of industrial applications. Optimal or optimizing controllers take into account, through a functional, the required objectives and incorporate the system operating constraints. In this sense, model-based predictive control uses a mathematical prediction model to predict future system responses and to apply the control strategy that best satisfies the desired objectives. Therefore, to design these control schemes, several aspects must be considered, including the required objectives, process model for prediction, imposed constraints, control law, length of the prediction horizon, among others. On the basis of the aspects mentioned above, this thesis focuses on the design, development and evaluation of model-based predictive control strategies applied to typical industrial processes. The proposed techniques aim to ensure the stability of the controlled system, compliance with operating constraints, and contemplate uncertainty in the prediction model. Uncertainty can be arise from the non-linear nature of the system or due to the lack of exact knowledge of the model parameters.
... On the other hand, the forward QP is used to obtain a control law regarding constraints on the states, inputs and outputs. To achieve this, the MPC for tracking method [6] is considered for regulation purposes. Specifically, this control design ensures that the controller asymptotically lead the process to a steadystate reference x s in an admissible trajectory from any feasible initial state x(0). ...
... Then the term x(N p ) − x a 2 P is an offset that penalises the final state deviation from this target operation and the offset term x a − x s 2 ϕ ensuring that the artificial variable tracks the real set-point variable, with the actual target goal p s . Note that the inclusion of this suitable penalisation of the terminal state x(N p ) can steer to asymptotic stability with good performances, as evidenced in [6]. ...
In this work a predictive controller formulation is developed within a linear parameter-varying formalism, which serves as a non-linear process model. The proposed strategy is an adaptive Model-based Predictive Controller (MPC), designed with terminal set constraints and considering the scheduling polytope of the model. At each sample time, two Quadratic Programming (QP) problems are solved: the first QP considers a backward horizon to find a virtual model-process tuning variable that defines the best LTI prediction model, considering the vertices of the polytopic system; then, the second QP uses this LTI model to optimise performances along a forward horizon. This paper ends with a realistic solar thermal collector process simulation, comparing the proposed MPC to other techniques from the literature. Discussions regarding the results, the design procedure and the computational effort are presented.
... Then the term x(N p ) − x a 2 P is an offset that penalises the final state deviation from this target operation and the offset term x a − x s 2 ϕ ensuring that the artificial variable tracks the real setpoint variable, with the actual target goal p s . Note that the inclusion of this suitable penalisation of the terminal state x(N p ) can steer to asymptotic stability with good performances, as evidenced in [102]. ...
... MPC design coupled to the use of control invariant set sequences is used to make sure that the algorithm guarantees asymptotic convergence [102]. Anyhow, to compute a reachable sequence of control invariant sets for the case of qLPV models, the bounds of the scheduling parameters and their derivatives must be taken into account. ...
Modern plants rely on sophisticated control systems to meet performance and stability requirements. In particular, a conventional feedback control design for a complex system may result in unsatisfactory performance, or even instability, in the event of malfunctions in actuators, sensors or other system components. Hence, in concordance with Gene Kranz's epic quote "Failure is not an option", specially in safety-critical systems. For that reason, in order to circumvent such weaknesses, control systems must be designed to mitigate component malfunctions while maintaining the required levels of stability and performance.
Accordingly, fault-tolerant control systems are control schemes that possess the ability to accommodate component faults automatically. They are capable of maintaining overall system stability and acceptable performance in case such faults occur. Thus, to design these kinds of systems, several aspects should be taken into account. Being these the fault type considered, its classification, the use or not of a mathematical model of the plant, the selection of a fault detection and diagnosis method, the adoption of a controller reconfiguration strategy, among others.
In view of these aspects, this thesis addresses the design, development and evaluation of fault-tolerant controllers for typical industrial processes, which ensure the compliance of operational constraints despite the presence of faults. To begin with, the current state-of-art and the main specific concepts are introduced. Then, two model-based strategies are presented. On the one side, the design of a novel observer-based fault detection and diagnosis scheme and the development of an adaptive predictive controller are combined to deploy a non-linear active fault-tolerant control system, on the basis of the linear parameter varying system representation. The controller stability conditions and the observers design are established on terms of linear matrix inequalities problems. This proposed scheme is evaluated on typical non-linear chemical industrial processes. On the other hand, an optimisation-based fault-tolerant predictive controller was proposed to develop a tertiary-level energy management system, based on a sugarcane distillery power plant. This strategy guarantees the uninterrupted and efficient energy generation on an industrial microgrid, choosing between different energy sources to overcome the fault effects.
Lastly, it is important to remark that for each proposed scheme a realistic simulation scenario was presented. Enabling vast discussions about its performance and effectiveness, via graphical observations and metric indices.
... Tracking is significantly different and merits its own discussion. Relevant works are those of [12,14,15,26] and [25], where reference is maintained within a feasible region of the system. ...
... Closing, it is worth mentioning that in an end-to-end UAV control scheme, FE constraints should be incorporated in all guidance and control layers, spanning from the low level control of the rotational and translational dynamics up to the high level path planning [12,14,15,25]. Hence, in the proposed scheme we also include a path planner aware of the FE, generating waypoint sequences which are always feasible for the aircraft. ...
In this work a vertically integrated fault-tolerant control scheme for fixed-wing Unmanned Aerial Vehicles (UAVs) is presented. At its core, an online approximate Trim Flight Envelope generator yields the motion constraints of the UAV. Given fault information, it remains always up-to-date in view of emerging faults. The controller stack comprises of Nonlinear Model Predictive Controllers for angular velocity, linear velocity and position. Path Planning is achieved by Simple Sparse Rapidly-exploring Random Trees (SST). Both the controllers and the planner are aware of the flight constraints and are hence tolerant to faults. A large set of sensor and actuator faults, common to UAVs are considered and the controllability of the UAV is examined. Detailed simulations using real-time implementations of the controllers are carried out.
... The application of MPC to control multi-agent and distributed systems has also been widely studied, see for example Scattolini (2009) and Negenborn et al. (2009). While there is a variety of techniques, the approach most relevant to this paper is the work presented by Ferramosca et al. (2009) and Limon et al. (2018), where the authors present a tracking MPC formulation for nonlinear constrained systems. Spanos and Murray (2004) were among the first to consider connectivity in a control problem. ...
In cooperative multi-agent robotic systems, coordination is necessary in order to complete a given task. Important examples include search and rescue, operations in hazardous environments, and environmental monitoring. Coordination, in turn, requires simultaneous satisfaction of safety critical constraints, in the form of state and input constraints, and a connectivity constraint, in order to ensure that at every time instant there exists a communication path between every pair of agents in the network. In this work, we present a model predictive controller that tackles the problem of performing multi-agent coordination while simultaneously satisfying safety critical and connectivity constraints. The former is formulated in the form of state and input constraints and the latter as a constraint on the second smallest eigenvalue of the associated communication graph Laplacian matrix, also known as Fiedler eigenvalue, which enforces the connectivity of the communication network. We propose a sequential quadratic programming formulation to solve the associated optimization problem that is amenable to distributed optimization, making the proposed solution suitable for control of multi-agent robotics systems relying on local computation. Finally, the effectiveness of the algorithm is highlighted with a numerical simulation.
... MPC design coupled to the use of control invariant set sequences is used to make sure the algorithm guarantees asymptotic convergence [22]. Anyhow, to compute a reachable sequence of control invariant sets for the case of qLPV models, the bounds of the scheduling parameters and their derivatives must be taken into account. ...
This paper explores the design of three different approaches of robust predictive control formulations for the case of multi-model system representations. The first one is an optimum multi-objective regulator with variable gain matrix that considers a continuous time multi-model system representation and an infinite horizon; the second one is a sub-optimal linear parameter varying model predictive controller based on a discrete time multi-model system representation with finite horizon and a sequence of contractive terminal set constraint; and, at last, an adaptive model predictive controller that considers a discrete time multi-model system representation, with finite horizon and a terminal invariant set, in common to all models within the system's polytope. Finally, these proposed methods are applied to a continuously-stirred tank reactor (CSTR) system, whose dynamic characteristics are well known and strongly non-linear. Through the simulation results, discussions are established on the design procedure, the online computational effort, the performance indexes and the application difficulties.
Many control tasks can be formulated as tracking problems of a known or unknown reference signal. examples are motion compensation in collaborative robotics, the synchronisation of oscillations for power systems or the reference tracking of recipes in chemical process operation. Both the tracking performance and the stability of the closed-loop system depend strongly on two factors: Firstly, they depend on whether the future reference signal required for tracking is known, and secondly, whether the system can track the reference at all. This paper shows how to use machine learning, i.e. Gaussian processes, to learn a reference from (noisy) data while guaranteeing trackability of the modified desired reference predictions within the framework of model predictive control. Guarantees are provided by adjusting the hyperparameters via a constrained optimisation. Two specific scenarios, i.e. asymptotically constant and periodic REFERENCES, are discussed.
Cooperative robotic problems often require coordination in space in order to complete a given task, important examples include search and rescue, operations in hazardous environments, and autonomous taxi deployment. Events can be quickly detected by partitioning the working environment and assigning one robot to each partition. However, a crucial factor that limits the effectiveness and usage of coverage algorithms is related to the ability of taking decisions in the presence of constraints. In this paper, we propose a coverage control algorithm that is capable of handling nonlinear dynamics, and state and input constraints. The proposed algorithm is based on a nonlinear tracking model predictive controller and is proven to converge to a centroidal Voronoi configuration. We also introduce a procedure to design the terminal ingredients of the model predictive controller. The effectiveness of the algorithm is then highlighted with a numerical simulation.
Terminal equality and inequality constraints along with terminal costs are known to be ingredients that grant stability in many Nonlinear Model Predictive Control (NMPC) approaches. Despite the availability of different methods for computing a suitable terminal set and cost, they usually rely on the linearization of the system and the design of terminal stabilizing control laws. Thus, approaches based on contracting constraints represent an alternative to circumvent the calculation of terminal sets and penalties. The present work proposes an NMPC based on a terminal state contracting constraint. This approach also avoids the need of large prediction horizon, helping to alleviate the computational burden usually associated with NMPC. Another contribution of this proposal is a formulation in terms of output zone control and input targets, designed for the common situation in the process industry where the number of degrees of freedom is not enough to independently track the setpoint of all controlled variables. A simulated case study is presented with the application of the proposed controller to the well-known quadruple-tank process.
Sliding mode control (SMC) is to keep the system to a stable differential manifold. Model predictive control (MPC) calculates the control input by solving an optimization problem on receding horizon. The method of receding horizon sliding control (RHSC) includes the predicted information into the SMC design by combining SMC and MPC. Considering the modeling error and measurement noise, there are model-mismatch and disturbance problems in control practice. This paper combines the demonstrated method of RHSC with a state-augmented Kalman filter addressing the model mismatch and disturbance problem. The proposed scheme has been applied to the air system of an advanced heavy-duty engine. The results have shown the capability of tracking the reference signal during a step-response test and the convergence rate to the target signal is 10% faster than MPC.
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. This controller extend a recently
presented MPC for tracking for constrained linear systems to the nonlinear case. The main characteristics of this controller
are: considering an artificial steady state as a decision variable, minimizing a cost that penalizes the error with the artificial
steady state, 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 considering an invariant set for tracking as extended terminal constraint. The properties of this controller has been
tested on a constrained CSTR simulation model.
In this paper, the attention is focused on the tracking of a piecewise constant reference for an LTI system in presence of state-control constraints and bounded disturbance inputs. Exploiting robust controlled invariant sets in the extended state-setpoint space, we propose a novel predictive tracking algorithm which yields robust constraint satisfaction and good tracking performance with a reasonable on-line computational burden. The main idea is to adopt a dual-mode controller that operates as a predictive regulator whenever the desired setpoint is feasible or in feasibility recovery mode otherwise. Simulation results demonstrate the improved tracking performance of the proposed method with respect to available techniques.
In this paper, we present an off-line approach for robust constrained MPC synthesis that gives an explicit control law using Linear Matrix Inequalities (LMIs). This off-line approach can address a broad class of model uncertainty descriptions with guaranteed robust stability of the closed-loop system and substantial reduction of the on-line MPC computation.
Studies on the theory of model predictive control include the assumption that the origin is in the interior of the feasible region (that is, the inequality constraints are not active at steady state). The reason for making this assumption is that without it one cannot guarantee feasibility of the control problem on the infinite horizon because of the finite horizon parameterization of the input with an unconstrained linear feedback law. As demonstrated in this article, however, this assumption often does not hold in practice. A strategy for handling inequality constraints active at steady state is presented by projecting the system onto the active constraints under the finite horizon parameterization of the input, as well as an algorithm for constructing the optimal linear feedback law that constrains the system to the active constraints. Feasibility is obtained using output admissible sets. For the steady-state target calculation, we propose an algorithm utilizing exact penalties that treats systems in a unified fashion with more inputs than outputs and vice versa. Assuming the system is detectable, it is proven that the algorithm yields a unique steady-state target.
The design of a dynamic state feedback receding horizon controller is addressed, which guarantees robust constraint satisfaction, robust stability and offset-free control of constrained linear systems in the presence of time-varying setpoints and unmeasured disturbances. This objective is obtained by first designing a dynamic linear offset-free controller and computing an appropriate domain of attraction for this controller. The linear (unconstrained) controller is then modified by adding a perturbation term, which is computed by a (constrained) robust receding horizon controller. The receding horizon controller has the property that its domain of attraction contains that of the linear controller. In order to ensure robust constraint satisfaction, in addition to offset-free control, the transient, as well as the limiting behavior of the disturbance and setpoint need to be taken into account in the design of the receding horizon controller. The fundamental difference between the results and the existing literature on receding horizon control is that the transient effect of the disturbance and set point sequences on the so-called “target calculator” is explicitly incorporated in the formulation of the receding horizon controller. An example of the control of a continuous stirred-tank reactor is presented. © 2005 American Institute of Chemical Engineers AIChE J, 2005
This paper presents an extension of recently developed robust, tube based, model predictive controllers to some classes of nonlinear discrete time systems. The simplicity of the proposed robust controller design methodology is obtained by constructing simple robust control invariant tubes corresponding to the parameterized feedback control policy. The proposed method has reduced on-line computational complexity compared with standard worst case or min-max based design methods. The resultant controller allows for fairly strong robustness properties such as robust exponential convergence to and stability of the potentially minimal robust positively invariant set for the uncertain controlled system. A simple example illustrates the proposed method
This paper proposes a new approach to reference governor design. As in prior literature, the governor accepts input commands and modifies their evolution so that specified pointwise-in-time constraints on state and control variables are satisfied. The new approach applies to general discrete-time and continuous-time nonlinear systems with uncertainties. It relies on safety properties provided by sublevel sets of equilibria-parameterized functions. These functions need not be Lyapunov functions, and the corresponding sublevel sets need not be positively invariant. Technical conditions that capture the bare essentials of what is needed are identified and the usual desirable properties of reference governors are established. The new approach significantly broadens the class of methods available for constructing the nonlinear function that is required in the implementation of the reference governors. This advantage is illustrated in a nonlinear control problem where off-line, computer-based simulation is the basis for constructing the nonlinear function.
The practicality of model predictive control (MPC) is partially limited by its ability to solve optimization problems in real time. Moreover, on-line computational demand for synthesizing a robust MPC algorithm will likely grow significantly with the problem size. In this paper, we use the concept of an asymptotically stable invariant ellipsoid to develop a robust constrained MPC algorithm which gives a sequence of explicit control laws corresponding to a sequence of asymptotically stable invariant ellipsoids constructed off-line one within another in state space. This off-line approach can address a broad class of model uncertainty descriptions with guaranteed robust stability of the closed-loop system and substantial reduction of the on-line MPC computation. The controller design is illustrated with two examples.