Showing papers in "Lecture Notes in Control and Information Sciences in 2009"

Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation

Abstract: This overview paper reviews numerical methods for solution of optimal control problems in real-time, as they arise in nonlinear model predictive control (NMPC) as well as in moving horizon estimation (MHE). In the first part, we review numerical optimal control solution methods, focussing exclusively on a discrete time setting. We discuss several algorithmic ”building blocks” that can be combined to a multitude of algorithms. We start by discussing the sequential and simultaneous approaches, the first leading to smaller, the second to more structured optimization problems. The two big families of Newton type optimization methods, Sequential Quadratic Programming (SQP) and Interior Point (IP) methods, are presented, and we discuss how to exploit the optimal control structure in the solution of the linear-quadratic subproblems, where the two major alternatives are “condensing” and band structure exploiting approaches. The second part of the paper discusses how the algorithms can be adapted to the real-time challenge of NMPC and MHE. We recall an important sensitivity result from parametric optimization, and show that a tangential solution predictor for online data can easily be generated in Newton type algorithms. We point out one important difference between SQP and IP methods: while both methods are able to generate the tangential predictor for fixed active sets, the SQP predictor even works across active set changes. We then classify many proposed real-time optimization approaches from the literature into the developed categories.

A survey on explicit model predictive control

Abstract: Explicit model predictive control (MPC) addresses the problem of removing one of the main drawbacks of MPC, namely the need to solve a mathematical program on line to compute the control action. This computation prevents the application of MPC in several contexts, either because the computer technology needed to solve the optimization problem within the sampling time is too expensive or simply infeasible, or because the computer code implementing the numerical solver causes software certification concerns,especially in safety critical applications.

Optimizing Process Economic Performance Using Model Predictive Control

Abstract: The current paradigm in essentially all industrial advanced process control systems is to decompose a plant’s economic optimization into two levels. The first level performs a steady-state optimization. This level is usually referred to as real-time optimization (RTO). The RTO determines the economically optimal plant operating conditions (setpoints) and sends these setpoints to the second level, the advanced control system, which performs a dynamic optimization. Many advanced process control systems use some form of model predictive control or MPC for this layer. The MPC uses a dynamic model and regulates the plant dynamic behavior to meet the setpoints determined by the RTO.

Input-to-State Stability: A Unifying Framework for Robust Model Predictive Control

Abstract: This paper deals with the robustness of Model Predictive Controllers for constrained uncertain nonlinear systems. The uncertainty is assumed to be modeled by a state and input dependent signal and a disturbance signal. The framework used for the analysis of the robust stability of the systems controlled by MPC is the wellknown Input-to-State Stability. It is shown how this notion is suitable in spite of the presence of constraints on the system and of the possible discontinuity of the control law.

TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations

Abstract: In the recent years the authors developed numerical schemes to detect the stability properties of different classes of systems involving delayed terms. The base of all methods is the use of pseudospectral differentiation techniques in order to get numerical approximations of the relevant characteristic eigenvalues. This chapter is aimed to present the freely available Matlab package TRACE-DDE devoted to the computation of characteristic roots and stability charts of linear autonomous systems of delay differential equations with discrete and distributed delays and to resume the main features of the underlying pseudospectral approach.

Stability Analysis and Synthesis for Linear Positive Systems with Time-Varying Delays

Abstract: This paper provides necessary and sufficient conditions for the asymptotic stability of linear positive systems subject to time-varying delays. It introduces and initiates an original method for solving directly the proposed stability and stabilization problems without using the well-known Lyapunov theory that is commonly used in the field of stability analysis. In that way and for readers convenience, the paper avoids possible long and tedious superfluous calculus.

Nonlinear Programming Strategies for State Estimation and Model Predictive Control

Abstract: Sensitivity-based strategies for on-line moving horizon estimation (MHE) and nonlinear model predictive control (NMPC) are presented both from a stability and computational perspective. These strategies make use of full-space interior-point nonlinear programming (NLP) algorithms and NLP sensitivity concepts. In particular, NLP sensitivity allows us to partition the solution of the optimization problems into background and negligible on-line computations, thus avoiding the problem of computational delay even with large dynamic models. We demonstrate these developments through a distributed polymerization reactor model containing around 10,000 differential and algebraic equations (DAEs).

On the Observability of Nonlinear and Switched Systems

Abstract: In this paper, new concept of observability are introduced for both nonlinear systems and switched systems. The new definitions are applicable to a much broader family of problems of estimation including unmeasured state variables, unknown input, and unknown parameters in control systems. It is also taken into account the notion of partial observability which is useful for complex or networked systems. For switched systems, the relationship between the observability and hybrid time trajectories is analyzed. It is proved that a switched system might be observable even when individual subsystems are not. Another topic addressed in this paper is the measure of observability, which is able to quantitatively define the robustness and the precision of observability. It is shown that a system can be perfectly observable in the traditional sense, but in the case of high dimensions, it is practically unobservable (or extremely weekly observable). Moreover, computational algorithm for nonlinear systems is developed to compute the observability with precision. Several examples are given to illustrate the fundamentals and the usefulness of the results.

Sequential Monte Carlo for Model Predictive Control

Abstract: This paper proposes the use of Sequential Monte Carlo (SMC) as the computational engine for general (non-convex) stochastic Model Predictive Control (MPC) problems. It shows how SMC methods can be used to find global optimisers of non-convex problems, in particular for solving open-loop stochastic control problems that arise at the core of the usual receding-horizon implementation of MPC. This allows the MPC methodology to be extended to nonlinear non-Gaussian problems. We illustrate the effectiveness of the approach by means of numerical examples related to coordination of moving agents.

The Maximum Weight Independent Set Problem for Data Association in Multiple Hypothesis Tracking

Abstract: Multitarget tracking (MTT) hinges upon the solution of a data association problem in which observations across scans are partitioned into tracks and false alarms so that accurate estimates of true targets can be recovered. In this chapter, we describe a methodology for solving this data association problem as a maximum weight independent set problem (MWISP). This MWISP approach has been used successfully for almost a decade in fielded sensor systems using a multiple hypothesis tracking (MHT) framework, but has received virtually no attention in the tracking literature, nor has it been recognized as an application in the clique/independent set literature. The primary aim of this chapter is to simultaneously fill these two voids. Second, we show that the MWISP formulation is equivalent to the multidimensional assignment (MAP) formulation, one of the most widely documented approaches for solving the data association problem in MTT. Finally, we offer a qualitative comparison between the MWISP and MAP formulations, while highlighting other important practical issues in data association algorithms that are commonly overlooked by the optimization community.

MPC for Tracking of Constrained Nonlinear Systems

Abstract: 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.

Stabilizing Nonlinear Predictive Control over Nondeterministic Communication Networks

Abstract: Networked control systems are systems in which distributed controllers, sensors, actuators and plants are connected via a shared communication network. The use of nondeterministic networks introduces two major issues: communication delays and packet dropouts. These problems cannot be avoided and they might lead to a degradation in performance, or, even worse, to instability of the system. Thus, it is important to take network effects directly into account. In this paper, nonlinear continuous time networked control systems are considered and a nonlinear model predictive controller that is able to compensate the network nondeterminism is outlined.

Nonlinear Model Predictive Path-Following Control

Abstract: In the frame of this work, the problem of following parametrized reference paths via nonlinear model predictive control is considered. It is shown how the use of parametrized paths introduces new degrees of freedom into the controller design. Sufficient stability conditions for the proposed model predictive path-following control are presented. The method proposed is evaluated via simulations of an autonomous mobil robot.

Tropical Scaling of Polynomial Matrices

Abstract: The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by an earlier work of Akian, Bapat, and Gaubert, relies on the computation of “tropical roots”. We give explicit bounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not, the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups.

A Flatness-Based Iterative Method for Reference Trajectory Generation in Constrained NMPC

Abstract: This paper proposes a novel methodology that combines the differential flatness formalism for trajectory generation of nonlinear systems, and the use of a model predictive control (MPC) strategy for constraint handling The methodology consists of a trajectory generator that generates a reference trajectory parameterised by splines, and with the property that it satisfies performance objectives The reference trajectory is generated iteratively in accordance with information received from the MPC formulation This interplay with MPC guarantees that the trajectory generator receives feedback from present and future constraints for real-time trajectory generation

Set Theoretic Methods in Model Predictive Control

Abstract: The main objective of this paper is to highlight the role of the set theoretic analysis in the model predictive control synthesis. In particular, the set theoretic analysis is invoked to: (i) indicate the fragility of the model predictive control synthesis with respect to variations of the terminal constraint set and the terminal cost function and (ii) discuss a simple, tube based, robust model predictive control synthesis method for a class of nonlinear systems.

Successive linearization NMPC for a class of stochastic nonlinear systems

Abstract: A receding horizon control methodology is proposed for systems with nonlinear dynamics, additive stochastic uncertainty, and both hard and soft (probabilistic) input/state constraints. Jacobian linearization about predicted trajectories is used to derive a sequence of convex optimization problems. Constraints are handled through the construction of tubes and an associated Markov chain model. The parameters defining the tubes are optimized simultaneously with the predicted future control trajectory via online linear programming.

Stabilization of Networked Control Systems by Nonlinear Model Predictive Control: A Set Invariance Approach

Abstract: The present paper is concerned with the robust state feedback stabilization of uncertain discrete-time constrained nonlinear systems in which the loop is closed through a packet-based communication network. In order to cope with model uncertainty, time-varying transmission delays and and packet dropouts which typically affect networked control systems, a robust control policy, which combines model predictive control with a network delay compensation strategy, is proposed.

OreMorphisms : A Homological Algebraic Package for Factoring, Reducing and Decomposing Linear Functional Systems

Abstract: The purpose of this paper is to demonstrate the symbolic package OREMORPHISMS which is dedicated to the implementation of different algorithms and heuristic methods for the study of the factorization, reduction and decomposition problems of general linear functional systems (eg, systems of partial differential or difference equations, differential time-delay systems) In particular, we explicitly show how to decompose a differential timedelay system (a string with an interior mass [15]) formed by 4 equations in 6 unknowns and prove that it is equivalent to a simple equation in 3 unknowns We finally give a list of reductions of classical systems of differential time-delay equations and partial differential equations coming from control theory and mathematical physics

Model Predictive Control of Vehicle Formations

Abstract: We propose a two-layer scheme to control a set of vehicles moving in a formation. The first layer, the trajectory controller, is a nonlinear controller since most vehicles are nonholonomic systems and require a nonlinear, even discontinuous, feedback to stabilize them. The trajectory controller, a model predictive controller, computes centrally a bang-bang control law and only a small set of parameters need to be transmitted to each vehicle at each iteration. The second layer, the formation controller, aims to compensate for small changes around a nominal trajectory maintaining the relative po- sitions between vehicles. We argue that the formation control can be, in most cases, adequately carried out by a linear model predictive controller accommodating input and state constraints. This has the advantage that the control laws for each vehicle are simple piecewise affine feedback laws that can be pre-computed off-line and implemented in a distributed way in each vehicle. Although several optimization problems have to be solved, the control strategy proposed results in a simple and efficient implementation where no optimization problem needs to be solved in real-time at each vehicle.

Particle Swarm Optimization Based NMPC: An Application to District Heating Networks

Abstract: Predictive control is concerned with the on-line solution of successive optimization problems. As systems are more and more complex, one of the limiting points in the application of optimal receding horizon strategy is the tractability of these optimization problems. Stochastic optimization methods appear as good candidates to overcome some of the difficulties. Indeed, these methods are not dependent on the structure of costs and constraints (linear, convex...), can escape from local minima and do not require the computation of local informations (gradient, hessian). In this paper, a Particle Swarm Optimization (PSO) is proposed to solve the receding horizon principle with an application to district heating networks. Tests of the approach are given for a network benchmark, showing that more than satisfactory results are achieved, compared with classical control laws for such systems.

Stability and D-stability for Switched Positive Systems

Abstract: We consider a number of questions pertaining to the stability of positive switched linear systems. Recent results on common quadratic, diagonal, and copositive Lyapunov function existence are reviewed and their connection to the stability properties of switched positive linear systems is highlighted. We also generalise the concept of D-stability to positive switched linear systems and present some preliminary results on this topic.

Multiple Model Predictive Control of Nonlinear Systems

Abstract: An augmented state space formulation for multiple model predictive control (MMPC) is developed to improve the regulation of nonlinear and uncertain process systems. By augmenting disturbances as states that are estimated using a Kalman filter, improved disturbance rejection is achieved compared to an additive output disturbance assumption. The approach is applied to a Van de Vusse reactor example, which has challenging dynamic behavior in the form of a right half plane zero and input multiplicity.

Fast Nonlinear Model Predictive Control with an Application in Automotive Engineering

Abstract: Although nonlinear model predictive control has become a well-established control approach, its application to time-critical systems requiring fast feedback is still a major computational challenge. In this article we investigate a new multi-level iteration scheme based on theory and algorithmic ideas from [2], and extending the idea of real-time iterations as presented in [4]. This novel approach takes into account the natural hierarchy of different time scales inherent in the dynamic model. Applications from aerodynamics and chemical engineering have been successfully treated. In this contribution we apply the investigated multi-level iteration scheme to fast optimal control of a vehicle and discuss the computational performance of the scheme.

Robust Track Association and Fusion with Extended Feature Matching

Abstract: In this work, we propose a new data processing architecture as well as track association and fusion algorithms to improve target classification and tracking accuracy using distributed and, possibly, legacy-sensor platforms. We present a robust data fusion algorithm that can incorporate target classes/types at the fusion center when receiving sensor reports and/or local tracks. We aim to tackle the following technical challenges in feature aided tracking. Unknown number of targets: When the fusion center does not have any prior knowledge on the number of targets in the surveillance area, track fusion becomes extremely difficult especially when targets are closely spaced. Measurement origin uncertainty: The local tracker does not know which measurement comes from which target and each local tracker may provide false tracks or incorrect target types. Consequently, the fusion center does not know which local tracks are from the same target and fusion has to be made based on imperfect data association. Tracks from legacy sensor systems: Existing trackers often have very different filter designs. Some may be based on the state-of-the-art multiple model algorithm while some on the fixed gain Kalman filter. Thus some trackers can report both target state estimate and the associated covariance to the fusion center, but others may only provide target state estimate without the covariance information. Those legacy sensor systems require special treatment in the development of fusion algorithms. Our track association framework can also incorporate tracks with extended feature points and kinematic constraints, which improves both data association and tracking accuracy.

Explicit Approximate Model Predictive Control of Constrained Nonlinear Systems with Quantized Input

Abstract: In this paper, a Model Predictive Control problem for constrained nonlinear systems with quantized input is formulated and represented as a multi-parametric Nonlinear Integer Programming (mp-NIP) problem. Then, a computational method for explicit approximate solution of the resulting mp-NIP problem is suggested. The proposed approximate mp-NIP approach is applied to the design of an explicit approximate MPC controller for a clutch actuator with on/off valves.

Frequency Stability Analysis of Linear Systems with General Distributed Delays

Abstract: A new method that reveals conditions under which the characteristic function of a linear time delay system has a root s o such that ― s o is also a root of the function is presented. The method is based on some recent results on the computation of the Lyapunov matrices for time delay systems. A general class of linear systems with distributed delays is studied. A number of examples are given to illustrate the approach and to show its strength.

Robust Stabilization and H ∞ Control of Uncertain Distributed Delay Systems

Abstract: Linear matrix inequality (LMI) conditions for the robust stabilization and robust H ∞ control of uncertain linear systems with distributed delays are presented. All system matrices including the delay kernel are uncertain. Yet, the nominal delay kernel is assumed to be a matrix of rational functions, i.e., it can be written in a linear fractional form. The synthesis conditions are derived using a Lyapunov-Krasovskii functional. Techniques from robust control, such as the full-block S-procedure, are used to transform the resulting parametric matrix inequality into an LMI. As an important feature, the controller synthesis algorithm uses explicitly the information about the continuous delay kernel.

LMI-Based Model Predictive Control for Linear Discrete-Time Periodic Systems

Abstract: This paper presents a newmodel predictive control (MPC) scheme for linear constrained discrete-time periodic systems. In each period of the system, a new periodic state feedback control law is computed via a convex optimization problem that minimizes an upper bound on an infinite horizon cost function subject to state and input constraints. The performance of the proposed model predictive controller, that stabilizes the discrete-time periodic system if it is initially feasible, is illustrated via an example.

Enlarging the Terminal Region of NMPC with Parameter-Dependent Terminal Control Law

Abstract: Nominal stability of a quasi-infinite horizon nonlinear model predictive control (QIH-NMPC) scheme is obtained by an appropriate choice of the terminal region and the terminal penalty term. This paper presents a new method to enlarge the terminal region, and therefore the domain of attraction of the QIH-NMPC scheme. The proposed method applies a parameter-dependent terminal controller. The problem of maximizing the terminal region is formulated as a convex optimization problem based on linear matrix inequalities. Compared to existing methods using a linear time-invariant terminal controller, the presented approach may enlarge the terminal region significantly. This is confirmed via simulations of an example system.