Multi-model predictive control of Hammerstein-Wiener systems based on balanced multi-model partition

ABSTRACT

Model analysis of Hammerstein-Wiener systems has been made, and it is found that the included angle is applicable to such systems to measure the non-linearity. Then, a dichotomy gridding algorithm is proposed based on the included angle. Supporting by the gridding algorithm, a balanced multi-model partition method is put forward to partition a Hammerstein-Wiener system into a set of local linear models. For each linear model, a linear model predictive controller (MPC) is designed. After that, a multi-MPC is composed of the linear MPCs via soft switching. Thus, a complex non-linear control problem is transformed into a set of linear control problems, which simplifies the original control problem and improves the control performance. Two non-linear systems are built into Hammerstein-Wiener models and investigated using the proposed methods. Simulations demonstrate that the proposed gridding and partition methods are effective, and the resulted multi-MPC controller has satisfactory performance in both set-point tracking and disturbance rejection control.     

Change-of-bases abstractions for non-linear hybrid systems

We present abstraction techniques that transform a given non-linear dynamical system into a linear system, or more generally, an algebraic system described by polynomials of bounded degree, so that invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change-of-bases transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change-of-bases transformation for a non-linear system can define an abstraction. Furthermore, the techniques developed here apply to continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems defined by transition systems and hybrid systems that combine continuous as well as discrete subsystems.The techniques presented here allow us to discover, given a non-linear system, if a change-of-bases transformation involving degree-bounded polynomials yielding an algebraic abstraction exists. If so, our technique yields the resulting abstract system, as well. Our techniques enable the use of analysis techniques for linear systems to infer invariants for non-linear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation.     

An online identification algorithm of unknown time-varying delay and internal multimodel control for discrete non-linear systems

In this paper, an online algorithm is proposed for the identification of unknown time-varying input delay in the case of discrete non-linear systems described by decoupled multimodel. This method relies on the minimization of a performance index based on the error between the real system and the partial internal models outputs. In addition, a decoupled internal multimodel control is proposed for the compensation of discrete non-linear systems with time-varying delay. This control scheme incorporates partial internal model controls. Each partial controller is associated to a specified operating zone of the non-linear system. The switching between these controllers is ensured by a supervisor that contains a set of local predictors. A simulation example is carried out to illustrate the significance of the proposed time-varying delay identification algorithm and the proposed internal multimodel control scheme.     

An approach for relating social structure to cognitive structure

There are three different reasons why non-linear functions between social macro-variables (aggregates) may arise. They can be related to three basic steps of a specific model of explanation of social phenomena, defined by Coleman: the logic of situation, the logic of selection and the logic of transition. Starting with a model of one linear difference equation, a change to three different non-linear system equations can generate stable cycles, bifurcations, and chaos. These non-linear system equations can be deduced from simple assumptions about individual or institutional social attributes. It is shown that a) non-linear individual reactions, b) different selection rules for actors having different social attributes and c) institutional constraints resulting in different transition processes are possible causes for non-linearity at the system level. Furthermore it is demonstrated that the assumption of non-linear but homogeneous reactions of all persons have a similar effect on non-linearity, like it is the case for different selection rules. However, despite of being able to show mathematically the possibility of chaos, it has to be said that chaos as a durable state of social systems is very improbable.     

New tools in non-linear modelling and prediction

In this paper we give an account of a new change of perspective in non-linear modelling and prediction as applied to smooth systems. The core element of these developments is the Gamma test a non-linear modelling and analysis tool which allows us to examine the nature of a hypothetical input/output relationship in a numerical data-set. In essence, the Gamma test allows us to efficiently calculate that part of the variance of the output which cannot be accounted for by the existence of any smooth model based on the inputs, even though this model be unknown. A key aspect of this tool is its speed: the Gamma test has time complexity O( ), where M is the number of data-points. For data-sets consisting of a few thousand points and a reasonable number of attributes, a single run of the Gamma test typically takes a few seconds. Around this essentially simple procedure a new set of analytical tools has evolved which allow us to model smooth non-linear systems directly from the data with a precision and confidence that hitherto was inaccessible. In this paper we briefly describe the Gamma test, its benefits in model identification and model building, and then in more detail explain and motivate the procedures which facilitate a Gamma analysis. We briefly report on a case study applying these ideas to the practical problem of predicting level and flow rates in the Thames valley river basin. Finally we speculate on the future development and enhancement of these techniques into areas such as datamining and the production of complex non-linear models directly from data via graphical representations of process charts and automated Gamma analysis of each input-output node.     

Pricing surplus server capacity for mean waiting time sensitive customers

We consider a queueing model wherein the resource is shared by two different classes of customers, primary (existing) and secondary (new), under a service level based pricing contract. This contract between secondary class customers and resource manager specifies unit admission price and quality of service (QoS) offered. We assume that the secondary customers’ Poisson arrival rate depends linearly on unit price and service level offered while the server uses a delay dependent priority queue management scheme. We analyze the joint problem of optimal pricing and operation of the resource with the inclusion of secondary class customers, while continuing to offer a pre-specified QoS to primary class customers. Our analysis leads to an algorithm that finds, in closed form expressions, the optimal points of the resulting non-convex constrained optimization problem. We also study in detail the structure and the non-linear nature of these optimal pricing and operating decisions.     

LPV identification of a turbocharged diesel engine

This work proposes a methodology of identifying linear parameter varying (LPV) models for nonlinear systems. First, linear local models in some operating points, by applying standard identifications procedures for linear systems in time domain, are obtained. Next, a LPV model with linear fractional dependence (LFR) with respect to measured variables is fitted with the condition of containing all the linear models identified in previous step (differential inclusion). The fit is carried out using nonlinear least squares algorithms. Finally, this identification methodology will then be applied to a nonlinear turbocharged diesel engine.     

The approximation of reachable sets of control systems with integral constraint on controls

In this paper an approximation method for the construction of reachable sets of control systems with integral constraints on the control is considered. It is assumed that the control system is non-linear with respect to the phase state vector and is linear with respect to the control vector. The admissible control functions are chosen from the ball centered at the origin with radius μ₀ in L_p, p > 1. The reachable set is replaced by the set which consists of finite number of points. The estimated accuracy of the Hausdorff distance between the reachable set and the set which is approximately constructed is obtained.     

Interpreting linear systems of equalities and inequalities. Application to the water supply problem

This paper shows how the mathematical and the engineering points of view are complementary and help to model real problems that can be stated as systems of linear equations and inequalities. The paper is devoted to point out these relations and making them explicit for the readers to realize about the new possibilities that arise when contemplating the compatibility conditions or the set of general solutions from the dual perspective. After reviewing an orthogonally based powerful algorithm to analyse the compatibility of linear systems of equations and solving them, a water supply problem is used to illustrate its mathematical and engineering multiple aspects, including the optimal statement of the problem in terms of an adequate selection and numbering of equations and unknowns, an analysis of the compatibility conditions and a physical interpretation of the general solution, together with that of each individual generators of the affine space. The possibilities of removing unknowns without altering the compatibility of the problem is also analysed. Next, the Γ‐algorithm to analyse the compatibility of linear systems of inequalities and solving them is described and then, the water supply problem is revisited adding some constraints, such as capacity limits for the pipes and retention valves, and discussed as to how they affect the resulting general solution and other aspects. Finally, some conclusions are derived. Copyright © 2005 John Wiley & Sons, Ltd.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Linear regularity, equirregularity, and intersection mappings for convex semi-infinite inequality systems

We study the mixed-integer rounding (MIR) closures of polyhedral sets. The MIR closure of a polyhedral set is equal to its split closure and the associated separation problem is NP-hard. We describe a mixed-integer programming (MIP) model with linear constraints and a non-linear objective for separating an arbitrary point from the MIR closure of a given mixed-integer set. We linearize the objective using additional variables to produce a linear MIP model that solves the separation problem exactly. Using a subset of these additional variables yields an MIP model which solves the separation problem approximately, with an accuracy that depends on the number of additional variables used. Our analysis yields an alternative proof of the result of Cook et al. (1990) that the split closure of a polyhedral set is again a polyhedron. We also discuss a heuristic to obtain MIR cuts based on our approximate separation model, and present some computational results. Andrea Lodi was supported in part by the EU projects ADONET (contract n. MRTN-CT-2003-504438) and ARRIVAL (contract n. FP6-021235-2).     

Semistability of switched dynamical systems,Part II: Non-linear system theory

In a companion paper [Q. Hui, W.M. Haddad, Semistability of switched dynamical systems. Part I: Linear system theory, Non-linear Anal. Hybrid Syst. 3 (3) (2009) 343–353] semistability and uniform semistability results for switched linear systems were developed. In this paper we develop semistability analysis results for non-linear switched systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. The main results of the paper involve sufficient conditions for semistability using multiple Lyapunov functions and integral-type inequalities.     

A model order reduction technique for systems with nonlinear frequency dependent damping

Frequency domain solution of systems with frequency dependent damping is a computationally expensive endeavour especially when dealing with large order three-dimensional systems. A moment-matching based reduced order model is proposed in this work which is capable of handling nonlinear frequency dependent damping in second-order systems. In the proposed approach, local linear systems with frequency independent matrices are derived from the original system, and using the principles of the Rational Krylov approach, orthogonal basis vectors are computed from these local systems through the second-order Arnoldi procedure. The system is then projected on to the basis set to obtain a numerically efficient reduced order model, accurate in the entire frequency domain of interest. The proposed approach is also shown to be more accurate than the popular modal projection based multi-model approach of the same order. The proposed tool is applied to the problem of determining the frequency response of an idealised centrifugal compressor impeller with non-viscous (frequency dependent) damping.     

Optimization of aircraft structural components by using nature-inspired algorithms and multi-fidelity approximations

In this work, a flat pressure bulkhead reinforced by an array of beams is designed using a suite of heuristic optimization methods (Ant Colony Optimization, Genetic Algorithms, Particle Swarm Optimization and LifeCycle Optimization), and the Nelder-Mead simplex direct search method. The compromise between numerical performance and computational cost is addressed, calling for inexpensive, yet accurate analysis procedures. At this point, variable fidelity is proposed as a tradeoff solution. The difference between the low-fidelity and high-fidelity models at several points is used to fit a surrogate that corrects the low-fidelity model at other points. This allows faster linear analyses during the optimization; whilst a reduced set of expensive non-linear analyses are run “off-line,” enhancing the linear results according to the physics of the structure. Numerical results report the success of the proposed methodology when applied to aircraft structural components. The main conclusions of the work are (i) the variable fidelity approach enabled the use of intensive computing heuristic optimization techniques; and (ii) this framework succeeded in exploring the design space, providing good initial designs for classical optimization techniques. The final design is obtained when validating the candidate solutions issued from both heuristic and classical optimization. Then, the best design can be chosen by direct comparison of the high-fidelity responses.     

A global supply chain model with transfer pricing and transportation cost allocation

We present a model for the optimization of a global supply that maximizes the after tax profits of a multinational corporation and that includes transfer prices and the allocation of transportation costs as explicit decision variables. The resulting mathematical formulation is a non-convex optimization problem with a linear objective function, a set of linear constraints, and a set of bilinear constraints. We develop a heuristic solution algorithm that applies successive linear programming based on the reformulation and the relaxation of the original problem. Our computational experiments investigate the impact of using different starting points. The algorithm produces feasible solutions with very small gaps between the solutions and their upper bound (UB).     

Location of an Obnoxious Route

This problem deals with the location of a route or path through a set of given points in order to maximize the smallest weighted distance from the given points to the route. Applications may include the planning of pipelines carrying noxious material, and also certain problems in robotics. The first algorithm finds a non-linear path by iteratively solving network minimal-cut problems. A second algorithm solves the case where the route is restricted to be linear.     

System Identification with piecewise linear models in the frequency domain – application to an servopneumatic actuator

By means of a real world application a system identification method was investigated for nonlinear systems from input‐output measurements. This approach is based on a blended multiple model structure, which describes the global behaviour of the system over the whole operating range. Depending on the operating point twenty local linear blackbox models were identified in the frequency domain from a finite number of measurements of the inputs and outputs. A comparative study was made of a model, which have been derived using physical laws [4] and measurements of several process states to estimate unknown parameters.     

Rank-deficient matrices as a computational tool

Rank-deficient matrices arise naturally in many applications. Detecting rank changes and computing parameter values for which a matrix has a prescribed (low) rank deficiency is a fundamental task in computing least squares and minimum norm solutions to systems of linear equations. We describe an approach that originates from numerical continuation and bifurcation theory but has a wider applicability. It uses only linear solves with a bordered extension of the rank-deficient matrix and the transpose of that extension. We discuss the basic methods and their application in fundamental problems such as minimization and in more advanced problems in non-linear analysis. We present extensive numerical evidence in instructive test cases as well as in chemical model (one-dimensional PDE) and a biological model (using the software package CONTENT for dynamical systems). © 1997 by John Wiley & Sons, Ltd.     

A derivative-free algorithm for linearly constrained optimization problems

Based on the NEWUOA algorithm, a new derivative-free algorithm is developed, named LCOBYQA. The main aim of the algorithm is to find a minimizer $x^{*} \in\mathbb{R}^{n}$ of a non-linear function, whose derivatives are unavailable, subject to linear inequality constraints. The algorithm is based on the model of the given function constructed from a set of interpolation points. LCOBYQA is iterative, at each iteration it constructs a quadratic approximation (model) of the objective function that satisfies interpolation conditions, and leaves some freedom in the model. The remaining freedom is resolved by minimizing the Frobenius norm of the change to the second derivative matrix of the model. The model is then minimized by a trust-region subproblem using the conjugate gradient method for a new iterate. At times the new iterate is found from a model iteration, designed to improve the geometry of the interpolation points. Numerical results are presented which show that LCOBYQA works well and is very competing against available model-based derivative-free algorithms.