On the nonuniqueness of singular value functions and balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced by Scherpen in 1993. Analogous to the linear case, the so-called singular value functions of a system describe the relative importance of each state component from an input–output point of view. In this paper it is shown that the procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, distinct sets of singular value functions and balanced realizations are possible.     

Admissible factorizations of Hankel operators induce well-posed linear systems

One of the basic axioms of a well-posed linear system says that the Hankel operator of the input–output map of the system factors into the product of the input map and the output map. Here we prove the converse: every factorization of the Hankel operator of a bounded causal time-invariant map from L² to L² which satisfies a certain admissibility condition induces a stable well-posed linear system. In particular, there is a one-to-one correspondence between the set of all minimal stable well-posed realizations of a given stable causal time-invariant input–output map (or equivalently, of a given H^∞ transfer function) and all minimal stable admissible factorizations of the Hankel operator of this input–output map.     

Model order selection of nonlinear input–output models––a clustering based approach

Selecting the order of an input–output model of a dynamical system is a key step toward the goal of system identification. The false nearest neighbors algorithm (FNN) is a useful tool for the estimation of the order of linear and nonlinear systems. While advanced FNN uses nonlinear input–output data-based models for the model-based selection of the threshold constant that is used to compute the percentage of false neighbors, the computational effort of the method increases along with the number of data and the dimension of the model. To increase the efficiency of this method, in this paper we propose a clustering-based algorithm. Clustering is applied to the product space of the input and output variables. The model structure is then estimated on the basis of the cluster covariance matrix eigenvalues. The main advantage of the proposed solution is that it is model-free. This means that no particular model needs to be constructed in order to select the order of the model, while most other techniques are ‘wrapped' around a particular model construction method. This saves the computational effort and avoids a possible bias due to the particular construction method used. Three simulation examples are given to illustrate the proposed technique: estimation of the model structure for a linear system, a polymerization reactor and the van der Vusse reactor.     

Fliess operators on Lp spaces: convergence and continuity

Fliess operators as input–output mappings are particularly useful in a number of fundamental problems concerning nonlinear realization theory. In the classical analysis of these operators, certain growth conditions on the coefficients in their series representations insure uniform and absolute convergence, provided every input is uniformly bounded by some fixed upperbound. In some emerging applications, however, it is more natural to consider other classes of inputs. In this paper, L_p function spaces are considered. In particular, it is shown that the classic growth conditions also provide sufficient conditions for convergence and continuity when the admissible inputs are from a ball in L_p[t₀,t₀+T], where T is bounded and p1. In addition, stronger global growth conditions are given that apply even for the case where T is unbounded. When the coefficients of a Fliess operator have a state space representation, it is shown that the state space model will always locally realize the corresponding input–output map on L_p[t₀,t₀+T] for sufficiently small T>0. If certain well-posedness conditions are satisfied then the state space model will globally realized the input–output mapping for unbounded T when the coefficients satisfy the global growth condition.     

On converse Lyapunov theorems for ISS and iISS switched nonlinear systems

In this paper we present converse Lyapunov theorems for ISS and integral input to state stable (iISS) switched nonlinear systems. Their proofs are based on existing converse Lyapunov theorems for input–output to state stable and iISS nonlinear systems, and on the association of the switched system with a nonlinear system with inputs and disturbances that take values in a compact set.     

Nonlinear feedback control of multivariable non-minimum-phase processes

A new method of controlling nonlinear processes with a non-minimum-phase delay-free part is presented. Two control laws are derived for stable, multiple-input multiple-output processes. They are obtained by requesting an approximately linear, input–output response and exploiting the connections between model-predictive control and input–output linearization. Conditions under which the closed-loop system is asymptotically stable are given. The application and performance of the control laws are illustrated using numerical simulation of two chemical reactor examples that exhibit non-minimum-phase behavior.     

Irreducibility, reduction and transfer equivalence of nonlinear input–output equations on homogeneous time scales

The purpose of this paper is to present a necessary and sufficient condition for irreducibility of nonlinear input–output delta differential equations. The condition is presented in terms of the common left divisor of two differential polynomials describing the behaviour of the system defined on a homogenous time scale. The concept of reduction is explained. Subsequently, the definition of transfer equivalence based upon the notion of an irreducible differential form of the system is introduced, inspired by the analogous definition for continuous-time systems.     

Control of reactive distillation production of high-purity isopropanol

The process characteristics and control strategy of a high-purity IPA reactive distillation column were investigated. A robust nominal operation was found by maintaining an excess of propylene feed to the column and recycling the unreacted propylene to the feed instead of the top stage. Stage temperature and propylene composition with one-to-one relationship with reboiler duty and propylene feed are selected as controlled variables for maintaining bottom purity and feed ratio in the presence of possible measurement bias respectively. High nonlinearity between selected input–output pair was reduced by using variable transformation. Dynamic simulations demonstrated that such a control scheme with nonlinear transformed variable was capable of providing much superior control performance than the one using natural variable.     

A class of optimal solutions of the matrix Nehari-extension problem

The degrees of freedom that are available in the solution set of the multivariable Nehari-extension problem are used to minimize an ??²-type cost and an ‘entropy-like’ cost associated with the smaller singular values of the (optimal) error system. The optimal extensions are constructed via a diagonalization procedure based on a (normalized) Schmidt pair of the Hankel operator induced by the function which is approximated.     

Robust controller design for temperature tracking problems in jacketed batch reactors

There is an increasing trend to employ advanced instrumentation and control strategies for batch processes where expensive products are being manufactured. In this paper, a robust nonlinear control strategy is developed for temperature tracking problems in batch reactors in the presence of parametric uncertainty. The controller has a multi-loop feedback configuration. An inner loop is designed for approximate input–output linearization of a nominal plant. The outer loop is designed for stability and robust performance by utilizing results from structured singular values (μ-synthesis). It is shown via simulation of a temperature tracking problem in batch synthesis that the controller provides excellent tracking despite parametric uncertainty.     

Input–output decoupling of nonlinear systems by static measurement feedback

This paper gives necessary and sufficient conditions for solvability of the strong input–output decoupling problem by static measurement feedback for nonlinear control systems.     

Asymptotic characterizations of input–output-to-state stability for discrete-time systems

We establish the equivalence between global detectability and output-to-state stability for difference inclusions with outputs, and we present equivalent asymptotic characterizations of input–output-to-state stability for discrete-time nonlinear systems. These new stability characterizations for discrete-time systems parallel what have been developed for continuous-time systems in Angeli et al. [Uniform global asymptotic stability of differential inclusions, J. Dynamical Control Systems 10 (2004) 391–412] and Angeli et al. [Seperation principles for input–output and integral-input-to-state stability, SIAM J. Control Optim. 43 (2004) 256–276].     

On the computation of the gap metric for LTV systems

In this paper, we consider the robust stabilization problem for linear discrete time-varying (LTV) systems using the gap metric. In particular, we show that the time-varying (TV) directed gap reduces to an operator with a TV Hankel plus Toeplitz structure. Computation of the norm of such an operator can be carried out using an iterative scheme involving a TV Hankel operator defined on a space of Hilbert–Schmidt causal operators. The “infimization” in the TV directed gap formula is shown to be, in fact, a minimum by using duality theory. The latter holds as well in the time-invariant case.     

Multi-objective constraint optimizing IOL control of distillation column with nonlinear observer

Performance of input–output linearizing (IOL) controllers suffers due to constraints on input and output variables. This problem is successfully tackled by augmenting IOL controllers with quadratic dynamic matrix controller (QDMC). However, this has created a constraint-mapping problem for coupled MIMO systems like distillation column. A multi-objective optimization problem needs to be solved to map the constraints on inputs. A suitable transformation technique is proposed to convert this multi-objective optimization problem to a single objective one. This makes the controller less computationally intensive and easy to implement. This controller (IOL-QDMC) along with nonlinear observer is implemented on a binary distillation column for dual composition control. Its performance is evaluated against a quadratic dynamic matrix controller (QDMC) and input–output linearization with PI controller (IOL-PI).     

Connectivity and the measurement of operational risk: an input–output approach

In this article I propose a framework for the analysis of the interdependencies within a financial institution that is based on the input–output framework originally developed by Leontiev (1941). After discussing the state of the art of operational risk measurement, I briefly review the foundations of input–output analysis and explain how to build an input–output model at the business unit level for a financial institution. I also discuss the suitability of an input–output model in capturing the impact on operational risk losses of the interdependencies within a financial institution and then present, through some numerical examples, how to implement the model within a quantitative framework for the measurement of operational risk. In Ersilia, to establish the relationships that sustain the city's life, the inhabitants stretch strings from the corners of the houses, white or black or gray or black-and-white according to whether they mark a relationship of blood, of trade, authority, agency. When the strings become so numerous that you can no longer pass among them, the inhabitants leave: the houses are dismantled; only the strings and their supports remain. From a mountainside, camping with their household goods, Ersilia's refugees look at the labyrinth of taut strings and poles that rise in the plain. That is the city of Ersilia still, and they are nothing. Italo Calvino, Invisible Cities (1972)     

On a switching control scheme for nonlinear systems with ill-defined relative degree

This paper discusses the applicability of a switching control scheme for a nonlinear system with ill-defined relative degree. The control scheme switches between exact and approximate input–output linearisation control laws. Unlike a linear system under a switching control scheme, the equilibria of a nonlinear system may change with the switching. It is pointed out that this is not sufficient to cause instability. When the region of the approximate linearisation control law is attractive to the exact zero dynamics, it is possible that the closed-loop system under the switching control scheme is still stable. The results in this paper shows that the switching control scheme proposed in Tomlin and Sastry (Systems Control Lett. 35(3) (1998) 145) is applicable for a wider class of nonlinear systems.     

Fuzzy control of an ANFIS model representing a nonlinear liquid-level system

This paper aims to serve two main objectives; one is to demonstrate the modelling capabilities of a neuro-fuzzy approach, namely ANFIS (adaptive-network based fuzzy inference system) to a nonlinear system; and the other is to design a fuzzy controller to control such a system. The nonlinear system, which is a liquid-level system, is represented first by its mathematical model and then by ANFIS architecture. The ANFIS model is formed by means of input–output data set taken from the mathematical model. Then a PID-type fuzzy controller, which linguistically approximates the classical three-term compensation, was designed to control the system represented by both its mathematical and ANFIS models in order to perform an agreement comparison between them. It is shown that the ANFIS architecture can model a nonlinear system very accurately by means of input–output pairs obtained either from the actual system or its mathematical model. It is also shown that such a system can be controlled effectively by a fuzzy controller.     

Input–output pairing analysis for uncertain multivariable processes

Decentralized control structure is widely employed in many industrial multivariable processes. In this approach, control structure design and in particular input–output pairing is a vital stage in the design procedure. There are several powerful methods to select the appropriate input–output pair in linear multivariable plants. However, in the face of plant uncertainties, the input–output pairs can change. Input–output pairing problem, in the presence of uncertainties, and its consequences on the pairing problem have not been widely addressed. In this paper, Hankel interaction index array is used to choose the appropriate input–output pair and a new method is proposed to compute Hankel interaction index array, which reduces the computational load. Also, a theorem will be presented to show the effect of additive uncertainties on input–output pairing of the process. An upper bound on the element variations of Hankel interaction index array of the additive uncertainties in state space framework is given to show the possible change in input–output pairing. Finally, two typical processes are employed to show the main points of the proposed methodology.     

Modeling Connectionist Neuro-Fuzzy network and Applications

The main objective of this paper is to propose a Neuro-Fuzzy network, which can model a system from input–output data by automatically dividing the input–output space and extracting fuzzy if-then rules from numerical data. The structure of the network is simple with input, rule and output layers only. The connections between input and rule layer is used to derive the membership functions of the fuzzified part. Kohonens self-organizing learning algorithm is applied to partition the pattern space. Using this algorithm, similar rules are mapped close by and extraction of if-then rules is made easy. It can also adapt to a number of rules automatically. The proposed network is verified for three benchmark applications. Experimental results show that the adaptive method discussed in this paper not only trains in a few hundred iterations but also provides better performance measures when compared to conventional methods.