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.     

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.     

A note on the stability of habituating process control

For a class of single-output processes with at least two control inputs, this note proposes a simple control design and study its stability. To this end, the process is factorized into a stable and non-minimum-phase factor and a possible unstable minimum-phase factor. By the addition of a (low-pass) filter of suitable dimension, one can obtain a process factorization where the minimum-phase factor has a proper and stable inverse. Once the plant has been factorized in this form, the proposed control synthesis method is composed of two main steps; namely, output regulation based on the non-minimum-phase factor and design of an optimization-based divider to compute the value corresponding to the multiple inputs. In this way, one obtains the control transfer function of each control input in a parallel control architecture. An analysis of the control structure is provided, giving sufficient conditions for the stability of the control loop.     

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).     

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.     

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.     

A noncausal approach for PID control

A new approach for the improvement of the set-point following performances achieved by a PID controller is presented. Basically, it consists of applying a suitable command signal to the closed-loop control system in order to achieve a desired transient response when the process output is required to assume a new value. This command signal is determined by means of a stable input–output procedure for which a closed-form solution is presented, thus making the technique suitable to implement in the industrial context. Simulation and experimental results show that high performances are obtained despite the presence of model uncertainties and, above all, almost independently on the PID tuning. Thus, the PID gains can be selected in order to guarantee good load rejection performances without impairing the set-point transient responses.     

Robust nonlinear feedback–feedforward control of a coupled kinetic Monte Carlo–finite difference simulation

Robust nonlinear feedforward–feedback controllers are designed for a multiscale system that dynamically couples kinetic Monte Carlo (KMC) and finite difference (FD) simulation codes. The coupled codes simulate the copper electrodeposition process for manufacturing on-chip copper interconnects in electronic devices. The control objective is to regulate the current density subject to the condition that the steady-state fluctuation of the overpotential remains bounded within ±0.01 V. The controller designs incorporate a low-order stochastic model that captures the input–output behavior of the coupled KMC–FD code. The controllers achieve the objectives and the closed-loop responses implemented on the low-order model and the coupled KMC–FD code match well within stochastic variations. The nonlinear feedforward control reduces the rise time of the controller response while the feedback control ensures robustness in the presence of model uncertainty.     

Robust control with uncertainty estimation for feedback linearizable systems: application to control of distillation columns

The goal of this paper is to describe a linearizing feedback adaptive control structure which leads to a high quality regulation of the output error in the presence of uncertainties and external disturbances. The controller consists of three elements: a nominal input–output linearizing compensator, a state observer and an uncertainty estimator, which provides the adaptive part of the control structure. In this way, the feedback controller, based on the disturbance observer, compensates for external disturbances and plant uncertainties. The effectiveness of the controller is demonstrated on a distillation column via numerical simulations. ©     

Switching through singularities

Asymptotic tracking is studied for systems which are not regular, that is, the relative degree is not well defined. For these systems, the input–output linearizing control law has singularities. We propose a tracking control law which switches between an approximate tracking law (Hauser et al., IEEE Trans. Automat. Control 37 (3) (1992) 392–398) close to the singularities, and an exact tracking law away from the singularities, and we study the applicability of this law based on the behavior of the system’s zero dynamics at the switching boundary. As in Hauser et al. (IEEE Trans. Automat. Control 37 (3) (1992) 392–398), the ball and beam example is used to motivate the study.     

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.     

Relationship between poles and zeros of input–output and chain-scattering systems

A system is frequently represented by transfer functions in an input–output characterization. However, such a system (under mild assumptions) can also be represented by transfer functions in a port characterization, frequently referred to as a chain-scattering representation. Due to its cascade properties, the chain-scattering representation is used throughout many fields of engineering. This paper studies the relationship between poles and zeros of input–output and chain-scattering representations of the same system.     

Model matching for discrete-time periodic systems with time-varying relative degree and order

This paper extends the well-known solution for the linear time invariant model matching problem to discrete-time periodic systems with time-varying relative degree and order. It is shown that a key step to the design of a periodic output feedback controller is to compute the stable inverse of the periodic system. Using input–output equations, this problem is solved and model matching is achieved with system internal stability.     

Intelligent adaptive control of bioreactors

Supervised model-based self-tuning control of fermentation processes is addressed. The diversity, nonlinearity and time-varying nature of these processes make their control a challenging task. Conventional linear (PID) controllers with fixed parameters cannot meet the increasing performance requirements over the whole operating range. In the approach pursued in this research, a local linear model is identified at the current working point by using a limited amount input–output data obtained through an identification experiment. A linear controller is then tuned on the basis of this model. To minimize the intervention into the process operation, this tuning procedure is only initiated if the performance of the current controller deteriorates. To this end, a supervisory expert system is designed whose tasks are to monitor the process performance, design an appropriate identification experiment, validate the obtained model and tune the controller. The supervisory system is based on a combination of a state automaton with a rule-based fuzzy inference system. Experimental results have demonstrated the feasibility of this approach.     

Locomotion control of a four-legged robot embedding real-time reasoning in the force distribution

This paper discusses the application of rule-based reasoning to manage in real time the force distribution computation within a locomotion control of quadruped robots. The control uses input–output linearization in the attitude subsystem, and optimal linear control in the overall locomotion system. The force distribution approach provides more adaptability and flexibility to the locomotion control, because the system is capable of fast adaptation to a wide variety of situations. Rules defining the knowledge about how to deal with walk events and feet forces calculation are presented. The rule-based reasoning is made using the system (LAAS).     

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)     

A note on a new method based on the dispersion of weights in data envelopment analysis

In a very recent paper by Bal et al. (Bal, H., Örkcü, H. H., & Çelebioğlu, S. (2008). A new method based on the dispersion of weights in data envelopment analysis. Computers & Industrial Engineering, 54(3), 502–512), a data envelopment analysis (DEA) model which incorporates the coefficients of variations (CVs) of input–output weights was proposed to improve the discrimination power of DEA and balance input–output weights. This note points out that the input and output weights in DEA are of different dimensions and units. The weights with different dimensions and units cannot be simply added together and averaged. In other words, the DEA model with the inclusion of CVs of input–output weights, which was referred to as CVDEA model for short, makes no sense if input and output data are not normalized to eliminate their dimensions and units. This note also illustrates the facts that the CVDEA model can cause significant efficiency changes when a scale transformation is performed for an input or output and may produce multiple local optimal solutions due to its nonlinearity, leading to totally different assessment conclusions. These facts reveal that the CVDEA model suffers from serious drawbacks and its applications for efficiency assessment should be very cautious.     

Piecewise regression for fuzzy input–output data with automatic change-point detection by quadratic programming

To handle the large variation issues in fuzzy input–output data, the proposed quadratic programming (QP) method uses a piecewise approach to simultaneously generate the possibility and necessity models, as well as the change-points. According to Tanaka and Lee [H. Tanaka, H. Lee, Interval regression analysis by quadratic programming approach, IEEE Transactions on Fuzzy Systems 6 (1998) 473–481], the QP approach gives more diversely spread coefficients than linear programming (LP) does. However, their approach only deals with crisp input and fuzzy output data. Moreover, their method is weak in handling fluctuating data. So far, no method has been developed to cope with the large variation problems in fuzzy input–output data. Hence, we propose a piecewise regression for fuzzy input–output data with a QP approach. There are three advantages in our method. First, the QP technique gives a more diversely spread coefficient than does a linear programming technique. Second, the piecewise approach is used to detect the change-points in the estimated model automatically, and handle the large variation data such as outliers well. Third, the possibility and necessity models with better fitness in data processing are obtained at the same time. Two examples are presented to demonstrate the merits of the proposed method.     

A covariance matching approach for identifying errors-in-variables systems

The errors-in-variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise. Several estimation methods have been proposed for identifying dynamic errors-in-variables models. In this paper a covariance matching approach is proposed to solve the identification problem. It applies for general types of input signals. The method utilizes a small set of covariances of the measured input–output data. This property applies also for some other methods, such as the Frisch scheme and the bias-eliminating least squares method. Algorithmic details for the proposed method are provided. User choices, for example specification of which input–output covariances to utilize, are discussed in some detail. The method is evaluated by using numerical examples, and is shown to have competitive properties as compared to alternative methods.