Stable polyhedra in parameter space

A typical uncertainty structure of a characteristic polynomial is P(s)=A(s)Q(s)+B(s) with A(s) and B(s) fixed and Q(s) uncertain. In robust controller design Q(s) may be a controller numerator or denominator polynomial; an example is the PID controller with Q(s)=K_I+K_Ps+K_Ds². In robustness analysis Q(s) may describe a plant uncertainty. For fixed imaginary part of Q(jω), it is shown that Hurwitz stability boundaries in the parameter space of the even part of Q(jω) are hyperplanes and the stability regions are convex polyhedra. A dual result holds for fixed real part of Q(jω). Also σ-stability with the real parts of all roots of P(s) smaller than σ is treated.Under the above conditions, the roots of P(s) can cross the imaginary axis only at a finite number of discrete “singular” frequencies. Each singular frequency generates a hyperplane as stability boundary. An application is robust controller design by simultaneous stabilization of several representatives of A(s) and B(s) by a PID controller. Geometrically, the intersection of convex polygons must be calculated and represented tomographically for a grid on K_P.     

Modeling and parameter identification of microbial bioconversion in fed-batch cultures

In this study, the nonlinear dynamical system of glycerol bio-dissimilation to 1,3-propanediol by Klebsiella pneumoniae in fed-batch cultures is investigated. Because of the abrupt changes of reactant concentrations in fed-batch cultures, the continuous dynamical system previously used is inappropriate to describe the process of fed-batch cultures. In this respect we present a nonlinear impulsive dynamical system of glycerol bioconversion to 1,3-propanediol in fed-batch cultures based on the dynamical system of batch cultures. Then we study some properties of the piecewise continuous solutions to the proposed impulsive system. More important, observing the bigger errors between the experimental results and computational ones obtained from the initial parameters of the dynamical system of batch cultures which was derived from hundreds of practical chemical experiments, we establish a parameter identification model for the impulsive system. Subsequently the existence of the optimal solutions of model is demonstrated by using the compactness of the set of the system solutions and an optimization algorithm is constructed to seek to the optimal parameters of the proposed system by means of the genetic optimization. Finally an illustrative numerical example shows the appropriateness of the nonlinear impulsive system and the validity of optimization algorithm.     

Redesign of adaptive observers for improved parameter identification in nonlinear systems

We propose a method for redesigning adaptive observers for nonlinear systems. The redesign uses an adaptive law that is based on delayed observers. This increases the computational burden, but gives significantly better parameter identification and robustness properties. In particular, given that a special persistency of excitation condition is satisfied, we prove uniform global asymptotic stability and semi-global exponential stability of the origin of the state and parameter estimation error, and give explicit lower bounds on the convergence rate of both the state and parameter estimation error dynamics. For initial conditions with a known upper bound, we prove tunable exponential convergence rate. To illustrate the use of the proposed method, we apply it to estimate the unmeasured flow rate and the uncertain friction parameters in a model of a managed pressure drilling system. The simulation results clearly show the improved performance of the redesigned adaptive observer compared to a traditional design.     

Regularized kernel-based BRDF model inversion method for ill-posed land surface parameter retrieval

In this paper, we consider the direct solution of the kernel-based bidirectional reflectance distribution function (BRDF) models for the retrieval of land surface albedos. This is an ill-posed problem due to nonuniqueness of the solution and the instability induced by error/noise and small singular values of the linearized system or the linear BRDF model. A robust inversion algorithm is critical for the BRDF/albedo retrieval from the limited number of satellite observations. We propose a promising algorithm for resolving this kind of ill-posed problem encountered in BRDF model inversion using remote sensing data.New techniques for robust estimation of BRDF model parameters are needed to cope with the scarcity of the number of observations. We are reminded by Cornelius Lanczos' dictum: “Lack of information cannot be remedied by mathematical trickery.” Thus identifying a priori information or appropriate constraints, and the embedding of the information or constraints into the regularization algorithm, are pivotal elements of a retrieval algorithm. We develop a regularization method, which is called the numerically truncated singular value decomposition (NTSVD). The method is based on the spectrum of the linear driven kernel, and the a priori information/constraint is based on the minimization of the l₂ norm of the parameters vector. The regularization algorithm is tested using field data as well as satellite data. Numerical experiments with a subset of measurements for each site demonstrate the robustness of the algorithm.     

Fault isolation for nonlinear dynamic systems based on parameter intervals

The study of fault detection and isolation for nonlinear dynamic systems has been receiving significant attention. Up to now few literatures pay attention to the speed of fault isolation. However, it is a crucial problem for the design of the fault-tolerant control (FTC) of the nonlinear dynamic systems. In this article a new method of fault isolation for nonlinear dynamic systems is proposed. The method is based on the monotonous characteristic of the prediction error of the observer with respect to singular parameter difference between the system and the observer. The proposed method has the advantage of the methods based on adaptive observers that fits a large kind of nonlinear dynamic systems, while it does not have their disadvantage that take a long time to identify the system parameter: Therefore the fault isolation of this method is quicker. The performance of the method is illustrated by simulation results using a nonlinear dynamic model of an alcoholic fermentation process.     

Advanced control with parameter estimation of batch transesterification reactor

The objective of this work is to enhance the economic performance of a batch transesterification reactor producing biodiesel by implementing advanced, model based control strategies. To achieve this goal, a dynamic model of the batch reactor system is first developed by considering reaction kinetics, mass balances and heat balances. The possible plant-model mismatch due to inaccurate or uncertain model parameter values can adversely affect model based control strategies. Therefore, an evolutionary algorithm to estimate the uncertain parameters is proposed. It is shown that the system is not observable with the available measurements, and hence a closed loop model predictive control cannot be implemented on a real system. Therefore, the productivity of the reactor is increased by first solving an open-loop optimal control problem. The objective function for this purpose optimizes the concentration of biodiesel, the batch time and the heating and cooling rates to the reactor. Subsequently, a closed-loop nonlinear model predictive control strategy is presented in order to take disturbances and model uncertainties into account. The controller, designed with a reduced model, tracks an offline determined set-point reactor temperature trajectory by manipulating the heating and cooling mass flows to the reactor. Several operational scenarios are simulated and the results are discussed in view of a real application. With the proposed optimization and control strategy and no parameter mismatch, a revenue of 2.76 $ min⁻¹ can be achieved from the batch reactor. Even with a minor parameter mismatch, the revenue is still 2.01 $ min⁻¹. While these values are comparable to those reported in the literature, this work also accounts for the cost of energy. Moreover, this approach results in a control strategy that can be implemented on a real system with limited online measurements.     

Output regulation of nonlinear output feedback systems with exponential parameter convergence

This paper revisits the global robust output regulation (GROR) problem of nonlinear output feedback systems with uncertain exosystems by error output feedback control. The problem was conventionally tackled by employing a linear canonical internal model and as a result, suitable adaptive stabilization has to be done for the augmented system to achieve output regulation. Distinguished from that, a novel nonlinear internal model approach is developed in the present study that successfully converts the GROR problem into a robust non-adaptive stabilization problem for the augmented system. The feature of the new approach is two-fold. On one hand, stabilization of augmented system is disentangled from any extra adaptive control law and thus the procedure is simplified. On the other hand, it leads to explicit strict Lyapunov characterization for the closed-loop system and consequently assures exponential parameter convergence.     

Stable inversion of Abel equations: Application to tracking control in DC–DC nonminimum phase boost converters

Stable inversion plays a key role in the solution of the exact tracking control problem in nonminimum phase systems. However, the general methods developed so far for the computation of stable inverses require backwards time numeric integration of the internal dynamics equation, which yields high sensitivity to external disturbances and/or structured uncertainties. This article introduces an iterative technique that provides periodic, closed-form analytic expressions uniformly convergent to the exact periodic solution of a certain class of Abel ODE written in the normal form. The method is then applied to the output voltage tracking of periodic references in DC–DC boost power converters through a state feedback indirect control scheme. The procedure lies on a number of assumptions for which sufficient conditions involving system parameters and reference candidates are derived. It also allows one to attenuate the effect of bounded, piecewise constant load disturbances using dynamic compensation. Simulation results validate the proposed algorithm.     

Stable on-line parameter identification with general knowledge on level of information noise discrete-time case

An on-line parameter identification problem is posed and solved for discrete-time systems with general knowledge on the level of the inherent information noise. The knowledge can be the bound on either the magnitude or the finite-index ^p norm, pε[1, ∞), of the noise. Based on the knowledge, a switching type gradient algorithm (or called gradient algorithm with dead zone) is proposed to estimate the parameters of the system from the available input-output data. In spite of the existence of the noise, this on-line algorithm guarantees that the estimation error is monotonically decreasing, and the parameter estimate is convergent to a steady-state value under a mild condition. Furthermore, the algorithm is stable in the sense that the estimation error will converge to zero as the bound on the noise gradually diminishes.     

Multi-objective parameter estimation via minimal correlation criterion

The paper deals with the problem of parameter estimation using two different sources of information, namely a time series with dynamic data and steady-state data. The new estimator is based on a two-step procedure: first a multi-objective optimization is performed, leading to a set of Pareto-optimal vectors of parameter estimates and, second, a single model is chosen based on the free-run simulation error which is required to be minimally correlated with the model output. The procedure is general in nature and can be applied to any model representation, but for the sake of simplicity, the new procedure is illustrated using NARX polynomial models for which closed formulae for generating the Pareto-set are readily available. Monte Carlo simulation studies suggest that the new estimator, which does not assume any particular noise model, is fairly unbiased even when the conventional least-squares estimator is biased.     

Constrained RHC for LPV systems with bounded rates of parameter variations

This paper studies a stabilization problem of polytopically uncertain linear parameter varying systems with input constraints and bounded rates of parameter variations. In the framework of finite receding horizon control (RHC), a system containing “parameter” uncertainties is modified into a system with “parameter-incremental” uncertainties within each horizon. For the system modified in this manner, a robust RHC is designed by solving an optimization problem at each time instant. Based on the feasibility of the problem and the optimality of its solution, the closed-loop system stability is guaranteed. A numerical example is included to illustrate the validity of the results.     

Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems

This paper studies the parameter identification problem of nonlinear abstract parabolic distributed parameter systems via variational method ^[1]. Based on the fundamental optimal control theory and the transposition method studied in ^[2], the existence of optimal parameter is proved, and the necessary condition for the optimal parameter is established.     

An approach to the selection of optimal sensor locations in distributed parameter systems

This paper presents an approach to the selection of optimal sensor locations in distributed parameter systems, which distinguishes the purposes of state estimation from the purposes of parameter estimation. In the first case, the optimality criterion is based on a measure of independence between the sensor responses, while in the second case, it is based on a measure of independence between the parameter sensitivity functions. The procedure, which is general and can be applied to models with any degree of complexity, is illustrated with the optimal placement of temperature sensors in a catalytic fixed-bed reactor. Some numerical results for the on-line estimation of temperature and concentration profiles as well as for the estimation of unknown model parameters are discussed.     

Nonlinear averaging theorems, and the determination of parameter convergence rates in adaptive control

The paper presents nonlinear averaging theorems for two-time scale systems, where the dynamics of the fast system are allowed to vary with the slow system. The results are applied to the Narendra-Valavani adaptive control algorithm, and estimates of the parameter convergence rates are obtained which do not rely on a linearization of the system around the equilibrium, and therefore are valid in a larger region in the parameter space.     

On the aesthetics of inversion and osculation

The goal of this article is to present an informal introduction and tutorial on certain graphical aspects of inversion and osculation for the non-mathematician and to present artistic renditions using these methods.     

Set membership state and parameter estimation for systems described by nonlinear differential equations

This paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous-time systems, in a bounded-error context. A state estimator based on a prediction-correction approach is given, where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation (IVP for ODE) using interval analysis and high-order Taylor models, while the correction step uses a set inversion technique. The state estimator is extended to solve the parameter estimation problem. An illustrative example is presented for each part.     

On the modular inversion hidden number problem

We give a rigorous deterministic polynomial time algorithm for the modular inversion hidden number problem introduced by D. Boneh, S. Halevi and N.A. Howgrave-Graham in 2001. For our algorithm, we need to be given about 2/3 of the bits of the output, which matches one of the heuristic algorithms of D. Boneh, S. Halevi and N.A. Howgrave-Graham and answers one of their open questions. However their more efficient algorithm that requires only 1/3 of the bits of the output still remains heuristic.     

Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation

In this paper, we present a design procedure of composite nonlinear feedback control for general multivariable systems with actuator saturation. We consider both the state feedback case and the measurement feedback case without imposing any restrictive assumption on the given systems. The composite nonlinear feedback control consists of a linear feedback law and a nonlinear feedback law without any switching element. The linear feedback part is designed to yield a closed-loop system with faster rise time, while at the same time not exceeding the actuator limits for the desired command input levels. The nonlinear feedback law is used to reduce overshoot and undershoot caused by the linear part. As such, a highly desired tracking performance with faster settling time and smaller overshoot can be obtained. The result is illustrated by a numerical example, which shows that the proposed design method yields a very satisfactory performance.