New techniques for structural identifiability for large linear and nonlinear compartmental systems

This paper introduces methods which are directed towards large linear and nonlinear systems. The methods are based on digraph properties and decomposition into subsystems (large linear systems), asymptotics and a sensitivity condition (nonlinear systems). Examples are drawn from the literature and they represent actual (nonfabricated) systems.     

New results in global stabilization for stochastic nonlinear systems

This paper presents new results on the robust global stabilization and the gain assignment problems for stochastic nonlinear systems. Three stochastic nonlinear control design schemes are developed. Furthermore, a new stochastic gain assignment method is developed for a class of uncertain interconnected stochastic nonlinear systems. This method can be combined with the nonlinear small-gain theorem to design partial-state feedback controllers for stochastic nonlinear systems. Two numerical examples are given to illustrate the effectiveness of the proposed methodology.     

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.     

State elimination and identifiability of the delay parameter for nonlinear time-delay systems

The identifiability of the delay parameter for nonlinear systems with a single constant time delay is analyzed. We show the existence of input–output equations and relate the identifiability of the delay parameter to their form. Explicit criteria based on rank calculations are formulated. The identifiability of the delay parameter is shown not to be directly related to the well-characterized identifiability/observability of the other system parameters/states.     

Parameter identifiability of nonlinear systems: the role of initial conditions

Identifiability is a fundamental prerequisite for model identification; it concerns uniqueness of the model parameters determined from the input-output data, under ideal conditions of noise-free observations and error-free model structure. In the late 1980s concepts of differential algebra have been introduced in control and system theory. Recently, differential algebra tools have been applied to study the identifiability of dynamic systems described by polynomial equations. These methods all exploit the characteristic set of the differential ideal generated by the polynomials defining the system. In this paper, it will be shown that the identifiability test procedures based on differential algebra may fail for systems which are started at specific initial conditions and that this problem is strictly related to the accessibility of the system from the given initial conditions. In particular, when the system is not accessible from the given initial conditions, the ideal I having as generators the polynomials defining the dynamic system may not correctly describe the manifold of the solution. In this case a new ideal that includes all differential polynomials vanishing at the solution of the dynamic system started from the initial conditions should be calculated. An identifiability test is proposed which works, under certain technical hypothesis, also for systems with specific initial conditions.     

Stability results for nonlinear feedback systems

This paper presents an approach towards deriving sufficient conditions for the stability of nonlinear feedback systems. The central features of the approach are twofold. Firstly, useful stability tests are obtained for the case when the subsystems have nonlinear dynamics; secondly, a unifying set of general stability criteria are given, from which known situations can be treated as special cases and new ones are handled with equal ease. The results are obtained by use of a recently developed theory of dissipative systems.     

New results on robust output regulation of nonlinear systems with a nonlinear exosystem

The global robust output regulation problem for nonlinear plants subject to nonlinear exosystems has been a challenging problem and has not been well addressed. The main difficulty lies in finding a suitable internal model. The existing internal model for handling the nonlinear exosystem is not zero input globally asymptotically stable, and can only guarantee a local solution for the output regulation problem. In this paper, we first propose a new class of internal models, which is guaranteed to exist under the generalized immersion condition. An advantage of this internal model is that it is zero input globally asymptotically stable. This fact will greatly facilitate the global stabilization of the augmented system associated with the given plant and the internal model. Then we will further utilize this class of internal models to solve the global robust output regulation problem for output feedback systems with a nonlinear exosystem. Copyright © 2011 John Wiley & Sons, Ltd.     

New results on explicit adaptive control design for nonlinear systems with polynomial conditions

This paper discusses the adaptive control problem for a class of nonlinear systems with polynomial growing conditions. By introducing a Lyapunov function which carries the quadratic terms and higher‐order terms of the transformed variables and constructing an appropriate dynamic gain, a new control strategy is presented to design an adaptive controller using only one parameter estimation. Compared with the existing results, the presented method avoids the over‐parameterization and significantly simplifies the control design. A modified version of the method which adopts no parameter estimation is also provided. It shows that all the signals of the closed‐loop system are bounded and the solution of the original system converges to zero. A simulation example is given to show the validness of the method.     

New results on global output-feedback stabilization for nonlinear systems with unknown growth rate

In this paper, the global asymptotic stabilization by output feedback is investigated for a class of uncertain nonlinear systems with unmeasured states dependent growth. Compared with the closely related works, the remarkableness of the paper is that either the growth rate is an unknown constant or the dimension of the closed-loop system is significantly reduced, mainly due to the introduction of a distinct dynamic high-gain observer based on a new updating law. Motivated by the related stabilization results, and by skillfully using the methods of universal control and backstepping, we obtain the design scheme to an adaptive output-feedback stabilizing controller to guarantee the global asymptotic stability of the resulting closed-loop system. Additionally, a numerical example is considered to demonstrate the effectiveness of the proposed method.     

New results on robust tracking control for a class of high-order nonlinear time-delay systems

This paper considers the global tracking control problem for a class of nonlinear systems. Compared with the existing results, one significant distinction of this work is that it allows the high-order powers, zero dynamics, time-delay and external disturbances. By constructing a new Lyapunov–Krasovskii (L–K) functional and using the modified adding a power integrator method, we successfully design a non-smooth tracking controller which guarantees that all the signals of the closed-loop system are bounded. The tracking error can be adjusted small enough using the designed parameters. As a practical application, in the simulation, the single-link robot system is studied to show the validness of the strategy.     

Experimental design and identifiability for non-linear systems

The design of experiments for the identification of non-linear systems is discussed. In open-loop operation the optimum input for a completely unknown non-linear system is derived and practical rules to be followed in closed-loop operation are described. It is shown that the use of PRBS inputs to excite non-linear systems can cause loss of identinability. The results are illustrated by simulated examples.     

Blind identifiability of IIR systems

It is shown that, with no prior information, a necessary and sufficient condition for an nth order IIR system to be blindly identifiable is that the over-sampling ratio p?n+1.     

Structural identifiability of dynamical systems

Deterministic identifiability of finite dimensional dynamical systems is analysed using a generalized concept of structural invariants. Results applied to time-invariant bilinear and time-variable linear systems yield algebraic conditions for identifiability which are natural generalizations of the well known identifiability criteria for time-invariant linear systems.     

Moving-horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes

A moving-horizon state estimation problem is addressed for a class of nonlinear discrete-time systems with bounded noises acting on the system and measurement equations. As the statistics of such disturbances and of the initial state are assumed to be unknown, we use a generalized least-squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding-window strategy. For the resulting estimator, the existence of bounding sequences on the estimation error is proved. In the absence of noises, exponential convergence to zero is obtained. Moreover, suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value. The approximate solution can be determined either on-line by directly minimizing the cost function or off-line by using a nonlinear parameterized function. Simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended Kalman filter.     

一类不确定非线性系统自适应输出反馈跟踪控制的新结果

研究了一类不确定非线性系统的自适应输出反馈实际跟踪控制问题. 解决该控制问题的困难主要源于此类系统控制系数不确定, 并具有依赖于不可测状态的增长且其增速是关于输出的多项式函数. 首先, 通过推广现有的K–滤波器, 引入了新的动态高增益K–滤波器, 并基于此构造了状态观测器. 然后, 应用反推技术, 成功的设计了系统的自适应输出反馈跟踪控制器. 主要结果表明, 通过设计参数的适当选择, 所构造的控制器能保证闭环系统的所有状态全局有界, 并且当时间足够大时, 跟踪误差收敛到零点的既定小邻域内.     

On identifiability of linear time-delay systems

Identifiability analysis is developed for linear time-delay systems with delayed states, control inputs, and measured outputs, all with a finite number of lumped delays. These systems are governed by linear functional differential equations with uncertain time-invariant parameters and delays. It is shown that the transfer function of such a system admits the online identification if a sufficiently nonsmooth input signal is applied to the system. Sufficiently nonsmooth signals are constructively defined by imposing different smoothness properties on the control input and the state of the system. The required nonsmoothness property is verified independently of any underlying time-delay system.     

Structural identifiability of linear compartmental systems

In biology and mathematics compartmental systems are frequently used. System identification of systems based on physical laws often involves parameter estimation. Before parameter estimation can take place, we have to examine whether the parameters are structurally identifiable. In this paper tests for the structural identifiability of linear compartmental systems are proposed. The method is based on the similarity transformation approach. New contributions in the theory are the conditions for structural identifiability of structured positive linear systems. In addition, structural identifiability from the Markov parameters is extended to structural identifiability from the input-output data, in which the initial condition is (partially) unknown and nonnegligible. Finally, conditions are presented for structural identifiability of a sampled continuous-time linear dynamic system     

Parameter identifiability for distributed parameter systems of hyperbolic type

In this paper we study questions regarding parameter identifiability for distributed parameter systems of hyperbolic type. The unknown parameters are input distribution functions. We consider the systems with continuous-time input-output data and the systems with discrete-time inputs-output data. For these systems we present the necessary and sufficient conditions for identifiability, in the case of a finite number of sensors. We investigate the relations between the continuous-time input-output systems and the discrete-time input-output systems from the viewpoint of identifiability. Moreover we give the sets of input distribution functions which are N-step identifiable, for the discrete-time input-output systems.     

Two results for adaptive output feedback stabilization of nonlinear systems

The problem of (adaptive) stabilization by means of output feedback of a class of nonlinear systems is addressed and solved. The proposed method relies on the asymptotic reconstruction of a stabilizing state feedback control law, does not require stable zero dynamics nor the construction of a Lyapunov function for the closed loop system, and treats in a unified way unknown parameters and unmeasured states. The applicability of the proposed method is discussed via theoretical examples. Finally, it is shown that the proposed method yields a solution to the problem of output feedback regulation for a DC-to-DC power converter and the efficacy of the resulting controller is verified via experiments.     

New nonlinear residual feedback observer for fault diagnosis in nonlinear systems

The increased complexity of plants and the development of sophisticated control systems have necessitated the parallel development of efficient Fault Detection and Isolation (FDI) systems. This paper discusses a model based technique, viz., observers for detecting and isolating parametric and sensor faults. In this paper, a novel diagonal nonlinear residual feedback observer is proposed which is valid for a certain class of nonlinear systems where, subject to other conditions, the state depends nonlinearly on the fault. A number of typical chemical engineering systems can be represented by models of this form. The structure of the observer ensures that the residuals are diagonally affected by the faults. Conditions for exact decoupling of residuals are presented and convergence of the observer in the presence of step faults is proved using Lyapunov like analysis. Multiple observers and a decision logic module are used for FDI when there are un-monitored faults. Results are presented from numerical simulations of an illustrative example and a typical chemical engineering system: a counter-current heat exchanger.