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.     

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 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.     

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.     

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.     

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.     

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     

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.     

New algebraic criteria for absolute stability of nonlinear systems

In this paper, new and simple algebraic criteria are derived via elementary proofs to provide easy sufficient conditions for the standard absolute stability problems of nonlinear systems, i.e. Lur'e problems. These criteria are equivalent to the famous graphical circle criteria and Popov criterion. By means of the Sturm theorem and the Euclidean division algorithm, a Routh-Hurwitz-like Sturm criterion is obtained. No graphical technique is needed. Only basic numerical manipulations are involved in the new criteria.     

Further results on input-to-state stability for nonlinear systems with delayed feedbacks

We consider a class of nonlinear control systems for which stabilizing feedbacks and corresponding Lyapunov functions for the closed-loop systems are available. In the presence of feedback delays and actuator errors, we explicitly construct input-to-state stability (ISS) Lyapunov-Krasovskii functionals for the resulting feedback delayed dynamics, in terms of the available Lyapunov functions for the original undelayed dynamics, which establishes that the closed-loop systems are input-to-state stable (ISS) with respect to actuator errors. We illustrate our results using a generalized system from identification theory and other examples.     

Global approaches to identifiability testing for linear and nonlinear state space models

Whenever a model is not identifiable, there exist several parameter vectors $\text{θ}$ which yield the same input-output behavior, so that any further interprtation of $\text{θ}$ is questionable. In this paper two methods which can be used to test state-space models for identifiability with a global approach are presented. While the first one is useful for nonlinear and/or time-varying models, the second one is suitable for linear time-invariant models. An appendix provides tools for the solution of he sets of polynomial equations involved in both methods.     

Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using several examples.     

Further results on adaptive state-feedback stabilization for stochastic high-order nonlinear systems

This paper aims to relax the results in Xie and Tian (2009) from the following two aspects: completely removing the power order restriction and largely relaxing the growth conditions of nonlinear functions. By using the backstepping design method and homogeneous domination technique, this paper investigates the problem of adaptive state-feedback stabilization for a class of stochastic high-order nonlinear systems with nonlinear parameterization. The closed-loop system can be proved to be globally stable in probability and the states can be regulated to the origin almost surely. The efficiency of the adaptive state-feedback controller is demonstrated by a simulation example.