Constrained state-space system identification with application to structural dynamics

Constrained identification of state-space models representing structural dynamic systems is addressed. Based on physical insight, transfer function constraints are formulated in terms of the state-space parametrization. A simple example shows that a method tailored for this application, which utilizes the non-uniqueness of a state-space model, outperforms the classic sequential quadratic programming method in terms of robustness and convergence properties. The method is also successfully applied to real experimental data of a plane frame structure.     

Identification of low-order parameter-varying models for large-scale systems

In this paper we propose a novel procedure for obtaining low-dimensional models of large-scale multi-phase, non-linear, reactive fluid flow systems. Our approach is based on the combination of methods of proper orthogonal decompositions, black-box system identification techniques and non-linear spline based blending of local linear black-box models to create a reduced order linear parameter-varying model. The proposed method, which is of empirical nature, gives computationally very efficient low-order process models for large-scale processes. The proposed method does not need Galerkin type of projections on equation residuals to obtain the reduced order models and the proposed method is of generic nature. The efficiency of the proposed approach is illustrated on a benchmark problem of an industrial glass manufacturing process where the process non-linearity and non-linearity arising due to the corrosion of refractory materials is approximated using a linear parameter varying model. The results show good performance of the proposed framework.     

Identification of composite local linear state-space models using a projected gradient search

An identification method is described to determine a weighted combination of local linear state-space models from input and output data. Normalized radial basis functions are used for the weights, and the system matrices of the local linear models are fully parameterized. By iteratively solving a non-linear optimization problem, the centres and widths of the radial basis functions and the system matrices of the local models are determined. To deal with the non-uniqueness of the fully parameterized state-space system, a projected gradient search algorithm is described. It is pointed out that when the weights depend only on the input, the dynamical gradient calculations in the identification method are stable. When the weights also depend on the output, certain difficulties might arise. The methods are illustrated using several examples that have been studied in the literature before.     

Control of nonlinear systems represented by Galerkin models using adaptation-based linear parameter-varying models

This paper studies the control of nonlinear Galerkin systems, which are an important class of nonlinear systems that arise in reduced-order modeling of infinite-dimensional systems. A novel approach is proposed in which a linear parameter-varying (LPV) model representing the Galerkin model is built, where the parameter variation is dictated by a specially designed adaptation scheme. The controller design is then carried out on the simpler LPV model, instead of dealing directly with the complicated nonlinear Galerkin system. An automatically scheduled H-infinity controller is designed using the LPV model, and it is proven that this controller will indeed achieve the desired stabilization when applied to the nonlinear Galerkin model. The approach is illustrated with an example on cavity flow control, where the design is seen to produce satisfactory results in suppressing unwanted oscillations.     

Stability robustness bounds for linear state-space models with structured uncertainty

In this note, we consider the robust stability analysis problem in linear state-space models. We consider systems with structured uncertainty. Some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived. These bounds are shown to be less conservative than the existing ones.     

New approach for the stability analysis of linear time-varying state-space models

A new approach is proposed to establish stability bounds on the parameters of linear time-varying systems. In particular, stability of the Mathieu equation is investigated and stability boundaries are established. The proposed method is computationally simple to apply and can be calculated easily by computer. Examples are given to illustrate the improvement of the proposed method over that reported by Yedavalli and Kolla (1988).     

Identification of a class of nonlinear state-space models using RPEtechniques

The RPE (recursive prediction error) method in state-space form is developed in the nonlinear systems and extended to include the exact form of a nonlinearity, thus enabling structure preservation for certain classes of nonlinear systems. Both the discrete and the continuous-discrete versions of the algorithm in an innovations model are investigated, and a nonlinear simulation example shows a quite convincing performance of the filter as combined parameter and state estimator     

Finite-time control of linear parameter-varying systems with norm-bounded exogenous disturbance

In this paper, a new approach is presented for finite-time control problems for linear systems subject to time-varying parametric uncertainties and exogenous disturbance. The disturbance is assumed to be time varying and bounded. Sufficient conditions are obtained for the existence of a linear parameter-dependent state feedback gain, which can ensure that the closed-loop system is finite-time bounded (FTB). The conditions can be reduced to feasibility problems involving LMIs. Numerical examples show the validity of the proposed methodology.     

Set theoretic approach to fault-tolerant control of linear parameter-varying systems with sensor reintegration

In this paper, we present a fault-tolerant control scheme for linear parameter-varying systems that utilises multiple sensor switching to compensate for sensor faults. The closed-loop scheme consists of an estimator-based feedback tracking controller and sensor-estimate switching strategy which allows for the reintegration of previously faulty sensors. The switching mechanism tracks the transitions from faulty to healthy behaviour by means of set separation and pre-computed transition times. The sensor-estimate pairings are then reconfigured based on available healthy sensors. Under the proposed scheme, preservation of closed-loop system boundedness is guaranteed for a wide range of sensor fault situations. An example is presented to illustrate the performance of the fault-tolerant control strategy.     

From general state-space to VARMAX models

We propose two new algorithms to go from any state-space model to an output equivalent and invertible Vector AutoRegressive Moving Average model with eXogenous regressors (VARMAX). As the literature shows how to do the inverse transformation, these results imply that both representations, state-space and VARMAX, are equally general and freely interchangeable. These algorithms are useful to solve three practical problems: (i) discussing the identifiability of a state-space model, (ii) performing its diagnostic checking, and (iii) calibrating its parameters so that it realizes, exactly or approximately, a given reduced-form VARMAX. These applications are illustrated by means of practical examples with real data.     

Subspace identification of multivariable linear parameter-varying systems

A subspace identification method is discussed that deals with multivariable linear parameter-varying state-space systems with affine parameter dependence. It is shown that a major problem with subspace methods for this kind of system is the enormous dimension of the data matrices involved. To overcome the curse of dimensionality, we suggest using only the most dominant rows of the data matrices in estimating the model. An efficient selection algorithm is discussed that does not require the formation of the complete data matrices, but processes them row by row.     

Application of optimization methods to helicopter rotor blade design

The minimum weight design of helicopter rotor blades with constraints on multiple coupled flap-lag natural frequencies is studied in this paper. A constraint is also imposed on the minimum value of the autorotational inertia of the blade to ensure sufficient autorotational inertia to autorotate in case of an engine failure. A stress constraint is used to guard against structural failure due to blade centrifugal forces. Design variables include blade taper ratio, dimensions of the box beam located inside the airfoil and magnitudes of the nonstructural weights. The program CAMRAD is used for the blade modal analysis and the program CONMIN for the optimization. In addition, a linear approximation involving Taylor series expansion is used to reduce the analysis effort. The procedure contains a sensitivity analysis which consists of analytical derivatives of the objective function, the autorotational inertia constraint and the stress constraints. A central finite difference scheme is used for the derivatives of the frequency constraints. Optimum designs are obtained for both rectangular and tapered blades. The paper also discusses the effect of adding constraints on higher frequencies and stresses on the optimum designs. b box beam width - c chord - f ₁,f ₃,f ₄ first three lead-lag dominated frequencies (elastic modes) - f ₂,f ₅ first two flapping dominated frequencies (elastic modes) - g constraint function - h box beam height - h(z) box beam height variation along blade span - n number of blades - r _j distance from the root to the center of thej-th segment - t ₁,t ₂,t ₃ box beam wall thicknesses - x, y, z reference axes - A box beam cross-sectional area - AI autorotational inertia - E Young's modulus - F objective function - FS factor of safety - GJ torsional stiffness - I _x ,I _y total principal area moments of inertia about reference axes - L _j length ofj-th segment - M _j total mass ofj-th segment - N total number of blade segments - NDV number of design variables - R blade radius - W total blade weight - W() blade weight as a function of design variable - W _b box beam weight - W _o nonstructural blade weight (weight of skin, honeycomb, etc. along with tuning/lumped weights) - prescribed autorotational inertia - design variable increment - _h taper ratio inz direction - _i i-th design variable - _j mass density of thej-th segment - _j stress inj-th segment - _max maximum allowable stress - blade RPM - r root value - t tip value - L lower bound - U upper bound - ^ approximate value This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States     

Finite-time control of linear parameter-varying systems with norm-bounded exogenous disturbance

In this paper, a new approach is presented for finite-time control problems for linear systems subject to time-varying parametric uncertainties and exogenous disturbance. The disturbance is assumed to be time varying and bounded. Sufficient conditions are obtained for the existence of a linear parameter-dependent state feedback gain, which can ensure that the closed-loop system is finite-time bounded （FTB）. The conditions can be reduced to feasibility problems involving LMIs. Numerical examples show the validity of the proposed methodology.     

Identification of scalar errors-in-variables models with dynamics

The paper considers the task of identifying a causal linear dynamic system excited by stationary Gaussian zero mean noise of unknown spectrum, and given measurements of the system input and output contaminated by independent additive stationary noise signals of unknown spectra. While the solution is normally not unique, finite-dimensional parameterizations of the solution set are given (in some cases involving just one parameter), even though the various spectra may not be rational.     

Iterative algorithm for improved measures of stability robustness for linear state-space models

The problem of robust stability of linear lime-invariant systems in state-space models is considered. An iterative algorithm based on frequency domain approach is proposed which leads to new stability robustness measures. The case of structural perturbations is considered and the new bounds are shown to be a significant improvement over recent ones reported. In addition, it is shown that the directional information on structured perturbations can easily be incorporated in the new robustness criterion. Several illustrative examples are worked out.     

Exponential stabilisation of nonlinear parameter-varying systems with applications to conversion flight control of a tilt rotor aircraft

The exponential stabilisation problem of nonlinear parameter-varying (NPV) systems with input constraints is investigated. Unlike the existing NPV-related works, this paper directly addresses NPV models to conduct the stabilisation problem, without hiding nonlinearities or ignoring the time-varying nature. Existence conditions of a nonlinear time-varying (NTV) controller to render the uniformly exponential stability of an NPV system are given in terms of state-and-parameter-dependent linear matrix inequalities (SPLMIs). Specifically, a new controller structure and a novel Lyapunov functional are adopted such that the exact variation rates of the state and parameters can be incorporated into the SPLMIs. The generalised S-procedure is used to convexify the input constraints such that the resulting closed-loop system satisfies input constraints for any state starting from an admissible set. The derived SPLMIs can then be efficiently solved via sum-of-squares programming. The proposed approach is applied to the conversion flight control of a tilt rotor aircraft.     

Robust stability of state-space models with structureduncertainties

A method for robust eigenvalue location analysis of linear state-space models affected by structured real parametric perturbations is proposed. The approach, based on algebraic matrix properties, deals with state-space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters. A method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed. The method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space. This problem, which can be reduced to finding the real solutions of a system of polynomial equations, simplifies considerably when cases with one or two uncertain parameters are considered. For these cases, necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems     

Controller reduction for linear parameter-varying systems with a priori bounds

In this paper we develop a controller reduction procedure for linear parameter-varying (LPV) systems. The method uses synthesis Riccati inequalities for the normalized robust stabilization problem as a basis for the approximation. The technique provides a priori error bounds which are used to obtain closed-loop stability conditions and performance degradation level. We also generalize the relative model reduction method to LPV systems and give an energy interpretation to the controller reduction procedure. To illustrate the method, a reduced order controller is synthesized and tested on a nonlinear missile model.     

Gain-scheduled L-one control for linear parameter-varying systems with parameter-dependent delays

This paper deals with the problem of gain-scheduled L-one control for linear parameter-varying (LPV) systems with parameter-dependent delays. The attention is focused on the design of a gain-scheduled L-one controller that guarantees being an asymptotically stable closed-loop system and satisfying peak-to-peak performance constraints for LPV systems with respect to all amplitude-bounded input signals. In particular, concentrating on the delay-dependent case, we utilize parameter-dependent Lyapunov functions (PDLF) to establish peak-to-peak performance criteria for the first time where there exists a coupling between a Lyapunov function matrix and system matrices. By introducing a slack matrix, the decoupling for the parameter-dependent time-delay LPV system is realized. In this way, the sufficient conditions for the existence of a gain-scheduled L-one controller are proposed in terms of the Lyapunov stability theory and the linear matrix inequality (LMI) method. Based on approximate basis function and the gridding technique, the corresponding controller design is cast into a feasible solution problem of the finite parameter linear matrix inequalities. A numerical example is given to show the effectiveness of the proposed approach.