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.     

Data driven local coordinates for multivariable linear systems and their application to system identification

In this paper we introduce a new parametrization for state-space systems: data driven local coordinates (DDLC). The parametrization is obtained by restricting the full state-space parametrization, where all matrix entries are considered to be free, to an affine plane containing a given nominal state-space realization. This affine plane is chosen to be perpendicular to the tangent space to the manifold of observationally equivalent state-space systems at the nominal realization. The application of the parametrization to prediction error identification is exemplified. Simulations indicate that the proposed parametrization has numerical advantages as compared to e.g. the more commonly used observable canonical form.     

Canonical forms for the identification of multivariable linear systems

The advantage of using a unique parameterization in a numerical procedure for the identification of a system from operating records has been well established. In this paper several sets of canonical forms are described for state space models of deterministic multivariable linear systems; the members of these sets having therefore the required uniqueness property within the equivalence classes of minimal realizations of the system. In the identification of a stochastic system, it is shown how the problem depends also upon determining a unique factorization of the spectral density matrix of the system, and the sets of canonical forms obtained for the deterministic system are extended to this case.     

Optimal instrumental-variable methods for identification of multivariable linear systems

The identification of multivariable linear systems using instrumental-variable (IV) methods is discussed. The emphasis is on accuracy properties. The IV estimates of multi-input-multi-output system parameters are proved to be asymptotically Gaussian distributed under weak conditions. An explicit expression for the covariance matrix of the parameter estimates is given. It is then shown how this matrix can be optimized by appropriately choosing the IV variant. The optimal accuracy so obtained is for some model structures equal to that corresponding to a prediction error method.     

Structural identification and software package for linear multivariable systems

This paper is concerned with the structural identification of linear multivariable systems and an interactive identification package. The structural identification is done by taking the time-invariant subsystem from the realizations of the input-output relations identified using data of disjoint time intervals, and the statistical hypothesis test is employed to determine the order, where the input-output relation is identified based on the generalized least squares method using the possibly larger model for the plant. The identification package is for the identification of the input-output relation of a linear multivariable system, for the structural identification based on the realization and for data management.     

Determination of the structure of multivariable stochastic linear systems

The problem of determination of the structure of a multivariable linear system from given input/output data is considered, for both deterministic and stochastic cases. A new canonical input/output form is proposed for this study. A systematic procedure, normalized residual technique, for the estimation of the canonical form is presented. A comparison of our result with others is discussed.     

Robust real-time algorithms for identification of linear multivariable time-varying systems

The paper presents a discussion on the problem of the robust real-time identification of linear multivariable time-varying dynamic systems working in a noisy environment. Two methodologically different approaches to the design of such algorithms are presented. The first is based on the one-step estimation, optimal in the sense of the minimal conditional mean-square error, combined with convenient approximations of the underlying error covariance matrix. The second is based on the general formulation of robustified stochastic approximation algorithms, characterized by a suitable non-linear transformation of normalized residuals. Particular algorithms are derived on the basis of step-by-step optimization with respect to the weighting matrix of the algorithm. Monte Carlo simulation results illustrate the discussion, and show the efficiency of the proposed robust algorithms in the presence of large disturbance realizations, the so-called outliers.     

A new on-line identification method for discrete multivariable linear systems

An improved method for on-line identification of discrete data linear multivariable systems based on stability theory is presented. This method makes use of data collected at the present and past instants and exhibits fast convergence.     

On multivariable linear systems

A general result in linear system theory concerning the controllability and observability of multi-input multi-output systems is presented. Specifically, it is shown that any controllable observable linear system can be made controllable from any input and observable from any output using output feedback.     

A parameter adaptive control structure for linear multivariable systems

This paper presents an adaptive control structure which can be used to assign all poles and zeros of a continuous-time linear multivariable system represented by an (m times m) strictly proper transfer matrixT(s), providedT(s)has no right-half plane zeros. The controller parameters can be directly estimated from input-output data. The paper also serves to point out the type of a priori information necessary for multivariable adaptive controller design. This information is a natural extension of that required in the scalar case.     

On-line structured subspace identification with application to switched linear systems

This article is concerned with the identification of switched linear multiple-inputs–multiple-outputs state-space systems in a recursive way. First, a structured subspace identification scheme for linear systems is presented which turns out to have many attractive features. More precisely, it does not require any singular value decomposition but is derived using orthogonal projection techniques; it allows a computationally appealing implementation and it is closely related to input–output models identification. Second, it is shown that this method can be implemented on-line to track both the range space of the extended observability matrix and its dimension and thereby, the system matrices. Third, by making use of an on-line switching times detection strategy, this method is applied to blindly identify switched systems and to label the obtained submodels. Simulation results on noisy data illustrate the abilities and the benefits of the proposed approach.     

Near-decoupling of linear multivariable systems

Available exact-decoupling conditions by constant compensation do not address the important class of ‘nearly decouplable’ systems. For practical systems, parameter perturbations, measurement errors and system-modelling inaccuracies may conceal the system decouplability by constant- or low-order compensation, thus calling for unnecessarily high-order dynamics. In this paper, state-space decoupling conditions formulated as rank-degeneracy conditions are used to detect how close a system is to one that is exactly decouplable. The procedure simultaneously provides the compensators necessary to achieve the indicated level of decoupling. The conditions are extended to cases where it is required to use a fixed constant compensator for decoupling under different sets of perturbations     

Observers for linear multivariable systems with applications

This paper presents an algorithm for the design of asymptotic state estimators (observers) for index-invariant uniformly observable time-varying linear finite-dimensional multivariable systems. The results obtained indicate that asymptotic estimators can be employed in optimally designed regulators provided an increase from the optimal cost is tolerable. It is also shown that any uniformly observable and uniformly controllable plant with index-invariant observability and controllability matrices can be stabilized with an observer.     

Time-scale structure assignment in linear multivariable systems using high-gain feedback

A systematic design method for assigning a required time-scale structure to a given multivariable system is developed. The time-scale structure assignment is carried out using high-gain feedback of either the state or the output. The method developed decomposes a given multivariable system into several single-input single-output subsystems, each of which is designed separately. A standard method of design for single-input single-output systems is developed, and this forms a building block for the multivariable system design. The state feedback design depends only on the stage sequence of the given system, while the output feedback design depends only on the set of integers that specifies the order of infinite zeros. In this way, our design is made robust with respect to certain parameter uncertainties. High-gain parameter(s) can be adjusted on line, and this provides a tool for the fine tuning of the desired performance characteristics of a control system.     

Perturbation of multivariable linear quadratic systems with jumpparameters

Considers the problem of the perturbation of a class of linear quadratic systems where the change from one structure (for the dynamics and costs) to another is governed by a finite-state Markov process. The problem above leads to the analysis of some perturbed linearly coupled sets of Riccati equations. We show that the matrix obtained as the solution of the equations, which determines the optimal value and control, has a Taylor expansion in the perturbation parameter. We compute explicitly the terms of this expansion     

The range error test in the structural identification of linear multivariable systems

Among the approaches that have been proposed in recent years for the identification of discrete-time linear multivariable systems, very efficient procedures are based on the introduction of a set of input-output canonical models and on the separate determination of the process structure (structural identification) and parameters (parametric identification). This paper introduces a new simple test which allows us to perform the structural identification without the drawbacks usually associated with singularity tests. The proposed test gives an a priori prediction of the model performance in order to avoid trial and error procedures in the determination of a suitable structure for the process to be identified.     

An effective direct closed loop identification method for linear multivariable systems with colored noise

Multivariable control systems with colored noise widely exist in the most industrial fields, while the system identification under the closed loop conditions is needed in many cases. In view of the above two situations, it needs to find a convenient and effective method to solve the problem. Firstly, the design of the external input signals ensures the identifiability of closed loop system. Secondly, to make the direct method feasible for closed loop identification, the noise model selected is reasonably flexible and independently parameterized. On this basis, this paper proposes an improved method combining the direct closed loop identification approach with the iterative least squares parameter estimation algorithm, which can be an practical solution to the closed loop identification of multivariable systems with colored noise. The presented algorithm based hierarchical identification principle has a strong anti-jamming capability to effectively deal with colored noise existed in the system. Finally, the illustrative examples are given to demonstrate the effectiveness and accuracy of the proposed algorithm.