Nonparametric estimation for normalized coprime factors of a MIMO system

This paper extends to multi-input multi-output (MIMO) systems the results on frequency response estimation for normalized coprime factors (NCF) under a stochastic framework from closed-loop frequency domain experimental data. Under the condition that the covariance matrices are available for the external disturbances and measurement errors, an analytic solution has been obtained for the maximum likelihood estimate (MLE), on the basis of a linear fractional transformation (LFT) representation for all the plant possible normalized right coprime factors (NRCF). It is proved that the estimate can be expressed by a linear combination of a normalized random matrix with all its columns having independent complex normal distributions. Some methods are suggested to reduce the estimation bias when high-quality experimental data can be obtained. A numerical example is included that confirms the theoretical results.     

Identification of normalized coprime factors through constrained curve fitting

This paper deals with identifying finite dimensional normalized right coprime factors (NRCFs) under a stochastic framework from a sequence of their frequency response estimates. A two-step procedure is suggested for the estimation in which two cost functions are sequentially minimized. These cost functions are constructed on the basis of weighted curve fitting and the conditions on NRCFs. A numerical procedure is proposed for computing the estimate. Moreover, it is proved that under some conditions, the estimate converges with probability 1 (w.p.1) to its actual value with increasing the data length, and in general the optimal estimate is asymptotically approximately Gaussian. The efficiency of the proposed method in NRCF identification is illustrated by numerical simulations.     

Robust stabilizability of normalized coprime factors: the infinite-dimensional case

The problem of robustly stabilizing a linear system subject to H_∞-bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite-dimensional systems. This class has possible unbounded, finite-rank input and output operators, which include many delay and distributed systems. The optimal stability margin is expressed in terms of the solutions of the control and filter algebraic Riccati equations.     

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S^−1/2B^*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H_∞ for positive real transfer functions of the form D+S^−1/2B^*(author−A)⁻¹,B.     

A connection between normalized coprime factorizations and linear quadratic regulator theory

Given a transfer matrix described by a minimal state-space triple, a method is given for computing state-space realizations for the numerator and denominator of a normalized, stable, right coprime factorization for the transfer matrix. The method involves the solution of an algebraic Riccati equation. It allows the use of existing computational state-space algorithms in finding normalized stable right coprime factorizations, and avoids explicit calculations of spectral factors.     

Guaranteed cost LQG control for uncertain systems with a normalized coprime factor uncertainty structure

The paper extends the robust minimax LQG control design methodology to stochastic uncertain systems with a general uncertainty structure, which includes normalized coprime factor uncertainty, passive uncertainty and sector norm-bounded uncertainty as special cases. The derivation of the result uses a special parameter-dependent Girsanov measure transformation. The efficacy of the proposed methodology is illustrated using the problem of frequency locking of an optical cavity which occurs in the area of experimental quantum optics.     

Remarks on computing coprime factors for distributed parametersystems

Presents an indirect method for computing coprime factors for a class of linear time-invariant distributed parameter systems. The method overcomes the numerical instability of an existing direct decomposition method. It is shown that coprime factorization can be reduced to constructing simultaneously stabilizing finite-dimensional controllers, for which an efficient iterative procedure is developed     

A counterexample on continuous coprime factors

In Georgiou and Smith (1992), the following question was raised: Consider a linear, shift-invariant system on L₂[0, ∞). Let the graph of the system have Fourier transform (MN)H² (i.e., the system has a transfer function P=N/M) where M, N are elements of C_A={f∈H^∞: f is continuous on the compactified right-half plane}. Is it possible to normalize M and N (i.e., to ensure |M|²+|N|²=1) in C_A? The author shows by example that this is not always possible     

Explicit characterization of decentralized coprime factors

This paper is concerned with parametrization of all decentralized stabilizing controllers. The auxiliary diagonal system, which is defined by the diagonal elements of Bezout factors, plays an important role in the parametrization of decentralized controllers. This paper gives an explicit characterization of the auxiliary system.     

Approximation of unstable infinite-dimensional systems using coprime factors

The effectiveness of comprime factor techniques in L₂ and L_∞ model reduction of unstable linear systems is analysed. Asymptotic estimates are given of the achievable error in the stable and unstable parts of the approximate system, measured in a number of different norms, some involving the associated Hankel operators. The chordal metric is introduced as a measure of approximation and is shown to yield the graph topology. Finally it is deduced that asymptotically optimal L₂ and L_∞ convergence rates can be obtained for a large class of unstable systems.     

Frequency response estimation for NCFs of an MIMO plant from closed-loop time-domain experimental data

Frequency response estimation from closed-loop time-domain experimental data is investigated in this paper for normalized coprime factors (NCF) of a multiple-input-multiple-output (MIMO) plant. Based on a linear fractional transformation (LFT) representation for all the NCFs of plants internally stabilizable by a known controller, this estimation problem is converted into the open-loop nonparametric estimation of an inner transfer function matrix (TFM). An estimate is derived through constrained data-matching. It is proved that when the probing signals are periodic and the NCFs of the auxiliary plant are appropriately selected, the estimate is asymptotically unbiased and with a normal/complex normal distribution. It has been made clear that the estimation bias and the estimation variance are always finite. Computationally tractable procedures are suggested for choosing the desirable auxiliary TFMs.     

Robust control with respect to coprime factors ofinfinite-dimensional positive real systems

It is shown that infinite-dimensional positive real systems can be robustly stabilized with respect to coprime factor perturbations with a robustness margin of at least 1/√2. This result is applied to dissipative colocated systems which arise in PDE models of flexible structures     

Robust stabilization of normalized coprime factor plantdescriptions with H∞-bounded uncertainty

The problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by H _∞ bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system. This problem can be reduced to a Nehari extension problem directly and gives an optimal stability margin. All controllers satisfying a suboptimal stability margin are characterized, and explicit state-space formulas are given     

Normalised coprime factorizations for nonstrictly proper systems

Some earlier results of D.G. Meyer and G.F. Franklin (see ibid., vol.AC-32, pp.227-8, March 1987) are extended, and a procedure is derived for obtaining directly a normalized right-coprime factorization of a system that may not be strictly proper     

Normalized coprime factorizations for linear time-varying systems

In this paper we show that a finite dimensional linear time-varying continuous-time system admits normalized coprime factorizations if and only if it admits a stabilizable and detectable realization. We construct state-space formulas for these factorizations using the stabilizing solutions to standard Riccati differential equations. In the process, we give a simple proof that stabilizability and detectability are sufficient to ensure the existence of such solutions. Based on these results, and on recent advances in the theory of _∞ optimization, we present an algorithm to compute the distance between two systems in the gap metric.     

Performance bounds for coprime factor controller reductions

This paper investigates the performance of coprime factor based controller reduction methods. We show that under some mild conditions closed-loop performance bounds can be derived for observer-based controllers. The results in this paper lay a mathematical foundation for using coprime factor controller reduction methods in certain classes of controller reduction problems.     

Frequency domain sample maximum likelihood estimation for spatially dependent parameter estimation in PDEs

The identification of the spatially dependent parameters in Partial Differential Equations (PDEs) is important in both physics and control problems. A methodology is presented to identify spatially dependent parameters from spatio-temporal measurements. Local non-rational transfer functions are derived based on three local measurements allowing for a local estimate of the parameters. A sample Maximum Likelihood Estimator (SMLE) in the frequency domain is used, because it takes noise properties into account and allows for high accuracy consistent parameter estimation. Confidence bounds on the parameters are estimated based on the noise properties of the measurements. This method is successfully applied to the simulations of a finite difference model of a parabolic PDE with piecewise constant parameters.     

On the balanced truncation and coprime factors reduction of Markovian jump linear systems

The paper deals with the balanced truncation and coprime factors reduction of Markovian jump linear (MJL) systems, which can have mode-varying state, input, and output dimensions. We develop machinery for balancing mean square stable MJL system realizations using generalized Gramians and strict Lyapunov inequalities, and provide an improved a priori upper bound on the error induced in the balanced truncation process. We also generalize the coprime factors reduction method and, in doing so, extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable MJL systems. We provide tools to establish mean square stabilizability and detectability of the considered MJL systems. In addition, a notion of right-coprime factorization of MJL systems and methods to construct such factorizations are given. The error measure in the coprime factors reduction approach, while still norm-based, does not directly capture the mismatch between the nominal system and the reduced-order model, as is the case in the balanced truncation approach where mean square stable models are considered. Instead, the error measure is given in terms of the distance between the coprime factors realizations, and thus has an interpretation in terms of robust feedback stability. The paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach.