Parameter identification in linear distributed parameter systems

Perturbation analysis is used to identify unknown but constant parameters in a linear distributed parameter system. Noisy observations are assumed to be available at a finite number of spatial locations. A numerical example is solved to illustrate the proposed method.     

Output regulation for linear distributed parameter systems

This work extends the geometric theory of output regulation to linear distributed parameter systems with bounded input and output operator, in the case when the reference signal and disturbances are generated by a finite dimensional exogenous system. In particular it is shown that the full state feedback and error feedback regulator problems are solvable, under the standard assumptions of stabilizability and detectability, if and only if a pair of regulator equations is solvable. For linear distributed parameter systems this represents an extension of the geometric theory of output regulation developed in Francis (1997) and Isidori and Byrnes (1990). We also provide simple criteria for solvability of the regulator equations based on the eigenvalues of the exosystem and the system transfer function. Examples are given of periodic tracking, set point control, and disturbance attenuation for parabolic systems and periodic tracking for a damped hyperbolic system     

Reduced-order filters for limited state estimation from measurements containing coloured noise

The design of optimal fixed-configuration filters for estimating, without bias, a linear transformation of the state of a dynamic system in the presence of coloured measurement noise is studied. The performance index employed is the integral of error variance, and this leads to a matrix variational problem in the filter gains. It is demonstrated that the optimal time-varying filter gains may be computed by solving this variational problem using standard conjugate gradient algorithms. Consideration is also given to both algebraic and differential approximation of the optimal filter gains. The problem is formulated as one of ordinary minimization in parameters defining the approximation, and the required gradient expressions are derived for its solution using standard function minimization algorithms. An illustrative numerical example is presented.     

A method of parameter identification for linear distributed parameter systems

A method is presented for estimating unknown parameters in distributed parameter systems. The system considered is assumed to be modeled by a stochastic partial differential equation whose form is known to be linear and unknown parameters are contained in exciting terms. Unknown parameters are assumed to be a set of random constants whose a priori probabilities are known. First, the estimation process of unknown parameters is given by the Bayesian approach in the Markovian framework. The dynamics of the state estimation is also given, which is simultaneously required in the parameter identification scheme. Secondly, the computing procedure is presented, circumventing tedious calculations of the covariance function between the system state and unknown parameters. Finally, two numerical examples are shown, emphasizing that the dynamics of the observation mechanisms adopted plays an important role in both the state estimation and parameter identification.     

Discrete time control of linear distributed parameter systems

The problem of optimum discrete time control for linear distributed parameter systems with quadratic cost function is considered. The system behaviour at discrete instants is experssed in terms of recursive functional expressions involving Green's function matrices. For the case of point controls the technique of dynamic programming is used to derive an expression for feedback control in terms of an auxiliary spatially dependent variable. This variable is shown to satisfy a Ricatti type functional equation with known final value. Using orthogonal series expansion, this equation can be transformed to a recursive algebraic equation in the coefficients of the expansion. A computational comparison of the straight discretization in time and space and the use of Green's function orthogonal expansion for a simple case is included.     

Stability robustness of linear normal distributed parameter systems

This paper considers the stability robustness analysis problem for linear distributed parameter systems containing known perturbation operators multiplied by uncertain parameters. The nominal system operators are assumed to be normal, but allowed to be unbounded. The perturbation operators are confined to some relative bounded set, but may be unbounded also. By using the Lyapunov stability criterion, simple bounds on uncertain parameters are derived to ensure the stability of the perturbed systems. Examples are provided to illustrate the usage of the theoretical results.     

Exponential stability of linear distributed parameter systems with time-varying delays

Exponential stability analysis via the Lyapunov-Krasovskii method is extended to linear time-delay systems in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is admitted to be unknown and time-varying with an a priori given upper bound on the delay. Sufficient delay-dependent conditions for exponential stability are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation and to a wave equation, these conditions are reduced to standard Linear Matrix Inequalities (LMIs). The proposed method is expected to provide effective tools for stability analysis and control synthesis of distributed parameter systems.     

Parameter identification of distributed parameter systems in the presence of measurement noise

A new approach to recursive parameter identification of second-order distributed parameter systems in the presence of measurement noise under unknown initial and boundary conditions is proposed. A two-dimensional low-pass filter is introduced to pre-filter the observed data corrupted by measurement noise. The low-pass filter is designed in the continuous time-space domain and discretized by bilinear transformation. Thus a discrete estimation model of the system under study is easily constructed with filtered input-output data for recursive identification algorithms. The recursive least squares method is still efficient in the presence of low measurement noise if the filter parameters are designed so that the noise effects are reduced sufficiently. Using filtered input data as instrumental variables, a recursive instrumental variable method is also presented to obtain consistent estimates when the digital low-pass filters are not designed successfully or when the output data is corrupted by high measurement noise. Illustrative examples are given to demonstrate the applicability of the proposed methods.     

On Recursive identification of linear and non-linear systems: case of coloured noise

Recursive identification algorithms based on least square estimation procedure have been presented for both linear and a class of nonlinear systems. The linear system is represented by an impulse response model while the non-linear system is a cascade combination of a. linear part and u zero memory non-linear part. It has been shown that even in the presence of coloured noise, for the model selected, ordinary least square estimation gives unbiased estimates of the system parameters. Results are illustrated through suitable numerical examples.     

Stochastic suboptimal pointwise regulation control for linear discrete-time distributed parameter systems

Based on an examination of the concept of open-loop-optimal feedback control of Dreyfus (1964), a suboptimal stochastic pointwise regulation control scheme for linear distributed parameter systems with unknown plant and measurement noise covariances is presented. By enforcing the term coupling the control and estimation costs in the open-loop cost functional to zero, the suboptimal control scheme retains the controller-estimator in a cascade structure. An on-line algorithm is presented to adaptively select the filter parametrized by a scalar gain so as to minimize the equivalent cost functional derived from the open-loop case. Bounds on the filter gain that ensure filter tracking of the state are obtained. The control scheme is efficient and reasonably simple to use. A numerical example is presented.     

Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems

Motion planning and control of a boundary controlled quasilinear parabolic partial differential equation in one spatial variable is considered. The approach relies on a flatness property of the system; namely, that the system solution can be differentially parameterized in terms of a flat output which, in the case considered, is a boundary value. Such a parameterization allows straightforward motion planning and computation of a control law. The approach is based on power series in the spatial variable, and the convergence of these series is ensured by choosing the flat output to be a nonanalytic, smooth function of appropriate Gevrey class.     

Linear filtering for time-varying systems using measurements containing colored noise

The Kalman-Bucy filter for continuous linear dynamic systems assumes all measurements contain "white" noise, i.e. noise with correlation times short compared to times of interest in the system. It is shown here that if correlation times are not short, or if some measurements are free of noise, the optimal filter is a modification of the Kalman-Bucy filter which, in general, contains differentiators as well as integrators. It is also shown for this case that the estimate and its covariance matrix are, in general, discontinuous at the time when measurements are begun. The case of random bias errors in the measurements is shown by example to be a limiting case of colored noise.     

Analysis and identification of linear distributed systems via Chebyshev series

A double Chebyshev series is introduced to approximate functions of two independent variables and then applied to analyse and identify linear distributed systems. The solution for the coefficient matrices can be obtained directly from a Kronecker product formula. In addition, the algorithm for formulating the algebraic equations to estimate unknown parameters is derived, with the method used being algebraic and computer-oriented. Two illustrative examples are given and excellent results are obtained owing to the rapid convergence property of the Chebyshev series     

Hierarchical optimal control for distributed parameter systems via block pulse operator

This paper presents a development of the hierarchical optimal control for distributed parameter systems via block pulse operator. The proposed algorithm enables one to solve optimal control of distributed parameter systems with hierarchical structure and as a result, the resulting solutions are piecewise constant with minimal mean-square error, The algorithm is simple in form and computationally advantageous. An illustrative example shows the applicability of the proposed method in practice.     

Gain of optical distributed sensing in distributed parameter systems

A real time holographic sensing technique is introduced and its advantages arc investigated from the filtering and control point of view. The feature of holographic sensing is its capability to make distributed measurements of the position and velocity of moving objects, such as a vibrating flexible space structure. This study is based upon the distributed parameter models of linear time-invariant systems, particularly including the linear oscillator equations describing the vibration of large flexible space structures. The general conclusion is that application of optical distributed sensors brings gain in the situation where Kalman filtering is necessary for state estimation. In this case, both transient and steady stale filtering error covariance becomes smaller. This in turn results in smaller cost in the LQG problem.     

On sampled-data optimization in distributed parameter systems

A system of parabolic partial differential equations is transformed into ordinary differential equations in a Hilbert space, where the system operator is the infinitesimal generator of a semigroup of operators. A sampled-data problem is then formulated and converted into an equivalent discrete-time problem. The existence and uniqueness of an optimal sampled-data control is proved. The optimal control is given by a linear sampled-states feedback law where the feedback operator is shown to be the bounded seff-adjoint positive semidefinite solution of a Riccati operator difference equation.     

On quadratic optimization in distributed parameter systems

Systems described by parabolic partial differential equations are formulated as ordinary differential equations in a Hilbert space. Quadratic cost criteria are then formulated as inner products on this Hilbert space. Existence of an optimal control is proved both in the case where the system operator is "coercive" and in the case where the system operator is the infinitesimal generator of a semigroup of operators. The optimal control is given by a linear state feedback law in which the feedback operator is shown to be the bounded positive self-adjoint solution of a nonlinear operator equation of the Riccati type.     

Sensors and observers in distributed parameter systems

Luenberger observer theory is extended to distributed parameter systems. This extension is based on the consideration of sensors. For systems with infinite dimensional state spaces, it is possible to construct the state vector asymptotically (or a part of the state vector) by a ‘good’ choice of sensors. We show that the link between detectability and sensor structure may be of some interest in the construction of observers.     

On-line identification of distributed parameter systems

A method is suggested for the identification of distributed parameter systems. The method is suitable for on-line identification and uses non-linear programming to minimize the integral of the weighted error squared over an observation interval. An exponential form for the weighting function is used. The factors which affect the identification are studied, and an optimal identifier is determined.A closed form expression for the estimated parameters is derived in the case of systems with slowly varying parameters. The effect of noise is studied. Experimental results are given to show the applicability of the methods suggested.