LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C₀-semigroup generation property of such an operator is proven and it is shown that the generated C₀-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.     

The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs

Many problems arising in different fields of sciences and engineering can be reduced, by applying some appropriate discretization, either to a system of integrodifferential algebraic equations or to a sequence of such systems. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of integrodifferential algebraic systems of temporal two-point boundary value problems. Two extended inner product spaces W[0, 1] and H[0, 1] are constructed in which the boundary conditions of the systems are satisfied, while two smooth kernel functions R _t (s) and r _t (s) are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.

     

DeMorgan systems on the unit interval

Logical connectives on fuzzy sets arise from those on the unit interval. The basic theory of these connectives is cast in an algebraic spirit with an emphasis on equivalence between the various systems that arise. Special attention is given to DeMorgan systems with strict Archimedean t-norms and strong negations. A typical result is that any DeMorgan system with strict t-norm and strong negation is isomorphic to one whose t-norm is multiplication. © 1996 John Wiley & Sons, Inc.     

Conservative linear systems,unitary colligations and Lax–Phillips scattering: multidimensional generalizations

Connections between conservative linear systems, Lax–Phillips scattering, and operator model theory are well known. A common thread in all the theories is a contractive, analytic, operator-valued function on the unit disc T(z) having a representation of the form T(z)?=?D?+?z C (I???zA)^?1 B, known as the transfer or frequency-response function in the system-theory community, the scattering function in the mathematical physics community, and the characteristic operator function in the operator theory community. Here we consider analogues of this circle of ideas in the more general setting of multidimensional systems/multi-evolution scattering systems/multivariable function-theoretic operator theory. Three particular extensions are discussed; from the point of view of system theory, these involve (1) a multidimensional linear system with transfer function a contractive analytic operator function on the unit polydisc in complex Euclidean space, (2) a non-commutative multidimensional linear system with evolution along a free semigroup and with transfer function equal to a formal power series in non-commuting indeterminants, and (3) an overdetermined multidimensional linear system with transfer function identified as a bundle mapping between two Hermitian vector bundles over an algebraic curve embedded in complex projective space. This survey updates an earlier survey by the first author appearing in 2000.     

Local system equivalence

The important concept of the strict system equivalence of polynomial realisations has been extended to realisations over an arbitrary principal ideal domainR. We use the algebraic concept of the localisation of a ring at a prime ideal to study the local properties, as opposed to the global properties, of realisations over arbitrary principal ideal domains. This allows us to gain a new understanding of strict system equivalence. We show that twoR-realisations of aK-matrix, whereK is the field of fractions of the principal ideal domainR, are strictly system equivalent if and only if they are locally system equivalent at every prime ideal (p) ofR. 1980 Mathematics Subject Classification Number: 93B99     

Lanczos <Emphasis Type="Italic">τ</Emphasis>-method regularization algorithm and its algebraic-programming implementation

An algebraic algorithm is developed for computing an algebraic polynomial y _n of order n ∈ N in computer algebra systems. This polynomial is the optimal approximation of the solution y = y(x), x ∈ [a,b], to a system of linear differential equations with polynomial coefficients and initial conditions at a regular singular zero point of this equation in a space C_{[ a,b ]}^k C_{\left[ {a,b} \right]}^k .     

New results on stability and L∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays

This article addresses the problems of stability and L_∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays for the first time. First, we consider the stability problem of a class of positive linear differential‐algebraic equations with unbounded time‐varying delays. A new method, which is based on the upper bounding of the state vector by a decreasing function, is presented to analyze the stability of the system. Then, by investigating the monotonicity of state trajectory, the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is characterized. It is shown that the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is also independent of the delays and fully determined by the system matrices. Two numerical examples are given to illustrate the obtained results.     

Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burgers’ equation

A method to solve weakly non-linear partial differential equations with Volterra series is presented in the context of single-input systems. The solution x(z,t) is represented as the output of a z-parameterized Volterra system, where z denotes the space variable, but z could also have a different meaning or be a vector. In place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems, the problem turns into solving linear differential equations. This paper introduces the method on an example: a dissipative Burgers'equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments. The kernels are computed analytically in the Laplace domain. As a new result, writing the Volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics. Furthermore, the ability of the Volterra system to treat other signals constitutes an improvement for the sound synthesis. It allows the simulation for any regime, including attacks and transients. Numerical simulations are presented and their validity are discussed.     

System Equivalence for AR-Systems over Rings—with an Application to Delay-Differential Systems

In this paper the notion of autoregressive systems over an integral domain ? is introduced, as a generalization of AR-systems over the rings ℝ[s] and ℝ[s,s ⁻¹]. The interpretation of the dynamics represented by a matrix over ? is fixed by the choice of a module ℳ over ?, consisting of all time-trajectories under consideration. In this setup the problem of system equivalence is studied: when do two different AR-representations characterize the same behavior? This problem is solved using a ring extension of ?, that explicitly depends on the choice of the module ℳ of all time-trajectories. In this way the usual divisibility conditions on the system defining matrices can be recovered. The results apply to the class of delay-differential systems with (in)commensurable delays. In this particular application, the ring extension of ? is characterized explicitly. Date received: May 13, 1998. Date revised: December 22, 1998.     

On the sensitivity of the linear state regulator†

The sensitivity of the optimal response of a linear system with quadratic performance index to changes in the weighting factors in the performance index is determined. This necessitates finding the sensitivity of the feedback gain matrix K to changes in the weighting factors. It is shown that the sensitivities of K can be obtained as the solution of a set of linear matrix differential equations. Further, for time-invariant systems and an infinite time interval, it is seen that the sensitivities of K are found more simply by solving a set of linear algebraic matrix equations.     

Convolution algebra representation of systems described by linear hyperbolic partial differential equations

The representation of the input-output operator in convolution algebra B(σ₀) is obtained for distributed parameter systems described by linear hyperbolic partial differential equations. Three kinds of system are considered depending on the kind of coefficient matrix corresponding to the space variable derivative, and their properties are studied. Necessary and sufficient conditions for stability are obtained in terms of factorization of the transition matrix. The obtained results allow the use of modern algebraic methods for analysis of such systems.     

Parallel Iterative Schemes of Linear Algebra with Application to the Stability Analysis of Solutions of Systems of Linear Differential Equations

Parallel modifications of linear iteration schemes are proposed that are used to solve systems of liner algebraic equations and to achieve the time complexity equal to T = log₂ k · O(log₂ n), where k is the number of iterations of an original scheme and n is the dimension of a system. Such schemes are extended to the case of approximate solution of systems of linear differential equations with constant coefficients. Based on them and using a program, the stability of solutions in the Lyapunov sense is analyzed.     

Stopping criteria for iterative methods:¶applications to PDE's

We show that, when solving a linear system with an iterative method, it is necessary to measure the error in the space in which the residual lies. We present examples of linear systems which emanate from the finite element discretization of elliptic partial differential equations, and we show that, when we measure the residual in H ⁻¹(Ω), we obtain a true evaluation of the error in the solution, whereas the measure of the same residual with an algebraic norm can give misleading information about the convergence. We also state a theorem of functional compatibility that proves the existence of perturbations such that the approximate solution of a PDE is the exact solution of the same PDE perturbed. Received: March 2000 / Accepted: October 2000     

Feedback equivalence for general control systems

We discuss the solution to the problem of local equivalence of control systems with n states and p controls in a neighbourhood of a generic point, under the Lie pseudo-group of local time independent feedback transformations. We have shown earlier that this problem is identical with the problem of simple equivalence of the time optimal variational problem. Here we indicate the way in which this identification may be used to obtain closed loop time critical controls for general systems. We show that the classification of general nonlinear systems depends on the classification of all n−p dimensional affine subspaces of the space of symmetric forms in p variables and that the case of control linear systems depends on the classification of all n−p dimensional affine subspaces of the space of skew forms in p variables. We show that in the latter case the G-structure is the prolongation of one determined by a state space transformation group. We give a complete list of normal forms for control linear systems in the case p=n−1.     

Dissipative control systems synthesis with full state feedback

In this paper we present a general framework for synthesizing state feedback controllers to achieve any desired closed loop dissipative behavior. Special cases include positive real andH _∞ controller synthesis. We show that the solution to this dissipative controller synthesis problem is equivalent to the existence of a solution to a partial differential inequality (nonlinear systems case) or an algebraic Riccati inequality (linear systems case). Stability results can be obtained under appropriate detectability assumptions, and a generalization of the strict bounded real lemma is given. We also present an application of the results to a robust stabilization problem. This work was supported by the Cooperative Research Centre for Robert and Adaptive Systems (CRASys), Australia.     

Optimal control of systems with a unitary semigroup and with colocated control and observation

We solve the quadratic optimal control problem on an infinite time interval for a class of linear systems whose state space is a Hilbert space and whose operator semigroup is unitary. The difficulty is that the systems in this class, having unbounded control and observation operators, may be ill-posed. We show that there is a surprisingly simple solution to the problem (the optimal feedback turns out to be output feedback). Our approach is to use a change of variables which transforms the system into a one which, according to recent research, is known to be conservative. We show that, under a mild assumption, the transfer function of this conservative system is inner, and then it follows that the optimal control of this conservative system is trivial. We give an example with the wave equation on an n-dimensional domain, with Neumann control and Dirichlet observation of the velocity.     

Transfer-function matrix synthesis by matrix generalized inverses

The matrix generalized inverse provides a symbolism for establishing the consistency and finding the solution of a system of linear algebraic equations. This concept is employed to find a control law for a linear time-invariant control system which causes the transfer-function matrix of the resulting system to be a specified function of s.     

State-space approach to nonlinear predictive generalised minimum variance control

A nonlinear predictive generalised minimum variance control algorithm is introduced for the control of nonlinear discrete-time multivariable systems. The plant model is represented by the combination of a very general nonlinear operator and also a linear subsystem which can be open-loop unstable and is represented in state-space model form. The multi-step predictive control cost index to be minimised involves both weighted error and control signal costing terms. The solution for the control law is derived in the time domain using a general operator representation of the process. The controller includes an internal model of the nonlinear process, but because of the assumed structure of the system, the state observer is only required to be linear. In the asymptotic case, where the plant is linear, the controller reduces to a state-space version of the well-known GPC controller.     

Spectral characteristics and eigenvalues computation of the harmonic state operators in continuous-time periodic systems

Spectral characteristics of what we call the harmonic state operators in finite-dimensional linear continuous-time periodic (FDLCP) systems are examined by means of the Fredholm theory for the first time. It is shown that the harmonic state operator is a closed, densely defined Fredholm operator on the Hilbert space l₂, and its spectrum contains only eigenvalues of finite type in any bounded neighborhood of the origin of the complex plane. These spectral characteristics lead us to an asymptotic computation algorithm for the eigenvalues of the harmonic state operator via truncations on its Fredholm regularization. Furthermore, the truncation approach also gives us a necessary and sufficient stability criterion in the FDLCP setting, which only involves the Fourier coefficients of the state matrix. Asymptotic stability of the lossy Mathieu differential equation is investigated to illustrate the results.