The paper studies the output observer design problem for a linear infinite-dimensional control plant modelled as an abstract boundary input/output control system. It is known that such models lead to an equivalent state space description with unbounded control (input) and observation (output) operators. For this class of infinite-dimensional systems we use the Cayley transform to approximate the sophisticated infinite-dimensional continuous-time model by a discrete-time infinite-dimensional one with all involved operators bounded. This significantly simplifies mathematical aspects of the observer design procedure. As is well known, the essential feature of the Cayley transform is that it preserves various system theoretic properties of the control system model, which may be useful in analysis. As an illustration, we consider an example of designing an output observer for the one-dimensional heat equation with measured controls (inputs) in the Neumann boundary conditions, measured outputs in the Dirichlet boundary conditions and an unmeasured output at a fixed point within the domain. Numerical simulations of this example show that the interpolated continuous-time signal, obtained from the discrete-time observer, can be successfully used for tracking the continuous-time plant output.
A new, state space, discrete-time, and memory-efficient model of a one-dimensional heat transfer process is proposed. The model is derived directly from a time-continuous, state-space semigroup one. Its discrete version is obtained via a continuous fraction expansion method applied to the solution of the state equation. Fundamental properties of the proposed model, such as decomposition, stability, accuracy and convergence, are also discussed. Results of experiments show that the model yields good accuracy in the sense of the mean square error, and its size is significantly smaller than that of the model employing the well-known power series expansion approximation.
The paper presents a survey of recent results in the area of controllability of second order dynamical systems. Controllability problem for finite and infinite dimensional, linear, semilinear, deterministic and stochastic dynamical systems (with delays and undelayed) is taken into consideration. Different types of controllability are discussed.
A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.
In the paper presented the methodology of investigation of the controllability of an infinite dimensional second order dynamical systems with damping term. Following this aim spectral theory for linear unbounded operators is involved. In the first part of the paper the problem is stated and the methodology of transforming the second order equation to the set of the first order equations is reminded. Next the theorem on transforming considered infinite dimensional dynamical system to infinite series of finite dimensional systems is proved. Finally the theorem on necessary and sufficient conditions of constrained approximate controllability of considered system is formulated and proved.
PL
W ramach pracy przedstawiono metodykę badania sterowalności nieskończenie wymiarowych układów dynamicznych rzędu drugiego z czynnikiem tłumiącym. Do tego celu wykorzystana została spektralna teoria liniowych operatorów nieograniczonych. W pierwszej części pracy został sformułowany problem i przypomniana została metodyka sprowadzenia rozpatrywanego układu drugiego rzędu do układu równań pierwszego rzędu. Następnie udowodniono twierdzenie o sprowadzeniu wyjściowego układu nieskończenie wymiarowego do nieskończonego ciągu układów skończenie wymiarowych. Na koniec zostało sformułowane i udowodnione twierdzenie podające warunki konieczne i wystarczające aproksymacyjnej sterowalności z ograniczeniami rozpatrywanego układu.
In the paper presented the methodology of investigation of the controllability of an infinite dimensional second order dynamical systems with damping term. Following this aim spectral theory for linear unbounded operators is involved. In the first part of the paper the problem is stated and the methodology of transforming the second order equation to the set of the first order equations is reminded. Next the theorem on transforming considered infinite dimensional dynamical system to infinite series of finite dimensional systems is proved. Finally the theorem on necessary and sufficient conditions of constrained approximate controllability of considered system is formulated and proved.
PL
W ramach pracy przedstawiono metodykę badania sterowalności nieskończenie wymiarowych układów dynamicznych rzędu drugiego z czynnikiem tłumiącym. Do tego celu wykorzystana została spektralna teoria liniowych operatorów nieograniczonych. W pierwszej części pracy został sformułowany problem i przypomniana została metodyka sprowadzenia rozpatrywanego układu drugiego rzędu do układu równań pierwszego rzędu. Następnie udowodniono twierdzenie o sprowadzeniu wyjściowego układu nieskończenie wymiarowego do nieskończonego ciągu układów skończenie wymiarowych. Na koniec zostało sformułowane i udowodnione twierdzenie podające warunki konieczne i wystarczające aproksymacyjnej sterowalności z ograniczeniami rozpatrywanego układu.
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This paper presents an analysis of some class of bilinear systems that can be applied to biomedical modelling. It combines models that have been studied separately so far, taking into account both the phenomenon of gene amplification and multidrug chemotherapy in their different aspects. The mathematical description is given by an infinite dimensional state equation with a system matrix whose form allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the other is infinite dimensional and tridiagonal. A methodology of the analysis of such models, based on system decomposition, is presented. An optimal control problem is defined in the l1 space. In order to derive necessary conditions for optimal control, the model description is transformed into an integro-differential form. Finally, biomedical implications of the obtained results are discussed.
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The aim of this investigation is to construct an adaptive observer and an adaptive compensator for a class of infinite-dimensional plants having a known exogenous input and a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and the convergence of the estimator error. For the case of a known periodic exogenous input we design an adaptive compensator which forces the system to converge to a unique periodic solution. We illustrate our approach with a delay example and a diffusion example for which we obtain convincing numerical results.
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