Domination in Controlled and Observed Distributed Parameter Systems

ABSTRACTThis study is an analysis of A Beka textbooks, a leading fundamentalist Christian curriculum in the United States and the ways that it reproduces the concept of Christian privilege and dominance in its discourse. Data is presented that illustrate underlying themes and the implications of the findings are discussed for their contribution to diversity tensions among conservative Christians and liberals. Using a critical lens, the study unveils significant themes in the textbooks that serve to marginalize those living outside its dominant parameters and reinforce preconceived notions of privilege and dominance. Finally, in countering the reproductive process, a reflective approach to the curriculum is urged which facilitates educational critique and a dialogue of inquiry.

We review the popular LQG controller design for DPS (distributed parameter systems) and its shortcomings. In particular, its applicability is essentially restricted to exponentially stabilizable and detectable DPS with finite-rank bounded input and output operators. As an alternative, we propose a new practical control design that is applicable to DPS with unbounded, finite-rank input and output operators, provided that the generator has finitely many unstable eigenvalues and satisfies the spectrum determined growth assumption

In the paper, a novel method for identification of distributed parameter system (DPS) with recurrent trajectory via deterministic learning is proposed. The system dynamics rather than the system parameters or system structure is identified, which is completely different from the existing literature. The DPS, which is often described by partial differential equation (PDE), is first transformed into a set of ordinary differential equations (ODEs) by using a special case of finite difference method, method of lines. Then, the system dynamics at each discrete node is identified by using deterministic learning. Without loss of generality, the DPS dynamics at any spatial point in the spatial domain Ω can also be accurately identified by using the proposed method. It will be more appropriate for DPS identification in many physical systems whose both structure and parameters are unknown. Simulation results have shown the feasibility and efficient of the proposed method.

We propose to use the framework of backpropagation through time (BPTT) to create optimal feedback neurocontrollers for distributed parameter systems (DPS). DPS are systems distributed in space while evolving in time. Unlike the lumped parameter systems, DPS are represented by a set of partial differential equations in the state space. Our neurocontrollers obtained for discretized DPS in the infinite-horizon regulator setting are applicable to a broad set of initial states (an envelope of initial state profiles). We compare our technique and results with another approach to synthesizing optimal DPS neurocontrollers introduced.

The distributed parameter systems (DPS) optimal boundary control problem described by the partial differential equations (PDEs) has been a thorny problem and a focus of some researchers recently. Based on Haar orthogonal wavelets approximate approach, this paper attempts to propose a new approach for a class of DPS boundary control. With the help of Haar wavelets transforms and its operational matrixes, by converting it into that of lumped parameter systems (LPS), the DPS' optimal boundary control problem can be solved. Compared with other orthogonal function approximate methods, the proposed method has advantages of little computation, simple algorithm and high approximate precision. The simulation results also proved that it is an efficient algorithm for DPS with the above merits.

The aim of this paper is twofold: The first objective is to study the degree of dominance of fuzzy choice functions, a notion that generalizes Banerjee’s concept of dominance. The second objective is to use the degree of dominance as a tool for solving multicriteria decision making problems. These types of problems describe concrete economic situations where partial information or human subjectivity appears. The mathematical modelling is done by formulating fuzzy choice problems where criteria are represented by fuzzy available sets of alternatives.

A number of stability results for general systems of linear and nonlinear partial differential equations that describe distributed parameter networks and systems are obtained by means of Liapunov's direct method. Conditions for the asymptotic stability of the null solution of linear time-invariant distributed parameter systems are given. This result is then extended to a class of distributed parameter systems consisting of a linear system and a zero-memory nonlinear elment described by a set of Lurie-like partial differential equations. Bounded input-bounded output stability for the above two classes of systems is established by extension of a result of Goldwyn, Chao, and Chang for lumped parameter systems. Finally, it is shown how Malkin's theorem on the stability under persistent disturbances can be extended to provide a solution to the dynamic range problem for systems described by a general set of nonlinear partial differential equations.

A new idea to control of distributed parameter systems is to induct the boundary control idea into predictive control of distributed parameter systems. Because of complexity of distributed parameter systems, control system design is the difficulty of control theory. In this paper, discrete-time boundary predictive control algorithm of first-order linear modulus-varying distributed parameter systems based on orthogonal wavelets transformation is proposed. First-order linear modulus-varying distributed parameter systems in boundary control is approximated in Haar wavelets transformation. So the predictive control proposition of distributed parameter systems has been transformed into the predictive control issue of lumped parameter systems. The boundary predictive controller is designed for the input sequence returning to the boundary predictive control rule of the original system. Simulation studies of the proposed algorithm, as well as the system robustness under uncertainty such as the parameters perturbations, and a disturb occurring to the system output are showed. The results have verified the control effectiveness of the proposed algorithm.

4.1 Introduction During the past decades, considerable advances have been made in the numerical simulation of controlled distributed parameter systems (DPS). In the opinion of this author, this numerical sophistication has not been matched by the theoretical understanding of the approximation processes involved. The aim of this chapter is to shed a little light on some of the system theoretic properties which determine the suitability of an approximation scheme for control design of DPS. At the same time, a new robust control design is proposed which leads to robust, low-order controllers. It is shown that, at least for the class of exponentially stabilizable and detectable state linear systems with bounded and finite-rank input and output operators, this design always leads to a low-order controller which stabilizes not only the original system but also a large class of perturbations. This robustly stabilizing controller also guarantees bounds on the main performance indices. The class of systems considered in this chapter is that of the exponentially stabilizable and detectable state linear systems Σ(A, B, C) on the Hilbert space Z, where A is the infinitesimal generator of the strongly continuous semigroup T(t) on Z, and the operators B and C are finite-rank and bounded; B ∈ ℒ(ℂm, Z), C ∈ ℒ(Z,ℂk). The basic properties required of a finite-dimensional controller for this system are: (P1) The controller stabilizes Σ(A, B, C). (P2) The controller is robust so that it will have a chance of stabilizing the actual physical plant.

In this manuscript, we address discrete-time state and error feedback output regulator designs for a class of linear distributed parameter systems (DPS) with bounded input and output operators. By utilising the Cayley–Tustin bilinear transform, a linear infinite-dimensional discrete-time system is obtained without model spatial approximation or model order reduction. Based on the internal model principle, discrete state and error feedback regulators are designed. In particular, discrete Sylvester regulator equations are formulated, and their solvability is proved and linked to the continuous counterparts. To ensure the stability of the closed-loop system, the design of stabilising feedback gain and its dual problem of stabilising output injection gain design are provided in the discrete-time setting. Finally, three simulation examples including a first-order hyperbolic partial differential equation model and a 1-D heat equation with considerations of step-like, ramp-like and harmonic exogenous signals are shown to demonstrate the effectiveness of the proposed method.

This chapter provides a systematic overview of the distributed parameter system (DPS) modeling and its classification. Three different problems in DPS modeling are discussed, which includes model reduction for known DPS, parameter estimation for DPS, and system identification for unknown DPS. All approaches are classified into different categories with their limitations and advantages briefly discussed. This overview motivates us to develop new methods for DPS modeling.

Consider a nonlinear distributed parameter system (DPS) described by ẋ (t) = Ax(t)+Bu(t)+B N N(x(t), t) for all t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group of operators 𝕋 on a Hilbert space X, B and B N , defined on Hilbert spaces U and U N , respectively, are admissible control operators for 𝕋 and N : X × [0,∞) ↦ U N is continuous in t and Lipschitz in x, with Lipschitz constant L N independent of t. Thus B and B N can be unbounded as operators from U and U N to X, in which case the nonlinear term B N N(x(t), t) in the DPS is in general not a Lipschitz map from X ×[0,∞) to the state space X. Our goal is to find conditions under which this DPS is exactly controllable in some time τ, which means that for any initial state x(0) ∈ X, we can steer the final state x(τ) of the DPS to any chosen point in X by using an appropriate input function u ∈ L2([0, τ];U). We suppose that there exist linear operators F and F b such that (A, [B B N ], F) and (−A; [B B N ], F b ) are regular triples and A + BF Λ and −A + BF b,Λ are generators of strongly continuous semigroups 𝕋f and 𝕋b on X such that ∥𝕋 t f∥ · ∥𝕋 t b∥ decays to zero exponentially. We prove that if L N is sufficiently small, then the nonlinear DPS is exactly controllable in some time τ > 0. Our proof is constructive and provides a numerical algorithm for approximating the required control signal. We illustrate our approach using a simple example.

Modeling of distributed parameter systems for applications—A synthesized review from time–space separation

In the last decade, the Computational Intelligence tools (CI), including Artificial Neural Networks (ANN) and Fuzzy Systems (FS), applying soft computing, became universal means for many applications. Because of their approximation and learning capabilities, the ANNs have been widely employed to dynamic process modeling, identification, prediction and control, (Boskovic & Narendra, 1995; Haykin, 1999; Bulsari & Palosaari, 1993; Deng & Li, 2003; Deng et al., 2005; Gonzalez-Garcia et al., 1998; Padhi & Balakrishnan, 2003; Padhi et al., 2001; Ray,1989). Many applications have been done for identification and control of biotechnological plants too, (Padhi et al., 2001). Among several possible neural network architectures the ones most widely used are the Feedforward NN (FFNN) and the Recurrent NN (RNN), (Haykin, 1999). The main NN property namely the ability to approximate complex non-linear relationships without prior knowledge of the model structure makes them a very attractive alternative to the classical modeling and control techniques. Also, a great boost has been made in the applied NN-based adaptive control methodology incorporating integral plus state control action in the control law, (Baruch et al., 2004; Baruch & Garrido, 2005; Baruch et al., 2007). The FFNN and the RNN have been applied for Distributed Parameter Systems (DPS) identification and control too. In (Pietil & Koivo, 1996), a RNN is used for system identification and process prediction of a DPS dynamics an adsorption column for wastewater treatment of water contaminated with toxic chemicals. In (Deng & Li, 2003; Deng et al., 2005) a spectral-approximation-based intelligent modeling approach, including NNs for state estimation and system identification, is proposed for the distributed thermal processing of the snap curing oven DPS that is used in semiconductor packaging industry. In (Bulsari & Palosaari, 1993), it is presented a new methodology for the identification of DPS, based on NN architectures, motivated by standard numerical discretization techniques used for the solution of Partial Differential Equation (PDE). In (Padhi & Balakrishnan, 2003), an attempt is made to use the philosophy of the NN adaptivecritic design to the optimal control of distributed parameter systems. In (Padhi et al., 2001) the concept of proper orthogonal decomposition is used for the model reduction of DPS to form a reduced order lumped parameter problem. The optimal control problem is then solved in the time domain, in a state feedback sense, following the philosophy of adaptive critic NNs. The control solution is then mapped back to the spatial domain using the same basis functions. In (Pietil & Koivo, 1996), measurement data of an industrial process are

Software sensor as distributed parameter system for the control of secondary cooling in the continuous casting of steel