Modeling of Distributed Parameter Systems: Overview and Classification

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

Spatiotemporal kernel-local-embedding modeling approach for nonlinear distributed parameter systems

This paper focuses on the problem of boundary control for a distributed parameter system (DPS) under denial-of-service (DoS) attacks. Initially, a DPS model is employed. Considering the incomplete measurement of the DPS's state, a novel boundary observer is then proposed, which only relies on the right boundary state instead of full-domain information to achieve accurate state estimation, significantly reducing the measurement cost. Subsequently, an anti-DoS observer-based boundary controller is designed, which is applied only to the spatial boundary to lower actuator deployment costs while improving robustness to intermittent DoS attacks. In addition, a Lyapunov-Krasovskii functional is introduced, and the design methods for the controller and observer are derived by solving linear matrix inequalities. Finally, the feasibility of the control strategy is verified through an example.

Modeling distributed parameter systems (DPSs) are usually challenging due to their infinite dimension nature and strong nonlinearity. As a result, the commonly used DPS modeling methods often do not represent this kind of DPSs well due to model reduction and its neglect of nonlinear dynamics. Here, a novel spatiotemporal recurrent neural network (SRNN) modeling method was proposed for nonlinear DPSs. Generally, the space neighboring the points in a DPS interact each other by means of energy transfer, also known as spatial dynamics. In this SRNN model, its hidden layer at each time is designed to represent the spatial dynamics using a bidirectional RNN (BRNN). The BRNN has the ability to represent this complex interaction since its neighboring hidden layers are used to represent these adjacent spatial points and using a forward step and a backward step represents the interaction between neighboring hidden layers. Then, with the combination of all hidden layers of the SRNN over time, the temporal dynamics of the snapshots is exhibited and represented. In this way, this SRNN integrates the spatial/temporal dynamics together and is without requirement of model reduction. A solving approach is then proposed to find its solution, and a convergence analysis further proves that the proposed method can effectively reconstruct the nonlinear spatiotemporal dynamics of the nonlinear DPS. The article not only demonstrate the effectiveness of the proposed method, but also demonstrate its superior modeling performance as compared to several common methods.

Error Compensation-based Time-Space Separation Modeling Method for Complex Distributed Parameter Processes

Physics-reserved spatiotemporal modeling of battery thermal process: Temperature prediction, parameter identification, and heat generation rate estimation

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

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.

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.

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

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.

Techniques of state estimator design for a class of non-linear distributed parameter systems by discrete-time models are presented. Two construction methods of discrete-time models for non-linear stochastic distributed Darameter svstems are presented. One is a method by weighted-average finite-difference approximation and the other is a discrelization method by partial inversion using Green's function. For these discrete-time models, me stability preservation is examined. Based on these sample data discrete-time models, a state filtering algorithm for a non-linear stochastic distributed parameter system is derived by means of the bayesian approach combined with local linearization. The study shows that the discrete-time models of non-linear distributed parameter systems can better be treated in the form of non-linear integral equations, and the state filtering algorithm can be derived from these integral models.

Complex distributed parameter systems (DPSs) are prevalent in numerous industrial processes. However, the nonlinear spatiotemporal dynamics inherent in DPS present significant challenges for accurate modeling. In this paper, an innovative online three-dimensional (3D) fuzzy multi-output support vector regression learning method is proposed for DPS modeling. The proposed method employs spatial fuzzy basis functions from the 3D fuzzy model as kernel functions, enabling direct construction of a comprehensive fuzzy rule base. Parameters C and ε in the 3D fuzzy model adaptively adjust according to data sequence variations, effectively responding to system dynamics. Furthermore, a stochastic gradient descent algorithm has been implemented for real-time updating of learning parameters and bias terms. The proposed method was validated through two typical DPS and an actual rotary hearth furnace industrial system. The experimental results show the effectiveness of the proposed modeling method.