Linear Distributed Parameter Systems Research Articles

Adaptive fault estimation and accommodation for distributed parameter systems with coupled parabolic partial differential equations

This paper presents a novel model-based approach for fault detection, estimation and accommodation in linear distributed parameter systems (DPS) governed by parabolic partial differential equations (PDEs), prone to sensor and actuator faults. Unlike traditional methods, our approach utilises a filter-based observer directly employing the PDE representation and relies solely on boundary measurements, eliminating the need for extensive sensor arrays. By generating fault detection residuals from a comparison of measured and observer outputs, our method offers efficient fault detection. Innovative parameter update laws facilitate accurate estimation of fault parameters, enabling precise adjustment of the nominal controller to mitigate fault effects. Rigorous Lyapunov analysis ensures the robustness of our approach. Additionally, we derive a formula for estimating time-to-accommodation (TTA), providing valuable insights into fault resolution timelines. Simulations on a linearised diffusion process validate the effectiveness of our scheme, highlighting its superiority over existing approaches.

Constrained Receding Horizon Output Estimation of Linear Distributed Parameter Systems

We address constrained output estimation of discrete-time linear distributed parameter systems in the presence of plant and measurement disturbances. Sufficient conditions are proposed for the strong stability of the proposed moving horizon estimator. We further show that the discrete-time estimation results can be linked to continuous-time infinite-dimensional systems described by partial differential equations (PDEs) with unbounded disturbance and output operators by using Cayley-Tustin transformation. The theoretical results are illustrated with numerical examples on a one-dimensional (1D) wave equation and a 1D diffusion.

Filter-Based Fault Detection and Isolation in Distributed Parameter Systems Modeled by Parabolic Partial Differential Equations

This paper covers model-based fault detection and isolation for linear and nonlinear distributed parameter systems (DPS). The first part mainly deals with actuator, sensor and state fault detection and isolation for a class of DPS represented by a set of coupled linear partial differential equations (PDE). A filter based observer is designed based on the linear PDE representation using which a detection residual is generated. A fault is detected when the magnitude of the detection residual exceeds a detection threshold. Upon detection, several isolation estimators are designed using filters whose output residuals are compared with predefined isolation thresholds. A fault on a linear DPS is declared to be of certain type if the corresponding isolation estimator output residual is below its isolation threshold while the other fault isolation estimator output residual is above its threshold. Next, the fault location is determined when a state fault is identified. The second part of this paper focuses on fault detection and isolation of nonlinear DPS by using a Luenberger type observer. Here fault isolation framework is introduced to isolate actuator, sensor and state faults with isolability condition by using additional boundary measurements and without filters. Finally, the effectiveness of the proposed fault detection and isolation schemes for both linear and nonlinear DPS are demonstrated through simulation.

Optimal Sensor Placement for Observer Design

This paper is motivated by sensor location problems for infinite dimensional linear distributed parameter systems (DPS) defined by partial differential equations (PDEs). In this setting the term “sensor location” usually refers to a spatial location. However, the problem is also meaningful in finite dimensional models and results in a parameterization of the system output matrix. The problem of “optimal” sensor (and actuator) location has a long history and plays an important role in controller design, system identification and state estimation and, depending on the application, can be formulated in many ways. Any optimal location problem implies that there is a cost function to be minimized and the choice of this cost function places specific requirements on the computational methods needed to solve the problem. In addition, as shown in the papers Yang and Morris [2014, 2015] choices such as “optimizing observability” can lead to ill-posed problems and result in non convergence of suboptimal placements. On the other hand, minimum error variance methods (Kalman filtering) can be shown be well-posed. Efficient computational methods for solving the corresponding Riccati partial differential equation have been developed and are readily available. In the classic paper by Luenberger [1964], he introduced the term “observer” for deterministic state estimation to distinguish it from the Kalman filter. In this paper we focus on state estimators based on the Luenberger observer and suggest some natural cost functions for sensor location. The framework is fairly general and applies to both finite and infinite dimensional systems. Examples are given to illustrate the ideas.

Discrete output regulator design for linear distributed parameter systems

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.

Analysis of the error in an iterative algorithm for asymptotic regulation of linear distributed parameter control systems

Applications of regulator theory are ubiquitous in control theory, encompassing almost all areas of systems and control engineering. Examples include active noise suppression [Banks et al., Decision and Control, Active Noise Control: Piezoceramic Actuators in Fluid/structure Interaction Models, IEEE, Los Alamitos, CA (1991) 2328–2333], design and control of energy efficient buildings [Borggaard et al., Control, Estimation and Optimization of Energy Efficient Buildings. Riverfront, St. Louis, MO (2009) 837–841.] and control of heat exchangers [Aulisa et al., IFAC-PapersOnLine 49 (2016) 104–109.]. Numerous other examples can be found in [Aulisa and Gilliam, A Practical Guide to Geometric Regulation for Distributed Parameter Systems. Chapman and Hall/CRC, Boca Raton (2015).]. In the geometric approach to asymptotic regulation the main object of interest is a pair of operator equations called the regulator equations, whose solution provides a control solving the tracking/disturbance rejection regulation problem. In this paper we present an iterative algorithm, called the β-iteration method, which is based on the geometric methodology, and delivers accurate control laws for approximate asymptotic regulation. This iterative scheme has been successfully applied to a wide range of linear and nonlinear multi-physics examples and in practice only one or two iterations are usually required to deliver sufficiently accurate results. One drawback to these research efforts is that no proof was given of the convergence of the method. This work contains a detailed analysis of the error in the iterative scheme for a large class of linear distributed parameter systems. In particular we show that the iterative errors converge at a geometric rate. We demonstrate our estimates on three control problems in multi-physics applications.

Discrete-Time Kalman Filter Design for Linear Infinite-Dimensional Systems

As the optimal linear filter and estimator, the Kalman filter has been extensively utilized for state estimation and prediction in the realm of lumped parameter systems. However, the dynamics of complex industrial systems often vary in both spatial and temporal domains, which take the forms of partial differential equations (PDEs) and/or delay equations. State estimation for these systems is quite challenging due to the mathematical complexity. This work addresses discrete-time Kalman filter design and realization for linear distributed parameter systems. In particular, the structural- and energy-preserving Crank–Nicolson framework is applied for model time discretization without spatial approximation or model order reduction. In order to ensure the time instance consistency in Kalman filter design, a new discrete model configuration is derived. To verify the feasibility of the proposed design, two widely-used PDEs models are considered, i.e., a pipeline hydraulic model and a 1D boundary damped wave equation.

Observer and filter design for linear transport-reaction systems

The present work extends known finite dimensional Luenberger observer and Kalman filter designs to the realm of linear transport-reaction systems frequently present in chemical engineering practice. A unified modelling framework for distributed parameter systems (DPS) which does not account for any type of spatial approximation or order reduction is developed. The Cayley–Tustin transformation of continuous linear distributed parameter system yields structure and properties preserving discrete distributed parameter models, amenable to observer and filter design developments. Designs presented here explore well known state reconstruction methodologies starting from least square estimation, continuous and discrete Luenberger observers and one-step predictor Kalman filter realization. Simple implementation and realization account for the appealing nature of the discrete system observers and filter designs for linear transport-reaction systems. The simulation scenarios cover the majority of representative examples found in the common engineering process control practice.

Iterative Learning Control for a Class of Fractional Order Distributed Parameter Systems

AbstractThis paper concerns a second‐order P‐type iterative learning control (ILC) scheme for a class of fractional order linear distributed parameter systems. First, by analyzing of the control and learning processes, a discrete system for P‐type ILC is established and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by using generalized Gronwall inequality, which is less conservative than the existing one. By incorporating the convergent condition obtained into the original system, the ILC scheme is derived. Finally, the validity of the proposed method is verified by a numerical example.

Necessary condition of linear distributed parameter systems with exact controllability

In this paper, we studied the necessary condition of distributed parameter system with exact controllability. Let Φ0t be the control mapping. We introduce new a class of control operators that is called the I-class control which satisfy R(Φ0t)∩R(T(t)) is closed set for t>0. If the system is exactly controllable in finite time τ, then the semigroup T(t) must have closed range. In particular, if the generator of T(t) has compact resolvent, its spectra distribute in a strip parallel to imaginary axis. As an application we assert that the systems associated with the immediately norm-continuous semigroups are never exact controllable for zero-class or I-class controls.

ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

This paper presents a second order P-type iterative learning control (ILC) scheme with initial state learning for a class of fractional order linear distributed parameter systems. First, by analyzing the control and learning processes, a discrete system for P-type ILC is established, and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by introducing a new norm and using the generalized Gronwall inequality, which is less conservative than the existing one. Finally, the validity of the proposed method is verified by a numerical example.

Pointwise exponential stabilization of a linear parabolic PDE system using non-collocated pointwise observation

This paper deals with the problem of exponential stabilization for a linear distributed parameter system (DPS) using pointwise control and non-collocated pointwise observation, where the system is modeled by a parabolic partial differential equation (PDE). The main objective of this paper is to construct an output feedback controller for pointwise exponential stabilization of the linear parabolic PDE system using the non-collocated pointwise observation. The observer-based output feedback control technique is utilized to overcome the design difficulty caused by the non-collocation pointwise observation. We construct a Luenberger-type PDE state observer to exponentially track the state of the PDE system. A collocated pointwise feedback controller is proposed based on the estimated state. A variation of Poincaré – Wirtinger inequality is derived and the spatial domain is decomposed into multiple sub-domains according to the number of the actuators and sensors. By using Lyapunov’s direct method, integration by parts, and the variation of Poincaré – Wirtinger inequality at each sub-domain, sufficient conditions for the existence of the observer-based out feedback controller are developed such that the resulting closed-loop system is exponentially stable, and presented in terms of standard linear matrix inequalities (LMIs). Furthermore, the closed-loop well-posedness analysis is also provided by the C0-semigroup approach. Extensive numerical simulation results are presented to show the performance of the proposed output feedback controller.

Delay-Dependent Exponential Stabilization for Linear Distributed Parameter Systems With Time-Varying Delay

This paper extends the framework of Lyapunov–Krasovskii functional to address the problem of exponential stabilization for a class of linearly distributed parameter systems (DPSs) with continuous differentiable time-varying delay and a spatiotemporal control input, where the system model is described by parabolic partial differential-difference equations (PDdEs) subject to homogeneous Neumann or Dirichlet boundary conditions. By constructing an appropriate Lyapunov–Krasovskii functional candidate and using some inequality techniques (e.g., spatial integral form of Jensen's inequalities and vector-valued Wirtinger's inequalities), some delay-dependent exponential stabilization conditions are derived, and presented in terms of standard linear matrix inequalities (LMIs). These stabilization conditions are applicable to both slow-varying and fast-varying time delay cases. The detailed and rigorous proof of the closed-loop exponential stability is also provided in this paper. Moreover, the main results of this paper are reduced to the constant time delay case and extended to the stochastic time-varying delay case, and also extended to address the problem of exponential stabilization for linear parabolic PDdE systems with a temporal control input. The numerical simulation results of two examples show the effectiveness and merit of the main results.

Transfer function models for distributed-parameter systems with impedance boundary conditions

ABSTRACTA transfer function description is derived for a general class of linear distributed parameter systems dependent on time and one spatial variable. Suitable functional transformations are the Laplace transformation for the time variable and the Sturm– Liouville transformation for the space variable. A practical problem is the determination of the eigenfunctions of the Sturm– Liouville transformation since these depend on the type and the parameters of the boundary conditions. This contribution shows that the design of a transfer function model can be separated from the correct treatment of the boundary conditions. The presented approach exhibits strong parallels to state feedback techniques from control theory. Examples for an electrical transmission line demonstrate how terminations with arbitrary complex impedances can be considered without redesigning the transmission line model.

On approximation and implementation of transformation based feedback laws for distributed parameter systems

This paper presents an approach for the efficient numerical implementation of transformation based state feedback laws for linear distributed parameter systems. The control laws considered may originate from backstepping or flatness-based design methods. They are, therefore, directly based on the underlying distributed parameter systems. In general, they are given as unbounded functionals on the infinite-dimensional state-space. By approximating a carefully chosen bounded part, the implementation of these feedback-operators can be considerably simplified. This is achieved by approximating the state in appropriate finite-dimensional sub-spaces of the state-space. The choice of these sub-spaces as well as the controller implementation are discussed for both, a particular motivating example specific and the general case. For the implementation and validation of the obtained approximated controllers the python-based software toolbox PyInduct is introduced.

Output and error feedback regulator designs for linear infinite-dimensional systems

This manuscript addresses the output regulation problem for linear distributed parameter systems (DPSs) with bounded input and unbounded output operators. We introduce novel methods for the design of the output feedback and error feedback regulators. In the output feedback regulator design, the measurements available for the regulator do not belong to the set of controlled outputs. The proposed output feedback regulator with the injection of the measurement ym(t) and reference yr(t) can realize both the plant and the exosystem states estimation, disturbance rejection and reference signal tracking, simultaneously. Moreover, new design approach provides an alternative choice for seeking the output injection gain in a traditional error feedback regulator design. The regulator parameters are easily configured to solve the output regulation problems, and to ensure the stability of the closed-loop systems. The results are demonstrated via computer simulation in two types of representative systems: the parabolic partial differential equation (PDE) system and the first order hyperbolic PDE system.

Observer design for a nonlinear diffusion system based on the Kirchhoff transformation

This paper deals with the state estimation for a nonlinear diffusion system. An observer that reconstructs the whole state, from the available measurements, is proposed based on an equivalent linear diffusion model obtained using the Kirchhoff tangent transformation. This bijective mapping allows to apply the available and powerful state estimation theory of linear distributed parameter systems and simplifies the observer design. Hence, an observer can be designed for the obtained equivalent linear diffusion system and by using the Kirchhoff transformation, the whole state of the original nonlinear diffusion system is recovered. The observability analysis of the nonlinear diffusion system and the convergence of the proposed observer are also investigated based on the equivalent linear diffusion system. The effectiveness of the proposed observer is shown, through numerical simulation runs, in the case of a heated steel rod by considering both an uniformly distributed and a punctual boundary sensing.

Approximation of linear distributed parameter systems by delay systems

The present work addresses continuous-time approximation of distributed parameter systems governed by linear one-dimensional partial differential equations. While approximation is usually realized by lumped systems, that is finite dimensional systems, we propose to approximate the plant by a time-delay system. Within the graph topology, we prove that, if the plant admits a coprime factorization in the algebra of BIBO-stable systems, any linear distributed parameter plant can be approximated by a time-delay system, governed by coupled differential–difference equations. Considerations on stabilization and state-space realization are carried out. A numerical method for constructive approximation is also proposed and illustrated.

Model-based fault detection, estimation, and prediction for a class of linear distributed parameter systems

This paper addresses a new model-based fault detection, estimation, and prediction scheme for linear distributed parameter systems (DPSs) described by a class of partial differential equations (PDEs). An observer is proposed by using the PDE representation and the detection residual is generated by taking the difference between the observer and the physical system outputs. A fault is detected by comparing the residual to a predefined threshold. Subsequently, the fault function is estimated, and its parameters are tuned via a novel update law. Though state measurements are utilized initially in the parameter update law for the fault function estimation, the output and input filters in the modified observer subsequently relax this requirement. The actuator and sensor fault functions are estimated and the time to failure (TTF) is calculated with output measurements alone. Finally, the performance of detection, estimation and a prediction scheme is evaluated on a heat transfer reactor with sensor and actuator faults.

A State-Space Modeling via the Galerkin Approximation for a Boundary Control System

For linear distributed parameter systems with a finite number of boundary inputs, we propose a framework to implement the method of weighted residuals using candidate trial functions without boundary homogenization. Proposed scheme utilizes inner product matrix, or Grammian, of the trial functions to separate appropriate homogenized basis functions and the other trial functions matching inhomogeneous boundary conditions. The finite-dimensional approximate model by using the proposed scheme is represented in descriptor form and it is proved to be straightforwardly transformed into state space form. Feasibility of the method is illustrated by a brief controller design example using the approximate model of a heat conduction rod with Dirichlet boundary input.