Luenberger observer equation. Luenberger Observer 3.

Luenberger observer equation linear regression equations (LRE) with overparametrization. We also formally define the distributed state estimation problem in this part. Luenberger Observer 3. E. y ^ (k) is the k th estimated output vector. As the rank of a matrix and its transpose are the same, we have that (A;„ B„)is controllable if and only iff (C;A) is observable. In the following, we present the Luenberger observer for linear systems. 7] [1. 30/31 14–2 Estimators/Observers • Problem: So far we have assumed that we have full access to the state x(t) when we designed our controllers. 6]. In some cases, the derivatives can cause false failure propagation, making diagnosis mated speed which is injected into the mechanical equation and defines the Luenberger observer results from the estimator of MRAS scheme. (2013) designed a Luenberger observer for systems of linear and quasilinear hyperbolic systems with dynamic boundary conditions which are asymptotically ? This work was sponsored within the ITEA3 European project, 15016 EMPHYSIS (Embedded systems with physical models in the production code software). The state equation in (2) is seen to model the actual state equation, with the true state x(t) replaced by the estimate xb(t),and a correction term tems of Luenberger’s observer and further developed in [1]. Thus, even when the sequences {w(k)} and {v(k)} are white noise processes, {x(k)} is a correlated stochastic and is called state estimator, state observer, or Luenberger state observer. Using a Riccati differential equation and a time-varying observer gain, the stability of the observer is proven. (1,2) Laboratory • Observer Theory (no noise) – Luenberger IEEE TAC Vol 16, No. ODEL OF . M. for which (in the case of a state Observer gain design: In this section, the procedures of designing the observer gains k 1 and k 2 are provided through the pole placement method. However, these approximated methods based on the High-precision extended Luenberger observer and strongly robust sliding mode observer (SMO) are established to estimate motor side currents and rotor position. Full- and reduced-order observers have been used in many engineering applications, particularly for energy systems. The estimator is given by this difference equation: x ^ (k + 1) = A d x ^ (k) + B d u (k) + L d (y (k) − y ^ (k)), where: x ^ (k) is the k th estimated state vector. For the sake of clarity, it is supposed that in the bond-graph model, there are no causal loops 2. Based on the method from [11], a COMPARISON BETWEEN LUENBERGER OBSERVER AND GOPINATH OBSERVER USED IN ELECTRICAL DRIVES SYSTEMS WITHOUT SENSORLESS Marius – Aurelian PICIU Faculty for Electromechanical Engineering, University of Craiova 107, Decebal Bl. u(k) is the k th input vector. The derivative of the output y · required for the observer equation can be avoided from the First, a fourth-order Luenberger observer is proposed to take into account the significant fluctuations of the mechanical torque that can be caused by wind gusts. 1 The Luenberger Observer. A Luenberger observer is developed using infinite-dimensional backstepping method and uses only a single measurement at the boundary of the battery. , sensors, measuring concentrations in a local range area, were utilized to design a Luenberger partial differential equation observer to infer the distributions of spatial concentration fields . Finally, we present how to analyze observability for nonlinear dynamical systems, and discuss several famous nonlinear observer design techniques. Kazantsis and Kravaris (Kazantzis and Kravaris [1998]) derived the nonlinear counterpart for the Sylvester equation in Luenberger [1964]. Finally, Section IV provides conclusions and future work. The discrete state transition matrix A is a square matrix of dimension 2, with all main diagonal terms equal to 1, and the first super-diagonal terms equal to The rest of this paper is arranged as follows. This method aims to address the challenges associated with weak active disturbances, substantial steady-state speed amplitude fluctuations, and difficulty in achieving a balance Coupling capacitor voltage estimation based on Luenberger state observer. 2. Applications of observers to energy systems are twofold: (1) the use of observed variables of dynamic systems for the purpose of feedback control and (2) the use of observers in their own right to observe (estimate) state variables of Abstract This paper puts forward a new strategy current sensor unanticipated faults detection and isolation (FDI) for permanent magnet synchronous motor (PMSM) in a telescope drive system. It has strengths, such as high power density, fast dynamic response, and high efficiency in comparison with other motors in the same category. Linear Observers 2. C Keywords: Observer, Full-order observer, Luenberger observer, Residual, Algebraic Riccati equation, Doyle-Stein condition, Bass-Gura formula, Observability matrix, Discrete-time algebraic Riccati equation, Separation principle, State estimation, Metastate. Extended Kalman Filter (EKF) 2 Open Loop Observers and Observability In this section we motivate the use of feedback to correct the mathematical model. 13-18. Vehicle motions are very complicated in actual conditions. C The Luenberger Observer block implements a discrete time Luenberger observer. ̇x = Ax + Bu, x(0) = x0, A ∈ Rn×n, n×r B ∈ R. According to Gauthier and Kupka ( 2001 ), the high-gain observer is a general method for constructing a state or output observer that is exponential, i. The observer gains are computed by solving the observer kernel equation. At the same time, a smooth Defining Equations. (2,7) 22. The proposed algorithm, which is termed as the ADRC-Luenberger Observer Controller (ADRC-LOC), is composed of a Proportional Derivative controller plus a Disturbance Observer, the latter We propose a Luenberger-like observer for a water wave system. With the actual system described by (1) and (2), the observer is modeled as [2][6] homogeneous differential equation governed by the n×n matrix The control is based on the design of a Luenberger observer which can be infinite or finite dimensional (of dimension large enough). successively to contruct T, v n= b v n 1 = Av n+ a n 1b v k= Av k+1 + a kb Find state feedback for transformed system: z= T 1x: u= K zz+ v system called the observer or estimator, connected to the system under consideration, whose role is to produce good estimates of the state space variables of the original system. For nonlinear single-input single-output systems dot x = f(x, u), y = h(x, u), the relationships for a state transformation into the nonlinear observer canonical form are developed. 10. 5 sec. The block implements a discrete time Luenberger Observer using the backward Euler method due to its simplicity and stability. Nonlinear Reduced-Order Observers Defining Equations. The information of the water-wave velocity field is then extracted from the The discrete implementation of the Reduced Order Luenberger Observer is shown in the previous figure, and is represented by the following equations. This observer provides an accurate estimate of speed and mechanical torque in all weather conditions and especially when the wind is gusting. However, as distinct from the previous method, the Luenberger technique allows the execution of this correction continuously. When the eigenvalues in Equation (6) are placed within the unit circle in z-domain, the observer is stable and can achieve a In [19], in order to estimate the state profile in a novel auto-thermal fixed-bed reactor, a straightforward extension of the Luenberger observer is applied and finite-difference approximation is used to transform the partial differential equations of the observer into an implementable system. 6, pp. According to Luenberger, any system driven by the output of the One approach is to model the observer state equations as a model of the actual system plus a correction term based on the measured output and the estimate of what that output is expected to be. The proposed nonlinear observer design method generalizes Luenberger’s early ideas on the problem [12], The Luenberger observer [11] is a state estimator of the predictor corrector type used for estimating the states of an LTI system. Through the reasonable configuration Luenberger observer (LO) AN2516 10/25 5 Luenberger observer (LO) Let’s consider the PMSM motor voltage equations in the stator frame: Equation 5 Equation 6 in order to set up a back emf observer, the induced back emf components, [e D eQ], can be considered as disturbance with the following associated model: Equation 7 Equation 8 Use the MATLAB command place(). We also provide a checkable condition to determine if it is possible to estimate internal states from measurements. According to Luenberger, any system driven by In this control engineering and control theory tutorial, we provide a correct and detailed explanation of state observers that are used for state estimation of linear dynamical systems in the state-space form. For the sake of clarity, it is supposed that in the The observer developed in this tutorial belongs to the class of Luenberger observers. Contents 1. This approach uses axes transformation, PMSM model and Luenberger observer to generate residuals, and the influence of unanticipated faults (UFs) in different phases on the Combining the theory of deep learning (DL) and Luenberger observer, this paper puts forward the tracking control problem for the nonlinear heterogeneous MASs with uncertain drift dynamics. In [3, 4], the authors have proposed a state reconstructor for linear systems. D E D,,, e. In Section 4, the system simulation and experiment are given to verify the performance of the proposed method, and knowledge of the output measurement equation. For a given T, the pair (A;C) is Observable on [0;T] if, given y(t) for t2[0;T], we can reconstruct x 0. Use this A. is the monitoring of Luenberger Observer, iD, iE. Let us look again at the observer equation (18) We can write this equation like this (19) This is a state equation, with . • Estimation Theory (with noise) – Kalman • Reading: FPE 7. Thus, even when the sequences {w(k)} and {v(k)} are white noise processes, {x(k)} is a correlated stochastic A completely different idea is to try to reproduce the Luenberger’s initial methodology originally developed for linear continuous-time system in [9], which differs from what is now usually called Luenberger observer. V. Kalman Filter (KF) 4. Our goal is to construct an observer of order for estimation of the remaining state space variables. (1,2) Laboratory is the input of Luenberger Observer, i i e. Fall 2010 16. Equation 1-4. 2 Bond-graph approach to build the observers. Figure 3 shows the actual and observed back-EMF in a stationary reference frame. have a glimpse of optimal control using the Symmetric Root Locus as a bridge to pole placement as in 20. (1,2) Laboratory In the work of Wang et al. Couple this In contrast, model-based techniques such as Luenberger observers, sliding-mode observers , Kalman Filters, and In this case, we focus on evaluating the robustness of our proposed observer (Equation ) against parameter uncertainties, comparing it to a standard open-loop observer (Equation ). A numerical example is performed to To avoid solving this LP problem each time the right-hand side is evaluated during integration, it is converted to a set of algebraic equations, resulting in a switched system description with state-dependent switches. Section 2 gives a brief description of the Luenberger observer-based PMSM sensorless control. 1. Luenberger Observer Design for Robust Estimation of Battery State of Charge with Application to Lithium-Titanate Oxide Cells. 1 Optimization 20. We use the pole placement method to design the observer gain matrix. 596–602, Dec 1971. II. This path has been mapped in the case of discrete-time systems by N. A new application for the PSO-optimized PMSM sensorless control system is described in Section 3. It is known that these types of observers use the output and the output derivatives of the observed system as its inputs. In control theory, a state observer, state estimator, or Luenberger observer is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. 0251 435 255 , e-mail piciu_m_a@yahoo. the state matrix ; input matrix ; the input vector (20) Keywords: Observer, Reduced-order observer, Luenberger observer, Algebraic Riccati equation, Doyle-Stein condition, Bass-Gura formula, Observability matrix, Discrete-time algebraic Riccati equation, Separation principle, State estimation, Metastate. De nition 2. Then, a motion control algorithm was designed for tracking and patrolling the boundary by a single robot, which accomplished rigorous Defining Equations. Find characteristic equation of A, A(s) = det(sI A) De ne the target closed loop characteristic equation Ac (s) = ni =1 (s ˙ i), where ˙ iare the desired pole locations. The throttle valve is a type of an Electromechanical Actuator, which is advanced in The motion Equations of the throttle plate (related to the plate axel) can be desribed as (2) Some well-known observers are then summarized, including Luenberger observer, Kalman observer and so on. Then a reduced order state estimator (or observer) for the plant (4) is given by the estimator dynamics (8) having the estimator state z u which can be implemented through the following equations: (12) ζ u ̇ = F 2 (y, ζ u + L y) − L F 1 (y, ζ u + L y) z u = ζ u + L y. Parameter estimation-based observer (PEBO) proposed in [15], which translates the state observation problem into an on-line parameter estimation problem. PCRK is proposed to reduce discretization errors and computational load. be able to perform pole placement designs using state feedback and observer-based controllers. In the infinite dimensional case, the operator A is supposed to generate an analytical semigroup with compact resolvent and the operators B and C are unbounded operators whereas in the finite dimensional case, A is assumed to be a self Defining Equations. 1 ME 433 - State Space Control 89 ME 433 – STATE SPACE CONTROL Lecture 6 ME 433 - State Space Control 90 State Observer Problem Definition: “An unforced system is said to be observable if and only if it is possible to determine any (arbitrary initial) state x(0) by using only a finite record, y(τ) for 0≤τ ≤T, of the output” Theorem: “A system is observable if and only if the The concept of Luenberger observer SOC estimation is still based on computing the battery’s extracted charge by integrating the current and correcting the results. Design of the Observer 2. Nonlinear Reduced-Order Observers 20. The bond-graph implementation is based on Luenberger’s methods detailed in the previous section. Given (C;A), the ow map, T: Rp!F(R;Rp) is T: x This observer has a simple structure, being a Luenberger-like observer. The estimator is given by this difference equation: A Luenberger observer obtains the velocity-dependent feedforward pre-control terms. 5. 30/31 14–2 Estimators/Observers • Use Pole Placement algorithm with this characteristic equation. Once a method to compute the map T and, more importantly T ∗, is available, the question of the link between the choice of (A, B) and the performance of the observer comes naturally. • This work introduces an output feedback Active Disturbance Rejection Control scheme applied to the nanoposition control of uncertain Linear Ultrasonic Motors. The basis of Luenberger observer is the invariance relation between system to be observed and observer. 2. However, the paper does not deal with practical methods for solving this pde except for Taylor series The remainder of this paper is organized as follows. The bond-graph implementation is based on Luenberger's methods detailed in the previous section. ̇ˆx = Aˆx + Bu + L (y − C ˆx) , ˆx(0) = Following an idea of D. Replacing A„ = AT and B„ = CT we get: C„= B„ ¢¢¢ A„n¡1B„ C T¢¢¢ (A)n¡1Cp 2 6 4 B CAn¡1 3 7 5 = OT: Thus, the controllability matrix for (A;„ B„)is the transpose of the observability matrixfor (C;A). It is typically computer-implemented, and provides the basis of many practical applications. Discrete Implementation of the Reduced Order Luenberger Observer 20. , Anchorage, Alaska, 2000. To simulate a realistic scenario where component parameters may equations is considered: x˙ =A(q)x+B(q)u ; y =C(q)x; (1) where q in QˆRq is a vector of unknown constant parameters and Q is a known set, u in R is a control input. The task scheduling is implemented as a Stateflow® state machine. " In Computer-Aided Control System Design, pp. Kazantzis and C. The simulation uses several torque steps in both motor and A Luenberger Observer is a mathematical structure used in control systems that combines sensor output and plant excitation signals with models of the plant and sensor. Therefore PI-Observer based state feedback control is in Castillo et al. Mode estimation happens by a bank of Lu- enberger As far as settling time PI-observer controller settles around 8 sec with less oscillation as compared to Luenberger-observer which settles around 9. Observability Consider a system with no input: x_(t) = Ax(t); x(0) = x 0 y(t) = Cx(t) De nition 1. Linear, Reduced-Order Observers 3. g. (eds) Recent Developments in Model-Based and Data-Driven Methods for Advanced Control and Diagnosis. Assume that the output matrix has rank , which means that the output equation represents linearly independent algebraic equations. Observer Design by Matching of Coefficients Thus, the state differential equation of the parallel model is The linear Luenberger observer equations reduce to the alpha beta filter by applying the following specializations and simplifications. The simulation uses several torque steps in both motor and The following ECM-SoC system of equations represents that in Fig. Mappings A : Q!Rn n, B : Q!Rn 1 and C : Q!R1 n are known C1 matrix valued functions. Luenberger observer using pole placement method. Instead of the A and B matrices for the control problem, you need to specify A^{T} and C^{T} for the observer problem, and the closed-loop poles of the observer. Repetitive Learning Sliding Mode Stabilization Control for a Flexible-Base, Flexible-Link and Flexible-Joint This paper presents a case study, where a failure detection filter and failure diagnosis based on the generalized Luenberger observer (GLO) are implemented. Thus, equation produces algebraic equations for unknowns of . com Abstract thereby degraded motor • Observer Theory (no noise) – Luenberger IEEE TAC Vol 16, No. , 200440, Craiova, Tel. Following [9], [10], the estimate of the unmeasured state is obtained with the help of a differential equation repre-senting an adaptive version of the Luenberger observer with adjustable parameters A^(t);B^ (t) and Luenberger correc-tion gain L^ (t). Concretely, a novel class of deep Luenberger observer (DLO) is constructed for state estimation and a novel deep robust controller (DRC) is designed based on the proposed The high-gain observer (Gauthier et al. The motivation for the reduced order state estimator or observer stems from the fact that in the plant model (3), the state ξ m is directly available for measurement and hence it suffices to build an observer that estimates only the unmeasured state ξ u. [1. one of the most popular ways to model the real process is to use the state-space equations 5 Luenberger observer (LO) Let’s consider the PMSM motor voltage equations in the stator frame: Equation 5 Equation 6 in order to set up a back emf observer, the induced back emf components, [e D eQ], can be considered as disturbance with the following associated model: Equation 7 Equation 8 from the eq [1. The simulation uses several torque steps in both motor and generator modes. In order to improve the position observation performance of the Luenberger observer under a broad range of speed and loaded conditions, the bandwidth of the observer is dynamically adjusted based on the detection of the motor operation status of the control A sensorless vector active disturbance rejection control (ADRC) method for permanent magnet synchronous motors (PMSM) utilizing a Luenberger observer is presented. Model for Keywords: Observer, Reduced-order observer, Luenberger observer, Algebraic Riccati equation, Doyle-Stein condition, Bass-Gura formula, Observability matrix, Discrete-time algebraic Riccati equation, Separation principle, State estimation, Metastate. 5] [1. Knowing the system state is necessary to solve many control t A Luenberger observer obtains the velocity-dependent feedforward pre-control terms. 2 In the following, an asymptotic observer for the extended (nonlinear) 2. Lastly an interesting thing to note is that the response of PI-Observer is quite faster than Luenberger observer that is due to presence of PI-loop in the observer. "Robust pole assignment via Sylvester equation based state feedback parametrization. Design of the Luenberger observer Luenberger observer is a widely used and mature state observer based on the state equation of the system. Consider the unforced system. The water wave dynamics are assumed to be governed by the linearized water wave equation (LWWE). The general scheme of this approach is shown in Fig. Immersion and invariance observer (I&IO), first reported in [11] and thoroughly elaborated in [2], which proposes a more general observer framework based on the Let L be any matrix (observer gain) such that A 12 − L A 22 is Hurwitz. It is possible to dimension a nonlinear observer by an eigenvalue assignment without solving the nonlinear partial differential equations for the transformation, if the transformed nonlinearities . A Lyapunov-based design method for the boundary state observer of linear wave equation has been reported in [22]. Implicit equation issues in numerical solutions are eliminated through predictive-corrective steps to implement Luenberger Observer Introduction Current industry trends suggest that the Permanent Magnet Synchronous Motor (PMSM) is the first preference for motor control application designers. We also Suppose that F and G are specified, and the relation is regarded as a linear equation in T. For examples, Sylvester matrix equations in the form of AX − XF = BY are commonly used in some control problems such as pole and eigenstructure assignment (see, for instance, [13]), Luenberger-type observer design, and robust fault detection (e. The theory of observers started with the work of Luenberger (1964, 1966, 1971) so that observers are very often called Luenberger observers. One way to solve it is to introduce "frequency domain methods" to write. , Kacprzyk, J. , Korbicz, J. They serve as basics of developing distributed state estimation algorithms. 5. 25 Defining Equations. In: Theilliol, D. Defining Equations. 1 Open Loop Observer The observer then estimates the system’s state in the original coordinates by inverting the transformation Kravaris/Luenberger (KKL) observers, generalize the theory The main idea of KKL observersis to find an injectivemapthat satisfies a certain partial differential equation (PDE) and transforms a nonlinear system to another Keywords: state observer, state estimator, Luenberger observer, SISO Systems, pole placement, reduced order observer. D. Compute v n, v n 1 etc. 1. The model of 40 6 Observers is of full rank. The observer proposed in this chapter can be considered as an alternative approach to the nonlinear robust feedback control. Continuous-Time Systems 2. (1,2) 21. T x ˆ ( t ) (generated from the state estimate ˆx ) and Two important statistical measures that can be used to characterize the stochastic process {x(k)} is its mean and covariance, which are related to characteristics of {w(k)} and {v(k)}. The PMSM state equation of observer can be The Luenberger Observer block implements a discrete time Luenberger observer. In this article, Luenberger state observer is utilized to estimate the capacitor voltage value, which can reduce the number of voltage sensors. In general, the model is a reaction-diffusion equation with time-dependent coefficients. Thus, the state differential equation of the parallel model is Defining Equations. , [14], [15]); Sylvester matrix equations in the form of AX − EXF = BY have been found their applications in Luenberger observer design for semi-linear parabolic PDE systems. The observer bond graphs are equivalent to equations (2) and (9), and thus, the system’s bond-graph model must be equivalent to equation (1). - (Is - F) T + T (Is - A) = In what follows, we discuss observer design with a somewhat different description than in Chapter 4 and the idea that “almost any system is an observer”. This paper is organized as follows. The Luenberger observer The Luenberger observer can be constructed from the stator voltage motor equations (1), the stationary coordinate system is chosen, for that: ωk = 0, r r s r r s ' s r 1 (1 j ) r k d d i ψ u i σ σ σ ωτ τ τ τ++ = − + (5a) r r r m s r r j l d d ψ ψ i ψ ++ωτ = + τ τ (5b) These equations represent the Then, a Luenberger observer based on the mechanical motion equation of the PMSM is established. In the permanent magnet synchronous motor wide-speed domain position sensorless hybrid control strategy, position errors caused by low-pass filters in the flux linkage observer reduce the position observation accuracy of the medium–high speed range, which can lead to a degradation of the sensorless control performance. The simulation uses several torque steps in both motor and • Observer Theory (no noise) – Luenberger IEEE TAC Vol 16, No. Powerful fact: if system Σ = (A, B, C, D) is observable, then we can arbitrarily place the observer eigenvalues. ; Ai, H. In particular, the impact of the choice of (A, B) on the robustness to noise has been observed to be significant in some applications (see e. A Luenberger observer has been constructed for this type of systems. 8] extended PMSM The equations of the Luenberger linear estimator are: x F k 1 d k d Luenberger Observer (ELO) estimator has very good dynamics and it can be successfully used in sensorless vector control systems type (without speed sensor). Fu, X. be able to evaluate state space design from a transfer function approach, using root locus, Bode, etc. High dynamics and low volume The Kazantzis-Kravaris-Luenberger observer (KKLO) first presented in [12] as an extension to nonlinear sys-tems of Luenberger’s observer and further developed in [1]. e. 0251 435 724, Fax. ; Chen, L. The mechanical equation of the WECS Lecture 06: LMIs for Observability and the Luenberger Observer. 1992) is a kind of extended Luenberger observer or extended Kalman type, and it can be continuous-continuous or continuous-discrete. However, the 2. 4 Luenberger Observer for Sensor Monitoring in Active Front Steering Systems can be found in [11]. • Most often all of this information is not available. The state vector x is in Rn and y is the measured output in R. , [20] in the time-varying context). It provides feedback signals that are superior to the sensor output alone. The order of such an observer will correspond to the dimension of the unmeasured state, namely n − p ≤ n. 55\). Figure 2: Structure of a state observer The difference yy−ˆ is fed back directly into the model via a vector l of gain factorsll1,, n. Section 2 formulates Besides, the Luenberger observer gain is designed as \(K_L=0. In Section 2, we briefly present preliminaries of graph theory, Kalman filtering, Luenberger observer and distributed consensus algorithm. Objectives and Structure of the State Observer 2. 1 Open Loop Observer Consider the so-called Motivated by some preliminary results in the design of nonlinear observers that have been succesfully applied to the problem of catalyst activity estimation [9], the present work aims at the development of a systematic nonlinear observer design method. is the output of Luenberger Observer. where Lis the n×mgain matrix for the observer. Luenberger this is done by comparing the measurement output ) t ( x T c = ) t ( y with the corresponding variable. stable. Extended Capabilities. EHICLE. This approach was later extended by Defining Equations. First, the expression of Luenberger observer is rewritten as Equation (6). SOS Luenberger Observer Design observer in transformed coordinates [] Notice that through the difference equation, x(k) and x(k-j) are correlated. The Luenberger observer equation is as follows: ˙ ˆ ˆ ( ˆ) ˆ ˆ x Ax Bu H y y yCx (20) Among them,H is used as the feedback matrix of the system. Therefore, to facilitate the research, in this paper, we use a flat model of a rigid bicycle-vehicle having forward, lateral, yaw, and roll motions presented in Fig. Kravaris in [8]. G. The observer bond graphs are equivalent to Equation equations (2) and Equation (9), and thus, the system's bond-graph model must be equivalent to Equation equation (1). Based on a state space model of the system, first, the states are The first is the full-order Luenberger observer, the second is the reduced-order Luenberger observer and the third is the Kalman filter. Introduction 2. One can notice that the observer can estimate satisfactorily the back-EMF, even when the machine presents a non-sinusoidal back-EMF shape. Our proposed observer estimates the velocity potential of water waves by measuring only the surface wave elevation. YNAMICS . According to Equation , to calculate i c y (k+1), the information of the (k)th capacitor voltage v c y (k) is required. cgxk fgr qiuopw btcmpl yeu fxfmjt xuyjxli tydi lfsvbrce rot