Auxiliary model based least squares parameter estimation algorithm for feedback nonlinear systems using the hierarchical identification principle

This paper presents a decomposition based least squares estimation algorithm for a feedback nonlinear system with an output error model for the open-loop part by using the auxiliary model identification idea and the hierarchical identification principle and by decomposing a system into two subsystems. Compared with the auxiliary model based recursive least squares algorithm, the proposed algorithm has a smaller computational burden. The simulation results indicate that the proposed algorithm can estimate the parameters of feedback nonlinear systems effectively.     

Hierarchical gradient- and least squares-based iterative algorithms for input nonlinear output-error systems using the key term separation

This paper considers the parameter identification problems of the input nonlinear output-error (IN-OE) systems, that is the Hammerstein output-error systems. In order to overcome the excessive calculation amount of the over-parameterization method of the IN-OE systems. Through applying the hierarchial identification principle and decomposing the IN-OE system into three subsystems with a smaller number of parameters, we present the key term separation auxiliary model hierarchical gradient-based iterative algorithm and the key term separation auxiliary model hierarchical least squares-based iterative algorithm, which are called the key term separation auxiliary model three-stage gradient-based iterative algorithm and the key term separation auxiliary model three-stage least squares-based iterative algorithm. The comparison of the calculation amount and the simulation analysis indicate that the proposed algorithms are effective.     

Combined state and parameter estimation for a bilinear state space system with moving average noise

This paper considers the identification problem of bilinear systems with measurement noise in the form of the moving average model. In particular, we present an interactive estimation algorithm for unmeasurable states and parameters based on the hierarchical identification principle. For unknown states, we formulate a novel bilinear state observer from input-output measurements using the Kalman filter. Then a bilinear state observer based multi-innovation extended stochastic gradient (BSO-MI-ESG) algorithm is proposed to estimate the unknown system parameters. A linear filter is utilized to improve the parameter estimation accuracy and a filtering based BSO-MI-ESG algorithm is presented using the data filtering technique. In the numerical example, we illustrate the effectiveness of the proposed identification methods.     

Filtering based parameter estimation for observer canonical state space systems with colored noise

This paper surveys the identification of observer canonical state space systems affected by colored noise. By means of the filtering technique, a filtering based recursive generalized extended least squares algorithm is proposed for enhancing the parameter identification accuracy. To ease the computational burden, the filtered regressive model is separated into two fictitious sub-models, and then a filtering based two-stage recursive generalized extended least squares algorithm is developed on the basis of the hierarchical identification. The stochastic martingale theory is applied to analyze the convergence of the proposed algorithms. An experimental example is provided to validate the proposed algorithms.     

Parameter estimation for an input nonlinear state space system with time delay

This paper researches parameter estimation problems for an input nonlinear system with state time-delay. Combining the linear transformation and the property of the shift operator, the system is transformed into a bilinear parameter identification model. A gradient based and a least squares based iterative parameter estimation algorithms are presented for identifying the state time-delay system. The simulation results confirm that the proposed two algorithms are effective and the least squares based iterative algorithm has faster convergence rates than the gradient based iterative algorithm.     

Statistical information based two-layer model predictive control with dynamic economy and control performance for non-Gaussian stochastic process

In this paper, a two-layer model predictive control (MPC) hierarchical architecture of dynamic economic optimization (DEO) and reference tracking (RT) is proposed for non-Gaussian stochastic process in the framework of statistical information. In the upper layer, with state feedback and dynamic economic information, the economically optimal trajectories are estimated by entropy and mean based dynamic economic MPC, which uses the nonlinear dynamic model instead of the steady-state model. These estimated optimal trajectories from the upper layer are then employed as the reference trajectories of the lower layer control system. A survival information potential based MPC algorithm is used to maintain the controlled variables at their reference trajectories in the nonlinear system with non-Gaussian disturbances. The stability condition of closed-loop system dynamics is proved using the statistical linearization method. Finally, a numerical example and a continuous stirred-tank reactor are used to illustrate the merits of the proposed economic optimization and control method.     

Identification of cement rotary kiln using hierarchical wavelet fuzzy inference system

Rotary kiln is the central and the most complex component of cement production process. It is used to convert calcineous raw meal into cement clinkers, which plays a key role in quality and quantity of the final produced cement. This system has complex nonlinear dynamic equations that have not been completely worked out yet. In conventional modeling procedure, a large number of the involved parameters are crossed out and an approximation model is presented instead. Therefore, the performance of the obtained model is very important and an inaccurate model may cause many problems for designing a controller. This study presents hierarchical wavelet TS-type fuzzy inference system (HWFIS) for identification of cement rotary kiln. In the proposed method, wavelet fuzzy inference system (WFIS) with two input variables is used as sub-model in a hierarchical structure and gradient descent (GD) algorithm is chosen for training parameters of antecedent and conclusion parts of sub-models. The results show that the proposed method has higher performance in comparison with the other models. The data collected from Saveh White Cement Company is used in our simulations.     

State space model identification of multirate processes with time-delay using the expectation maximization

This paper presents the problems of state space model identification of multirate processes with unknown time delay. The aim is to identify a multirate state space model to approximate the parameter-varying time-delay system. The identification problems are formulated under the framework of the expectation maximization algorithm. Through introducing two hidden variables, a new expectation maximization algorithm is derived to estimate the unknown model parameters and the time-delays simultaneously. The effectiveness of the proposed algorithm is validated by a simulation example.     

Data filtering-based multi-innovation forgetting gradient algorithms for input nonlinear FIR-MA systems with piecewise-linear characteristics

The piecewise-linear characteristics often appear in the nonlinear systems that operate in different ways in different input regions. This paper studies the identification issue of a class of block-oriented systems with piecewise-linear characteristics. The asymmetric piecewise-linear nonlinearity is expressed as a linear parametric representation through introducing an appropriate switching function, then the identification model of the system is derived by using the key term separation technique. On this model basis, a multi-innovation forgetting gradient algorithm is presented to estimate the unknown parameters. To further enhance the identification accuracy, the filtering identification model of the system is derived by changing the structure of the system without changing the relationship between the input and output. Further, a data filtering-based multi-innovation forgetting gradient algorithm is proposed through the use of the data filtering technique. A simulation example is employed to illustrate that the proposed approaches are effective for parameter estimation and the data filtering-based multi-innovation forgetting gradient algorithm has better estimation performance.     

Recursive parameter estimation algorithm for multivariate output-error systems

In this paper, we consider the parameter estimation issues of a class of multivariate output-error systems. A decomposition based recursive least squares identification method is proposed using the hierarchical identification principle and the auxiliary model idea, and its convergence is analyzed through the stochastic process theory. Compared with the existing results on parameter estimation of multivariate output-error systems, a distinct feature for the proposed algorithm is that such a system is decomposed into several sub-systems with smaller dimensions so that parameters to be identified can be estimated interactively. The analysis shows that the estimation errors converge to zero in mean square under certain conditions. Finally, in order to show the effectiveness of the proposed approach, some numerical simulations are provided.     

Model recovery for multi-input signal-output nonlinear systems based on the compressed sensing recovery theory

This paper considers the parameter and order estimation for multiple-input single-output nonlinear systems. Since the orders of the system are unknown, a high-dimensional identification model and a sparse parameter vector are established to include all the valid inputs and basic parameters. Applying the data filtering technique, the input-output data are filtered and the original identification model with autoregressive noise is changed into the identification model with white noise. Based on the compressed sensing recovery theory, a data filtering-based orthogonal matching pursuit algorithm is presented for estimating the system parameters and the orders. The presented method can obtain highly accurate estimates from a small number of measurements by finding the highest absolute inner product. The simulation results confirm that the proposed algorithm is effective for recovering the model of the multiple-input single-output Hammerstein finite impulse response systems.     

Iterative dynamic linearization and identification of a nonlinear learning controller: A data-driven approach

In this article, a nonlinear iterative learning controller (NILC) is developed using an iterative dynamic linearization (IDL) and a parameter iterative learning identification technique. First, the ideal NILC is transformed into a linear parameterized form by using a controller-oriented compact form IDL (controller-CFIDL) technique. Then an iterative learning identification approach is presented for tuning the parameters of the proposed controller using real-time I/O data. For the sake of analysis, a linear data model of the nonlinear plant is obtained by using the system-oriented IDL technology and a corresponding system parameter identification algorithm is developed in iteration domain. The convergence analysis is provided for the dynamically linearized nonlinear and nonaffine discrete-time system. The results are further extended by using a controller-oriented partial form iterative dynamic linearization (controller-PFIDL) method to gain a higher-order NILC utilizing additional control information from previous iterations. Simulations of two examples show the effectiveness of the proposed methods.     

Discrete-time recurrent high order neural networks for nonlinear identification

This paper focuses on the problem of discrete-time nonlinear system identification via recurrent high order neural networks. It includes the respective stability analysis on the basis of the Lyapunov approach for the NN training algorithm. Applicability of the proposed scheme is illustrated via simulation for a discrete-time nonlinear model of an electric induction motor.     

Hierarchical recursive generalized extended least squares estimation algorithms for a class of nonlinear stochastic systems with colored noise

This paper considers the parameter identification problem of a bilinear state space system with colored noise based on its input-output representation. An input-output representation of a bilinear state-space system is derived for the parameter identification by eliminating the state variables in the model, and a recursive generalized extended least squares algorithm is presented for estimating the parameters of the obtained model. Furthermore, a three-stage recursive generalized extended least squares algorithm is proposed for reducing the computational cost. The validity of the proposed method is evaluated through a numerical example.     

Nonlinear curve fitting-based fast robust MPC algorithm for nonlinear system

In the existing efficient robust model predictive control (ERMPC) algorithms (see e.g. [14,31,32]), through offline optimization and online lookup table calculation, a fixed state feedback control law or a linear interpolated control law is applied to a system when the system state lies between two adjacent polyhedrons, which undoubtedly will result in conservativeness of the controller. Faced with this issue, an improved ERMPC algorithm is proposed in this paper, which considers the nonlinearity between the state feedback control laws with respect to polyhedrons and the norm distance from system state to origin, and can provide continuously variable state feedback control law varying with the state. First, a set of polyhedral parameters and their corresponding state feedback control law sequences are obtained offline by solving a set of LMIs optimization problems. Next, for each state feedback control law sequence, a nonlinear fitting function is established offline between the state feedback control law and its serial number. Then a simplified lookup table is constructed offline to save memory space and shorten online computation time of the controller. According to the simplified lookup table and information of the norm distance from system state to origin, we online establish the coordinate of current state in the nonlinear fitting curve for getting current feedback control law, which changes continuously with the state. The proposed ERMPC algorithm is successfully applied to an actual fast-responding linear one stage inverted pendulum (LOSIP) system to verify its effectiveness.     

Particle filtering based parameter estimation for systems with output-error type model structures

The output-error model structure is often used in practice and its identification is important for analysis of output-error type systems. This paper considers the parameter identification of linear and nonlinear output-error models. A particle filter which approximates the posterior probability density function with a weighted set of discrete random sampling points is utilized to estimate the unmeasurable true process outputs. To improve the convergence rate of the proposed algorithm, the scalar innovations are grouped into an innovation vector, thus more past information can be utilized. The convergence analysis shows that the parameter estimates can converge to their true values. Finally, both linear and nonlinear results are verified by numerical simulation and engineering.     

Gradient iterative algorithm for dual-rate nonlinear systems based on a novel particle filter

This paper proposes a novel particle filter based gradient iterative algorithm for the identification of dual-rate nonlinear systems. The novel particle filter is applied to estimate the missing outputs, and the measurable outputs are utilized to adjust the weights of particles during each interval of the slow sampled rate. Then the missing outputs and the unknown parameters can be estimated iteratively by the novel particle filter based gradient iterative algorithm. The simulation results indicate that the proposed method is more effective than the classical auxiliary model method.     

Hierarchical estimation of vehicle state and tire forces for distributed in-wheel motor drive electric vehicle without previously established tire model

In this paper, a hierarchical estimator combined with the nonlinear observer and particle filter (PF) is proposed to accurately estimate the vehicle state and tire forces of distributed in-wheel motor drive electric vehicles (DIMDEVs) when the traditional tire models are not available. The proposed estimator consists of lower and upper layers. The lower layer, i.e. longitudinal tire force nonlinear observer (LTFNO) aims at estimating the longitudinal force based on the available drive/brake torques and rotational speed of wheels. The convergence of LTFNO is proved by the invariant set principle. The upper layer receives these estimated longitudinal tire forces from LTFNO and estimates the vehicle state including lateral tire forces based on an expert model (EM). The designed EM utilizes basic knowledge and rules about tire characteristics to approximate the unknown lateral tire force. The upper estimator combines with EM (EEM) to further improve the accuracy. The EEM takes the modeling errors and disturbances into account and avoids the usage of complex established tire models. Then PF is applied in the upper layer to complete the estimation, which only needs measurable longitudinal/lateral accelerations and yaw rate signals. Finally, the effectiveness of the designed hierarchical estimator is verified by Carsim and Simulink co-simulations. The results show the proposed strategy can accurately estimate the vehicle state and tire forces in real-time.     

Adaptive-critic-based model reference control for unknown nonlinear systems with input constraints

In this paper, the optimal model reference adaptive control (MRAC) problem is studied for the unknown discrete-time nonlinear systems with input constraint under the premise of considering robustness to uncertainty. Through an input constraint auxiliary system, a new adaptive-critic-based MRAC algorithm is proposed to transform the above problem into the optimal regulation problem of the auxiliary error system with lumped uncertainty. In order to realize the chattering-free sliding model control for the auxiliary error system, an action-critic variable is introduced into the adaptive identification learning. In this case, the closed-loop control system is robust to the disturbance and the neural network approximation error. The uniformly ultimate bounded property is proved by the Lyapunov method, and the effectiveness of the algorithm is verified by a simulation example.     

Stable model predictive control for a nonlinear system

In this paper, a stable model predictive control approach is proposed for constrained highly nonlinear systems. The technique is a modification of the multistep Newton-type control strategy, which was introduced by Li and Biegler. The proposed control technique is applied on a constrained highly nonlinear aerodynamic test bed, the twin rotor MIMO system (TRMS) to show the efficacy of the control technique. Since the accuracy of the plant model is vital in MPC techniques, the nonlinear state space equations of the system are derived considering all possible effective components. The nonlinear model is adaptively linearized during the prediction horizon. The linearized models of the system are employed to form a linear quadratic objective function subject to a set of inequality constraints due to the system input/output limits. The stability of the control system is guaranteed using the terminal equality constraints technique. The satisfactory performance of the proposed control algorithm on the TRMS validates the effectiveness and the reliability of the approach.