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Improving time series forecasting accuracy is an important work for decision makers. A single model applied on a data series may not obtain satisfactory prediction accuracy. Both theoretical and empirical findings have indicated that integration of linear model and nonlinear model may provide more accurate prediction than an individual linear or nonlinear model. This paper presents a hybrid modeling approach that combines a linear autoregressive (AR) model and a nonlinear deep belief network-based autoregressive (DBN-AR) model for time series forecasting. The proposed modeling approach first applies an AR model to fit time series data, and the error between the original date and the AR model forecast data as a nonlinear component is considered, and then the error is modeled by a DBN-AR model. DBaN-AR model is a modeling method for nonlinear time series, which uses a set of deep belief networks to approximate the state-dependent functional coefficients of state dependent auto-regressive model. The proposed hybrid model can overcome limitation of an individual model and obtain more general and more accurate forecasting result than some existing hybrid models. The experiment results demonstrate that the MSE of modeling residuals using the proposed hybrid model is largely reduced compared with the results of some single prediction models and some hybrid models for one-step-ahead and multistep-ahead forecast.

Liquid level control of the liquid tank is a basic control problem in the process industry. The liquid level control system usually has the characteristics of strong coupling, nonlinearity and large lag. Up to now, PID control is the main method of liquid level control in industry, but it is difficult to obtain high precision control performance. The difficulty of model-based liquid level control such as Robust Model Predictive Control (RMPC) is hard to obtain an accurate model of the plant or need accurate steady-state information which is hard to be obtained in practice. This paper provides a systematic modeling and RMPC method to implement a Coupled Water Tank (CWT) plant output-tracking control in the case without whole steady-state information of the system. First, a data-driven modeling technique based on the Radial Basis Function (RBF) net-type coefficients Multi-Input/Multi-Output (MIMO) AutoRegressive model with eXogenous variable (ARX) is applied to build the RBF-ARX model of the CWT plant. Using the model, a locally linearized model and a polytopic uncertain linear parameter varying (LPV) model are constructed to represent the current and future behavior of the nonlinear CWT system. Based on the two models, an RMPC method is designed to implement the system’s output tracking control under the condition without whole steady-state knowledge of the CWT plant. The stability of the RMPC strategy is guaranteed by using the time-varying parameter-dependent Lyapunov function and feasibility of the Linear Matrix Inequalities (LMIs). Real-time control experiments and comparison with other control methods are carried out on the CWT system, and the results show that the presented method is superior.

There is a class of typical nonlinear industrial process, which can be characterized by a first-order inertia plus pure delay model in an operating range, but the model parameters are different in different operating ranges. Usually, a set of Proportion Integration Differentiation (PID) controllers may be used to control the process, and the controllers have different parameters in different operating areas. However, the adjustment process of the PID controllers’ parameters is not an easy job in practice, and the control performance may also be not perfect. The Dahlin algorithm may provide very good control performance for the process, but its control performance may become very poor if the model parameters are not accurate and/or the input is constrained. Faced with this issue, this paper proposes an Operating-Range Scheduled Robust Dahlin Algorithm (ORSRDA) for the process control with input constraint, which is designed on the basis of a nominal first-order inertia plus pure delay model and the given parameters uncertainty. The process operating ranges are divided into pre-designed several zones according to the difference between output setting value and current output when a new setting appears. For each operating range, the parameters of ORSRDA are obtained by solving a min-max problem offline to guarantee the closed-loop system’s robust stability and acquire the best step-response control performance. To eliminate the steady state error, the integration control action is added into the ORSRDA when the system output is close to its setting value. The proposed method is applied to temperature control of an experimental electric furnace to demonstrate its effectiveness and implement procedure.

The mechanical model of a plant in real industry is usually difficult to obtain. This paper integrates the data-driven RBF-ARX modeling method and a fast Robust Model Predictive Control (RMPC) approach to achieving output-tracking control of a nonlinear system with unknown steady status information. Considering the large online computational burden of online RMPC, this paper proposes a RBF-ARX model-based efficient robust predictive control (RBF-ARX-ERPC) approach. First, based on the RBF-ARX model, a polytopic uncertain linear parameter varying (LPV) state-space model is built to represent the dynamic behavior of the system; next, two convex polytopic sets are constructed to wrap the globally nonlinear behavior of the system. Then, an optimization problem including several linear matrix inequalities (LMIs) is formulated, which is solved offline to synthesize a sequence of explicit control laws corresponding to a sequence of asymptotically stable invariant ellipsoids in the state space, of which all the optimization results are stored in a look-up table. For the real-time control online, it only involves simple state-vector computation and bisection search. Two simulation examples, i.e. the modeling and control of a widely used continuously stirred tank reactor (CSTR) and a linear one-stage inverted pendulum (LOSIP) system, and the real-time control experiments on an actual LOSIP plant are provided to demonstrate the effectiveness of the proposed RBF-ARX model-based efficient RPC approach.

For a class of nonlinear time series whose dynamic behavior smoothly changes with the system state, a state-dependent auto-regressive (SD-AR) model is proposed to characterize the nonlinear time series. A set of deep belief networks (DBNs) is used to build the state-dependent functional coefficients of the SD-AR model, and the proposed model is called DBN-AR model, which combines the advantage of DBN in function approximation and the merit of SD-AR model in nonlinear dynamics description. The DBN-AR model is driven by the state signal changing with time. Based on the least squares solution with minimum norm and the pseudo inverse matrix approach, the initial target values of the DBNs are determined in pre-training stage. In fine tuning stage, all parameters of DBN-AR model is finally tuned by the back propagation (BP) algorithm designed for fine-tuning of DBN-AR model. Through experiment and comparative study on the sunspot data, the electricity load demand data sets from Australian Energy Market Operator (AEMO), the weekly British Pound/US dollar (GBP/USD) exchange rate data and the daily electricity generation data of the Three Gorges dam right bank power station, it is shown that the DBN-AR model is superior to some existing models or methods in prediction accuracy.

The Underactuated, Fast-responding, Nonlinear and Unstable (UFNU) system is a typical hard-to-control plant, such as multi-stage inverted pendulum (IP). This paper considers the modelling and stabi...

Time series forecasting has important theoretical significance and engineering application value. A number of studies have shown that hybrid modelling is very successful in various modelling applications, and both theoretical and empirical findings have shown that hybrid modelling is an effective method to improve the accuracy of time series models. This paper proposes a hybrid model that combines a linear regression (LR) model and deep belief network (DBN) model for the prediction of time series data. In the hybrid model, the linear AR (auto-regression) LR model or ARIMA (auto-regressive integrated moving average) model and the nonlinear DBN model are explored to capture the linear and nonlinear behaviours of a time series, respectively. We first use an LR model to fit the original data and obtain the LR model residuals between the original data and the predicted data of the LR model. Then, the residuals are regarded as the nonlinear component and are used as inputs into the DBN model. The LR model prediction and the output of the DBN model are the final forecasting value for the time series, which takes full advantage of the two models for predicting time series. The proposed hybrid model and other existing models are applied to four well-known time series for comparison, and the results show that the proposed hybrid model has a high prediction accuracy and may be a useful tool for time series forecasting.

With directly considering the unknown and bounded disturbance, a RBF-ARX model-based two-stage scheduling quasi-min–max robust predictive control (RBF-ARX-TRPC) algorithm for output-tracking control is proposed for a class of smooth nonlinear systems with unknown steady-state knowledge. Firstly, from the RBF-ARX model that is identified using input/output data of the system, the two local linearization state-space models that consider the bounded disturbance and a polytopic uncertain LPV state-space model are built to approximate the present and future system’s nonlinear dynamics, respectively. Based on the state-space models, the RBF-ARX-TRPC algorithm is designed without relying on the system steady-state knowledge. In the RBF-ARX-TRPC algorithm, the future nonlinear behavior of the system is forced to vary within the region constructed by the polytopic uncertain LPV state-space model. Closed-loop stability is guaranteed when the algorithm is implemented in a receding horizon fashion by including a Lyapunov constraint in the formulation. The comparative experiments demonstrate the effectiveness of the proposed strategy on a continuously stirred tank reactor (CSTR) simulator.

Considering the unknown and bounded disturbance, a RBF-ARX model-based Robust Predictive Control (RBF-ARX-RPC) algorithm for output tracking control without relying on steady state knowledge is proposed for a class of smooth nonlinear systems. From the identified RBF-ARX model, a local linearization state-space model that considers the modeling error and bounded uncertain disturbance is obtained to represent the current behavior of the nonlinear system, and a polytopic uncertain linear parameter varying (LPV) state-space model is built to represent the future system's nonlinear behavior between the output deviation and input increment. Based on the two state-space models, a quasi-min–max robust MPC algorithm is designed for output tracking control of the nonlinear system under the unknown and bounded disturbance, which does not rely on steady state knowledge. Optimization problem of the RBF-ARX-RPC algorithm is finally converted to a convex linear matrix inequalities (LMIs) optimization problem, and the stability is guaranteed by the use of parameter-dependent Lyapunov function and feasibility of the LMIs. The comparative experiments demonstrate the effectiveness of the proposed RBF-ARX-RPC strategy on a continuously stirred tank reactor (CSTR) simulator.

A RBF-ARX modeling and robust model predictive control (MPC) approach to achieving output-tracking control of the nonlinear system with unknown steady-state knowledge is proposed. On the basis of the RBF-ARX model with considering the system time delay, a local linearization state-space model is obtained to represent the current behavior of the nonlinear system, and a polytopic uncertain linear parameter varying (LPV) state-space model is built to represent the future system’s nonlinear behavior. Based on the two models, a quasi-min–max MPC algorithm with constraint is designed for output-tracking control of the nonlinear system with unknown steady state knowledge. The optimization problem of the quasi-min–max MPC algorithm is finally converted to the convex linear matrix inequalities (LMIs) optimization problem. Closed-loop stability of the MPC strategy is guaranteed by the use of parameter-dependent Lyapunov function and feasibility of the LMIs. Two examples, i.e. the modeling and control of a continuously stirred tank reactor (CSTR) and a two tank system demonstrate the effectiveness of the RBF-ARX modeling and robust MPC approach.

In general, the online computation burden of robust model predictive control (RMPC) is very heavy, and the mechanical model of a plant, which is used in RMPC, is hard to obtain precisely in real industry. These issues may largely restrict the applicability of RMPC in real applications. This paper proposes a RBF-ARX (state-dependent Auto-Regressive model with eXogenous input and Radial Basis Function network type coefficients) model-based efficient robust predictive control (RBF-ARX-ERPC) approach to an inverted pendulum system, which is a complete and systematic method for designing robust MPC controller because it integrates the RBF-ARX modeling method and a fast RMPC approach. First, based on the offline identified RBF-ARX model without offset term, two convex polytopic sets are constructed to wrap the globally nonlinear behavior of the system. Then, the optimization problem of implementing a quasi-min-max MPC algorithm including several linear matrix inequalities (LMIs) is formulated, and it is solved offline to synthesize a sequence of explicit control laws that correspond to a sequence of asymptotically stable invariant ellipsoids, of which all the optimization results are stored in a look-up table. During the online real-time control, the controller only needs to carry out a simple state-vector computation and bisection search. The proposed approach is applied to an actual linear one-stage inverted pendulum (LOSIP), which is a fast-responding and nonlinear plant. The real-time control experiments demonstrate the effectiveness of the proposed RBF-ARX model-based efficient RMPC approach.

T0629. This paper considers modeling and stabilization control of a Linear Two-Stage Inverted Pendulum (LTSIP). To avoid the potential problems resulted using first principle model-based control to the plant, such as that some model parameters are unknown or inaccurate, this paper uses a data-driven modeling approach. A State-Dependent AutoRegressive eXogenous (SD-ARX) model without offset term, whose state-dependent functional coefficients are approximated by Gaussian radial basis function (RBF) neural networks, is built to describe the dynamic behavior of the LTSIP system. Based on the identification model, an infinite horizon model predictive control (MPC) strategy is proposed to implement stabilization control of the LTSIP plant, which is designed by using the locally linearized state-space model and obtaining the locally optimal state feedback control low via solving an algebra Riccati equation online. The real-time control experiment results of the proposed approach and the comparisons with traditional Linear Quadratic Regulator (LQR) method to the plant demonstrate that the modeling and control method proposed in this paper are very effective and superior in modeling and controlling the underactuated, fast-responding and nonlinear system.