Multi-dimensional Taylor Network Optimal Control For Nonlinear Time-varying Systems With Noise Disturbances

As the most common phenomenon in the engineering technology,the stability analysis and controller design of nonlinear systems have the important theoretical value and practical significance.The pursuit of a simple direct control and processing method for nonlinear systems is a goal that has always been pursued in the field of automatic control.Based on this problem,from the perspective of engineering implementation,this thesis is based on the stability proof,takes the improvement of dynamic performance as the core and focuses on the improvement of computational complexity as the key.The purpose is to provide a theoretical basis for the application of the multi-dimensional Taylor network(MTN)in nonlinear systems.Although there are several developments on the control of nonlinear systems,they do not involve the overall consideration of the coupling,uncertainty,time-varying characteristics and measurement noise commonly.How to minimize the influence of system coupling,randomness,time variation and uncertain nonlinearity together and how to improve the real-time performance are of great significance.In recent years,MTN optimal control method has been widely used in nonlinear control systems,which provides a new solution for systematic design and stability analysis of nonlinear systems controllers.Using the MTN's excellent properties,this dissertation is devoted to propose the MTN based control scheme to address the problems of tracking control for several classes of nonlinear time-varying systems with noise disturbances by combining the approaches of system identification,adaptive control and nonlinear filtering,etc.Moreover,the stability of the closed-loop systems is proved by employing Lyapunov theory.The main contributions of this thesis are summarized as follows:1.An inverse control scheme based on MTN is proposed for the real-time tracking control of single-input/single-output(SISO)nonlinear time-varying systems with noise disturbances.Utilized in this scheme are the three MTNs: the adaptive model identifier for system modeling,the adaptive inverse controller for inverse modeling,and the adaptive nonlinear filter for eliminating the noise disturbance.To avoid “compromise”,this scheme is designed into a structure wherein controlling the object dynamic response and eliminating the noise disturbance are divided into two relativelyindependent processes.Furthermore,the weight-elimination algorithm is adopted for choice of effective regression items to avoid the dimension explosion,thus overcoming the shortcoming that the number of middle nodes needs to be determined before using the traditional neural network.After a certain number of training,the more streamlined MTNs are observed to contribute to satisfying the real-time requirements of software implementation and engineering application.To ensure that MTN inverse control is strict in theory,the general conditions for the existence of SISO nonlinear inverse systems are identified.Simulation of the MTN inverse control is conducted to confirm the effectiveness of the control scheme.2.A stable adaptive control approach based on MTN is proposed to control the SISO uncertain nonlinear time-varying systems with noise disturbances.Firstly,an MTN filter(MTNF)is developed to eliminate the control interference and measurement noise,so that the model output without stochastic disturbance can be obtained.Then,an MTN identifier(MTNI)is so designed as to be capable of dynamic mapping and require fewer weights than traditional neural networks.On the basis of the above,the feed-forward MTN controller(MTNC)is developed to realize the precise tracking control of the system.The uncertain nonlinear time-varying system is identified by MTNI,which then provides sensitivity information of the plant to MTNC to make it adaptive.Furthermore,the skeletonization algorithm is adopted to remove redundant inputs and redundant regression items from MTNI and MTNC for concise MTNs.Successful convergence and faster learning are guaranteed using the Lyapunov theorem,and the optimal learning rates are identified.Simulation results demonstrate that the proposed approach features its accurate identification,excellent tracking and better anti-interference capability for the adaptive real-time control of uncertain,stochastic and time-varying nonlinear systems.3.An adaptive control approach based on multi-input/multi-output(MIMO)MTN is presented for tracking control of MIMO uncertain nonlinear time-varying systems with noises in real time,where two MTNs are proposed to formulate the optimal control and nonlinear filtering approaches.Firstly,the MIMO MTNC is proposed to realize the precise tracking control.The closed-loop errors between directly measured outputs(which have been filtered)and expected values are chosen to be the MTNC's inputs.The proposed MTNC can update its weights online according to errors caused by system's uncertain factors and fast time-varying characteristics.The stability of closed-loop system is proved based on stable learning rate.The resilient back-propagation algorithm and adaptive variable step size algorithm via linear reinforcement are utilized to update the MTNC's weights.Secondly,the MIMO MTNF is proposed to eliminate measurement noises and other stochastic factors.The proposed adaptive MTN filtering system possesses the distinctive properties of the Lyapunov-theory-based adaptive filtering system and MTN,where a Lyapunov function of the errors between the desired signals and the MTNF's outputs is first defined.By properlychoosing the weights update law in the Lyapunov sense,the MTNF's outputs can asymptotically converge to the desired signals.The LAF MTN filter is independent of the stochastic properties of the input disturbances.The convergence and stability are proved by the Lyapunov stability theory.Finally,the simulation demonstrates excellent tracking ability and better anti-disturbance capability with improved control performance over that of RBF neural network.And the results show that the MTN based control system is very promising for real-time applications.4.Based on above three contents,to further explore the performance optimization considering system coupling,randomness,fast time variation and uncertain nonlinearity,an effective MIMO MTN based optimal control scheme is presented.Firstly,the direct method of inverse modeling is introduced to initialize the MTNC for improving the convergent speed and preventing weights trapping into local optima.To adapt the initially uncertain and fast time-varying parameters in the control system,we introduce an improved gradient descent method to adjust the MTNC parameters.The stability of our closed-loop system is proved according to the Lyapunov method.Secondly,MTNF weight adaptation scheme is designed based on the Lyapunov stability theory to iteratively update the weights.In the design,the Lyapunov function has to be well selected to construct an energy space with a single global minimum.Analysis and discussion on Lyapunov properties of the proposed MTNF are included.Finally,the simulation of complex nonlinear MIMO system is presented with strong coupling,uncertainty,fast time-varying characteristics,measurement noises and other stochastic factors.Empirical results illustrate that the proposed controllers can obtain good precision with shorter time compared with the other considered methods.