Tensor Product Based Fuzzy Adaptive Control For A Class Of Under-actuated Nonlinear Systems

Under-actuated systems are widely used in some important areas such as transportation, metallurgy, national defense and equipment manufacturing, as economy grows and economy competition became increasingly intensive in these fields, the higher requirement of equipment is needed, thus, the demanding of higher performance of control system also became intensive, especially the adaptiveness of controller. In the under-actuated system control field such as inverted pendulum system control, variable universe fuzzy adaptive control theory has been verified by many authors that the controller has high adaptiveness, which is one of the most successful fuzzy theory. In this paper, we set about to analyse the limitation that variable uni-verse fuzzy adaptive control is constructed based on semi-strict pure feedback model, a deep study of avoiding variable universe fuzzy adaptive controller's semi-strict pure feedback model is carried out firstly, and we found the factors which limit the application of variable universe fuzzy adaptive control to more general models, such as quasi-linear parameter-varying model, and a tensor product based adaptive fuzzy control algorithm is proposed; To the pure feedback model with structure uncertainty, a tensor product based adaptive integral fuzzy sliding control method is proposed for the structure uncertainty pure feedback model with the property that tensor product model's approximation error is a bounded function of deserted singular value of quasi-linear parameter-varying system; Second, for the problem that under-actuated systems have multiple control target and coupled variables, with the idea adopted from variable universe adaptive controller's contraction-expansion factors which are designed with two factors, that is, error and error derivative, all the target factors are incorporated into hybrid sliding surface which is constructed by main sliding surface and auxiliary sliding surface, an adaptive control method is designed directly based on tensor product model's state variables; Finally, in order to enhance the extensibility of variable universe fuzzy adaptive controller, furthermore, to unclose the adap-tive mechanism of variable universe fuzzy adaptive control, an A2-C1 fuzzy adaptive control method is proposed for under-actuated system with unknown dynamics, the controller has the ability to enhance the high frequency reference signals tracking ability of variable universe fuzzy adaptive controller. The main work of this thesis is presented as below:1) Based on the background of tensor product model transformation and approximation the-ory, in order to eliminate the limitation that variable universe fuzzy adaptive controller's design must use the semi-strict pure feedback model, we found that tensor product model can make the design of universe fuzzy adaptive controller need semi-strict pure feedback model any longer, and tensor product parallel distributed compensator can afford the universe fuzzy adaptive con-troller with synthesis gains also, based on the above mentioned limitation, a tensor product model transformation based adaptive controller is proposed. To resolve the problem, quasi-linear parameter-varying model of under-actuated system is obtained firstly, combined with error and error derivative gains and decay factor calculated by tensor product parallel distributed compen-sator, two fuzzy adaptive controllers are designed. Fuzzy adaptive controller's error and error derivative are dynamically calculated by weighting membership functions. Moreover, decay factor obtained by tensor product parallel distributed compensator is used to construct a new decaying type Lyapunov function. Tensor product model transformation based adaptive control eliminates the constraint of known upper bound of system disturbance with the help of ?-adaptive strategy. To enhance the tensor product model transformation based adaptive control's ability to tackle the structure uncertain under-actuated system, for a class of-nth order uncertain pure feedback under-actuated systems, tensor product model is used to approximate the system, the nth order uncertain pure feedback under-actuated system then be transformed into two parts, one is the nominal nonlinear plant in the form of quasi-linear parameter-varying model, the other is the lumped uncertainty, the nominal quasi-linear parameter-varying model is a polytopic model and the combinatorial coefficient satisfies normalizing condition. Finally, with the property that integral sliding surface can reach the sliding mode at start, and also with the help of ?-adaptive strategy and linear matrix inequality, a tensor product model transformation based adaptive inte-gral sliding mode controller is designed, the time to equilibrium is reduced, the proposed control method resolves the problem that adaptive controller's control gains will always increase when the ideal sliding surface does not exist.2) Based on the background of tensor product model approximation theory, to the prob-lem that under-actuated systems have multiple control target and coupled variables, for under-actuated system's quasi-linear parameter-varying model, a hybrid sliding surface design scheme which intends to incorporate multiple control targets into one sliding surface is proposed based on the idea adopted from variable universe fuzzy adaptive controller's contraction-expansion factors, an adaptive controller is proposed with tensor product model's state variables. To SIMO under-actuated nonlinear system, under-actuated nonlinear system's quasi-linear parameter-varyi model is obtained by tensor product model transformation. The obtained model is in numerical quasi-linear parameter-varying form, the model is a precise model of under-actuated nonlin-ear system while all the singular values are reserved for quasi-linear parameter-varying model. Different sliding surfaces are designed for each subsystem, finite time to reach sliding mode is guaranteed. Design steps of sliding surfaces and control laws do not need analytical model any-more for tensor product model transformation based decoupled terminal sliding mode control. All design steps are designed by tensor product model's state variables, the transformation ability of tensor product model transformation method is shown by the design step.3) In order to enhance the control precision of variable universe fuzzy adaptive controller, type-1 fuzzy set is replaced with type-2 fuzzy set, variable universe fuzzy adaptive controller's adaptive tracking ability of high frequency reference signal is enhanced further, then adaptive mechanism of variable universe fuzzy adaptive control is unclosed. For semi-strict pure feed-back system, variable universe fuzzy adaptive method and type-2 fuzzy set are utilized to design A2-C1 evolutionary adaptive fuzzy controller, its feedback controller's contraction-expansion factors can be adaptively regulated by error and error derivative. Quantum inspired bacterial foraging algorithm is employed to optimize interval type-2 fuzzy set offline, the proposed A2-C1 evolutionary adaptive fuzzy system can be used as one kind of direct adaptive fuzzy controller. To verify the effectiveness of the idea of hybrid sliding surface, second order under-actuated system with unknown nonlinear dynamics is researched, and dynamical fuzzy neural network is used to estimate unknown nonlinear dynamics. The system is divided into two subsystems based on the idea of hybrid sliding surface, each subsystem's convergence property is guaranteed while different sliding surface is designed for each subsystem. An indirect adaptive decoupled slid-ing mode control method is proposed based on hybrid sliding surfaces, this method overcomes the difficult that system presents unknown nonlinear dynamics, and the coupled variables be in-corporated into hybrid sliding surface design, thus the adaptiveness of under-actuated system's decoupled sliding mode controller is improved.For the above mentioned control methods, the proposed methods on cart-pole inverted pen-dulum, parallel-type double inverted pendulum, TORA system are verified by simulations and experiments. Results show the effectiveness of the proposed methods.