In this paper, stable adaptive neural network (NN) control, a combination of weighted one-step-ahead control and adaptive NN is developed for a class of multi-input-multi-output (MIMO) nonaffine nonlinear discrete-time systems. The weighted one-step-ahead control is designed to stabilize the nominal linear system, while the adaptive NN compensator is introduced to deal with the nonlinearities. Under the assumption that the inverse control gain matrix has an either positive definite or negative definite symmetric part, the obstacle in NN weights tuning for the MIMO systems is transformed to unknown control direction problem for single-input-single-output (SISO) system. Discrete Nussbaum gain is introduced into the NN weights adaptation law to overcome the unknown control direction problem. It is proved that all signals of the closed-loop system are bounded, while the tracking error converges to a compact set. Simulation result illustrates the effectiveness of the proposed control.
here $X(\cdot)$ is a $n$-state vector of the RTTN; $U(\cdot)$ is a $m$-input vector; $Y(\cdot)$ is a $l$-output vector; $Z(\cdot)$ is an auxiliary vector variable with $l$ dimension; $S(\cdot)$ is a vector-valued smooth activation function (sigmoid, $tanh$, saturation) with appropriate dimensions; $J$ is a weigh-state block-diagonal matrix with $(1 \times 1)$ and $(2 \times 2)$ blocks; $J_i$ is an $i-th$ block of $J$ and $|J_i|<1$ is a stability condition.
The model of the MAGLEV system is unstable and nonlinear
$$
m\ddot{x}=mg-\dfrac{K_{c}V^{2}}{x^{2}},
$$
where $x$ is the metal ball position being the system output, $V$ is the system input as the voltage. Other parameters are $m$ as the mass of the metal ball, $K_c$ as constant for magnet circuit, and $g$ is the gravitational acceleration of 9.8 m/s$^2$. A free-body diagram is shown also in the attached image.
Identification of a practical process, especially if unstable, is challenging as its model is generally stochastic and nonlinear. In this work we consider a class of unstable processes where the model is identified in a closed-loop operating regime. Important issues in identification are addressed, namely: identification scheme, the closed loop identification of unstable plants, choice of sampling period, and constraints on the estimated model parameters. Further the structure of the identified model may not be identical to that of the physical system due to noise artifacts, and inability to capture faster dynamics. Generally least-squares identification is employed to estimate the parameters of the system wherein all the coefficients of numerator and the denominator coefficients of system transfer function are estimated. In many practical system there are constraints on the model parameters. The identified coefficients using the conventional scheme may not obey the constraint. In this work a novel constrained least-squares identification scheme is proposed where in a priori known structural constraint is factored in parameter estimation. This scheme is evaluated on a physical magnetic lévitation system.
where $x_1(t)$ : angle difference between truck and trailer. $x_2(t)$ : angle of trailer. $x_3(t)$ : vertical position of rear of trailer, $u(t)$ : steering angle, $T$ : sampling time. In this example, the parameters are $T=2.0s$, $l=2.8m$, $L=5.5m$, $v=-1.0m/s$.
This paper deals with the stabilization problem for discrete-time nonlinear systems that are represented by the Takagi - Sugeno fuzzy model. By the multiple fuzzy Lyapunov function and the three-index algebraic combination technique, a new stabilization condition is developed. The condition is expressed in the form of linear matrix inequalities (LMIs) and proved to be less conservative than existing results in the literature. Finally, a truck-trailer system is given to illustrate the novelty of the proposed approach.
