A tuning method of multi variable PID gains using Jacobian

Generally, a PID controller is not suitable to control a multi variable system because it is very difficult to tune their gains. However, this paper shows that it is not hard to tune the PID gains if we can find a Jacobian matrix of the system. Because the Jacobian matrix expresses the ratio of output variations according to input variations, it is possible to adjust the input values in order to reduce the output error. If the cost function is composed of error related terms, the gradient approach can tune the PID gains to minimize the function. In simulation, a hydrofoil catamaran with two inputs and two outputs is considered as a multi variable system. We can easily obtain the multi variable PID controller by the proposed tuning method. When it was compared with a LQR controller with a modelling equation, its performance was very good.