Fig 1.
Neural PID architecture and example neural network.
A neural network is integrated into standard closed loop control (a). The neural network receives the system output and the error as input and outputs the three PID parameters KP, KI, and KD. The double-lined arrows indicate that the associated variable could be a vector, while the single-lined arrows indicate scalar variables. An example neural network (b) shows one possible set of connections. All networks have 9 neurons in three layers. The output of each input neuron is fed back into the input layer with a delay, denoted as g−1, of one time step.
Fig 2.
This nonlinear system is widely used to study nonlinear behaviour in control engineering systems. The controller can adjust the amount of water being put into the first tank. The goal is to keep the water level in the lower tank (v(t) = x2(t) at the setpoint.
Fig 3.
This system is a nonlinear system with an unstable equilibrium. The control task is to move the cart to a predefined position, while keeping the pole up.
Fig 4.
System with non-negligible time delay.
The Figure shows the systems response (including the time delay TD) to an input.
Fig 5.
Chaotic thermal convection loop.
This system is an example for chaotic behaviour. The control task is to maintain a constant flow in the inner torus—the flow is measured at the points A and B. Half of the torus that contains the fluid is surrounded by a heating element, the other half is surrounded by a cooling jacket. The control variable is the heating power, applied on the lower half of the torus.
Table 1.
Control results for the four benchmark systems.
Fig 6.
Control performance for the disturbed chaotic thermal convection loop.
Subfigure (a) shows the setpoint (x1 = 0, which corresponds to a steady flow) and the system output for all controllers. Subfigure (b) shows the controller outputs for all three controllers. In subfigure (c), the PID parameters, applied by the neural network are shown.
Fig 7.
Stability analysis for the chaotic thermal convection loop with disturbance.
The dashed line shows the systems output, when controlled by the neural PID controller. The closed loop transfer function is not guaranteed to be stable within the grey areas, despite the algorithm stabilizing the system. As the system approaches its steady state, the system becomes input-output stable with the chosen PID parameters. The second subplot shows the real values for all four poles. The system is only stable (white background colour) if all poles have real values smaller than zero.