In this tutorial, we shall illustrate the use of the FOMCON toolbox in connection with the problem of designing a controller for a system with a significant input-output delay. For this purpose, a fractional-order first-order plus dead time (FO FOPDT) system is considered. We treat the continuous-time case.
1 Control design goal
Consider a system, that exhibits fractional-order dynamics, as well as a comparatively significant input-output delay, and is accurately described by a fractional-order transfer function
We assume, that the actuator of this system saturates at $u=\pm 1$. Suppose also that the "ideal" transfer function (1) is not known. However, experimental data from a step experiment is available:
In what follows, we use the tools available in the FOMCON toolbox to identify the system and design a suitable controller by means of constrained optimization.
2 System identification
Our first task is to identify the system from the experimental dataset shown in Fig. 1. To identify the system, we may use the fractional-order transfer function time-domain identification tool (
From observation of the registered transient response, we can conclude, that the system may be described by a first-order model with the static gain being close to $7$. Additionally, an input-output delay is observed and is close to $3$ seconds. Thus, our initial guess model is
and may be entered into the tool text fields as shown in Fig. 2. With this initial model, the system is identified in several iterations, and after a simple transformation1 using the
normalize() command the system becomes
In the next section we shall use this model to obtain a fractional-order PID controller for the plant in (1).
3 Controller design
The first step in controller design is choosing a suitable controller structure. In this case, we are concerned with the design of a fractional-order PID controller. However, the plant to be controlled exhibits a significant input-output delay. Therefore, we consider using a Smith predictor . The basic structure of the predictor is given in the next figure.
The design of the controller $C(s)$ is carried out in two steps. First we formulate an integer-order PID controller, and then optimize the parameters of the obtained controller, including the orders of the integrator and differentiator, finally arriving at a fractional-order PID controller working in a Smith predictor based control scheme. Note, that we use the parallel form of the PID controller.
3.1 Conventional PID controller tuning
We now describe the procedure for tuning a classical, integer-order PID controller for a fractional-order plant in FOMCON. We can use the
iopid_tune graphical tool to first approximate the fractional-order model by a conventional FOPDT model, and then apply classical tuning formulae to get the PID controller parameters.
Suppose, that the identified plant has a workspace name of
Gp. Then, we can use the integer-order PID tuning tool and fill the data field of the Fractional plant model box as shown in the following figure.
The fractional-order model is thence identified:
The result of the approximation is illustrated on the following figure.
Now we use the Ziegler-Nichols tuning formula to obtain PID controller parameters:
We shall use these parameters as the point of departure for our fractional PID controller design.
3.2 Fractional-order PID controller optimization
The design is carried out using the
fpid_optim tool. Since a custom controller structure is used, we shall use Simulink for evaluating the characteristics of the transient response of our system. We set the parameters of the optimization as shown in Fig. 6.
You may notice, that we use an LTI system
Gp1 instead of the aforementioned
Gp, which is exactly the same system with the delay term removed, since it is compensated for by the predictor.
Now we need to build the Simulink model. We start by creating a new model
fpid_optimize_Gp.mdl, which is a copy of the default model, and then clicking the Edit button. The model should open in a Simulink editor window as shown in the next figure.
Our task now is to modify this model and to add the predictor in Fig. 3. The identified model is added as part of the predictor, while the original, "ideal" model is added as the plant replacing the block in the inital model. With this setup it is also possible to evaluate the impact of the difference of the reference and original model on the control system. The resulting model is shown in Fig. 8.
We can now proceed to optimize the controller. After 100 iterations, the following controller is obtained:
In this tutorial, we have illustrated the application of the FOMCON toolbox to the problem of designing a fractional-order controller for a process, exhibiting both fractional-order dynamics and a significant delay. The obtained results are considered satisfactory. However, in case of a real system the described procedure may need corrections, depending on particular performance specifications.
|||N. Abe and K. Yamanaka, “Smith predictor control and internal model control – a tutorial,” in SICE 2003 Annual Conference, vol. 2, 2003, pp. 1383–1387.|