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 (`fotfid`

).

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 transformation^{1} 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 [1]. 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:

The results^{2} of the optimization are presented in Fig. 9. A clear improvement of the control system performance can be observed.

## Conclusions

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.

## References

[1] | 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. |

^{1}This transformation is not strictly necessary^{2}Your results may be slightly different

## 17 Comments

Hi ALEKSEI,

I am trying to repeat this exercise. Up to point 2, I don’t have any issue.

Could you please share figure 8 model with needed LTI model?

I want to learn FPID for Input/Output System.

Thanking You,

Hi Pritesh,

Sorry; the model supplied in the figure description was the one without the predictor implementation. I have corrected that now.

Hi ALEKSEI,

Thanks for your kind reply.

In model given by you, there is block called Fractional PID Controller.

Inside that block, it is FPID LTI. In that there is LTI function fpid_optimize_fpid.

How should I create the same? Is it Fractional Transfer function of FPID?

I have created ft type of Fractional Controller. Error: The “LTI system variable” parameter of the LTI block must be set to a valid LTI model

Please advice me…..

Once again Thanks

This model should be used with the FPID Optimization tool (Figure 6). You may set the model in the

Simulation parametersoptions box. The model must be present in the current workspace. Otherwise, if you just want to simulate the transient response, you may replace the controller and the plant with those described in the tutorial using blocks from the FOMCON Simulink library.Hi ALEKSEI,

I agree with your comment. FPID Optimization tool will give me fotf type of object. But in simulation, Fractional PID Controller block needs LTI Block.

How should I get LTI block from FPID Optimization tool ?

Once again thanks for your kind reply.

You may use the fractional PID contorller block from the FOMCON Simulink library.

sir,after several iteration for initial guess model we are doing the normalize for fractional order transfer function ,but here it is showing same initial model????????.can u help me please.,.,.,.,.

Please provide a screenshot depicting this situation.

sir,

wht is the function to open fomcon Simulink library…

Hi Aleksei,

In this tutorial why we use saturation block ? what is function of this block in this model ?

Thanking You

The block models actuator saturation. This is a purely physical phenomenon caused by limitations of what the actuator can actually produce (for example, a pump with max. displacement capacity of 100L/min cannot pump 200L/min).

sir,

Can this tool be used for an integer order plant transfer function..should the plant transfer function changed into an fractional order ..or else is there any way to change it from the toolbox.please suggest..

Thanking you

Yes, the tool supports conventional (i.e. integer-order systems) as well. You can also convert

`tf`

objects into`fotf`

objects, since the former is technically a subclass of the latter.sir,

can you just give me an example of using this toolkit with an integer order plant transfer function…I will be very thankful as I tried a lot…

thanking you

See, for example, page 126 of the “Fractional-order Systems and Controls: Fundamentals and Applications” book in http://fomcon.net/fo/overview-of-literature/

i dont have fopid matlab block.what can i do?

I do not understand what the problem is.