Model-free control of non-minimum phase systems and switched systems

The model-free control methodology, originally proposed by [1], has been widely successfully applied to many mechanical and electrical processes. The model-free control provides good performances in disturbances rejection and an efficient robustness to the process internal changes. The control of non-minimum phase systems has been deeply studied and successful methods have been proposed (e.g. [2] [3] [4] [5] [6] [7] [8] [9]). Since the model-free control can not a priori stabilize a non-minimum phase system [1], we propose a possible derivation of the original model-free control law, dedicated to the control of non-minimum phase systems. The dynamic performances are especially tested in the case of switched systems.

The paper is structured as follows. Section II presents an overview of the modelfree control methodology including its advantages in comparison with classical methodologies. Section III discusses the application of the modified model-free control, called NM-model-free control, for non-minimum phase systems. Some concluding remarks may be found in Section IV.

2 Model-free control: a brief overview 2.1 General principles

We only assume that the plant behavior is well approximated in its operational range by a system of ordinary differential equations, which might be highly nonlinear and time-varying. * The system, which is SISO, may be therefore described by the inputoutput equation E(t, y, ẏ, . . . , y (ι) , u, u, . . . , u (κ) ) = 0

where

• u and y are the input and output variables,

• E, which might be unknown, is assumed to be a sufficiently smooth function of its arguments.

Assume that for some integer n, 0 < n ≤ ι, ∂E ∂y (n) ≡ 0. From the implicit function theorem we may write locally y (n) = E(t, y, ẏ, . . . , y (n-1) , y (n+1) , . . . , y (ι) , u, u, . . . , u (κ) )

By setting E = F + αu we obtain the ultra-local model. Definition 2.1 [1] If u and y are respectively the variables of input and output of a system to be controlled, then this system admits the ultra-local model defined by:

where

• α ∈ R is a non-physical constant parameter, such that F and αu are of the same magnitude;

• the numerical value of F , which contains the whole “structural information”, is determined thanks to the knowledge of u, α, and of the estimate of the derivative y (n) .

In all the numerous known examples it was possible to set n = 1 or 2.

Let us emphasize that one only needs to give an approximate numerical value to α.

It would be meaningless to refer to a precise value of this parameter.

Definition 2.2 [1] we close the loop via the intelligent PI controller, or i-PI controller,

where • y * is the output reference trajectory, which is determined via the rules of flatnessbased control ( [11,12]);

• e = y * -y is the tracking error;

• C(ε) is of the form K P ε + K I ε. K P , K I are the usual tuning gains.

Equation ( 3) is called model-free control law or model-free law.

The i-PI controller 3 is compensating the poorly known term F . Controlling the system therefore boils down to the control of a precise and elementary pure integrator. The tuning of the gains K P and K I becomes therefore quite straightforward.

See [13] for a comparison with classic PI controllers.

Introduce as in [1,10] the stable transfer function

We apply the well known method due to Broïda (see, e.g., [14]) by approximating System 4 via the following delay system Ke -τ s (T s + 1) K = 4, T = 2.018, τ = 0.2424 are obtained thanks to graphical techniques. The gain of the PID controller are then deduced [14]:

We are employing ẏ = F + u and the i-PI controller

where

• y is a reference trajectory,

• C(ε) is an usual PI controller.

Figure 1(a) shows that the i-PI controller behaves only slightly better than the classic PID controller (Fig. 1(b)). When taking into account on the other hand the ageing process and some fault accommodation there is a dramatic change of situation: Figure 1(c) indicates a clear cut superiority of our i-PI controller if the ageing process corresponds to a shift of the pole from 1 to 1.5, and if the previous graphical identification is not repeated (Fig. 1(d)).

• It might be useless to introduce delay systems of the type

for tuning classic PID controllers, as often done today in spite of the quite involved identification procedure.

• This example demonstrates also that the usual mathematical criteria for robust control become to a large irrelevant.

• As also shown by this example some fault accommodation may also be achieved without having recourse to a general theory of diagnosis.

We explain in this section, how to derive the model-free control law (3) in order to stabilize and guarantee certain performances for non-minimum phase systems. We will show that the proposed control law is also robust to disturbances and switched models.

Firstly, consider the discretized model-free control law, which is typically used for a digital implementation.

Definition 3.1 [15] For any discrete momen

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