The generalised isodamping approach for robust fractional PID controllers design

ABSTRACT

In this paper, we present a novel methodology to design fractional-order proportional-integral-derivative controllers. Based on the description of the controlled system by means of a family of linear models parameterised with respect to a free variable that describes the real process operating point, we design the controller by solving a constrained min–max optimisation problem where the maximum sensitivity has to be minimised. Among the imposed constraints, the most important one is the new generalised isodamping condition, that defines the invariancy of the phase margin with respect to the free parameter variations. It is also shown that the well-known classical isodamping condition is a special case of the new technique proposed in this paper. Simulation results show the effectiveness of the proposed technique and the superiority of the fractional-order controller compared to its integer counterpart.

KEYWORDS:

• Fractional control
• robust control
• PID control
• isodamping
• optimisation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Note that condition 2 reduces to $arg\left(C\left(j{\bar{\omega }}_c\right)\right)=arg\left(P(j{\bar{\omega }}_c;\bar{\Phi })\right)-\pi +{\bar{\varphi }}_m$.

2. Note that this condition is still more general than the classical isodamping one since no hypotheses on the function $\frac{d\Phi }{dx}$ have been considered. The very special case of pure uncertainty to gain variation is actually obtained when x = K, (i.e. when $\frac{d\Phi }{dx}={[0,...,0,1,0,...,0]}^T)$.

3. In this case, the zero steady-state error constraint (10(10) $$\underset{t\to \infty }{lim}e\left(t\right)=0$$(10) ) is not achievable (unless an integral process is considered) and should be removed.