Data-driven robust controller design by geometric constraints in frequency-domain

Abstract

This paper aims to propose an improved method for designing the robust controller that satisfies the mixed sensitivity criterion. In robust controller design using frequency-domain data, the important task is to find the appropriate constraints for the design parameters. In our method, the controller is designed with fixed-order linearly parameterized structure and the desired robust performance criteria are represented by the geometrical constraints for open-loop frequency characteristic. Then the constraints lead to the convex searching problem for the parameters of controller so that we obtain the optimal parameters by the numerical iterative algorithm for the finite number of the linear inequalities. The significance of our research is that the proposed method is more reliable for the robust performance criterion because it does not require the approximation for the constraint. The control simulation for the robust PID controller shows the improved transient response.

Appendix

Here we prove that a set of points that have the same ratio of distance from different two points becomes a circle on a two-dimensional plane.

For this purpose, we define axis x as a straight line passing two points and axis y as a line orthogonal to axis x and passing the left point, respectively.

Let the distance between the two points be a, the distances from these two points to an arbitrary point in the x–y plane, (x, y), are represented respectively as

$$ r_{1} = \sqrt {x^{2} + y^{2} } ,\quad r_{2} = \sqrt {(x - a)^{2} + y^{2} } $$

Given the ratio of the two distances as a constant $k \in (0,1)$,the relation $r_{1} = kr_{2}$ leads to the following equation:

$$ x^{2} + y^{2} = k^{2} (x - a)^{2} + k^{2} y^{2} $$

which is re-written as

$$ (1 - k^{2} )x^{2} + 2k^{2} ax + (1 - k^{2} )y^{2} = k^{2} a^{2}, $$

and dividing the two sides by $1 - k^{2}$ yields

$$ x^{2} + \frac{{2k^{2} a}}{{1 - k^{2} }}x + y^{2} = \frac{{k^{2} a^{2} }}{{1 - k^{2} }} $$

This equation can be written into the following circle equation:

$$ \left( {x + \frac{{k^{2} a}}{{1 - k^{2} }}} \right)^{2} + y^{2} = \left( {\frac{ka}{{1 - k^{2} }}} \right)^{2} $$

Consequently, it shows that a set of such points becomes a circle with a radius of $ka/(1 - k^{2} )$ centered on axis x at a distance of $k^{2} a/(1 - k^{2} )$ from the origin.