The mixed H₂/H_∞ problem is very difficult and remains mostly open over decades. Hence, it is desirable to consider special cases that may provide insights into solving the problem entirely. This thesis is concerned with a mixed H₂/H_∞ problem with degree constraint. For an n-th order plant, we aim to find the controller that is of degree no greater than n, satisfies the given robust stability constraint and gives the best transient performance....[ Read more ]

The mixed H₂/H_∞ problem is very difficult and remains mostly open over decades. Hence, it is desirable to consider special cases that may provide insights into solving the problem entirely. This thesis is concerned with a mixed H₂/H_∞ problem with degree constraint. For an n-th order plant, we aim to find the controller that is of degree no greater than n, satisfies the given robust stability constraint and gives the best transient performance.

Firstly, we reduce the robust stability constraint to an interpolation problem and solve it. Then, a set of desired controllers is characterized over a stable function set F(s). We have proved that the robust stability and degree constraints on controllers can be equivalently put onto F(s) ∈ F(s), and poles of F(s) turn out to be closed-loop poles of the feedback system. Since the set F(s) can be parameterized by n-th order stable polynomials via a quadratic map and a stable polynomial can be determined by the first column of its Routh table that consists of only positive numbers, we can characterize the set F(s) over a convex set.

Next, we explore the properties of the quadratic map. We have discovered and proved that this map is a bijection and its Jacobian matrix is invertible. Then, a numerical method is developed to solve the nonlinear equations to derive the controller for any given parameter. Finally, we present an algorithm to find the mixed H₂/H_∞ control solutions, and make comparisons with other approaches.