H2 Control Problem Research Articles

Output-feedback mixed H∞/H2 control— a dynamic game approach

We consider a mixed H∞/H2 control problem, in which a controller is required to achieve the best possible rejection of a stochastic disturbance with given statistics, subject to an H∞-performance constraint. We assume that the H∞-constraint rules out the use of the H2-optimal (LQG) control law. As a result, the H2-performance of the controller must be degraded. Moreover, no control law can be the best with respect to all the possible measurement signals. It is thus reasonable to look for a controller that will achieve the best H2-performance, subject to the worst possible measurement signal. An optimal controller, in the sense of this paper, minimizes the worst-case ratio between the resulting H2-performance degradation and the performance degradation that would result if one had used a standard H∞-central controller. In this setting, the mixed H∞/H2 problem reduces to a standard two-person, zero-sum, linear quadratic dynamic game, with a state-feedback information pattern. Both the continuous and the dis...

Full and reduced-order observer-based controller design forH2-optimization

In this paper the most general H2 control problem is considered. We derive necessary and sufficient conditions when the infimum is attained by state feedback. We do the same for the measurement feedback case where we derive necessary and sufficient conditions when the infimum is attained by proper dynamic compensators. We also investigate reduced-order compensators if some states are observable without noise. We discuss, for all of these cases, the freedom that the non-uniqueness of optimal compensators gives us in assigning the closed-loop eigenvalues. The second half of this paper investigates the case when the infimum cannot be attained. We give a constructive algorithm to find a minimizing sequence of stabilizing controllers and discuss the freedom in the asymptotic locations of the closed-loop eigenvalues. Again we do the above for three different cases: static-state feedback, full-order measurement feedback and reduced-order measurement feedback.

The generalized H2 control problem

In this paper we consider the problem of finding an internally stabilizing controller such that the controlled or regulated signals have a guaranteed maximum peak value in response to arbitrary (but bounded) energy exogenous inputs. More specifically, we give a complete solution to the problem of finding a stabilizing controller such that the closed loop gain from L 2[0, ∞) to L ∞[0, ∞) is below a specified level. We consider both state-feedback and output feedback problems. In the state-feedback case it is shown that if this synthesis problem is solvable, then a solution can be chosen to be a constant state-feedback gain. Necessary and sufficient conditions for the existence of solutions as well as a formula for a state-feedback gain that solves this control problem are obtained in terms of a finite dimensional convex feasibility program. After showing the separation properties of this synthesis problem, the output feedback case is reduced to a state-feedback problem. It is shown that, in the output feedback case, generalized H 2 controllers can be chosen to be observer based controllers. The theory is demonstrated with a numerical example.

The standard ℋ2-optimal control problem: a polynomial solution

In this paper we present a solution to the ‘standard’ ℋ2 control problem. The solution presented here is based upon the algebra of polynomials, and in particular we use the polynomial equation approach. In considering the scalar version of the standard ℋ2 problem we derive a couple of linear polynomial (diophantine) equations which define the optimal regulator. The coefficients of these equations are obtained from polynomial spectral factorization.

The singular H2 control problem

In this paper we discuss the standard H 2 control problem for linear, finite-dimensional, time-invariant systems without any assumptions on the system parameters. We give an explicit formula for the infimum over all internally stabilizing strictly proper compensators and give a characterization when the infimum is attained.