The discrete time H∞H_\infty control problem with measurement feedback

This paper is concerned with the discrete time $H_\infty$ control problem with measurement feedback. It turns out that, as in the continuous time case, the existence of an internally stabilizing controller which makes the $H_\infty$ norm strictly less than 1 is related to the existence of stabilizing solutions to two algebraic Riccati equations. However in the discrete time case the solutions of these algebraic Riccati equations have to satisfy extra conditions. Keywords: $H_\infty$ control, Discrete time, Algebraic Riccati equation, Measurement feedback

The singular minimum entropy H∞H_\infty control problem

In this paper we search for controllers which minimize an entropy function of the closed loop transfer matrix under the constraint of internal stability and under the constraint that the closed loop transfer matrix has $H_\infty$ norm less than some a priori given bound $\gamma$. We find an explicit expression for the infimum. Moreover, we give a characterization when the infimum is attained (contrary to the regular case, for the singular minimum entropy $H_\infty$ control problem the infimum is not always attained).Keywords: $H_\infty$ control, algebraic Riccati equation, quadratic matrix inequality, minimum entropy

The discrete time H∞H_\infty control problem : the full-information case

This paper is concerned with the discrete time, full-information $H_\infty$ control problem. It turns out that, as in the continuous time case, the existence of an internally stabilizing controller which makes the $H_\infty$ norm strictly less than 1 is related to the existence of a stabilizing solution to an algebraic Riccati equation. However the solution of this algebraic Riccati equation has to satisfy an extra condition. Moreover it is interesting to note that in general state feedbacks do not suffice and we have to include the disturbance in our feedback. Keywords: Discrete time, Algebraic Riccati equation, $H_\infty$ control, Full information, Static feedback

The singular zero-sum differential game with stability using H∞H_\infty control theory

In this paper we consider the time-invariant, finite-dimensional, infinite-horizon, linear quadratic differential game. We will derive sufficient conditions for the existence of (almost) equilibria as well as necessary conditions. Contrary to all classical references we allow for singular weighting on the minimizing player in the cost-criterion. It turns out that this problem has a strong relation with the singular $H_\infty$ problem with state feedback, i.e. the $H_\infty$ problem where the direct feed through matrix from control input to output is not necessarily injective

The singular H∞H_\infty control problem with dynamic measurement feedback

This paper is concerned with the $H_\infty$ problem with measurement feedback. The problem is to find a dynamic feedback from the measured output to the control input such that the closed loop system has an $H_\infty$ norm strictly less than some a priori given bound $\gamma$ and such that the closed loop system is internally stable. Necessary and sufficient conditions are given under which such a feedback exists. The only assumptions we have to make is that there are no invariant zeros on the imaginary axis for two subsystems. Contrary to recent publications no assumptions are made on the direct feed through matrices of the plant. It turns out that this problem can be reduced to an almost disturbance decoupling problem with measurement feedback and internal stability, i.e. the problem in which we can make the $H_\infty$ norm arbitrarily small. Keywords: Quadratic matrix inequality, Riccati equation, Almost disturbance decoupling, Measurement feedback, Internal stability

Mixed H2/H∞H_2/H_\infty control in a stochastic framework

This paper deals with a mixture of H_2 and H_\infty. We have two inputs and one output. One input signal is a white noise stochastic process, and represents errors e.g. resulting from measurement noise. The other input has a more deterministic character. If one has a reference signal (e.g. a step) as input one can not model this as white noise, but it fits nicely into this new class of inputs. The objective is to minimize the effect of these exogenuous signals on the output ofthe system. We define a cost function which enables us to combine the structural difference between these two exogenuous inputs. The analysis of this function leads to a standard H_\infty Riccati equation. We will motivate this cost function by looking at two theoretical applications: the derivation of robust performance bounds and a tracking problem

Continuity properties of solutions to H2H_2 and H∞H_\infty Riccati equations

In H2 and H8 optimal control (semi-) stabilizing solutions of algebraic Riccati equations play an essential role. It is well-known that these solutions might have discontinuities as a function of the system parameters. The paper shows that these discontinuities are directly linked to nonleft-invertibility and, in contrast to what one might think, unrelated to zeros on the imaginary axi

Sampled-data and discrete-time H2H_2 optimal control

This paper deals with the sampled-data H2 optimal control problem. Given a linear time-invariant continuous-time system, the problem of minimizing the H2 performance over all sampled-data controllers with a fixed sampling period can be reduced to a pure discrete-time H2 optimal control problem. This discrete-time H2 problem is always singular. Motivated by this, in this paper we give a treatment of the discrete-time H2 optimal control problem in its full generality. The results we obtain are then applied to the singular discrete-time H2 problem arising from the sampled-data H2 problem. In particular, we give conditions for the existence of optimal sampled data controllers. We also show that the H2 performance of a continuous-time controller can always be recovered asymptotically by choosing the sampling period sufficiently small. Finally, we show that the optimal sampled-data H2 performance converges to the continuous-time optimal H2 performance as the sampling period converges to zero