Nonlinear L1L_1 optimal controllers for linear systems

In this paper we study the $L_1$ optimal control problem for linear systems. We will show that by allowing the class of controllers to include nonlinear controllers we can make the closed loop $L_1$ norm strictly smaller then we could do using only linear controllers. Keywords: $L_1$ optimal control, Linear systems, Nonlinear controllers

Stabilizing solutions of the H∞H_\infty algebraic Riccati equation

The algebraic Riccati equation studied in this paper is related to the suboptimal state feedback $H_\infty$ control problem. It is parameterized by the $H_\infty$ norm bound $\gamma$ we want to achieve. The objective of this paper is to study the behaviour of the solution to the Riccati equation as a function of $\gamma$. It turns out that a stabilizing solution exists for all but finitely many values of $\gamma$ larger than some a priori determined boundary $\gamma_{-}$. On the other hand for values smaller than $\gamma_{-}$ there does not exist a stabilizing solution. The finite number of exception points turn out to be switching points where eigenvalues of the stabilizing solution can switch from negative to positive with increasing $\gamma$. After the final switching point the solution will be positive semi-definite. We obtain the following interpretation: the Riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exist a static feedback such that the closed loop transfer matrix has no more than k unstable poles and an $L_\infty$ norm strictly less than $\gamma$. Keywords: The $H_\infty$ control problem, The Algebraic Riccati Equation, J-spectral factorization, Wiener-Hopf factorization

l1l_1 state estimation for linear systems using non-linear observers

In this paper we study the l/sub 1/ optimal estimation problem for linear systems. It has been shown in the literature that in contrast with the state feedback problem it is in general sufficient to consider only linear controllers. However these estimators can have very high complexity. In this paper we study whether by using nonlinear estimators we can obtain estimators of a lower complexity

Stabilizing solutions of the H∞H_\infty algebraic Riccati equation

The algebraic Riccati equation studied in this paper is related to the suboptimal state feedback H/sub /spl infin// control problem. It is parameterized by the H/sub /spl infin// norm bound /spl gamma/ we want to achieve. The objective of this paper is to study the behaviour of the solution to the Riccati equation as a function of /spl gamma/. It turns out that a stabilizing solution exists for all but finitely many values of /spl gamma/ larger than some a priori determined boundary /spl gamma/*. On the other hand for values smaller than /spl gamma/* there does not exist a stabilizing solution. The finite number of exception points turn out to be switching points where eigenvalues of the stabilizing solution can switch from negative to positive with increasing /spl gamma/. After the final switching point the solution will be positive semi-definite. We obtain the following interpretation: the Riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exist a static feedback such that the closed loop transfer matrix has no more than k unstable poles and an L/sub /spl infin// norm strictly less than /spl gamma/

Robust stabilization of systems with multiplicative perturbations

In this paper we discuss robust stabilization of systems subject to multiplicative perturbations. For a given ball of perturbed systems around our nominal model, we find necessary and sufficient conditions under which we can find a controller which stabilizes each system in this ball. We give an explicit formula for a number \gamma* with the property: the radius of the ball is less than \gamma* if and only if there exists one controller which stabilizes all systems in the ball. Therefore \gamma* is the best we can do. Moreover, under an extra assumption and if they exist, we derive an explicit formula for a controller which stabilizes all systems in the given ball. Keywords: Roo control, algebraic Riccati equation, quadratic matrix inequality

The H∞H_\infty control problem with zeros on the boundary of the stability domain

In this paper we study the discrete and continuous time $H_\infty$ control problems without any assumptions on the system parameters. Our approach yields necessary and sufficient conditions for the existence of a suitable controller. Keywords: $H_\infty$ optimal control, Riccati equation, discrete time systems, continuous time systems

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 follows that, as in the continuous time case, the existence of an internally stabilizing controller that 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 must satisfy extra conditions

Completion of the squares in the finite horizon H∞H^\infty control problem by measurement feedback

In this paper we study the finite horizon version of the standard $H^\infty$ control problem by measurement feedback. Given a finite-dimensional linear, time-invariant system, together with a positive real number $\gamma$, we obtain necessary and sufficient conditions for the existence of a possibly time-varying dynamic compensator such that the $L_2 [0,T]$-induced norm of the closed loop operator is smaller than $\gamma$. These conditions are expressed in terms of a pair of quadratic differential inequalities, generalizing the well-known Riccati differential equations that were introduced recently in the context of finite horizon $H^\infty$ control

Sampled-data and discrete-time H2 optimal control

This paper deals with the sampled-data H/sub 2/ optimal control problem. Given a linear time-invariant continuous-time system, the problem of minimizing the H/sub 2/ performance over all sampled-data controllers with a fixed sampling period can be reduced to a pure discrete-time H/sub 2/ optimal control problem. This discrete-time H/sub 2/ problem is always singular. Motivated by this, in this paper the authors give a treatment of the discrete-time H/sub 2/ optimal control problem in its full generality. The results obtained are then applied to the singular discrete-time H/sub 2/ problem arising from the sampled-data H/sub 2/ problem. In particular, the authors give conditions for the existence of optimal sampled data controllers. It is also shown that the H/sub 2/ performance of a continuous-time controller can always be recovered asymptotically by choosing the sampling period sufficiently small. Finally, it is shown that the optimal sampled-data H/sub 2/ performance converges to the continuous time optimal H/sub 2/ performance as the sampling period converges to zero