2,814 research outputs found
Computation of H∞ controllers for infinite dimensional plants using numerical linear algebra
The mixed sensitivity minimization problem is revisited for a class of single-input-single-output unstable infinite dimensional plants with low order weights. It is shown that H∞ controllers can be computed from the singularity conditions of a parameterized matrix whose dimension is the same as the order of the sensitivity weight. The result is applied to the design of H∞ controllers with integral action. Connections with the so-called Hamiltonian approach are also established. © 2012 John Wiley & Sons, Ltd
System-level, Input-output and New Parameterizations of Stabilizing Controllers, and Their Numerical Computation
It is known that the set of internally stabilizing controller
is non-convex, but it admits convex
characterizations using certain closed-loop maps: a classical result is the
{Youla parameterization}, and two recent notions are the {system-level
parameterization} (SLP) and the {input-output parameterization} (IOP). In this
paper, we address the existence of new convex parameterizations and discuss
potential tradeoffs of each parametrization in different scenarios. Our main
contributions are: 1) We first reveal that only four groups of stable
closed-loop transfer matrices are equivalent to internal stability: one of them
is used in the SLP, another one is used in the IOP, and the other two are new,
leading to two new convex parameterizations of . 2)
We then investigate the properties of these parameterizations after imposing
the finite impulse response (FIR) approximation, revealing that the IOP has the
best ability of approximating given FIR
constraints. 3) These four parameterizations require no \emph{a priori}
doubly-coprime factorization of the plant, but impose a set of equality
constraints. However, these equality constraints will never be satisfied
exactly in numerical computation. We prove that the IOP is numerically robust
for open-loop stable plants, in the sense that small mismatches in the equality
constraints do not compromise the closed-loop stability. The SLP is known to
enjoy numerical robustness in the state feedback case; here, we show that
numerical robustness of the four-block SLP controller requires case-by-case
analysis in the general output feedback case.Comment: 20 pages; 5 figures. Added extensions on numerial computation and
robustness of closed-loop parameterization
Optimal Disturbance Rejection and Robustness for Infinite Dimensional LTV Systems
In this paper, we consider the optimal disturbance rejection problem for
possibly infinite dimensional linear time-varying (LTV) systems using a
framework based on operator algebras of classes of bounded linear operators.
This approach does not assume any state space representation and views LTV
systems as causal operators. After reducing the problem to a shortest distance
minimization in a space of bounded linear operators, duality theory is applied
to show existence of optimal solutions, which satisfy a "time-varying" allpass
or flatness condition. Under mild assumptions the optimal TV controller is
shown to be essentially unique. Next, the concept of M-ideals of operators is
used to show that the computation of time-varying (TV) controllers reduces to a
search over compact TV Youla parameters. This involves the norm of a TV compact
Hankel operator defined on the space of causal trace-class 2 operators and its
maximal vectors. Moreover, an operator identity to compute the optimal TV Youla
parameter is provided. These results are generalized to the mixed sensitivity
problem for TV systems as well, where it is shown that the optimum is equal to
the operator induced of a TV mixed Hankel-Toeplitz. The final outcome of the
approach developed here is that it leads to two tractable finite dimensional
convex optimizations producing estimates to the optimum within desired
tolerances, and a method to compute optimal time-varying controllers.Comment: 30 pages, 1 figur
H_2-Optimal Decentralized Control over Posets: A State-Space Solution for State-Feedback
We develop a complete state-space solution to H_2-optimal decentralized
control of poset-causal systems with state-feedback. Our solution is based on
the exploitation of a key separability property of the problem, that enables an
efficient computation of the optimal controller by solving a small number of
uncoupled standard Riccati equations. Our approach gives important insight into
the structure of optimal controllers, such as controller degree bounds that
depend on the structure of the poset. A novel element in our state-space
characterization of the controller is a remarkable pair of transfer functions,
that belong to the incidence algebra of the poset, are inverses of each other,
and are intimately related to prediction of the state along the different paths
on the poset. The results are illustrated by a numerical example.Comment: 39 pages, 2 figures, submitted to IEEE Transactions on Automatic
Contro
Approximate robust output regulation of boundary control systems
We extend the internal model principle for systems with boundary control and
boundary observation, and construct a robust controller for this class of
systems. However, as a consequence of the internal model principle, any robust
controller for a plant with infinite-dimensional output space necessarily has
infinite-dimensional state space. We proceed to formulate the approximate
robust output regulation problem and present a finite-dimensional controller
structure to solve it. Our main motivating example is a wave equation on a
bounded multidimensional spatial domain with force control and velocity
observation at the boundary. In order to illustrate the theoretical results, we
construct an approximate robust controller for the wave equation on an annular
domain and demonstrate its performance with numerical simulations.Comment: 29 pages, 4 figure
LMI approach to mixed performance objective controllers: application to Robust â„‹2 Synthesis
The problem of synthesizing a controller for plants subject to arbitrary, finite energy disturbances and white noise disturbances via Linear Matrix Inequalities (LMIs) is presented. This is achieved by considering white noise disturbances as belonging to a constrained set in â„“2. In the case of where only white noise disturbances are present, the procedure reduces to standard â„‹2 synthesis. When arbitrary, finite energy disturbances are also present, the procedure may be used to synthesize general mixed performance objective controllers, and for certain cases, Robust â„‹2 controllers
Linear Control Theory with an ℋ∞ Optimality Criterion
This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods
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