On the convexity of H∞ Riccati solutionsand its applications

The authors consider the two-Riccati-equation solution to a standard H^∞ control problem, which can be used to characterize all possible stabilizing optimal or suboptimal H^∞ controllers if the optimal H^∞ norm (or γ), an upper bound of a suboptimal H^∞ norm is given. Some eigen properties of these H^∞ Riccati solutions are revealed. The most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of γ on the domain of interest. Based on these properties, a quadratically convergent algorithm is developed to compute the optimal H^∞ norm     

LQG control with an H∞ performance bound:a Riccati equation approach

An LQG (linear quadratic Gaussian) control-design problem involving a constraint on H^∞ disturbance attenuation is considered. The H^∞ performance constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on L₂ performance. In contrast to the pair of separated Riccati equations of standard LQG theory, the H^∞-constrained gains are given by a coupled system of three modified Riccati equations. The coupling illustrates the breakdown of the separation principle for the H^∞ -constrained problem. Both full- and reduced-order design problems are considered with an H^∞ attenuation constraint involving both state and control variables. An algorithm is developed for the full-order design problem and illustrative numerical results are given     

Finite horizon H∞ control problems withterminal penalties

The standard H^∞ control problem is generalized to the finite horizon case with two (possibly singular) terminal penalties at the initial and final times. The major objective of the generalization is to increase flexibility of H^∞ controls. Transients in the finite horizon and the terminal penalties are taken into account within the framework of H^∞ control problems. A complete solution, a necessary and sufficient condition, and a parameterization to the finite horizon H^∞ control problem are given. The solution is a natural extension of the Riccati equation solution. In the special case, when all the terminal penalties vanish, the solution is reduced to the existing one and to the finite horizon standard H^∞ control problem. This approach to the problem is based on completing the square argument of a particular quadratic form, which is at least technically different from the previous ones     

Fast iterative computation of optimal two-blockH∞-norm

The two-block H^∞ optimization problem is discussed, in which the most computationally demanding work is the computation of the optimal H^∞ norm denoted by γ₀. The problem of computing γ₀ can be considered as that of finding a γ such that μ(γ) is equal to 1. Some new properties of μ(γ) are revealed and studied. Based on these properties and those found by C.-C. Chu and J. Doyle (1985) and Z.Z. Wang and J.B. Pearson (1984), a very fast computational algorithm for finding γ₀ is proposed     

Robust stabilization of uncertain linear systems: quadraticstabilizability and H∞ control theory

The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H^∞ optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H^∞ control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H ^∞ control theory-based methods     

A treatment of jw-axis model-matching transformation zerosin the optimal H2 and H∞control designs

A model-matching transformation (MMT) zero is defined as a rank-deficiency condition which prevents an H² or H^∞ optimal control problem from being transformed into an equivalent model-matching problem. By imposing saturation constraints and accounting for additive instrument noise in the sensor and actuator signals, all MMT zeros can be eliminated     

Recursive algorithm for the computation of theH∞-norm of polynomials

A recursive algorithm for computing the H^∞ -norm of polynomials is developed. The algorithm is shown to converge monotonically and the convergence rate is also established. Some examples are presented to illustrate the algorithm     

An H∞ approach to feedback design withtwo objective functions

The problem of tightly bounding and shaping the frequency responses of two objective functions T_i(s)( i=1,2) associated with a closed-loop system is considered. It is proposed that an effective way of doing this is to minimize (or bound) the function max {∥T₁(s)∥ _∞, ∥T₂(s)∥_∞} subject to internal stability of the closed-loop system. The problem is formulated as an H^∞ control problem, and an iterative solution is given     

Optimal H∞ norm computation formultivariable systems with multiple zeros

B. C.Chang and J.B. Peason's (see ibid., vol. AC-29, p.880-7, Oct. 1984) computation of the optimal H^∞ norm is generalized to the case with multiple right half-plane zeros. D. Sarason's (1967) interpolation theory is used to reduce the problem to a simple eigenvalue or singular-value computation     

The explicit structure of inner matrices and its application inH∞-optimization

An expression for inner matrices is obtained which allows a detailed investigation of their structure. The expression is used to solve the H^∞-interpolation problem in a simple way. A solution to this problem is obtained explicitly in terms of interpolation parameters without solving any equation. It is found that the resulting solution encounters a reduction in its order at the critical point, and the characteristics of the resulting reduced-order solution are investigated     

Data reduction in the mixed-sensitivityH∞ design problem

The mixed-sensitivity H^∞ design problem is discussed in the two-block case. The state-space approach is used to investigate the internal structure of this design problem, which inherits many pole-zero cancellations. The complete state-space solution is provided. Efficient mathematical software can then be implemented     

Asymptotic H∞ disturbance attenuationbased on perfect observation

A design method of controllers which ensure internal stability and attain asymptotically H^∞ disturbance attenuation is presented. The design procedure consists of two steps: (1) to design an H^∞ state feedback control via an algebraic Riccati equation approach; and (2) under a certain minimum-phase condition, to recover the achievable performance asymptotically by applying high-gain observers. It makes use of the perfect observation for the design of high-gain observers. It is shown that the asymptotic recovery can be attained by using reduced-order observers, provided there is no direct feedthrough of controls and disturbances in observations     

H∞-control by state-feedback and fastalgorithms for the computation of optimalH∞-norms

The suboptimality of some parameter for H_∞ -optimization by dynamic state-feedback is characterized in terms of the solvability of Riccati inequalities. This is done without restricting the finite zero structure of the plant. If there are no system zeros on the imaginary axis, the H_∞-problem can be treated in a complete and satisfactory way. Explicit characterizations optimum to be achieved are provided, and a closed formula for the optimal value is derived in terms of the H_∞-norm of some fixed transfer matrix. If the optimum is not attained, any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high-gain. A globally and quadratically convergent algorithm to compute the optimal value is proposed. This algorithm is generalized to the H_∞-optimization problem by measurement feedback     

Direct state-space solution to the discrete-timeH∞ one-block problem

A general state-space representation is used to allow a complete formulation of the H_∞ optimization problem without any invertibility condition on the system matrix, unlike existing solutions. A straightforward approach is used to solve the one-block H_∞ optimization problem. The parameterization of all solutions to the discrete-time H_∞ suboptimal one-block problem is first given in transfer function form in terms of a set of functions in H _∞ that satisfy a norm bound. The parameterization of all solutions is also given as a linear fractional representation     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H₂/H_∞ performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n×n and n ×q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H_∞ estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant     

A two-sided interpolation approach to H∞optimization problems

A solution to the two-sided interpolation problem which arises in H_∞ optimization theory is obtained. This solution is found in closed form, explicitly in terms of the required interpolation directions. It is simple to obtain and it does not require the application of the relatively complicated matrix Pick-Nevanlinna theory. The solution obtained is of minimum order; due to its simplicity, the order reduction, which occurs at the minimum value of the H_∞-norm, is clearly explained     

Approximation of infinite-dimensional systems

A Fourier series-based method for approximation of stable infinite-dimensional linear time-invariant system models is discussed. The basic idea is to compute the Fourier series coefficients of the associated transfer function T_d(Z) and then take a high-order partial sum. Two results on H^∞ convergence and associated error bounds of the partial sum approximation are established. It is shown that the Fourier coefficients can be replaced by the discrete Fourier transform coefficients while maintaining H^∞ convergence. Thus, a fast Fourier transform algorithm can be used to compute the high-order approximation. This high-order finite-dimensional approximation can then be reduced using balanced truncation or optimal Hankel approximation leading to the final finite-dimensional approximation to the original infinite-dimensional model. This model has been tested on several transfer functions of the time-delay type with promising results     

An efficient algorithm on optimal Hankel-norm approximation formultivariable systems

This note provides a novel methodology for Hankel approximation and H^∞-optimization problems, based on a new formulation of the one-step extension problem, which is solved by the Sarason interpolation theorem. The present method does not require an initial balanced realization, and the parameterization of all optimal solutions is given in terms of the eigenvalue decomposition of a Hermitian matrix composed directly from the coefficients of the given transfer function matrix     

H∞-minimum error state estimation oflinear stationary processes

A state estimator is derived which minimizes the H_∞-norm of the estimation error power spectrum matrix. Two approaches are presented. The first achieves the optimal estimator in the frequency domain by finding the filter transfer function matrix that leads to an equalizing solution. The second approach establishes a duality between the problem of H_∞-filtering and the problem of unconstrained input H_∞-optimal regulation. Using this duality, previously published results for the latter regulation problem are applied which lead to an optimal filter that possess the structure of the corresponding Kalman filter. The two approaches usually lead to different results. They are compared by a simple example which also demonstrates a clear advantage of the H_∞-estimate over the conventional l ₂-estimate