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     

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     

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     

A modification of the Nevanlinna algorithm on the solution of theH∞m×n-optimization problem

A modification of the Nevanlinna algorithm is presented for the solution of the H^∞_m×m-optimization problem. This is an essential modification when the Nevanlinna algorithm is applied to find the extremal function of the matrix H^∞ interpolation problem which deals with the case that the interpolation points have multiplicities. For the computation of the minimal H^∞-norm, a pure matrices computational algorithm is used where the formulation is solely in terms of the coefficients of the inner function     

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     

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     

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     

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     

CACSD using the state-space L∞ theory-adesign example

The L^∞ optimal control theory is used to achieve singular value loop-shaping design objectives in a multivariable aircraft pitch-axis control design example involving an unstable nonminimum phase plant. Comparison to the frequency-weighted LQG (Linear quadratic Gaussian) design shows that the L^∞ theory easily produced a superior design having both higher bandwidth and greater stability margin, with no need for designer interaction for iterative adjustment of weights, than LQG. The design example provides a vivid illustration of how the all-pass property of the L^∞ theory enables precise shaping of closed-loop singular-value Bode plots to conform pointwise to prespecified weighting functions     

On computing the spectral radius of the Hankel plus Toeplitzoperator

The square root of the spectral radius of the Hankel plus Toeplitz operator has been shown to be the achievable performance of the mixed-sensitivity H^∞ design. The computation of the spectral radius is the bottleneck in the synthesis of the H ^∞ controller. In this paper, the spectral properties of the Hankel plus Toeplitz operator are investigated. A finite procedure for computing the spectral radius of the Hankel plus Toeplitz operator is proposed     

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     

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     

On discrete spectral factorizations-a unified approach

State-space formulas for all key spectral factorizations appearing in the discrete-time H²/H^∞ optimization are summarized. The factorization problems are categorized into three groups. Solutions are constructed by finding special coprime factors of a given transfer matrix using the associated discrete algebraic Riccati equation. Solution procedures for the three groups are in general the same, which may lead to yield to a unified approach     

Minimization of a regulated response to a fixed input

The problem of minimizing a regulated output due to a specific bounded input is solved in the SISO (single-input single-output) discrete-time case. It is shown that, in most situations, the optimal solution will not exist, but given any ϵ>0, it is possible to find a suboptimal solution with performance within ϵ of the optimal performance. The proposed approach is extended to the problem of shaping the error transient response. It is shown how time-domain templates can be used in this problem in the same way that frequency-weighting functions are used in the H^∞ and H² problems     

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     

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     

H∞ control for nonlinear systems withoutput feedback

The basic question of nonlinear H^∞ control theory is to decide, for a given two-port system, when feedback that makes the full system dissipative and internally stable exists. This problem can also be viewed as a question about circuits, and, after translation, also has a game-theoretic statement. Several necessary conditions for solutions to exist are presented, and sufficient conditions for a certain construction to lead to a solution are given