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     

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     

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     

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 the structure of H∞ control systemsand related extensions

An existence condition of a H^∞ controller which achieves a prescribed norm bound of the closed-loop transfer function is derived in the frequency domain based on a generalization of the notion of J-lossless systems. This condition is regarded as a frequency-domain representation of the well-known existence condition in the state space represented in terms of the two algebraic Riccati equations. A notion of J-orthogonal complement, which is introduced as a generalization of the usual orthogonal complement, play an important role in clarifying the fundamental frequency domain structure of the model-matching problem and simplifying the computation of controllers. The results are extended to the nonstandard case where the direct feedthrough from the input to the error or from the exogenous signal to the output is no longer of full rank. It is shown that the proper controller achieving the prescribed norm bound exists even in this case. In nonstandard cases, the controller order can be smaller than the plant order     

Filtering and smoothing in an H∞ setting

The problems of filtering and smoothing are considered for linear systems in an H^∞ setting, i.e. the plant and measurement noises have bounded energies (are in L₂), but are otherwise arbitrary. Two distinct situations for the initial condition of the system are considered; the initial condition is assumed known in one case, while in the other the initial condition is not known but the initial condition, the plant, and measurement noise are in some weighted ball of RⁿXL₂. Finite-horizon and infinite-horizon cases are considered. Necessary and sufficient conditions are presented for the existence of estimators (both filters and smoothers) that achieve a prescribed performance bound, and algorithms that result in performance within the bounds are developed. In case of smoothers, the optimal smoother is also presented. The approach uses basic quadratic optimization theory in a time-domain setting, as a consequence of which both linear time-varying and time-invariant systems can be considered with equal ease. (In the smoothing problem, for linear time-varying systems, one considers only the finite-horizon case)     

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     

H2-optimal sampled-data control

The H₂-optimal control of continuous-time linear time-invariant systems by sampled-data controllers is discussed. Two different solutions, state space and operator theoretic, are given. In both cases, the H₂ sampled-data problem is shown to be equivalent to a certain discrete-time H₂ problem. Other topics discussed include input-output stability of sampled-data systems, performance recovery in digital implementation of analog controllers, and sampled-data control of systems with the possibility of multiple-time delays     

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     

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     

A state-space approach to super-optimalH∞ control problems

A numerically stable algorithm is presented for solving the strengthened (or super-optimal) model-matching problem. The steps of the algorithm follows closely those given by N.J. Young (1986) except that for each step a reliable implementation using state-space models is provided     

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     

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     

Robust H∞ control for linear systems withnorm-bounded time-varying uncertainty

Robust H_∞ control design for linear systems with uncertainty in both the state and input matrices is treated. A state feedback control design which stabilizes the plant and guarantees an H_∞-norm bound constraint on disturbance attenuation for all admissible uncertainties is presented. The robust H_∞ control problem is solved via the notion of quadratic stabilization with an H_∞ -norm bound. Necessary and sufficient conditions for quadratic stabilization with an H_∞-norm bound are derived. The results can be regarded as extensions of existing results on H_∞ control and robust stabilization of uncertain linear systems     

Robust parameter adjustment with nonparametricweighted-ball-in-H∞ uncertainty

Using a parameter adjustment law, robust monotonic parameter error reduction is proved for a linear parameter estimation problem in which the signal is corrupted by the presence of nonparametric dynamical uncertainty. The nonparameterized dynamics are characterized by a known frequency-domain magnitude bound. A nonconservative bound on the finite-time energy of the nonparametric-dynamics output is constructed in real time. When the error single energy is small enough to be due to nonparametric dynamics alone, the parameter adjustment is shut off to avoid misadjustment. The approach is an extension of existing adaptive control error-dead-zone ideas to the case of frequency-domain bounded uncertainty     

H∞-optimal control with state-feedback

A H_∞-optimal control problem in which the measured outputs are the states of the plant is considered. The main result shows that the infimum of the norm of the closed-loop transfer function using linear static state-feedback equals the infimum of the norm of the closed-loop transfer function over all stabilizing dynamic (even, nonlinear time-varying) state-feedback controllers     

One-step extension approach to optimal Hankel-norm approximationand H∞-optimization problems

A methodology is presented for Hankel approximation and H ^∞-optimization problems that is based on a new formulation of a one-step extension problem which is solved by the Sarason interpolation theorem. The parameterization of all optimal Hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an Hermitian matrix composed directly from the coefficients of a given transfer function matrix φ. Rather than starting with the state-space realization of φ, the authors use polynomial coefficients of φ as input data. In terms of these data, a natural basis is given for the finite dimensional Sarason model space and all computations involve only manipulations with finite matrices     

Some computational results on size reduction inH∞ design

The authors explore the properties of the algebraic Riccati equation. New results on the rank of the solution to the algebraic Riccati equation are obtained and lead to an efficient algorithm for reducing the size of the antistable transfer function matrix in the model-matching problem. In the 1-block problem, the method is taken further to obtain explicit formulas for a minimal realization of the matrix. The approach is direct and numerically reliable as it is based almost entirely on orthogonal transformations