Optimal solution and approximation to thel1/H∞ control problem

This paper addresses the l₁/H_∞ optimal control problem for a system described by linear time-invariant finite dimensional discrete-time equations. It is shown that a solution to this problem exists and can be approximated arbitrarily by real-rational transfer matrices. Perhaps more interesting from a computational point of view, a bound on the order of a δ-suboptimal solution is also given     

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     

Discrete Fourier transform and H∞ approximation

It is shown that uniform rational approximation of nonrational transfer functions can always be obtained by means of the discrete Fourier transform (DFT) as long as such approximants exist. Based on this fact, it is permissible to apply the fast Fourier transform (FFT) algorithm in carrying out rational approximations without being apprehensive of convergence. The DFT is used to obtain traditional approximations for transfer functions of infinite-dimensional systems. Justification is provided for using the DFT in such approximations. It is established that whenever a stable transfer function can be approximated uniformly on the right half-plane by a rational function, its approximants can always be recognized by means of a DFT     

H∞ optimal control as a weightedWiener-Hopf problem

It is shown that H_∞ optimization is equivalent to weighted H₂ optimization in the sense that the solution of the latter problem also solves the former. The weighting rational matrix that achieves this equivalence is explicitly computed in terms of a state-space realization. The authors do not suggest transforming H_∞ optimization problems to H₂ optimization problems as a computational approach. Rather, their results reveal an interesting connection between H_∞ and H₂ optimization problems which is expected to offer additional insight. For example, H₂ optimal controllers are known to have an optimal observer-full state feedback structure. The result obtained shows that the minimum entropy solution of H_∞ optimal control problems can be obtained as an H₂ optimal solution. Therefore, it can be expected that the corresponding H_∞ optimal controller has an optimal observer-full state feedback structure     

An L∞ error estimate for the Ritz-Trefftz approximation in optimal control

An a priori L_∞ error estimate is established for the control approximation generated by using the Ritz-Trefftz method on the state regulator problem. The approximations are taken from spaces of quasiuniform peicewise-polynomials. The estimates are the best possible in the sense that the order of convergence is optimal.     

Nonlinear H2 and H∞ optimal controllersfor current-fed induction motors

This paper presents a nonlinear control design for both the H₂ and H_∞ optimal control for current-fed induction motor drives. These controllers are derived using analytical stationary solutions that minimize a generalized convex energy cost function including the stored magnetic energy and the coil losses, while satisfying torque regulation control objectives. Explicit control expressions for both the H₂ and H_∞ optimal design are given. Furthermore, the optimal attenuation factor, i.e., the optimal H_∞ norm and the corresponding worst case disturbance, are both computed explicitly     

An optimal approximation algorithm for the rectilinearm-center problem

Given a set ofn points on the plane, the rectilinearm-center problem is to findn rectilinear squares covering all thesen points such that the maximum side length of these squares is minimized. In this paper we prove that there is no polynomial-time algorithm with an error ratio < 2 for the rectilinearm-center problem unless NP = P. A polynomial-time approximation algorithm with an error ratio of 2 is also proposed.This research work was partially supported by a grant from the National Science Council of the Republic of China under Grant NSC-77-0408-E007-03.     

Exact computation of infimum for a class of continuous-time H∞ optimal control problem with a nonzero direct feedthrough term from the disturbance input to the controlled output

A noniterative method for the computation of infimum for a class of continuous-time H_∞ optimal control problem is considered in this paper. The problem formulation is fairly general and does not place any restrictions on any direct feedthrough terms of the given systems. The method is applicable to systems where (i) the transfer function from the disturbance input to the measurement output is free of imaginary axis invariant zeros and left invertible, and (ii) the transfer function from the control input to the controlled output of the given system is free of imaginary axis invariant zeros and right invertible. The result presented in this paper is a continuation of the previous work of the author and his co-workers (Chen et al., 1992), in which the direct feedthrough term from the disturbance input to the measurement output of the given system are required to be zero.     

Robust synthesis in 1: a globally optimal solution

In this paper, a method to synthesize controllers that achieve globally optimal robust performance within any prespecified tolerance, against structured norm-bounded time-varying and/or nonlinear uncertainty is developed. The performance measure considered is the infinity to infinity induced norm of a system's transfer function. The method developed utilizes linear relaxation techniques to solve the infinite dimensional nonconvex problem via finite dimensional linear programming problems     

On L2- and L∞-stability approaches forthe robust control of robot manipulators

Robustness results were recently obtained for the linear and nonlinear control of robot manipulators using the L₂- and L _∞-stability approaches, respectively. Critical remarks on these approaches and further investigations are presented. In order to investigate L₂-stability, it is shown that additional assumptions have to be made and a reformulation of the control problem has to be carried out     

General polynomial solution to the optimal feedback/feedforward stochastic tracking problem

A general solution to the optimal stochastic tracking problem in the presence of both measurable and immeasurable disturbances is obtained using polynomial techniques. The plant under consideration includes a coloured measurement noise signal. In addition, all disturbance and reference sub-systems may be unstable. The problem formulation involves a cost-function which has dynamic weighting elements. A feedforward compensator is incorporated in the optimization procedure for the rejection of the measurable disturbance.     

On the static H∞ control problem

The problem of existence of static controller in standard H_∞ control problem or, equivalently, solvability of a couple of Riccati inequalities with respect to the same Hermitian matrix is reduced to solvability of a single Riccati inequality. This inequality is presented in a form which is bilinear with respect to unknown parameters. In a special case the inequality is transformed to LMI.     

An L∞error bound for the phase approximation problem

Given a random process spectral factorw(cdot), the phase approximation algorithm, initiated by Jonckheere and Helton [1], constructs a reduced-order spectral factorhat{w}(cdot)such thatparallel w/w^{ast}-hat{w}/ hat{w}^{ast}parallelis small in the Hankel-norm sense. In this note, we derive theL^{infty}error boundparallel w/w^{ast} - hat{w}/hat{w}^{ast}parallel_{infty} leq 4(sigma_{k+1} + ... +sigma_{N}), where the σ's are the canonical correlation coefficients. A similar result holds in the multivariable case.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

L∞ approximation and nuclearity of delay systems

The L_∞ convergence of partial-fraction decompositions of single-input-single-output delay systems is studied. Conditions for the nuclearity of such systems are derived, and examples given.     

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     

A simple solution to the optimal eigenvalue assignment problem

The problem of the optimal eigenvalue assignment for multi-input linear systems is considered. It is proven that for an n-order system with m independent inputs, the problem is split into the following two sequential stages. Initially, the n-m eigenvalues are assigned using an (n-m)-order system. This assignment is not constrained to satisfy optimality criteria. Next, an m-order system is used to assign the remaining m eigenvalues in such a way that linear quadratic optimal criteria are simultaneously satisfied. Therefore, the original n-order optimal eigenvalue assignment problem is reduced to an m-order optimal assignment problem. Moreover, the structure of the equivalent m-order system permits further simplifications which lead to solutions obtained by inspection     

Optimal ℒ1 approximation of the Gaussian kernelwith application to scale-space construction

Scale-space construction based on Gaussian filtering requires convolving signals with a large bank of Gaussian filters with different widths. We propose an efficient way for this purpose by ℒ₁ optimal approximation of the Gaussian kernel in terms of linear combinations of a small number of basis functions. Exploring total positivity of the Gaussian kernel, the method has the following properties: 1) the optimal basis functions are still Gaussian and can be obtained analytically; 2) scale-spaces for a continuum of scales can be computed easily; 3) a significant reduction in computation and storage costs is possible. Moreover, this work sheds light on some issues related to use of Gaussian models for multiscale image processing     

An optimal distributed solution to the dining philosophers problem

An optimal distributed solution to the dining philosophers problem is presented. The solution is optimal in the sense that it incurs the least communication and computational overhead, and allows the maximum achievable concurrency. The worst case upper bound for concurrency is shown to ben div 3,n being the number of philosophers. There is no previous algorithm known to achieve this bound.     

An optimal approximation algorithm for the rectilinearm-center problem

Given a set ofn points on the plane, the rectilinearm-center problem is to findn rectilinear squares covering all thesen points such that the maximum side length of these squares is minimized. In this paper we prove that there is no polynomial-time algorithm with an error ratio ? < 2 for the rectilinearm-center problem unless NP = P. A polynomial-time approximation algorithm with an error ratio of 2 is also proposed.