Showing papers in "International Journal of Robust and Nonlinear Control in 1991"

H∞ estimation for discrete-time linear uncertain systems

Abstract: SUMMARY This paper is concerned with the problem of Hm estimation for linear discrete-time systems with timevarying norm-bounded parameter uncertainty in both the state and output matrices. We design an estimator such that the estimation error dynamics is quadratically stable and the induced operator norm of the mapping from noise to estimation error is kept within a prescribed bound for all admissible uncertainties. A Riccati equation approach is proposed to solve the estimation problem and it is shown that the solution is related to two algebraic Riccati equations.

A first principles solution to the non-singular H∞ control problem

Abstract: This paper presents an elementary solution to the non-singular H∞ control problem. In this control problem, the underlying linear system satisfies a set of assumptions which ensures that the solution can be obtained by solving just two algebraic Riccati equations of the game type. This leads to the central solution to the H∞ control problem. The solution presented in this paper uses only elementary ideas beginning with the Bounded Real Lemma.

Branch and bound algorithm for computing the minimum stability degree of parameter‐dependent linear systems

Abstract: We consider linear systems with unspecified parameters that lie between given upper and lower bounds. Except for a few special cases, the computation of many quantities of interest for such systems can be performed only through an exhaustive search in parameter space. We present a general branch and bound algorithm that implements this search in a systematic manner and apply it to computing the minimum stability degree.

Adaptive motion control of robot manipulators: A unified approach based on passivity

Abstract: This paper presents a unified approach to direct adaptive motion control laws for robot manipulators that have been studied during the last few years by several authors. It provides a general approach based on sensivity to demonstrate the global asymptotic stability of adaptive schemes applied to rigid multilinked manipulators. It is shown that most of the schemes fit within this framework, which presents the advantage of being more systematic than other techniques and therefore will enable a unified presentation of the several schemes proposed to date and will increase our understanding of adaptive control of robot manipulators.

Robust stability—degree assignment and its applications to the control of flexible structures

Abstract: A new method of robust controller synthesis is introduced which is an extension of robust stabilization to robust stability-degree assignment. This method guarantees a lower bound of stability degree for all uncertainties within a prescribed magnitude band. The method is applied to the control of four types of flexible structures, namely, colocated beam, non-colocated beam, beam with moving base and large scale satellite model. Excellent performances have been obtained experimentally with respect to the response speed and the robustness.

Robust control of robotic manipulators without velocity feedback

Abstract: This study concerns the problem of robust control of robotic manipulators without joint velocity feedback. A robust lead + bias controller is studied. The bias signal is intended to compensate the nonlinear dynamics of the robot. The focus of this study is robustness when the nonlinear compensation is not perfect and the external disturbances are not negligible. A conservative polynomial bound is introduced to describe the worst feedback effect of the compensation error and the external disturbances. The polynomial bound covers a class of possible bias signals, synthesized according to the available knowledge about the robot dynamics. Based on the polynomial bound, the tracking errors of a lead + bias controller are proved to be uniformly bounded. They can be minimized by a proper design of the bias signal. In the ideal case where the bias signal compensates the robot dynamics perfectly, the tracking errors will converge to zero.

A factorization principle for stabilization of linear control systems

Abstract: By introducing a fictitious signal y0 if necessary we define a transform which generalizes the passage from the scattering to the chain formalism in circuit theory. Given a factorization ˜ = ⊖R of ˜ where R is a block matrix function with a certain key block equal to a minimal phase (or outer) matrix function, we show that a given compensator u = Ky is internally stabilizing for the system if and only if a related compensator K′ is stabilizing for ⊖. Factorizations ˜ = ⊖R with ⊖ having a certain block upper triangular form lead to an alternative derivation of the Youla parametrization of stabilizing compensators. Factorizations with ⊖ equal to a J-inner matrix function (in a precise weak sense) lead to a parametrization of all solutions K of the H∞ problem associated with . This gives a new solution of the H∞ problem completely in the transfer function domain. Computation of the needed factorization ˜ = ⊖R in terms of a state-space realization of leads to the state-space formulas for the solution of the H∞ problem recently obtained in the literature.

Stability radii of linear discrete‐time systems and symplectic pencils

Abstract: In this paper, we introduce and analyse robustness measures for the stability of discrete-time system x(t + 1) = Ax(t) under parameter perturbations of the form AA + BDC where B,C are given matrices. In particular we characterize the stability radius of the uncertain system x(t + 1) = (A + BDC)x(t), D an unknown complex perturbation matrix, via an associated symplectic pencil and present an algorithm for the computation of that radius.

On J-Lossless coprime factorizations and H∞ control

Abstract: This paper presents a relatively simple approach, based on chain scattering descriptions, to the synthesis of internally stabilizing controllers which result in a cost function with H∞ norm strictly less than a prespecified bound. State-space formulae for a generator of all suboptimal controllers are found by solving two coupled J-lossless coprime factorizations of a chain scattering matrix description. A mixed-sensitivity design problem is used to illustrate the solution procedure, and to demonstrate how in certain cases only one algebraic Riccati equation need be solved to get a solution.

H, optimal controller discretization

Abstract: A redesign method for discretizing a continuous-time controller is proposed. The resulting hybrid control system, for example with continuous-time plant and discrete-time controller, is stable, and performance including the system's inter-sampling behaviour can be optimized by approximating some chosen reference transfer function of the continuous-time control system. In order to obtain a tractable problem, the continuous-time part of the hybrid system and the reference transfer function are approximated by a discrete-time system with arbitrary fast sampling. After lifting the resulting periodic system, the approximation problem can be formulated as a standard H∞-problem which is solved using standard software for H∞-controller design.

Allowable parameter variations and robustness recovery in LQG regulators

Abstract: In this paper, the bounds of allowable parameter variations in the LQG regulator for fixed weighting matrices are obtained and the asymptotic properties of these bounds with respect to weighting matrices are analysed. It is shown that the guaranteed bounds of allowable parameter variations, which are independent of weighting matrices, are often small but not necessarily small under some conditions. It is also shown that under some conditions, the allowable bounds of the LQG regulator can become as large as those of the LQ regulator. In addition, a loop transfer recovery method for perturbed systems is derived under which the LQG regulator may possess the same robustness as the LQ regulator. Examples are given to validate these results.

H∞ generalized minimum sensitivity control design

Abstract: This paper presents a two-stage approach to the H∞ generalized sensitivity minimization problem (H∞-GSM) for SISO, continuous-time plants. Besides some possible advantages in the control law evaluation with respect to alternative polynomial methods, the proposed approach provides a direct link with the solution of an underlying generalized minimum variance (H2-GMV) problem and allows one to identify the class of the joint H2/H∞-GMV (equalizing H2-GMV) optimal controllers.

Best analytic approximation of rational functions of degree 2

Abstract: Let G be a rational function of degree 2. Explicit formulae are given for the distance from G to H in the L∞ norm, k = 0, 1, and for the unique Ǧ ϵ H at which the distance is attained.

Combined ℋ︁∞/LQG control via the optimal projection equations: On minimizing the LQG cost bound

Abstract: The Optimal Projection equations for combined 7Yo/LQG control are considered. Positive semidefiniteness of the associated Lagrange multiplier is shown to be necessary for the LQG cost boumd to be minimal. It follows that all four Optimal Projection equations have have a role to play, even in the full-order case.

Quadratic robust tracking control — A Polynomial Matrix Approach

Abstract: A quadratic robust tracking problem is solved using a polynomial matrix approach. Because of the possibly unstable mode of the control sequence we propose a new quadratic cost function of error and control sequences to facilitate the optimization. The optimal controller makes the plant output track the reference sequence robustly and minimizes the proposed quadratic cost function. We also present the parametrized set of suboptimal controllers which yield finite costs so that there exists the central optimal controller in the set when the parameter is set to zero. The usefulness of our proposed cost function is demonstrated by simulation examples. Our design procedure has advantages over two other possible approaches: the LQ state-space approach and the Wiener–Hopf approach which may be tried to tackle the same problem. Our approach obviates the need to construct a dynamic observer compared to the state-space approach and it is computationally attractive compared to the Wiener–Hopf approach.

Best analytic approximation of rational functions of degree 2: Symmetrization of formulae of N. J. Young

Abstract: Formulae, explicitly symmetric in the roots of the denominator, are derived for the best approximation to a rational function of degree 2 by an analytic function and by a function with at most one pole.

A class of optimal solutions of the matrix Nehari‐extension problem

Abstract: The degrees of freedom that are available in the solution set of the multivariable Nehari-extension problem are used to minimize an 2-type cost and an ‘entropy-like’ cost associated with the smaller singular values of the (optimal) error system. The optimal extensions are constructed via a diagonalization procedure based on a (normalized) Schmidt pair of the Hankel operator induced by the function which is approximated.