Optimal Synthesis of State-Estimate Feedback Controllers With Minimum $l_{2}$ -Sensitivity
07 Mar 2008-IEEE Transactions on Circuits and Systems I-regular Papers (Institute of Electrical and Electronics Engineers (IEEE))-Vol. 55, Iss: 8, pp 2402-2410
TL;DR: This paper investigates the problem of synthesizing the optimal structure of a state-estimate feedback controller with minimum l 2-sensitivity and no overflow by developing two iterative techniques for obtaining the coordinate transformation matrix.
Abstract: This paper investigates the problem of synthesizing the optimal structure of a state-estimate feedback controller with minimum l 2-sensitivity and no overflow. First, the l 2-sensitivity of a closed-loop transfer function with respect to the coefficients of a state-estimate feedback controller is analyzed. Next, two iterative techniques for obtaining the coordinate transformation matrix which constructs the optimal structure of a state-estimate feedback controller are developed so as to minimize an l 2-sensitivity measure subject to l 2-scaling constraints. One technique is based on a Lagrange function, some matrix-theoretic techniques, and an efficient bisection method. Another technique converts the problem into an unconstrained optimization formulation by using linear-algebraic techniques, and optimizes it by applying an efficient quasi-Newton method with closed-form formula for gradient evaluation. A numerical example is also presented to illustrate the utility of the proposed techniques.
Citations
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TL;DR: This paper presents a systematic study of pole and zero sensitivity minimization for state-space digital filters in several different yet related settings and develops efficient iterative techniques for minimizing this measure by employing a quasi-Newton algorithm and relying on a recursive matrix equation.
Abstract: This paper presents a systematic study of pole and zero sensitivity minimization for state-space digital filters in several different yet related settings. First, a new weighted measure for pole and zero sensitivity for state-space digital filters is proposed and the problem of minimizing this measure is investigated. To this end, two efficient iterative techniques for minimizing this measure are developed by employing a quasi-Newton algorithm and relying on a recursive matrix equation, respectively. Furthermore, minimization of the proposed sensitivity measure subject to $l_{2}$ -scaling constraints is examined by extending the two aforementioned solution methods—one converts the constrained optimization problem at hand into an unconstrained problem and solves it using a quasi-Newton algorithm, while the other relaxes the constraints into a single constraint on matrix trace and solves the relaxed problem with an effective matrix iteration scheme. In addition, a simple yet novel method for the minimization of a zero sensitivity measure subject to minimal pole sensitivity is explored by pursuing an optimal coordinate transformation matrix. Simulation studies are presented to demonstrate the validity and effectiveness of the proposed techniques.
6 citations
Additional excerpts
...Several techniques for minimizing the -sensitivity measure [1]–[6] and the -sensitivity measure [7]–[14] have been proposed....
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TL;DR: The joint optimization problem of high-order error-feedback and realization for minimizing roundoff noise at the output of a closed-loop system with a full-order observer feedback system subject to l2-scaling constraints is dealt with.
Abstract: This paper deals with the joint optimization problem of high-order error-feedback and realization for minimizing roundoff noise at the output of a closed-loop system with a full-order observer feedback system subject to l2-scaling constraints. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. The unconstrained optimization problem at hand is then solved iteratively by employing an efficient quasi-Newton algorithm with closed-form formulas for key gradient evaluation. A case study is presented to demonstrate the validity and effectiveness of the proposed technique.
3 citations
Cites methods from "Optimal Synthesis of State-Estimate..."
...Furthermore, the -sensitivity of a closed-loop system with respect to the coefficients of a full-order observer feedback system has also been analyzed, and two iterative techniques for constructing an optimal full-order observer feedback system which minimizes an -sensitivity measure subject to -scaling constraints have been explored [20]....
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01 Jan 2005
TL;DR: The problem of minimizing an L2-sensitivity measure subject to L2norm dynamic-range scaling constraints for state-space digital filters is formulated and it is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques.
Abstract: The problem of minimizing an L2-sensitivity measure subject to L2norm dynamic-range scaling constraints for state-space digital filters is formulated. It is shown that the problem can be converted into an unconstrained optimization problem by using linear-algebraic techniques. Next, the unconstrained optimization problem is solved by applying an efficient quasi-Newton algorithm with closed-form formula for gradient evaluation. The coordinate transformation matrix obtained is then used to construct the optimal state-space filter structure that minimizes the L2-sensitivity measure subject to the scaling constraints. A numerical example is presented to illustrate the utility of the proposed technique.
References
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9,153 citations
Book•
01 Jan 2009
TL;DR: The aim of this book is to provide a Discussion of Constrained Optimization and its Applications to Linear Programming and Other Optimization Problems.
Abstract: Preface Table of Notation Part 1: Unconstrained Optimization Introduction Structure of Methods Newton-like Methods Conjugate Direction Methods Restricted Step Methods Sums of Squares and Nonlinear Equations Part 2: Constrained Optimization Introduction Linear Programming The Theory of Constrained Optimization Quadratic Programming General Linearly Constrained Optimization Nonlinear Programming Other Optimization Problems Non-Smooth Optimization References Subject Index.
7,278 citations
"Optimal Synthesis of State-Estimate..." refers methods in this paper
...Applying a quasi-Newton algorithm to minimize in (36), in the th iteration the most recent point is updated to point as [ 20 ]...
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Book•
01 Oct 1987
1,529 citations
"Optimal Synthesis of State-Estimate..." refers methods in this paper
...Applying a quasi-Newton algorithm to minimize in (36), in the th iteration the most recent point is updated to point as [20]...
[...]
TL;DR: In this paper, an external specification of a digital filter is investigated via the internal structure of the filter using a state variable formulation, and conditions for minimizing this output noise are established and realizations which meet these conditions are constructed.
Abstract: Beginning with an external specification of a digital filter, structures which minimize roundoff noise are investigated. After fixing the probability of overflow through an l_{2} scaling procedure, roundoff noise is studied via the internal structure of the filter using a state variable formulation. An output noise variance formula in terms of the internal structure is derived. Conditions for minimizing this output noise are established and realizations which meet these conditions are constructed. A new set of filter invariants called second-order modes are defined and shown to play a definitive role in minimal noise realizations. From these invariants, for example, one can calculate the minimal output noise variance of a given external specification. Numerical results are given which compare these new filter structures with the usual parallel and cascade connections of second-order filters, both theoretically and through simulations. For narrow-band filters, these new structures can be orders of magnitude better (in terms of output noise variance). One drawback of these new structures is a large increase in the number of multipliers needed to realize them. However, by applying the theory to subfilters connected in parallel and cascade, a good compromise between output noise and number of multipliers is obtained.
774 citations
Book•
09 Nov 2021
TL;DR: Practical Optimization: Algorithms and Engineering Applications provides a hands-on treatment of the subject of optimization suitable for use in one or two semesters of a first-year graduate course or an advanced undergraduate course.
Abstract: Practical Optimization: Algorithms and Engineering Applications provides a hands-on treatment of the subject of optimization. A comprehensive set of problems and exercises makes the book suitable for use in one or two semesters of a first-year graduate course or an advanced undergraduate course. Each half of the book contains a full semesters worth of complimentary yet stand-alone material. The practical orientation of the topics chosen and a wealth of useful examples also make the book suitable as a reference work for practitioners in the field. Advancements in the efficiency of digital computers and the evolution of reliable software for numerical computation during the past three decades have led to a rapid growth in the theory, methods, and algorithms of numerical optimization. This body of knowledge has motivated widespread applications of optimization methods in many disciplines, e.g., engineering, business, and science, and has subsequently led to problem solutions that were considered intractable not too long ago. Key Features: extensively class-tested provides a complete teaching package with MATLAB exercises and online solutions to end-of-chapter problems includes recent methods of emerging interest such as semidefinite programming and second-order cone programming presents a unified treatment of unconstrained and constrained optimization uses a practical treatment of optimization accessible to broad audience, from college students to scientists and industry professionals provides a thorough appendix with background theory so non-experts can understand how applications are solved from point of view of optimization
584 citations