Sayed, A.H., V.H. Nascimento and S. Chandrasekaran (1998). Estimation and control with bounded data uncertainties. In: Linear Algebra and its Applications. Vol. 284. Elsevier. pp. 259--306.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....norms on both terms of (2.4) for the Frobenius norm re ects the fact that the 2 norm is self dual. This problem has been considered by Chandrasekaran et al. [1] 2] who give an ecient method of solution. Least squares orthogonality results, and the special case of Theorem 2. 5, are considered in [9]. When p = q = 2 and (2.4) is di erentiable, 2.7) becomes (A T A I)x = A T b; where from (2.8) krk 2 kxk 2 : Let the singular value decomposition of A be A = U 0 V T ; where U 2 R m m and V 2 R n n are orthogonal and = diagf 1 ; n g is the matrix ....

Sayed, A. H., Nascimento, V. H. and S. Chandrasekaran, Estimation and control with bounded data uncertainties, Lin. Alg. Appl. 284, pp. 259-306. G A Watson Department of Mathematics University of Dundee Dundee DD1 4HN Scotland gawatson@mcs.dundee.ac.uk

....As mentioned before, the special case Q = 0 and W = I was treated in [3, 4] by different methods; one uses SVD techniques while the other uses LMI techniques. For this special case, a geometric framework that is similar in nature to the geometry of least square problems was also developed in [11, 12]. 2.2. A Special Case: Uncertainties in Factored Form. Consider now a problem of the form ; 2.5) where the perturbations fffiA; ffibg are assumed to satisfy a model of the form (2.6) where S is an arbitrary contraction, kSk 1, and fH; E a ; E b g are known quantities of ....

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran. Estimation and control with bounded data uncertainties. Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

....can be. Observe further that the strength of the opponent can vary with the choice of x. We shall assume in the sequel that H and OE(x) are not identically zero, H 6= 0 and OE( Delta) 6= 0; 4) since if either is zero, the game problem (3) trivializes to (1) A special case of (3) was studied in [31] [34] with the choices W = I , Q = 0, H = I , and OE(x) jkxk. It turns out, however, that for treating the state space estimation problem of this paper one has to allow for nontrivial choices of fW; Q; Hg, as well as for more general choices of OE(x) The problem in this general case is ....

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran. Estimation and control with bounded data uncertainties. Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

....1. Both A and b are assumed known with A called the data matrix and b the measurementvector. The solution of (1) is x = Q A T WA] 1 A T Wb: 2) In practice, the nominal data fA# bg are often subject to uncertainties. Such errors can degrade the performance of the estimator (2) see [9]. This motivated us to introduce in [8] a generalization of (1) that can account for uncertainties in fA# bg. Thus let J(x# y) denote a cost function of the form J(x# y) x T Qx R(x# y)# where R(x# y) Delta = Ax ; b Hy T W Ax ; b Hy Here H is an N Thetam known matrix and y ....

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran. Estimation and control with bounded data uncertainties. Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

.... ffix k ) T w o v(k) 2) where ffix k denotes the uncertainty in x k . In this context, x k represents the nominal value of the actual regression vector and ffix k represents the unknown disturbance to x k . Motivated by the formulation in [1] for design problems with uncertainties (see also [2] [4] for related applications) we consider two models for the uncertainty fffix k g: 1. Uncertainty in factored form. In this first model we assume that the uncertainty lies along a certain direction, i.e. it satisfies a relation of the form ffix k = aE a for all k; 3) where E a is a known ....

.... simply minimizing a quadratic cost function, we are minimizing its worst possible residual in the presence of the uncertainties (which corresponds to solving a min max problem) To solve this problem we rely on the least squares estimation theory with uncertain data developed in [1] see also [2] [4] and [5] for related discussions and applications. To proceed, we introduce the quantities, w Delta = w k 1 Gamma w k e(k) Delta = d(k) Gamma x T k w k ; which are defined in terms of the given data fx k ; d(k)g and the weight estimates fw k ; w k 1 g, as well as the following ....

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran, "Estimation and control with bounded data uncertainties," Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

....information about bounds on the sizes of the uncertainties in the model. A key feature of the BDU formulation is that geometric insights (such as orthogonality conditions and projections) which are widely appreciated for classical quadratic cost designs, can be pursued in this new framework (see [1]) Consider the cost function J(x) x T Qx R(x) where x T Qx is a regularization term, while the residual cost R(x) is defined by (a more general case is treated 1 This material was based on work supported in part by the National Science Foundation under Award No. CCR 9732376. The work ....

....trying to pick an x that minimizes the cost while the opponents fffiA; ffibg try to maximize the cost. The game problem is constrained since it imposes a limit on how large (or how damaging) the opponents can be. The case Q = 0 and W = I , and variations thereof, were treated in detail in [1, 3, 4] with several applications in image processing, digital communications, and estimation in [1, 4] It turns out that, unlike standard least squares theory, solving a weighted problem of the form (1) is more complex (and also more rich) than solving the unweighted version (with W = I) 2 SOLUTION ....

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A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran. Estimation and control with bounded data uncertainties, Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

....A and b are assumed known with A called the data matrix and b the measurement vector. The solution of (1) is x = Q A T WA] Gamma1 A T W b: 2) In practice, the nominal data fA; bg are often subject to uncertainties. Such errors can degrade the performance of the estimator (2) see [9]. This motivated us to introduce in [8] a generalization of (1) that can account for uncertainties in fA; bg. Thus let J(x; y) denote a cost function of the form J(x; y) x T Qx R(x; y) where R(x; y) Delta = Ax Gamma b Hy T W Ax Gamma b Hy Here H is an N Thetam known ....

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran. Estimation and control with bounded data uncertainties. Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

....problem with multiple opponents of different strengths. While the solution can be obtained algebraically, we resort instead to geometric arguments (such as orthogonality conditions and projections) These arguments provide powerful insights into the nature of the solution (as explained in [7] for the single source 2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 1, PP. 680 692, MARCH 2000 case) and they also establish important connections with classical least squares theory where such geometric insights are prevalent. II. The Least Squares Method and the BDU Formulation In ....

....of unavoidable experimental errors when collecting data, or even due to unknown or unmodeled effects. Regardless of their source, they can degrade the performance of least squares designs. Several examples to this effect, and comparisons with alternative robust design methods are provided in [7] and also later in this paper (Sec. III) B. The BDU Formulation Motivated by the above discussion, we formulate below a new optimization problem. Thus let A be a given N Theta n matrix, which we shall refer to as the nominal matrix or the nominal data. Assume further that A is partitioned ....

[Article contains additional citation context not shown here]

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran, Estimation and control with bounded data uncertainties, Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

....are widely known and appreciated for classical quadratic cost designs, can be pursued in this new framework. More details on these aspects can This material was based on work supported in part by the National Science Foundation under Awards MIP 9796147, CCR 9732376, and CCR 9734290. be found in [1]. Also, algorithms for computing optimal solutions with the same computational effort as standard least squares solutions exist, thus making the new formulations attractive for practical use. An SVD based solution is developed rather fully in [2] In this paper, we introduce the following ....

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran, Estimation and control with bounded data uncertainties, Linear Algebra and Its Applications, vol. 284, pp. 259--306, Nov. 1998.

....of unmodeled or unknown effects. Regardless of their source, modeling errors can adversely affect the performance of otherwise optimal estimators. This fact has motivated recent works on robust leastsquares methods, especially in Chandrasekaran et al. 1997,1998) Ghaoui and Lebret (1997) and Sayed et al. 1998,1999,2000) In the works by Sayed et al. 1998,1999,2000) general cost functions that allow for different levels 1 Published in Proc. of the IFAC System Identification Symposium, Santa Barbara, CA, June 2000. This work was partially support by NSF ECS 9820765 and by ARO ....

....of their source, modeling errors can adversely affect the performance of otherwise optimal estimators. This fact has motivated recent works on robust leastsquares methods, especially in Chandrasekaran et al. 1997,1998) Ghaoui and Lebret (1997) and Sayed et al. 1998,1999,2000) In the works by Sayed et al. 1998,1999,2000) general cost functions that allow for different levels 1 Published in Proc. of the IFAC System Identification Symposium, Santa Barbara, CA, June 2000. This work was partially support by NSF ECS 9820765 and by ARO DAAH04 96 1 0176 P00005. The work of AlNaffouri was also supported by ....

Sayed, A. H., V. H. Nascimento, and S. Chandrasekaran (1998). Estimation and control with bounded data uncertainties. Linear Algebra and Its Applications, 284, pp. 259--306.

....A and b are assumed known with A called the data matrix and b the measurement vector. The solution of (1) is x = Q A T WA] Gamma1 A T W b: 2) In practice, the nominal data fA; bg are often subject to uncertainties. Such errors can degrade the performance of the estimator (2) see [7]. This motivated us to introduce in [8] a generalization of (1) that can account for uncertainties in fA; bg. Thus let J(x; y) denote a cost function of the form J(x; y) x T Qx R(x; y) where R(x; y) Delta = Ax Gamma b Hy T W Ax Gamma b Hy Here H is an N Thetam known ....

A. H. Sayed, V. H. Nascimento, and S. Chandrasekaran. Estimation and control with bounded data uncertainties. Linear Algebra and Its Applications, vol. 284, pp. 259-- 306, Nov. 1998.

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Sayed, A.H., V.H. Nascimento and S. Chandrasekaran (1998). Estimation and control with bounded data uncertainties. In: Linear Algebra and its Applications. Vol. 284. Elsevier. pp. 259--306.

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