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which a simple observability analysis shows to be observable Based on the measurement y = 2, the estimator would produce x1 = 2, which is incorrect and biased from its correct value by the amount H2 x2 In subsequent chapters it will be suggested that unobservable variables be combined or dropped in the quest to produce a near optimal estimation scheme with a reasonable amount of computation When this is done, the new combined variable should be used with caution In no case should this be interpreted as selection of the unobservable states (subspace) The unobservable subspace can only be a ected by changing either the actual system dynamics or the sensor suite One method to compute numerically a basis for the observable and unobservable subspaces is to use the singular value decomposition (SVD) Let the SVD of O be represented as O = V U The matrix is diagonal with ro non-zero elements Let V be decomposed as V = [V1 , V2 ] where V1 Rn ro and V2 Rn (n ro ) The columns of V1 form an orthogonal basis for the range space of O which is also an orthogonal basis for the rows of O Therefore, the columns of V1 serve as a basis for the observable subspace The columns of V2 are orthogonal to each other and to the columns of V1 Therefore, the columns of V2 serve as a basis for the unobservable subspace For exercises related to this topic, see Exercises 311 and 415 More detailed examples of the de nition of unobservable spaces are contained on pages 350, 427, and 449

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the antenna, located some distance from the transmitter This involves the use of an RF transmission line The most common type is coaxial cable Two-wire line is also sometimes used At ultrahigh and microwave frequencies, another kind of transmission line, known as a waveguide, is often employed

There are several good sources for the material from this chapter At the undergraduate level, two good references are [94, 105] At the graduate level, two good references are [38, 87] The main sources for Sections 362 and 363 of this chapter were [28] and Sections 824, 83, and 101 in [19] Observability, and its dual concept, controllability are discussed in relation to model reduction in [101]

Exercise 31 The purpose of this exercise is to supply the details of the derivation of the transfer function to state space transformation on p 67

CHAPTER 3 DETERMINISTIC SYSTEMS 1 Manipulate eqn (312) to have the form V (s)sn + d1 V (s)sn 1 + + dn V (s) = U (s) 2 Use inverse Laplace transforms to show that the di erential equation relating u(t) to v(t) is v (n) + d1 v (n 1) + + dn v = u (390)

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