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# 04.Automatic Control by John G. Webster (Editor)

By John G. Webster (Editor)

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If at some point x the BLS can move in ρ(x) directions, this must remain true at all points that the trajectories can reach from x. ) Due to the radial scaling properties of BLS, the Lie rank actually needs to be checked only on the unit sphere and is the same at antipodal points. If g satisﬁes the condition in equation 14, it is called transitive on Rn0 ; see the “Matrix Groups” section to see why the word transitive is used. To check a BLS for transitivity, ﬁnd the n × n minors of Bx; these are nth degree polynomials.

Uk ], and the transition k matrix satisﬁes ˙ = (A + u B ) . Concatenation is j=1 j j 4 Bilinear Systems deﬁned in the same way as for scalar controls: u o v is u followed by the translate of v. The time-invariance of the BLS leads to useful properties of the transition matrices. The transition matrix depends on the control u and its starting time, so the matrix should be labeled accordingly as (u; t, t0 ), and the state trajectory corresponding to u is x(t) = (u; t, t0 )x(0) Given two controls and their basic intervals {u, 0 < t < σ} and {v, 0 < t < τ}, the composition property for BLS transition matrices can be written in a nice form that illustrates concatenation (u followed by the translate of v) (vσ ; τ, σ) (u; σ, 0) = (u ◦ v; τ, 0) (8a) A transition matrix always has an inverse, but it is not always a transition matrix for the BLS.

Recent work on hybrid systems (ﬁnite-state machines interacting with continuous plants) also falls into the category of switched linear systems. STABILIZATION I: CONSTANT CONTROLS This section will introduce an important engineering design goal, stability, and the beginning of a running discussion of stabilization. Stabilization is an active area of research, in an effort to ﬁnd good design principles. , R(λi ) < − < 0. Then x˙ = F x is said to be exponentially stable (ES); as time increases, all solutions are bounded and ||x(t)|| < ||x(0)||e−t .