Download Control Systems with Saturating Inputs: Analysis Tools and by Maria Letizia Corradini, Andrea Cristofaro, Fabio Giannoni, PDF

By Maria Letizia Corradini, Andrea Cristofaro, Fabio Giannoni, Giuseppe Orlando

Saturation nonlinearities are ubiquitous in engineering structures: each actual actuator or sensor is topic to saturation because of its greatest and minimal limits. enter saturation is an working situation that's popular to the keep an eye on neighborhood for its “side effects”, which reason traditional controllers to lose their closed-loop functionality in addition to keep watch over authority in stabilization. consequently, the sensible software of keep watch over conception can't keep away from making an allowance for saturation nonlinearities in actuators, explicitly facing constraints on top of things design.

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19) ∗ θ j φ (k j , x− 0,M∗ , −M ), with j∗ ∈ [0, N + 1]. Note that the elements of S∞ obtained as limk→−∞ φ (k, x, M), for some x, belong to ∂ S∞ . 19) that x∗ (t) = + j∗ ∑ θ j φ (k + k j , x+0,M∗ , M∗ ) + j=1 N+1 ∑ j= j ∗ +1 ∗ θ j φ (k + k j , x− 0,M∗ , −M ) satisfies the system equations for initial datum x∗0 and control input u∗ (k) = M ∗ ∗ ∗ (∑ jj=1 θ j − ∑N+1 j= j ∗ +1 θ j ). This means that any point in ZM lies on a trajectory associated to a suitable admissible control input. As a consequence, from the invariance + ∗ ∗ of the closed curve Φ−M∗ (x− 0,M∗ ) ∪ ΦM (x0,M∗ ), the invariance of the whole set ZM can be deduced.

14) turn out to be the end-points of any reversed-time trajectory corresponding to u ≡ ±M also for systems having only complex eigenvalues. The points x± 0 play a central role in the description of the null controllable region BM . A unified approach for systems having both real and complex eigenvalues is presented. 4. 14) and consider the countable set of points {γ (k)}k≤0 = φ (k, x± , ±M) . For s ∈ N with s < n there is no s-dimensional 0 k≤0 affine subspace containing γ . Proof. , in ) = [wi1 wi2 ··· win ] .

T→−∞ Fixing u∗1 ≡ M, there are 2m−1 possible combinations for the other components of an extremal control u ∈ UM∗∗ ; in this way, we can define a set of 2m−1 points m xM,u = A−1 B1 + ∑ ±Bi M i=2 such that ∗∗ lim φ (t, ∓xM,u , ±u) = ±xM,u ∀u ∈ UM,+ t→−∞ ∗∗ := {u ∈ U ∗∗ : u = M}. For u ∈ U ∗∗ we define the where it has been set UM,+ 1 M M,+ family of curves γM,u (t) := φ (t, −xM,u , u), t ∈ (−∞, 0] and the corresponding supports ΓM,u := {x ∈ Rn : x = φ (t, −xM,u , u), t ∈ (−∞, 0]}. 1. There exists at least one extremal control u¯ ∈ UM,+ s < n there is no s-dimensional affine subspace containing γM,u¯ (t).

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