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 platforms: each actual actuator or sensor is topic to saturation as a result of its greatest and minimal limits. enter saturation is an working situation that's renowned to the keep an eye on group for its “side effects”, which reason traditional controllers to lose their closed-loop functionality in addition to regulate authority in stabilization. accordingly, the sensible software of keep watch over idea can't stay away from considering saturation nonlinearities in actuators, explicitly facing constraints on top of things design.

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Extra info for Control Systems with Saturating Inputs: Analysis Tools and Advanced Design

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This concludes the proof. 3 The MIMO Case As for continuous-time systems, the description of the null controllable for discretetime MIMO systems can be obtained with a direct method as well as by a method based on lower-order single input subsystems. 4, some details have been omitted here. e. , n, of the matrix A. We will use the following notation: ⎡ ⎤ u1 ⎢ u2 ⎥ ⎢ ⎥ B = [B1 B2 · · · Bm ], u = ⎢ . ⎥ , ⎣ .. , m. , m, as follows ⎧ ⎨ x(k + 1) = Ax(k) + Bi ui (k) Σi = ⎩ x(0) = x0 Let ni ∈ N be the dimension of the controllability subspace for the system Σi , ni := rank[Bi ABi A2 Bi · · · An−1 Bi ] ≤ n.

0 0 0 . (i) (i) (i) a1 a2 a3 · · · 0 0 .. 1 0 .. 0 0 .. ⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ 1 ⎦ (i) a ni ⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢ .. ⎥ ∗ ¯ Q−1 ⎥. 3, we can compute the null controllable region for ¯ i , B¯ i ); let us denote this region by B (i) . The description of B (i) with the system (A M M respect to the original coordinates of the system Σi is given by the inverse transformation (i) (i) Ri Hi BM × {0n−ni } := DM , where the matrix Hi ∈ Rn×n is defined as Hi := Qi 0(n−ni)×n 0ni ×(n−ni) . I(n−ni ) (i) By construction, the set DM is a convex set contained in a linear subspace Vi ⊂ Rn with dim(Vi ) = ni .

Moreover each component of v(ε ) has a finite number of switches in the interval [0, T ]. Proof. Using the explicit formula for the solution, the statement of the lemma is equivalent to the existence of a bang-bang control v(ε ) ∈ UM∗ with a finite number of switchings such that T J ε := 0 eA(T −s) B(u(s) − v(ε ) (s))ds ≤ ε . Consider now a general partition Πr = {0 = t1 < t2 < ... 4 The MIMO Case 27 where Hi are constant n × m matrices to be fixed and χ(tk ,tk+1 ) (t) are standard characteristic functions.

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