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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 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 an eye on authority in stabilization. for that reason, the sensible program of regulate concept can't steer clear of making an allowance for saturation nonlinearities in actuators, explicitly facing constraints up to the mark design.

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We will use the following notation: ⎡ ⎤ u1 ⎢ u2 ⎥ ⎢ ⎥ B = [B1 B2 · · · Bm ], u = ⎢ . ⎥ , ⎣ .. , m. , m, as follows ⎧ ⎨ x˙ = Ax + Bi ui 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. 24 2 Estimation of the Null Controllable Region: Continuous-Time Plants The system Σi can be transformed into a controllable/uncontrollable subsystems decomposition by a linear coordinates transformation; let Ri ∈ Rn×n the matrix associated to such transformation.

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 . 5. , m . 4 The MIMO Case 25 (i) Proof. The set ∑m i=1 DM , as it is defined as a finite sum of convex sets, is still con(i) (i) m vex (see for instance [33]).

Denote this set as FM ; let us point out that, if ν < n, then FM is an unbounded set. 12), if a trajectory starts from a point outside the set FM , there is no admissible control that can drive it inside. 3. , n. Then BM ⊆ FM . 14) where the sum represents a linear combination of the euclidean basis vectors ei . The solution of the reversed-time system (k ≤ 0) x∗ (k − 1) = A−1 x∗ (k) − A−1 Bu(k) with the control input set to the saturation level u(k) ≡ ±M verifies the identity |k| x∗ (k) = A−|k| x0 ∓ ∑ A−i BM.

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