On the Practical Stability of Hybrid Control Algorithm With Minimum Dwell Time for a DC–AC Converter

Abstract

This paper presents a control law based on Hybrid Dynamical Systems (HDS) theory for a dc-ac converter. This theory is very suited for analysis of power electronic converters, since it combines continuous (voltages and currents) and discrete (on-off state of switches) signals avoiding, in this way, the use of averaged models. Here, practical stability results are induced for this tracking problem, ensuring a minimum dwell-time associated with an LQR performance level during the transient response and an admissible chattering around the operating point. The effectiveness of the resultant control law is validated by means of simulations and experiments.

Global Practical Stabilization of Discrete-time Switched Affine Systems via Switched Lyapunov Functions and State-dependent Switching Functions

“…The problem of practical stabilization of switched systems without common equilibria or in particular case of switched affine systems has been investigated in the last years either in a general problem formulation or as in the case of particular problems most notably those related to the switching power converters [12], [35], [36]. There are very few works for the global practical stabilization of discrete-time switched affine systems via multiple Lyapunov functions and using state-dependent switching functions.…”

“…hj [(A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k)]) < 0, ∀h ∈ K(31) Since β hi ≥ 0, i, h ∈ K, and ∑ i∈K β hi > 0 according toLemma 10 and (31) one can conclude e(k) T P h e(k) > 1 ⇒ ∃i ∈ K such that ∑ j∈K λ hj [(A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k)] < 0, ∀h ∈ K(32)Again according to Lemma 10, since λ hj ≥ 0, j, h ∈ K and∑ j∈K λ hj > 0, Relation (32) implies e(k) T P h e(k) > 1 ⇒ ∃i, j ∈ K such that (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0, ∀h ∈ K (33)Now, according to the definition of the set V in (3), we havee(k) / ∈ V ⇒ ∃h ∈ K such that e(k) T P h e(k) > 1(34)From Relations(33) and(34), one can infere(k) / ∈ V ⇒ ∃i, j ∈ K such that (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0(35)since e(k) / ∈ V, according to Relation (35) and item 2) in the switching Algorithm 9, one can conclude there exist σ(e(k)), i, j ∈ K satisfying the following expression (A σ(e(k)) e(k) + l σ(e(k)) ) T P i (A σ(e(k)) e + l σ(e(k)) ) − e(k) T P σ(e(k)) e(k) ≤(A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k)(36)From Relations(35) and(36) one can conclude that there exist σ(e(k)), i ∈ K such thate(k) / ∈ V ⇒ ∃i, j, σ(e(k)) ∈ K such that (A σ(e(k)) e(k)+l σ(e(k)) ) T P i (A σ(e(k)) e(k)+l σ(e(k)) )−e(k) T P σ(e(k)) e(k) ≤ (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0 (37)or equivalentlye(k) / ∈ V ⇒ ∃i, j ∈ K such that v(e(k + 1)) − v(e(k)) = ∆v(e(k)) ≤ (A j e(k) + l j ) T P i (A j e(k) + l j ) − e(k) T P j e(k) < 0 (38) since v(e(k)) ≥ min i∈K λ min (P i )∥e(k)∥ 2 > 0 when e(k) ̸ = 0 n×1 , then according item (d) of Lemma 6 the continuous, nondecreasing and positive definite scalar function w(∥e(k)∥) can be taken as w(∥e(k)∥) = min i∈K λ min (P i )∥e(k)∥ 2 . Moreover, according to item (c) of Lemma 6 we still need to find a nondecreasing and positive definite function ϕ(∥e(k)∥) such that ∆v(e(k)) ≤ −ϕ(∥e(k)∥) < 0 when e(k) ∈ D − V. In this regard, we define ϕ i,j (e(k)) = e(k) T P j e(k) − (A j e(k) + l j ) T P i (A j e(k) + l j ), (i, j) ∈ S (39) S = {(i, j) ∈ K × K|∆v(e(k)) ≤ −ϕ i,j (e(k)) < 0} (40) According to Relation (38), |S| ≥ 1 and ϕ i,j (e(k)) > 0 when e(k) / ∈ V. Now we define the function ϕ(s) as ϕ(s) = inf s ≤ ∥e(k)∥ The function ϕ(∥e(k∥) is nondecreasing and positive definite when e(k) / ∈ V. Moreover, according to Eq.…”

Robust hybrid control law for a boost inverter

“…Proof. From the proof given in [16] and without loss of generality, we consider that the first jump occurs in t = t 0 , definingt := t − t 0 . Then, the trajectories of the error dynamics of (2), given byx = x − x e flowing in C, as follows,…”

“…To deal with this objective, we assume that the variation of the desired reference with respect to time is sufficiently slow, in order to obtain a small enough tracking error. Moreover, we consider a minimum dwell time in the hybrid control scheme, by introducing a time regulation [16]. Stability properties of the tracking error in a small neighbourhood of zero are guaranteed, applying the theory for hybrid system presented in [12].…”

“…[16] Consider that Property 1 is satisfied, then the eigenvalues of matrix P −1 Q are positive and e Avt ≤ λ −αt , whit α = λ m (P −1 Q).…”