Stabilization under a Strict Completeness Condition

The first step towards constructing a state-dependent switching rule for (5.5) is partitioning the state-space into appropriate subregions. In order to do so, the following definition is needed.

Definition 5.2.1. [177] The set of matrices {Φ_i}^m_i=1 is said to be strictly complete if for every x ∈ Rⁿ\ {0}, there exists a j ∈ P such that x^TΦ_jx < 0.

Remark 5.2.3. If there exist α_i ≥ 0 such that Pm

i=1α_i = 1 and Pm

i=1α_iΦ_i is negative definite² then the set {Φ_i} is strictly complete. If m = 2 then the condition is also necessary [177].

Then we construct the switching regions as in, for example, [71,72,153]:

Υ_i = {x ∈ Rⁿ : x^TΦ_ix < 0}.

The main idea behind such a partition is that if the solution x(t) of (5.5) is in Υ_i, then it is possible to show that the time-derivative of a Lyapunov functional along the solution is negative definite if the i^th subsystem is active. If this is true then the state-dependent switching rule associated with this partition should activate the i^thmode of (5.5) whenever the solution state is in the region Υ_i.

The strict completeness of the set {Φ_i} is sufficient and necessary to ensure that Rⁿ=

∪^m_i=1Υi [72]. That is, the union of the switching regions fully covers Rⁿ. However, there may be ambiguity here in how to switch the system if the regions Υ_i overlap. To deal with this, we follow the approach in the literature mentioned above by letting

Υ¯₁ = Υ₁, Υ¯_i = Υ_i\ ∪ⁱ⁻¹_j=1Υ¯_j, i = 2, 3, . . . , m.

In this construction, the regions ¯Υ_icannot overlap, and so we are in a position to construct the first state-dependent switching rule algorithm as follows.

Algorithm 5.2.1. (Strict completeness rule) Given an initial state x₀ = φ₀(0):

(SC1) Set σ(x₀) = i where i is the index such that x₀ ∈ ¯Υ_i.

(SC2) Remain in the i^th mode as long as the state x(t) remains in ¯Υ_i.

2The symmetric matrix A ∈ R^n×n is said to be negative definite if x^TAx <0 for all x ∈ Rⁿ\ {0}.

(SC3) If x(t) crosses the boundary of ¯Υ_i at t_c, set x₀ = x(t_c) and go to step (SC1).

Remark 5.2.4. In Algorithm 5.2.1, the initial mode is chosen by determining which of the switching regions ¯Υ_i the initial state lies in (this is unambiguous by construction of the regions). The state trajectory evolves according to the initial subsystem until a time t_c where x(t⁻_c) ∈ ¯Υ_σ(x0) and x(t_c) /∈ ¯Υ_σ(x0). This means the state trajectory has crossed the boundary (either by continuous dynamics or due to an impulse) and the next mode is chosen depending on which region the trajectory has entered. The process is repeated.

Since the focus of this section is on linear systems with nonlinear perturbations, the following nonlinearity assumption is made on the functionals F_i(t, x_t).

Assumption 5.2.1. Assume that there exist nonnegative constants η₁, η₂, and η₃ such that for t ≥ t₀ and ψ ∈ P C([−τ^∗, 0], Rⁿ),

kF_i(t, ψ)k² ≤ η₁kψ(0)k²+ η₂kψ(−r)k²+ η₃ Z 0

−τ

kψ(s)k²ds. (5.6)

A lemma is required, which follows from the Matrix Cauchy Inequality (for example, see [72]).

Lemma 5.2.1. For any w, z ∈ Rⁿ, the inequality w^Tz + z^Tw ≤ w^Tw + z^Tz always holds.

We are now in position to state and prove the first result.

Theorem 5.2.2. Suppose that Assumption 5.2.1 holds and suppose that there exist con-stants λ > 0, ρ_k > 0, α_i > 0 satisfying Pm

i=1α_i = 1, positive definite symmetric matrices P , R, S, and symmetric constant matrices D_k such that for k ∈ N,

(i) Pm

i=1α_iΦ_i is negative definite where

Φ_i =A^T_i P + P A_i+ λP + P²+ η₁I + R + τ S

+ P B_i(e^−λrR − η₂I)⁻¹B_i^TP + τ P C_i(e^−λτS − η₃I)⁻¹C_i^TP ; (5.7) (ii) (e^−λrR − η₂I) and (e^−λτS − η₃I) are positive definite matrices;

(iii) for each T_k and for all x ∈ Rⁿ,

2Q^T_k(T_k, x)P (I + E_k)x + Q^T_k(T_k, x)P Q_k(T_k, x) ≤ x^TD_kx; (5.8)

(iv) T_k− T_k−1≥ ρ_k and

ln δ_k− λρ_k < 0 (5.9)

where

δk= max{1, λmax[P⁻¹((I + Ek)^TP (I + Ek) + Dk)]}. (5.10) Then it follows that the trivial solution of system (5.5) is globally asymptotically stable under the state-dependent switching rule σ(x) following Algorithm5.2.1.

Proof. Define a Lyapunov functional V (x_t) = V₁+ V₂+ V₃ where, time-derivative along solutions of (5.5) for t 6= T_k,

V˙₁ =

Hence,

V = x˙ ^T(t)(A^T_ikP + P Ai_k)x(t) + 2x^T(t)P Bi_kx(t − r) + 2F_i^T_k(t, xt)P x(t) + 2x^T(t)P C_ik

Z t t−τ

x(s)ds + x^T(t)Rx(t) − e^−λrx^T(t − r)Rx(t − r) + τ x^T(t)Sx(t) −

Z t t−τ

e^{−λ(t−θ)}x^T(θ)Sx(θ)dθ − λ(V2+ V3), and so,

V = x˙ ^T(t)(A^T_ikP + P Ai_k + R + τ S)x(t) − e^−λrx^T(t − r)Rx(t − r) + 2x^T(t)P Bi_kx(t − r) + 2F_i^T_k(t, xt)P x(t)

− Z t

t−τ

e^−λ(t−s)x^T(s)Sx(s) − 2x^T(t)P C_ikx(s)
ds − λ(V₂+ V₃).

For a positive definite matrix S, constants τ > 0 and λ > 0, and t ≥ t₀,

−Rt

t−τe^{−λ(t−θ)}x^T(θ)Sx(θ)dθ ≤ − Z t

t−τ

e^−λτx^T(θ)Sx(θ)dθ,

= −e^−λτ Z t

t−τ

x^T(θ)Sx(θ)dθ.

Using this fact along with Lemma5.2.1, V ≤ x˙ ^T(t)(A^T_i

kP + P A_ik + R + τ S)x(t) − (e^−λrx^T(t − r)Rx(t − r)

− 2x^T(t)P B_ikx(t − r)) + F_i^T

k(t, x_t)F_ik(t, x_t) + x^T(t)P²x(t)

− Z t

t−τ

(e^−λτx^T(s)Sx(s) − 2x^T(t)P C_ikx(s))ds − λ(V₂+ V₃).

Applying the nonlinearity assumption in equation (5.6),

V ≤ x˙ ^T(t)(A^T_ikP + P Ai_k+ R + τ S)x(t) − (e^−λhx^T(t − r)Rx(t − r)

− 2x^T(t)P B_ikx(t − r)) + η₁x^T(t)x(t) + η₂x^T(t − r)x(t − r) + η₃

Z t t−τ

x^T(s)x(s)ds + x^T(t)P²x(t)

− Z t

t−τ

(e^−λτx^T(s)Sx(s) − 2x^T(t)P Ci_kx(s))ds − λ(V2+ V3),

= x^T(t)(A^T_ikP + P Ai_k + λP + P²+ η1I + R + τ S)x(t) + x^T(t)[P B_ik[e^−λrR − η₂I]⁻¹(P B_ik)^T

+ τ P C_ik[e^−λτS − η₃I]⁻¹(P C_ik)^T]x(t)

− [(e^−λrR − η₂I)x(t − r) − (P B_ik)^Tx(t)]^T[e^−λrR − η₂I]⁻¹

× [(e^−λrR − η₂I)x(t − r) − (P B_ik)^Tx(t)]

− Z t

t−τ

[(e^−λτS − η₃I)x(s) − (P C_ik)^Tx(t)]^T(e^−λτS − η₃I)⁻¹

× [(e^−λτS − η₃I)x(s) − (P C_ik)^Tx(t)]ds − λV.

Thus,

V ≤ x˙ ^T(t)(A^T_i

kP + P A_ik + λP + P²+ η₁I + R + τ S)x(t) − λV

+ x^T(t)[P B_ik(e^−λrR − η₂I)⁻¹(P B_ik)^T]x(t) (5.11) + x^T(t)[τ P Ci_k(e^−λτS − η3I)⁻¹(P Ci_k)^T]x(t),

= −λV + x^T(t)Φ_ikx(t).

According to the state-dependent switching rule, the solution x(t) ∈ ¯Υ_ik for t ∈ [t_k−1, t_k), t 6= T_k. Therefore x^TΦ_ikx < 0 and so

V ≤ −λV.˙ (5.12)

Define m(t) = V (x_t), then for t ∈ [T_k, T_k+1),

m(t) ≤ m(T_k) exp[−λ(t − T_k)]. (5.13) For any symmetric matrix Q, x^TQx ≤ λ_max(P⁻¹Q)x^TP x for any positive definite matrix

P . Using this fact, observe that immediately after an impulse is applied at t = T_k, V1(Tk) =x^T(T_k⁻)[(I + Ek)^TP (I + Ek)]x(T_k⁻)

+ Q^T_k(Tk, x(T_k⁻))P Q(Tk, x(T_k⁻)) + Q^T_k(T_k, x(T_k⁻))P (I + E_k)x(T_k⁻) + x^T(T_k⁻)(I + E_k)^TP Q^T_k(T_k, x(T_k⁻)),

≤x^T(T_k⁻)[(I + E_k)^TP (I + E_k) + D_k]x(T_k⁻)

≤δ_kV₁(T_k⁻),

Further, by the continuity of V2 and V3, V2(Tk) = V2(T_k⁻) and V3(Tk) = V3(T_k⁻). Since δ_k ≥ 1, it follows that m(T_k) ≤ δ_km(T_k⁻). Thus equation (5.13) implies

m(t) ≤m(T_k⁻)δ_kexp[−λ(t − T_k)] (5.14) for t ∈ [T_k, T_k+1). Apply equation (5.14) successively on subintervals:

m(T₁⁻) ≤ m(t₀) exp[−λ(T₁− t₀)]

so that m(T1) ≤ m(t0)δ1exp[−λ(T1− t0)]. Similarly,

m(T₂) ≤ m(t₀)δ₁δ₂exp[−λ(T₂− t₀)] = m(t₀)Cδ₂exp[−λ(T₂− T₁)], where C = δ₁exp[−λ(T₁− t₀)]. In general, for t ∈ [T_k, T_k+1),

m(t) ≤m(t₀)Cδ₂· · · δ_kexp[−λ(T₂− T₁) − . . . − λ(T_k− T_k−1) − λ(t − T_k)],

=m(t0)C exp[ln δ2+ . . . + ln δk− λ(T2− T1) − . . . − λ(t − Tk)],

≤m(t₀)C exp[ln δ₂− λρ₂+ . . . + ln δ_k− λρ_k] exp[−λρ^∗]

where ρ^∗ = min{p_k}. From equation (5.9), there exists a constant χ > 0 such that ln δ_k− λρ_k ≤ −χ.

Then,

m(t) ≤m(t₀)M ζ^k−1, (5.15)

where M = C exp[−λρ^∗] and ζ = exp[−χ] satisfies 0 < ζ < 1. Evaluated along the initial function φ₀,

V (φ₀) =φ₀(0)^TP φ₀(0) + Z t0

t0−r

e^−λ(t−s)φ^T₀(s)Rφ₀(s)ds +

Z τ 0

Z t0

t0−s

e^{−λ(t−θ)}φ^T₀(θ)Sφ₀(θ)dθds,

≤λ_max(P )kφ₀k²_τ^∗+ λ_max(R)kφ₀k²_τ^∗ Z t0

t0−r

e^λsds + λmax(S)kφ0k²_τ^∗

Z τ 0

Z t0

t0−s

e^λθdθds,

=(c2+ c3)kφ0k²_τ^∗ where c2 = λmax(P ) and

c₃ = 1 − e^−λr λ



λ_max(R) + λτ + e^−λτ − 1 λ²



λ_max(S) > 0. (5.16) Also, c₁kψ(0)k² ≤ V (ψ) for all ψ ∈ P C([−τ^∗, 0], Rⁿ) where c₁ = λ_min(P ). Hence equation (5.15) implies that for t ∈ [Tk, Tk+1),

kx(t)k² ≤ c₂+ c₃ c1

kφ₀k²_τ∗M ζ^k−1,

where 0 < ζ < 1. Therefore, the trivial solution of system (5.5) is globally asymptotically stable.

Remark 5.2.5. Since δ_k ≥ 1, Theorem 5.2.2 can be interpreted as stabilizing state-dependent switching control that is robust to impulsive disturbances. From equation (5.9), a bound on the total disturbance of the impulses can be found:

1 ≤ δ_k < exp[λρ_k].

Another way to interpret this result is by observing that the time between successive impulses must be sufficiently large:

T_k− T_k−1 > ln δ_k λ .

Remark 5.2.6. The switching rule σ : Rⁿ → P constructed according to Algorithm 5.2.1 can be written as σ : [t_k−1, t_k) → P where the switching times t_k = t_k(x) are state-dependent.

To establish a result for the stabilizing impulsive case (where the switching control may not be adequate in stabilizing the system by itself), a lemma is required.

Lemma 5.2.3. [124]

Assume that there exist V₁ ∈ ν₀, V₂ ∈ ν_{P C}^∗ , positive constants λ, T, ζ, c₁, c₂, c₃, and δ_k ≥ 0, such that for k ∈ N,

(i) c₁kxk² ≤ V₁(t, x) ≤ c₂kxk², 0 ≤ V₂(t, ψ) ≤ c₃kψk²_τ∗, for all t ≥ t₀, x ∈ Rⁿ, ψ ∈ P C([−τ^∗, 0], Rⁿ);

(ii) along solutions of (5.5) for t 6= T_k,

D⁺V (t, ψ) ≤ λV (t, ψ) where V (t, x_t) = V₁(t, x) + V₂(t, x_t);

(iii) V₁(T_k, (I + E_k)x + Q_k(T_k, x)) ≤ δ_kV₁(T_k⁻, x) for all x ∈ Rⁿ; (iv) τ^∗ ≤ T_k− T_k−1 ≤ T and ln(δ_k+ c₃/c₁) + λT ≤ −ζT .

Then the trivial solution of system (5.5) is exponentially stable under the switching rule σ.

Proof. Follows immediately from the proof of Theorem 3.1 in [124] with gk(t, x) = Ekx + Q_k(t, x) if the upper right-hand derivative satisfies D⁺V (t, ψ) ≤ λV (t, ψ) along the switch-ing rule σ.

We are now in a position to present the next result.

Theorem 5.2.4. Suppose that Assumption 5.2.1 holds and suppose that there exist con-stants λ > 0, ζ > 0, T > 0, α_i > 0 satisfying Pm

i=1α_i = 1, positive definite symmetric matrices P , R, S, and symmetric constant matrices D_k such that for k ∈ N,

(i) Pm

i=1αiΦi is negative definite where

Φ_i =A^T_i P + P A_i− λP + P²+ η₁I + R + τ S

+ P B_i(e^λrR − η₂I)⁻¹B_i^TP + τ P C_i(S − η₃I)⁻¹C_i^TP ; (5.17) (ii) (e^λrR − η₂I) and (S − η₃I) are positive definite matrices;

(iii) for each T_k and for all x ∈ Rⁿ, Then the trivial solution of system (5.5) is exponentially stable under the state-dependent switching rule σ(x) constructed from Algorithm 5.2.1.

Proof. Define a Lyapunov functional V (x_t) = V₁+ V₂+ V₃ where, assumption in equation (5.6), and the fact that for λ > 0, τ > 0, and a positive definite matrix S,

then similar to the proof of Theorem 5.2.2 it is possible to show that along solutions of (5.5),

From the state-dependent switching rule construction, x(t) ∈ ¯Υ_ik for t ∈ [t_k−1, t_k). Thus, 5.2.3are satisfied and hence the trivial solution of (5.5) is exponentially stable.

In document Qualitative Theory of Switched Integro-differential Equations with Applications (Page 142-151)