Now, let r 5 be a positive scalar, then using Fact 1 we have −2x  (t)PA d  0 −τ μ(t + s)B o Ix(t + s)ds = −2x  (t)PA d  0 −τ B o Iz(t + s)ds ≤ τ + r −1 5 x  (t)PA d A  d Px(t)+r 5  0 −τ z  (t + s)I  B  o B o Iz(t + s)ds. (35) Also, if r 6 is a positive scalar, then using Fact 1 we have −2x  (t)PA d  0 −τ E(x, t + s)ds ≤ τ + r −1 6 x  (t)PA d A  d Px(t) + r 6  0 −τ E  (x, t + s)E(x, t + s)ds. (36) It is known that 2μ(t)x  (t)PB o Ix(t) ≤ 2||PB o I|||μ(t) |||x(t)|| 2 . (37) Also, using Assumption 2.1, it can be shown that 2x  (t)PE(x, t) ≤ 2||P|| θ ∗ ||x(t)|| 2 . (38) Using equations (31)- (38) and equations (17)- (24) (with the fact that 0 ≤ τ ≤ τ + ) in (30), we have ˙ V a (x) ≤ x  (t)Ξ x(t)+τ + r 4 x  (t)ΔK  (t)B  o B o ΔK(t)x(t) + τ + r 5 z  (t)I  B  o B o Iz(t)+τ + r 6 E  (x, t)E(x, t)+2ρ ∗ ||PB o || ||x(t)|| 2 +2||PB o I|||μ(t)|||x(t) || 2 + 2θ ∗ ||P|| ||x(t)|| 2 + 2 μ(t) ˙ μ (t). (39) where Ξ = PA od + A  od P + PB o K + K  B  o P + τ + r 1 A  o A o + τ + r 2 A  d A d + τ + r 3 B o KK  B  o +τ +  r −1 1 + r −1 2 + r −1 3 + r −1 4 + r −1 5 + r −1 6  PA d A  d P. (40) To guarantee that x  (t)Ξ x(t) < 0, it sufficient to show that Ξ < 0. Let us introduce the linearizing terms, X = P −1 , Y = KX,andZ = XB o K. Also, let ε 1 = r −1 1 , ε 2 = r −1 2 , ε 3 = r −1 3 , ε 4 = r −1 4 , ε 5 = r −1 5 and ε 6 = r −1 6 . Now, by pre-multiplying and post-multiplying Ξ by X and invoking the Schur complement, we arrive at the LMI (25) which guarantees that Ξ < 0, and consequently x  (t)Ξ x(t) < 0. Now, we need to show that the remaining terms of (39) are negative definite. Using the definition of z (t)=μ(t)x(t), we know that τ + r 5 z  (t)I  B  o B o Iz(t) ≤ τ + r 5 ||I  B  o B o I||μ 2 (t) ||x(t)|| 2 . (41) Also, using Assumptions 2.1 and 2.2 , we have τ + r 6 E  (x, t)E(x, t) ≤ τ + r 6 ( θ ∗ ) 2 ||x(t)|| 2 , (42) and τ + r 4 x  (t)ΔK  (t)B  o B o ΔK( t)x(t) ≤ τ + r 4 ( ρ ∗ ) 2 ||B  o B o || ||x(t)|| 2 . (43) 149 Resilient Adaptive Control of Uncertain Time-Delay Systems Now, using (41)- (43), the adaptive law (26), and the fact that |μ(t)|≥1, equation (39) becomes ˙ V a (x) ≤ x  (t)Ξ x(t)+τ + r 4 ( ρ ∗ ) 2 ||B  o B o || ||x(t)|| 2 + τ + r 5 ||I  B  o B o I||μ 2 (t) ||x(t)|| 2 +τ + r 6 ( θ ∗ ) 2 ||x(t)|| 2 + 2ρ ∗ ||PB o || ||x(t)|| 2 + 2||PB o I|||μ(t) |||x(t)|| 2 +2θ ∗ ||P|| ||x(t)|| 2 + 2α 1 |μ(t)|||x(t)|| 2 + 2α 2 μ 2 (t) ||x(t)|| 2 . (44) It can be easily shown that by selecting α 1 and α 2 as in (27) and (28), we guarantee that ˙ V a (x) ≤ x  (t)Ξ x(t), (45) where Ξ < 0. Hence, ˙ V a (x) < 0 which guarantees asymptotic stabilization of the closed-loop system. 3.2 Adaptive control when θ ∗ is known and ρ ∗ is unknown Here, we wish to stabilize the system (6) considering the control law (3) when θ ∗ is known and ρ ∗ is unknown. Before we present the stability results for this case, let us define ˜ ρ(t)= ˆ ρ (t) −ρ ∗ ,where ˆ ρ(t) is the estimate of ρ ∗ ,and ˜ ρ(t) is error between the estimate and the true value of ρ ∗ . Let the Lyapunov-Krasovskii functional for the transformed system (6) be selected as: V b (x) Δ = V a (x)+V 9 (x), (46) where V a (x) is defined in equations (7), and V 9 (x) is defined as V 9 (x)= ( 1 + ρ ∗ )[ ˜ ρ (t) ] 2 , (47) where its time derivative is ˙ V 9 (x)=2 ( 1 + ρ ∗ ) ˜ ρ (t) ˙ ˜ ρ (t). (48) Since ˜ ρ (t)= ˆ ρ (t) −ρ ∗ ,then ˙ ˜ ρ(t)= ˙ ˆ ρ (t). Hence, equation (48) becomes ˙ V 9 (x)=2 ( 1 + ρ ∗ )[ ˆ ρ (t) −ρ ∗ ] ˙ ˆ ρ (t). (49) The next Theorem provides the main results for this case. Theorem 2: Consider system (6). If there exist matrices 0 < X = X  ∈ n×n , Y∈ m×n , Z∈ n×n , and scalars ε 1 > 0, ε 2 > 0, ε 3 > 0, ε 4 > ε, ε 5 > ε and ε 6 > ε (where ε is an arbitrary small positive constant) such that the LMI (25) has a feasible solution, and K = YX −1 ,andμ(t) and ˆ ρ (t) are adapted subject to the adaptive laws ˙ μ (t)=Proj  [ β 1 sgn ( μ(t) ) + β 2 μ(t)+ β 3 sgn ( μ(t) ) ˆ ρ (t) ] ||x(t)|| 2 , μ(t)  (50) ˙ ˆ ρ (t)=γ ||x(t)|| 2 , (51) where Proj {·} Krstic et al. (1995) is applied to ensure that |μ(t)|≥1 as follows: μ (t)= ⎧ ⎨ ⎩ μ (t) if |μ(t)|≥1 1 if 0 ≤ μ(t) < 1 −1 if −1 < μ(t) < 0, and the adaptive law parameters are selected such that β 1 < − 1 2  τ + r 6 ( θ ∗ ) 2 + 2 ||PB o I|| + 2θ ∗ ||P||  , β 2 < − 1 2 τ + r 5 ||I  B  o B o I||, γ > 1 2 τ + r 4 ||B  o B o ||, 150 Time-Delay Systems β 3 < −γ,and ˆ ρ(0) > 1, then the control law (3) will guarantee asymptotic stabilization of the closed-loop system. Proof The time derivative of V b (x) is ˙ V b (x)= ˙ V a (x)+ ˙ V 9 (x). (52) Following the steps used in the proof of Theorem 1 and using equation (49), it can be shown that ˙ V b (x) ≤ x  (t)Ξ x(t)+τ + r 4 ( ρ ∗ ) 2 ||B  o B o || ||x(t)|| 2 + τ + r 5 ||I  B  o B o I||μ 2 (t) ||x(t)|| 2 +τ + r 6 ( θ ∗ ) 2 ||x(t)|| 2 + 2ρ ∗ ||PB o || ||x(t)|| 2 + 2||PB o I|||μ(t) |||x(t)|| 2 +2θ ∗ ||P|| ||x(t)|| 2 + 2 μ(t) ˙ μ (t)+2 ( 1 + ρ ∗ )[ ˆ ρ (t) −ρ ∗ ] ˙ ˆ ρ (t), (53) where Ξ is defined in equation (40). Using the linearization procedure and invoking the Schur complement (as in the proof of Theorem 1), it can be shown that Ξ is guaranteed to be negative definite whenever the LMI (25) has a feasible solution. Using the adaptive laws (50)- (51) in (53) and the fact that |μ(t)|≥1, we get ˙ V b (x) ≤ x  (t)Ξ x(t)+τ + r 4 ( ρ ∗ ) 2 ||B  o B o || ||x(t)|| 2 + τ + r 5 ||I  B  o B o I||μ 2 (t) ||x(t)|| 2 +τ + r 6 ( θ ∗ ) 2 ||x(t)|| 2 + 2ρ ∗ ||PB o || ||x(t)|| 2 +2||PB o I|||μ(t)|||x(t) || 2 + 2θ ∗ ||P|| ||x(t)|| 2 +2β 1 |μ(t)|||x(t)|| 2 + 2β 2 μ 2 (t) ||x(t)|| 2 + 2β 3 ˆ ρ (t) |μ(t)|||x(t)|| 2 + 2γ ˆ ρ(t) ||x(t)|| 2 −2γρ ∗ ||x(t)|| 2 −2γρ ∗ ˆ ρ (t) ||x(t)|| 2 −2γ ( ρ ∗ ) 2 ||x(t)|| 2 . (54) Using the fact that |μ(t)| > 1 and arranging terms of equation (54), it can be shown that ˙ V b (x) < 0 if we select β 1 < − 1 2  τ + r 6 ( θ ∗ ) 2 + 2 ||PB o I|| + 2θ ∗ ||P||  , β 2 < − 1 2 τ + r 5 ||I  B  o B o I||,andβ 3 < −γ,whereγ needs to be selected to satisfy the following two conditions: γ > 1 2 τ + r 4 ||B  o B o ||, (55) and 2 ||PB o ||−2γ + 2γ ˆ ρ(t) < 0. (56) Hence, we need to select γ such that γ > max  1 2 τ + r 4 ||B  o B o || , ||PB o || 1 − ˆ ρ (t)  . (57) It is clear that when ˆ ρ (t) > 1, we only need to ensure that γ > 1 2 τ + r 4 ||B  o B o ||.Notethatfrom equation (51), ˆ ρ (t) > 1 can be easily ensured by selecting ˆ ρ(0) > 1andγ > 1 2 τ + r 4 ||B  o B o || to guarantee that ˆ ρ(t) in equation (51) is monotonically increasing. Hence, we guarantee that ˙ V b (x) ≤ x  (t) Ξ x(t), (58) where Ξ < 0. Hence, ˙ V b (x) < 0 which guarantees asymptotic stabilization of the closed-loop system. 151 Resilient Adaptive Control of Uncertain Time-Delay Systems 3.3 Adaptive control when θ ∗ is unknown and ρ ∗ is known Here, we wish to stabilize the system (6) considering the control law (3) when θ ∗ is unknown and ρ ∗ is known. Since θ ∗ is unknown, let us define ˜ θ(t)= ˆ θ (t) − θ ∗ ,where ˆ θ(t) is the estimate of θ ∗ ,and ˜ θ(t) is error between the estimate and the true value of θ ∗ . Also, let the Lyapunov-Krasovskii functional for the transformed system (6) be selected as: V c (x) Δ = V a (x)+V 10 (x), (59) where V 10 (x)= ( 1 + θ ∗ )  ˜ θ (t)  2 , (60) where its time derivative is ˙ V 10 (x)=2 ( 1 + θ ∗ ) ˜ θ(t) ˙ ˜ θ (t), = 2 ( 1 + θ ∗ )  ˆ θ (t) −θ ∗  ˙ ˆ θ (t). (61) The next Theorem provides the main results for this case. Theorem 3: Consider system (6). If there exist matrices 0 < X = X  ∈ n×n , Y∈ m×n , Z∈ n×n , and scalars ε 1 > 0, ε 2 > 0, ε 3 > 0, ε 4 > ε, ε 5 > ε and ε 6 > ε (where ε is an arbitrary small positive constant) such that the LMI (25) has a feasible solution, and K = YX −1 ,andμ( t) is adapted subject to the adaptive laws ˙ μ (t)=Proj  δ 1 sgn ( μ(t) ) ||x(t)|| 2 + δ 2 μ(t) ||x(t)|| 2 + δ 3 sgn ( μ(t) ) ˆ θ(t) ||x(t)|| 2 , μ(t)  ,(62) ˙ ˆ θ (t)=κ ||x(t)|| 2 , (63) where Proj {·} Krstic et al. (1995) is applied to ensure that |μ(t)|≥1 as follows μ (t)= ⎧ ⎨ ⎩ μ (t) if |μ(t)|≥1 1 if 0 ≤ μ(t) < 1 −1 if −1 < μ(t) < 0, and the adaptive law parameters are selected such that δ 1 < −  ||PB o I||+ τ + r 4 ( ρ ∗ ) 2 ||B  o B o ||+ ρ ∗ ||PB o ||  , δ 2 < − 1 2 τ + r 5 ||I  B  o B o I||, δ 3 < −κ, κ > 1 2 τ + r 6 and ˆ θ(0) > 1, then the control law (3) will guarantee asymptotic stabilization of the closed-loop system. Proof The time derivative of V c (x) is ˙ V c (x)= ˙ V a (x)+ ˙ V 10 (x). (64) Following the steps used in the proof of Theorem 1 and using equation (61), it can be shown that ˙ V c (x) ≤ x  (t)Ξ x(t)+τ + r 4 x  (t)ΔK  (t)B  o B o ΔK( t)x(t)+τ + r 5 z  (t)I  B  o B o Iz(t) + τ + r 6 E  (x, t)E(x, t)+2ρ ∗ ||PB o || ||x(t)|| 2 + 2||PB o I|||μ(t)|||x(t) || 2 +2θ ∗ ||P|| ||x(t)|| 2 + 2 μ(t) ˙ μ (t)+2 ( 1 + θ ∗ )  ˆ θ (t) − θ ∗  ˙ ˆ θ (t), (65) 152 Time-Delay Systems where Ξ is defined in equation (40). Using the linearization procedure and invoking the Schur complement (as in the proof of Theorem 1), it can be shown that Ξ is guaranteed to be negative definite whenever the LMI (25) has a feasible solution. Now, we need to show that the remaining terms of (65) are negative definite. Using the definition of z (t)=μ(t)x(t), we know that τ + r 5 z  (t)I  B  o B o Iz(t) ≤ τ + r 5 ||I  B  o B o I||μ 2 (t) ||x(t)|| 2 . (66) Also, using Assumptions 2.1 and 2.2 , we have τ + r 6 E  (x, t)E(x, t) ≤ τ + r 6 ( θ ∗ ) 2 ||x(t)|| 2 , (67) and τ + r 4 x  (t)ΔK  (t)B  o B o ΔK( t)x(t) ≤ τ + r 4 ( ρ ∗ ) 2 ||B  o B o || ||x(t)|| 2 . (68) Now, using (66)- (68), the adaptive laws (62)- (63), and the fact that |μ(t)|≥1, equation (65) becomes ˙ V c (x) ≤ x  (t)Ξ x(t)+τ + r 4 ( ρ ∗ ) 2 ||B  o B o || ||x(t)|| 2 + τ + r 5 ||I  B  o B o I||μ 2 (t) ||x(t)|| 2 +τ + r 6 ( θ ∗ ) 2 ||x(t)|| 2 + 2ρ ∗ ||PB o || ||x(t)|| 2 + 2||PB o I|||μ(t)|||x(t) || 2 6 + 2θ ∗ ||P|| ||x(t)|| 2 + 2δ 1 |μ(t)|||x(t)|| 2 + 2δ 2 μ 2 (t) ||x(t)|| 2 +2δ 3 |μ(t)| ˆ θ (t) ||x(t)|| 2 + 2κ |μ(t)| ˆ θ (t) ||x(t)|| 2 −2κθ ∗ ||x(t)|| 2 +2κθ ∗ ˆ θ(t) ||x(t)|| 2 −2κ ( θ ∗ ) 2 ||x(t)|| 2 . (69) It can be shown that ˙ V c (x) < 0 if the adaptive law parameters δ 1 , δ 2 ,andδ 3 are selected as stated in Theorem 3, and κ is selected to satisfy the following two conditions: κ > 1 2 τ + r 6 and ||P||−κ + κ ˆ θ(t) < 0. Hence, we need to select κ such that κ > max  1 2 τ + r 6 , ||P|| 1 − ˆ θ (t)  . (70) It is clear that when ˆ θ (t) > 1, we only need to ensure that κ > 1 2 τ + r 6 .Notethatfrom equation (63), ˆ θ (t) > 1 can be easily ensured by selecting ˆ θ(0) > 1andκ > 1 2 τ + r 6 to guarantee that ˆ θ (t) in equation (63) is monotonically increasing. Hence, we guarantee that ˙ V c (x) ≤ x  (t)Ξ x(t), (71) where Ξ < 0. Hence, ˙ V c (x) < 0 which guarantees asymptotic stabilization of the closed-loop system. 3.4 Adaptive control when both θ ∗ and ρ ∗ are unknown Here, we wish to stabilize the system (6) considering the control law (3) when both θ ∗ and ρ ∗ are unknown. Here, the following Lyapunov-Krasovskii functional is used V d (x)=V c (x)+V 11 (x), (72) where V c (x) is defined in equations (59), and V 11 (x) is defined as V 11 (x)= ( 1 + ρ ∗ )[ ˜ ρ (t) ] 2 , (73) 153 Resilient Adaptive Control of Uncertain Time-Delay Systems where its time derivative is ˙ V 11 (x)=2 ( 1 + ρ ∗ ) ˜ ρ (t) ˙ ˜ ρ (t). (74) Since ˜ ρ (t)= ˆ ρ (t) −ρ ∗ ,then ˙ ˜ ρ(t)= ˙ ˆ ρ (t). Hence, equation (74) becomes ˙ V 11 (x)=2 ( 1 + ρ ∗ )[ ˆ ρ (t) − ρ ∗ ] ˙ ˆ ρ (t). (75) The next Theorem provides the main results for this case. Theorem 4: Consider system (6). If there exist matrices 0 < X = X  ∈ n×n , Y∈ m×n , Z∈ n×n , and scalars ε 1 > 0, ε 2 > 0, ε 3 > 0, ε 4 > ε, ε 5 > ε and ε 6 > ε (where ε is an arbitrary small positive constant) such that the LMI (25) has a feasible solution, and K = YX −1 ,andμ( t) is adapted subject to the adaptive laws ˙ μ (t)=Proj  λ 1 sgn ( μ(t) ) ||x(t)|| 2 + λ 2 μ(t) ||x(t)|| 2 +λ 3 sgn ( μ(t) ) ˆ θ (t) ||x(t)|| 2 + λ 4 sgn ( μ(t) ) ˆ ρ (t) ||x(t)|| 2 , μ(t)  , (76) ˙ ˆ θ (t)=σ ||x(t)|| 2 , (77) ˙ ˆ ρ (t)=ς ||x(t)|| 2 , (78) where Proj {·} Krstic et al. (1995) is applied to ensure that |μ(t)|≥1 as follows μ (t)= ⎧ ⎨ ⎩ μ (t) if |μ(t)|≥1 1 if 0 ≤ μ(t) < 1 −1 if −1 < μ(t) < 0, and the adaptive law parameters are selected such that λ 1 < − [ ||PB o I|| ] , λ 2 < − 1 2 τ + r 5 ||I  B  o B o I||, λ 3 < −σ, λ 4 < −ς, σ > 1 2 τ + r 6 , ς > 1 2 τ + r 4 ||B  o B o ||, ˆ θ(0) > 1 and ˆ ρ (0) > 1, then the control law (3) will guarantee asymptotic stabilization of the closed-loop system. Proof The time derivative of V d (x) is ˙ V d (x)= ˙ V c (x)+ ˙ V 11 (x). (79) Following the steps used in the proof of Theorem 3 and using equation (75), it can be shown that ˙ V d (x) ≤ x  (t)Ξ x(t)+τ + r 4 x  (t)ΔK  (t)B  o B o ΔK(t)x(t) + τ + r 5 z  (t)I  B  o B o Iz(t)+τ + r 6 E  (x, t)E(x, t)+2ρ ∗ ||PB o || ||x(t)|| 2 +2||PB o I|||μ(t)|||x(t) || 2 + 2θ ∗ ||P|| ||x(t)|| 2 + 2 μ(t) ˙ μ (t) + 2 ( 1 + θ ∗ )  ˆ θ (t) −θ ∗  ˙ ˆ θ (t)+2 ( 1 + ρ ∗ )[ ˆ ρ (t) −ρ ∗ ] ˙ ˆ ρ (t), (80) where Ξ is defined in equation (40). Using the linearization procedure and invoking the Schur complement (as in the proof of Theorem 1), it can be shown that Ξ is guaranteed to be negative definite whenever the LMI (25) has a feasible solution. Using the adaptive laws (76)- (78) 154 Time-Delay Systems in (80) and the fact that |μ(t)|≥1, we get ˙ V b (x) ≤ x  (t)Ξ x(t)+τ + r 4 ( ρ ∗ ) 2 ||B  o B o || ||x(t)|| 2 + τ + r 5 ||I  B  o B o I||μ 2 (t) ||x(t)|| 2 +τ + r 6 ( θ ∗ ) 2 ||x(t)|| 2 + 2ρ ∗ ||PB o || ||x(t)|| 2 + 2||PB o I|||μ(t) |||x(t)|| 2 +2θ ∗ ||P|| ||x(t)|| 2 + 2λ 1 |μ(t)|||x(t)|| 2 + 2λ 2 μ 2 (t) ||x(t)|| 2 +2λ 3 |μ(t)| ˆ θ (t) ||x(t)|| 2 + 2λ 4 |μ(t)| ˆ ρ (t) ||x(t)|| 2 + 2σ |μ(t)| ˆ θ (t) ||x(t)|| 2 −2σθ ∗ ||x(t)|| 2 + 2σθ ∗ ˆ θ (t) ||x(t)|| 2 −2σ ( θ ∗ ) 2 ||x(t)|| 2 + 2ς |μ(t)| ˆ ρ (t) ||x(t)|| 2 −2ςρ ∗ ||x(t)|| 2 + 2ςρ ∗ ˆ ρ (t) ||x(t)|| 2 −2ς ( ρ ∗ ) 2 ||x(t)|| 2 . (81) Arranging terms of equation (81), it can be shown that ˙ V d (x) < 0 if the adaptive law parameters λ 1 , λ 2 , λ 3 ,andλ 4 are selected as stated in Theorem 4, and σ and ς are selected to satisfy the following conditions: σ > 1 2 τ + r 6 ,2||P||−σ + σ ˆ θ(t) < 0, ς > 1 2 τ + r 4 ||B  o B o ||, and ||PB o ||−ς + ς ˆ ρ(t) < 0. Hence, we need to select σ and ς such that σ > max  1 2 τ + r 6 , ||P|| 1 − ˆ θ(t)  , (82) ς > max  1 2 τ + r 4 ||B  o B o || , ||PB o || 1 − ˆ ρ (t)  . (83) It is clear that when ˆ θ (t) > 1and ˆ ρ(t) > 1, we only need to ensure that σ > 1 2 τ + r 6 and ς > 1 2 τ + r 4 ||B  o B o ||. Note that from equations (77)- (78), ˆ θ(t) > 1and ˆ ρ(t) > 1 can be easily ensured by selecting ˆ θ (0) > 1and ˆ ρ(0) > 1andσ and ς as stated in Theorem 4 to guarantee that ˆ θ (t) and ˆ ρ(t) are monotonically increasing. Hence, we guarantee that ˙ V d (x) ≤ x  (t) Ξ x(t), (84) where Ξ < 0. Hence, ˙ V d (x) < 0 which guarantees asymptotic stabilization of the closed-loop system. Remarks: 1. The results obtained in all theorems stated above are sufficient stabilization results, that is asymptotic stabilization results are guaranteed only if all of the conditions in the theorems are satisfied. 2. The projection for μ may introduce chattering for μ and control input u Utkin (1992). The chattering phenomenon can be undesirable for some applications since it involves high control activity. It can, however, be reduced for easier implementation of the controller. This can be achieved by smoothing out the control discontinuity using, for example, a low pass filter. This, however, affects the robustness of the proposed controller. 4. Simulation example Consider the second order system in the form of (1) such that A o =  21.1 2.2 −3.3  , B o =  1 0.1  , A d =  −0.5 0 0 −1.2  , (85) 155 Resilient Adaptive Control of Uncertain Time-Delay Systems 0 0.5 1 1.5 2 2.5 3 −2 −1 0 x 1 (t) Resilient delay−dependent adaptive control when both θ * and ρ * are known 0 0.5 1 1.5 2 2.5 3 −1 0 1 x 2 (t) 0 0.5 1 1.5 2 2.5 3 −2 0 2 Time μ(t) 0 0.5 1 1.5 2 2.5 3 −5 0 5 Time u(t) Fig. 1. Closed-loop response when both θ ∗ and ρ ∗ are known and τ ∗ = 0.1. Using the LMI control toolbox of MATLAB, when the following scalars are selected as ε 1 = ε 2 = ε 3 = ε 4 = ε 5 = ε 6 = 1, the LMI (25) is solved to find the following matrices: X =  0.7214 0.1639 0.1639 0.2520  , Y =  −1.7681 −1.1899  . (86) Using the fact that K = YX −1 , K is found to be K =  −1.6173 −3.6695  . Here, for simulation purposes, the nonlinear perturbation function is assumed to be E (x(t)) =  1.2 |x 1 (t)| ,1.2|x 2 (t)|   ,wherex(t)=  x 1 (t) , x 2 (t)   . Based on Assumption 2.1, it can be shown that θ ∗ = 1.2. Also, the uncertainty of the state feedback gain is assumed to be ΔK (t)=  0.1sin (t) 0.1cos(t)  . Hence, based on Assumption 2.2, it can be shown that ρ ∗ = 0.1. 4.1 Simulation results when both θ ∗ and ρ ∗ are Known For this case, the control law (3) is employed subject to the initial conditions x(0)= [ −1, 1 ]  and μ(0)=1.5. To satisfy the conditions of Theorem 1, the adaptive law parameters are selected as α 1 = −10 and α 2 = −0.5. The closed-loop response of this case is shown in Fig. 1, where the upper two plots show the response of the two states x 1 (t) and x 2 (t), and third and fourth plots show the projected signal μ (t) and the control u(t). 4.2 Simulation results when θ ∗ is known and ρ ∗ is unknown For this case, the control law (3) is employed subject to the initial conditions x(0)= [ −1, 1 ]  and μ(0)=1.5 and ˆ ρ(0)=1.1. To satisfy the conditions of Theorem 2, the adaptive law parameters are selected as β 1 = −10, β 2 = −0.5, β 3 = −0.2, and γ = 0.1. For this case, the closed-loop response is shown in Fig. 2, where the upper two plots show the response of the two states x 1 (t) and x 2 (t), third plot shows the projected signal μ(t), the fourth plot shows ˆ ρ (t) and the fifth plot shows the control u(t). 156 Time-Delay Systems 0 0.5 1 1.5 2 2.5 3 −2 −1 0 x 1 (t) Resilient delay−dependent adaptive control when θ * is known and ρ * is unknown 0 0.5 1 1.5 2 2.5 3 −1 0 1 x 2 (t) 0 0.5 1 1.5 2 2.5 3 −2 0 2 Time μ(t) 0 0.5 1 1.5 2 2.5 3 1.1 1.2 1.3 Time ρ(t) 0 0.5 1 1.5 2 2.5 3 −5 0 5 Time u(t) Fig. 2. Closed-loop response when θ ∗ is known and ρ ∗ is unknown 0 0.5 1 1.5 2 2.5 3 −2 −1 0 x 1 (t) Resilient delay−dependent adaptive control when θ * is unknown and ρ * is known 0 0.5 1 1.5 2 2.5 3 −1 0 1 x 2 (t) 0 0.5 1 1.5 2 2.5 3 −2 0 2 μ(t) 0 0.5 1 1.5 2 2.5 3 1 1.5 2 θ(t) 0 0.5 1 1.5 2 2.5 3 −5 0 5 Time u(t) Fig. 3. Closed-loop response when θ ∗ is unknown and ρ ∗ is known 4.3 Simulation results when θ ∗ is unknown and ρ ∗ is known For this case, the control law (3) is employed subject to the initial conditions x(0)= [ −1, 1 ]  and μ(0)=1.1 and ˆ θ(0)=1.1. To satisfy the conditions of Theorem 3, the adaptive law parameters are selected as δ 1 = −5, δ 2 = −2, δ 3 = −1.5 and κ = 1. For this case, the closed-loop response is shown in Fig. 3, where the upper two plots show the response of the two states x 1 (t) and x 2 (t), third plot shows the projected signal μ(t), the fourth plot shows ˆ θ (t) and the fifth plot shows the control u(t). 157 Resilient Adaptive Control of Uncertain Time-Delay Systems 0 0.5 1 1.5 2 2.5 3 −2 −1 0 x 1 (t) Resilient delay−dependent adaptive control when both θ * and ρ * are unknown 0 0.5 1 1.5 2 2.5 3 −1 0 1 x 2 (t) 0 0.5 1 1.5 2 2.5 3 −2 0 2 μ(t) 0 0.5 1 1.5 2 2.5 3 1 1.5 2 θ(t) 0 0.5 1 1.5 2 2.5 3 1 1.5 2 ρ(t) 0 0.5 1 1.5 2 2.5 3 −5 0 5 Time u(t) Fig. 4. Closed-loop response when both θ ∗ and ρ ∗ are unknown 4.4 Simulation results when both θ ∗ and ρ ∗ are unknown For this case, the control law (3) is employed subject to the initial conditions x(0)= [ −1, 1 ]  and μ(0)=1.1, ˆ θ(0)=1.1 and ˆ ρ(0)=1.1. To satisfy the conditions of Theorem 4, the adaptive law parameters are selected as λ 1 = −5, λ 2 = −1, λ 3 = −1.5, λ 4 = −1.5, σ = 1, and ς = 1. For this case, the closed-loop response is shown in Fig. 4, where the upper two plots show the response of the two states x 1 (t) and x 2 (t), third plot shows the projected signal μ(t),the fourth plot shows ˆ θ (t), the fifth plot shows ˆ ρ(t), and the sixth plot shows the control u(t). 5. Conclusion In this chapter, we investigated the problem of designing resilient delay-dependent adaptive controllers for a class of uncertain time-delay systems with time-varying delays and a nonlinear perturbation when perturbations also appear in the state feedback gain of the controller. It is assumed that the nonlinear perturbation is bounded by a weighted norm of the state vector such that the weight is a positive constant, and the norm of the uncertainty of the state feedback gain is assumed to be bounded by a positive constant. Under these assumptions, adaptive controllers have been developed for all combinations when the upper bound of the nonlinear perturbation weight is known and unknown, and when the value of the upper bound of the state feedback gain perturbation is known and unknown. For all these cases, asymptotically stabilizing adaptive controllers have been derived. Also, a numerical simulation example, that illustrates the design approaches, is presented. 6. References Boukas, E K. & Liu, Z K. (2002). Deterministic and Stochastic Time Delay Systems, Control Engineering - Birkhauser, Boston. Cheres, E., Gutman, S. & Palmer, Z. (1989). Stabilization of uncertain dynamic systems including state delay, IEEE Trans. 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