# Observer Design for Time Delay Nonlinear Triangular Systems

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### 𝑥

### 𝑥

### 𝑥

### where 𝑢(𝑡) ∈ ℝ

### ⎣ 𝑥

### ∈ ℝ

### 𝑥

### ⎣ 𝑥

### ∈ ℝ

### 𝑢

### 𝑓

### Assumption 1 (A1). The functions 𝑓 𝑖 ; 𝑖 = 1, . . . , 𝑛 are globally Lipschitz w.r.t. 𝑥 and 𝑥 𝜏 , then ∃ 𝑐 > 0 and ∀ 𝑥(𝑡), 𝑦(𝑡), 𝑥(𝑡 − 𝜏), 𝑦(𝑡 − 𝜏) ∈ ℝ

### 𝑖 = 1, . . . , 𝑛 − 1, ∀𝜉 ∈ 𝑅

### (4) Assumption 3 (A3). The variable delay 𝜏 (𝑡) is bounded, e.g. 0 < 𝜏 (𝑡) ≤ 𝜏

### 0 𝑎 1 0 . . . 0 0 0 𝑎 2 .. . .. . . . . 0 0 . . . 0 𝑎

### 𝑠

### 𝑠 1 𝑠 22 . . . 0 0 . . . .. . .. . . . . 𝑠

### where 𝑠

### ∙ Deﬁne the (𝑛 × 𝑛) diagonal matrix Λ 𝜌 as Λ

### 𝜌, 𝜌

### 𝐴

### 𝑥 = 𝑓 (𝑢(𝑡), 𝑢 𝜏 (𝑡), ˆ 𝑥(𝑡), 𝑥 ˆ 𝜏 (𝑡)) − Λ 𝜌 𝑆

### ˆ 𝑥

### .. . ˆ 𝑥

### 𝑥

### 𝑥(𝑡) and ˆ 𝑥 𝜏 (𝑡) are the estimated state vectors, without and with delay respectively; 𝑢 and 𝑦 are the input and the output of the nonlinear system (1) respectively. The matrix Λ

### Theorem 1. Suppose that system (1) satisﬁes As- sumptions (A1)-(A3). Then, for 𝜌 > 𝜌

### 𝜂(1 − 𝜇) 𝑐𝜆 𝑚𝑎𝑥 (𝑆), system (9) is an asymptotic observer for system (1). The observation error ˜ 𝑥 = ˆ 𝑥(𝑡) − 𝑥(𝑡) is asymptotically stable for ˆ 𝑥(0), 𝑥(0) ∈ ℝ

### ˜ 𝑥

### with ˜ 𝑥

### Let 𝑥

### 𝑥

### ⎠ , 𝑥

### 𝑥

### △𝑓

### 𝑓

### 𝑓

### 𝑓 𝑖 (𝑢, 𝑢 𝜏 , 𝑥, ˆ 𝑥 ˆ 𝜏 ) − 𝑓 𝑖 (𝑢, 𝑢 𝜏 , 𝑥, 𝑥 𝜏 ) = 𝑓 𝑖 (𝑢, 𝑢 𝜏 , 𝑥 ˆ

### − 𝑓 𝑖 (𝑢, 𝑢 𝜏 , 𝑥 ˆ

### ∂𝑥

### 𝑥 = 𝑓 (𝑢, 𝑢

### 𝑥(𝑡) = (𝐴 − Λ 𝜌 𝑆

### Let ¯ 𝑥 = Λ

### 𝑥(𝑡) = 𝜌(𝐴 − 𝑆

### 𝑉 (¯ 𝑥) = ¯ 𝑥

### ∣∣¯ 𝑥(𝑠)∣∣

### 𝑉 ˙ = 2𝜌¯ 𝑥

### 1 − 𝜇 ∣∣¯ 𝑥(𝑡)∣∣

### 1 − 𝜇 (1 − 𝜏 ˙ (𝑡))∣∣ 𝑥(𝑡 ¯ − 𝜏)∣∣

### = 𝜌¯ 𝑥(𝑡)

### 1 − 𝜇 ∣∣¯ 𝑥(𝑡)∣∣

### 1 − 𝜇 (1 − 𝜏 ˙ (𝑡))∣∣ 𝑥(𝑡 ¯ − 𝜏)∣∣

### 𝑉 ˙ ≤ −𝜌𝜂∣∣ 𝑥(𝑡)∣∣ ¯

### 1 − 𝜇 ∣∣¯ 𝑥(𝑡)∣∣

### 1 − 𝜇 (1 − 𝜏(𝑡))∣∣¯ ˙ 𝑥(𝑡 − 𝜏 )∣∣

### ∣∣Λ

### it follows that 𝑉 ˙ ≤ −𝜌𝜂∣∣ 𝑥(𝑡)∣∣ ¯

### 1 − 𝜇 ∣∣¯ 𝑥(𝑡)∣∣

### 1 − 𝜇 (1 − 𝜏(𝑡))∣∣¯ ˙ 𝑥(𝑡 − 𝜏 )∣∣

### Since ∣∣ 𝑥(𝑡)∣∣∣∣¯ ¯ 𝑥(𝑡 − 𝜏))∣∣ ≤

### 1 − 𝜇 ∣∣¯ 𝑥(𝑡)∣∣

### 1 − 𝜇 𝑐∣∣𝑆∣∣∣∣¯ 𝑥(𝑡 − 𝜏)∣∣

### 1 − 𝜇 𝑐𝜆

### (15) Let 𝜌

### 𝜂(1 − 𝜇) 𝑐𝜆 𝑚𝑎𝑥 (𝑆). If 𝜌 > 𝜌

### 𝑉 ˙ ≤ −𝛼(𝜌)∣∣¯ 𝑥(𝑡)∣∣

### 1 − 𝜇 𝑐𝜆 𝑚𝑎𝑥 (𝑆) and 𝛼(𝜌) > 0, which results in ˙ 𝑉 ≤ 0, that means 𝑉 (𝑡) ≤ 𝑉 (0). Because of 𝛼(𝜌)∣∣¯ 𝑥(𝑡)∣∣

### ∣∣¯ 𝑥(𝑠)∣∣

### 𝑡→+∞ 𝑥(𝑡) ¯ → 0, moreover ∣∣˜ 𝑥(𝑡)∣∣ ≤ 𝜌

### The observation error ˜ 𝑥 = ˆ 𝑥(𝑡) − 𝑥(𝑡) is asymptotically stable for ˆ 𝑥(0), 𝑥(0) ∈ ℝ

### Theorem 2. Suppose that system (1) satisﬁes Assump- tions (A1)-(A3), then for 𝜌 > 𝜌

### 𝜂 𝑐𝜆

### 𝑥 1 = 𝑓 1 (𝑥 1 , 𝑥

### 𝑥 2 = 𝑓 2 (𝑥 1 , 𝑥

### 𝑥

### 𝑥 𝑛−1 = 𝑓 𝑛−1 (𝑥 1 , 𝑥

### 𝑥 𝑛 = 𝑓 𝑛 (𝑥, 𝑥

### 𝑥

### 𝑥 𝑖 (𝑡 − 𝜏 1 ) 𝑥

### .. . 𝑥

### 𝑥

### ⎝ 𝑥

### 𝑢

### 𝑢(𝑡 − 𝜏

### where 𝜏

### 𝑖 = 1, . . . , 𝑙 is bounded, that means 0 < 𝜏 𝑖 (𝑡) ≤ 𝜏

### { 𝑥(𝑡) = ˙ 𝑓 (𝑢(𝑡), 𝑢

### 𝑓 1 (𝑥 1 , 𝑥

### 𝑓

### 𝑥 = 𝑓 (𝑢(𝑡), 𝑢

### ⎣ ˆ 𝑥

### and ˆ 𝑥

### ⎣ ˆ 𝑥

### .. . ˆ 𝑥

### Theorem 3. Suppose that system (18) satisﬁes Assump- tions (A1)-(A4), then for 𝜌 > 𝜌

### with 𝜌

### 1 − 𝜇

### 𝑥 = ˆ 𝑥(𝑡)−𝑥(𝑡) is asymptotically stable for ˆ 𝑥(0), 𝑥(0) ∈ ℝ

### 𝑥(𝑡) = 𝜌(𝐴 − 𝑆

### 𝑉 (¯ 𝑥) = ¯ 𝑥

### 1 1 − 𝜇

### ∣∣ 𝑥(𝑠)∣∣ ¯

### 𝑉 ˙ = 2𝜌¯ 𝑥

### 1 − 𝜇

### = 𝜌¯ 𝑥(𝑡)

### 1 − 𝜇

### 𝑉 ˙ ≤ −𝜌𝜂∣∣¯ 𝑥(𝑡)∣∣

### 1 1 − 𝜇

### ∣∣ 𝑥(𝑡 ¯ − 𝜏 𝑖 )∣∣

### ∣∣Λ

### ∣∣¯ 𝑥(𝑡 − 𝜏

### 𝑉 ˙ ≤ −𝜌𝜂∣∣¯ 𝑥(𝑡)∣∣

### − 𝑐𝜆

### 1 − 𝜏 ˙

### ∣∣¯ 𝑥(𝑡 − 𝜏

### + 2𝑐∣∣𝑆∣∣∣∣¯ 𝑥(𝑡)∣∣

### ∣∣ 𝑥(𝑡 ¯ − 𝜏

### 2 (𝑙∣∣¯ 𝑥(𝑡)∣∣

### ∣∣¯ 𝑥(𝑡 − 𝜏

### 𝑉 ˙ ≤ −𝜌𝜂∣∣¯ 𝑥(𝑡)∣∣

### 1 1 − 𝜇

### ∣∣¯ 𝑥(𝑡 − 𝜏 𝑖 )∣∣

### + (2 + 𝑙)𝑐∣∣𝑆∣∣∣∣¯ 𝑥(𝑡)∣∣

### ∣∣¯ 𝑥(𝑡 − 𝜏

### 𝑉 ˙ ≤ −𝜌𝜂∣∣ 𝑥(𝑡)∣∣ ¯

### 1 1 − 𝜇

### 𝜏

### ∣∣ 𝑥(𝑡 ¯ − 𝜏 𝑖 )∣∣

### )𝑐𝜆

### ) 𝑐𝜆

### For 𝜌 > 𝜌

### 𝑉 ˙ ≤ −𝜉(𝜌)∣∣¯ 𝑥(𝑡)∣∣

### )𝑐𝜆

### Because of 𝜉(𝜌)∣∣¯ 𝑥(𝑡)∣∣

### ∣∣¯ 𝑥(𝑠)∣∣

### 𝑡→+∞ 𝑥(𝑡) ¯ → 0, moreover ∣∣𝑥(𝑡)∣∣ ≤ 𝜌

### The observation error ˜ 𝑥 = ˆ 𝑥(𝑡) − 𝑥(𝑡) is asymptotically stable for ˆ 𝑥(0), 𝑥(0) ∈ ℝ

### 𝑥

### 1 + 𝑥

### The time-delay 𝜏 is considered constant. It can be seen that system (25) can be written in the form of system (1) with 𝑓

### 1 + 𝑥

### ∂𝑥

### 2𝛼 and 𝑠 22 > 𝑠

### 𝑠

### 𝑥 1 = 2ˆ 𝑥 2 (𝑡) + cos(ˆ 𝑥 2 (𝑡)) − 𝑥 ˆ 1 (𝑡 − 𝜏) 1 + ˆ 𝑥

### 𝑥

### − 4𝜌

### 𝑥

### the other hand the time proﬁle 𝑥 2 (𝑡) and ˆ 𝑥 2 (𝑡) are quite similar. Figure 3 shows the errors estimations 𝑒

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