Time-varying low gain feedback for linear systems with unknown input delay

Systems & Control Letters 123 (2019) 98–107

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Time-varying low gain feedback for linear systems with unknown input delay✩ Yusheng Wei, Zongli Lin

∗

Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743, USA

article

info

Article history: Received 3 November 2017 Received in revised form 6 June 2018 Accepted 14 October 2018 Available online xxxx Keywords: Time delay Low gain feedback Time-varying

a b s t r a c t In this paper, the traditional low gain feedback design with a time-invariant feedback parameter is generalized to time-varying parameter design for linear systems with delayed input. For an unknown delay with a known upper bound, a time-varying low gain feedback law, constructed by using the parametric Lyapunov equation based approach, globally regulates a system with all open loop poles at the origin as long as the time-varying low gain parameter has a continuous second derivative and approaches a sufficiently small constant with its derivative approaching zero as time goes to infinity. Improvement of the closed-loop performance is addressed in a convergence rate analysis and then observed in simulation compared with the traditional constant parameter low gain design. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Time delay in engineering such as computational time of control algorithms or ignition time of diesel engines causes destabilizing effect on the closed-loop system under feedbacks designed without considering the delay. Various delay-dependent controllers and control algorithms have been identified in the past few decades to achieve stabilization of delayed systems and improve the closed-loop performance (see, for example, [1–10] and [11]). The celebrated model reduction technique developed in [12] establishes systematic methods to design stabilizing controllers for linear systems with delayed input. In the case of linear systems with a constant single input delay, the resulting infinitedimensional controller involves an integral term requiring the information of the past input in a time window of the length of the delay, and has a simple form of the product of the feedback gain matrix and the prediction of the future state. Thus, controllers formulated by the model reduction technique are typically referred to as predictor feedback laws, which achieve the finite spectrum assignment of the closed-loop system. The prediction is made by the solution of the state equation, which consists of two parts, a zero input solution that is the product of the state transition matrix and the initial condition, and the zero state solution that is the convolution of the transition matrix and the input term of the system. However, it was pointed out in [13] that the implementation of the predictor feedback law by approximating the integral ✩ This work was supported in part by the US National Science Foundation under grant CMMI-1462171. ∗ Corresponding author. E-mail addresses: [email protected] (Y. Wei), [email protected] (Z. Lin). https://doi.org/10.1016/j.sysconle.2018.10.016 0167-6911/© 2018 Elsevier B.V. All rights reserved.

term with even highly accurate finite sums may cause instability of the closed-loop system. Therefore, reference [14] overcomes this difficulty by truncating the distributed term of the predictor feedback law and leaving only the linear state feedback term. It is shown that this truncated predictor feedback asymptotically stabilizes systems without exponentially unstable poles as long as the feedback gain matrix is constructed by using the low gain feedback design technique [15]. Furthermore, without the state transition matrix factor in the truncated predictor feedback law, delay independent truncated predictor feedback law achieves stabilization of systems with all open loop poles either in the open left-half plane or at the origin. The low gain design in [14] utilizes the eigenstructural assignment technique. An alternative low gain design, the parametric Lyapunov equation based approach, was adopted in [16] to arrive at the same results as in [14]. The low gain parameter embedded in the feedback gain matrix is a sufficiently small positive constant determined by the length of the delay (see [14] and [16]). The upper bound of the feedback parameter derived based on Lyapunov analysis is considerably conservative due to numerous estimations made in the derivation. Inspired by the spirit of the traditional low gain feedback design [15], we propose the concept of time-varying parameter for low gain feedback such that the feedback parameter is timedependent. To improve the closed-loop performance, we design the parameter that approaches a sufficiently small positive constant dependent on the known bound on the delay only as time goes to infinity, but starts from a relatively large value. Intuitively, such a time-varying parameter reduces the overshoot and increases the convergence rate in the early stage of the state evolution. After the closed-loop system reaches a state around zero, decreasing the parameter to a sufficiently small constant would not affect the closed-loop performance yet guarantee stability.

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

Lyapunov analysis requires V˙ ≤ −w (|X (t)|), where V is a Lyapunov function, w (·) is a positive scalar function and |X (t)| is the Euclidean norm of the current state of system. In the case of time-varying feedback laws, taking the time derivative of a typical Lyapunov function V = X T (t)P(t)X (t), where P(t) is a time-varying positive definite matrix, results in terms involving different time instants such as X T (s)P(θ )X (s) with s ̸ = θ . Thus, majorization of such terms by −w (|X (t)|) is difficult. However, the PDE representation of the closed-loop system, which has been extensively explored in [7], provides us with more freedom to construct a Lyapunov functional that facilitates such majorization. Indeed, this PDE representation has been exploited in [17] and [18] to carry out design and stability analysis of adaptive predictor feedback laws. Furthermore, as seen in [17] and [18], the Lyapunov analysis is direct, without resorting to Krasovskii or Razumikhin Stability Theorem. Recent advancements in designing predictor-based feedback laws have been made by employing the PDE method and various backstepping transformations for LTI systems with distinct input delays [19], and nonlinear systems with multiple input delays [20] or even distributed input delays [21]. In this paper, we will also adopt the PDE representation of the closed-loop system and direct stability analysis to obtain design requirements for the feedback parameter that guarantee closed-loop stability. This paper studies the regulation of linear systems whose open loop poles are at the origin with an unknown constant delay in the input by time-varying low gain feedback. With a known upper bound on the delay, we show that any time-varying low gain feedback law globally regulates the system as long as the feedback parameter has continuous second derivative and approaches a sufficiently small positive constant with its derivative approaching zero as time goes to infinity. The parametric Lyapunov equation based design is adopted to construct feedback gain matrix. Direct stability analysis is carried out, and the improvement in the closedloop convergence rate compared with that under the constant parameter design is analyzed and then observed in simulation. The remainder of the paper is organized as follows. In Section 2, the time-varying low gain feedback design based on the solution of a parametric Lyapunov equation is formulated and discussed. Section 3 recalls from [7] and [18] the representation of the open loop system as a cascade of an ODE with a transport PDE whose boundary value expresses the delayed input. Direct stability analysis and convergence rate study are then carried out in Sections 4 and 5, respectively. A numerical example is provided in Section 6 to demonstrate the advantages of the time-varying parameter design over the traditional constant parameter design. Section 7 concludes the paper. Notation: Throughout this paper, we use rather standard notations. The set of real numbers, positive numbers and natural numbers are denoted by R, R+ and N, respectively. The Euclidean norm of a vector v is represented by |v|. We follow the conventions of [7], and adopt the following norm of a general vector valued function f (x, t) ∈ Rd on x ∈ [0, 1],

√

∫

∥f (t)∥ =

99

where X ∈ Rn and U ∈ Rm are the state vector and the input vector, respectively, and τ ∈ R+ is an unknown constant delay with a known upper bound τ ≥ τ . The initial condition is given by X (θ ) = ψ (θ ), where ψ (θ ) ∈ C 0 [−τ , 0]. It is assumed that (A, B) is controllable with all eigenvalues of A at the origin. Remark 1. Alternatively, we can define the initial condition of the delayed system (2) as U(θ ) = φ (θ ), θ ∈ C 0 [−τ , 0], and X (0) = X0 ∈ Rn . Because we are considering the closed-loop system under a state feedback law, it is more convenient to define the initial condition uniformly in the state X (t). It was pointed out in Theorem 1 of [16] that the delay independent truncated predictor feedback law U(t) = K (γ )X (t) = −BT P(γ )X (t), with a constant feedback parameter γ , asymptotically stabilizes system (2) for an arbitrarily large delay τ if

( 1 γ ∈ 0, √ √

3 3n nτ

] ,

(3)

where P(γ ) is the unique positive definite solution to the parametric algebraic Riccati equation, AT P(γ ) + P(γ )A − P(γ )BBT P(γ ) = −γ P(γ ), γ > 0.

This low gain feedback design technique is typically referred to as the parametric Lyapunov equation based approach. Notice from the delay independent truncated predictor feedback law that the feedback gain matrix K (γ ) is constant when γ is fixed. Also, the theoretical bound of γ given by (3) is considerably smaller compared with the bound observed from simulation, which further shows that a smaller value of γ leads to a larger overshoot and a slower convergence rate of the closed-loop system. To overcome this conservativeness in determining the value of γ , we design a time-varying low gain feedback law whose feedback parameter is time-dependent, U(t) = K (γ (t))X (t) = −BT P(γ (t))X (t), t ≥ −τ .

|f (x, t)| dx, t ≥ 0. 2

(1)

0

Also, C k [t1 , t2 ] denotes the set of real scalar functions or vector valued functions having kth continuous derivative on t ∈ [t1 , t2 ], where k ∈ N, t2 ≥ t1 , and derivatives up to kth order at t2 or t1 are defined as the left or right limits of the same order derivative from inside the interval.

γ (t) =

h

τˆ (t)

.

(6)

Here, as will be established in Section 4, h is a sufficiently small positive constant, whose upper bound for the regulation of system (2) only depends on system dimension n, and τˆ (t) satisfies the following conditions,

τˆ (t) ∈ C 2 [−τ , ∞), τˆ (t) > 0, lim τˆ (t) = τ , lim τ˙ˆ (t) = 0. (7) t →∞

Because the feedback gain matrix in the time-varying low gain feedback law is constructed by using the parametric Lyapunov equation based approach, we recall some properties of the algebraic Riccati equation (4) from [16] in the following lemma. Lemma 1. For system (2), the unique positive definite solution P(γ ) to the parametric algebraic Riccati equation (4) satisfies,

2. Time-varying low gain feedback design

tr(BT P(γ )B) = nγ , P(γ )BBT P(γ ) ≤ nγ P(γ ),

We consider the regulation of a linear system with delayed input,

Ac T (γ )P(γ )Ac (γ ) ≤ ϖ (γ )P(γ ),

X˙ (t) = AX (t) + BU(t − τ ), t ≥ 0,

(2)

(5)

Considering the inverse proportional relationship between the upper bound of γ and τ as in (3), we design the time-varying feedback parameter as

t →∞

1

(4)

dP(γ ) dγ

where Ac (γ ) = A − BBT P(γ ) and ϖ (γ ) =

> 0, 1 n(n 2

+ 1)γ 2 .

100

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

3. PDE–ODE cascade representation As pointed out in [7], the delayed input U(t − τ ) in system (2) can be considered as the boundary value of u(x, t) = U(t +τ (x − 1)), x ∈ [0, 1], at x = 0, where u(x, t) is the solution of a transport PDE τ ut (x, t) = ux (x, t) with the boundary condition u(1, t) = U(t). Thus, system (2) is equivalent to the cascade of an ODE with a transport PDE, X˙ (t) = AX (t) + Bu(0, t),

τ ut (x, t) = ux (x, t), u(1, t) = U(t).

The following lemma establishes estimates of some cross terms between u˜ (x, t), w ˆ (x, t), w ˆ x (x, t), w ˆ xx (x, t) and X (t). Lemma 2. The following properties hold for system (2),

( )( ) 1 1 ≤ |τ˜ | + τ |τ˙ˆ | ϵ∥˜u(t)∥2 + ∥w ˆ x (t)∥2 , ∀ϵ > 0, 2 ϵ

(9)

(11)

1

∫ 0

1

≤ h 2 ∥w ˆ (t)∥2 +

(13)

(1 + x)w ˆ T (x, t)τˆ BT P(γ (t))

∂P τˆ w ˆ t (x, t) = w ˆ x (x, t)(1 + τ˙ˆ (x − 1)) + τˆ BT γ˙ (t)X (t) ∂γ (( ) + τˆ BT P(γ (t)) A − BBT P(γ (t)) X (t) ) + Bu˜ (0, t) + Bw ˆ (0, t) , w ˆ (1, t) = 0.

(17)

By (8), (14) and (16), the closed-loop system under the timevarying low gain feedback law (5) takes the form,

(

)

BBT

∂P X (t), ∂γ

((

(26)

A − BBT P(γ (t)) X (t)

)

3

+ τˆ n(n + 1)γ 2 (t)X T (t)P(γ (t))X (t) 2

ˆ (0, t)|2 ), + 3τˆ nγ (t)(|˜u(0, t)|2 + |w

(27)

1

∫

ˆ xx (x, t)dx (1 + x)(1 + τ˙ˆ (x − 1))w ˆ x T (x, t)w 0

1 ≤ |w ˆ x (1, t)|2 + (|τ˙ˆ | − 1)|w ˆ x (0, t)|2 2 ( 1) ∥w ˆ x (t)∥2 , + |τ˙ˆ | − 2

(18)

X˙ (t) = A − BBT P(γ (t)) X (t) + Bu˜ (0, t) + Bw ˆ (0, t).

∂γ

) + Bu˜ (0, t) + Bw ˆ (0, t) dx ≤ τˆ nγ (t)∥w ˆ (t)∥2

and

from which, along with (5), (8), (12), (13) and (14), we obtain the PDE for w ˆ (x, t),

h 2 X T (t)

0

(14)

(16)

τˆ

1

∫

(15)

u˜ (x, t) = u(x, t) − uˆ (x, t),

∂P γ˙ (t)X (t)dx ∂γ ( τ˙ˆ )2 3 ∂P

(25)

(1 + x)w ˆ T (x, t)τˆ BT

To measure the distance between τˆ (t) and the actual delay τ , we further define

τ˜ (t) = τ − τˆ (t),

( 1 ˙ 1) (|τˆ | − 1)|w ˆ (0, t)|2 + |τ˙ˆ | − ∥w ˆ (t)∥2 , 2 2

≤

(12)

In view of the expression for the time-varying low gain feedback law (5), let the difference between uˆ (x, t) and U(t) be defined as

w ˆ (x, t) = uˆ (x, t) − U(t) = uˆ (x, t) + BT P(γ (t))X (t).

(1 + x)(1 + τ˙ˆ (x − 1))w ˆ T (x, t)w ˆ x (x, t)dx 0

which can be verified to satisfy the PDE,

τˆ uˆ t (x, t) = (1 + τ˙ˆ (x − 1))uˆ x (x, t), uˆ (1, t) = U(t).

(24)

1

∫

Following the idea in analyzing an adaptive predictor feedback law for linear systems with unknown input delay [18], we define a signal associated with τˆ (t), uˆ (x, t) = U(t + τˆ (x − 1)),

(1 + x)(τ˜ + τ τ˙ˆ (x − 1))u˜ T (x, t)w ˆ x (x, t)dx 0

(8) (10)

1

∫ −

∂P ∂P |w ˆ x (1, t)|2 ≤ 2τˆ 2 γ˙ 2 (t)X T (t) BBT X (t) + 6τˆ 2 n2 γ 2 (t) ∂γ ∂γ (n + 1 γ (t)X T (t)P(γ (t))X (t) × 2 ) + |˜u(0, t)|2 + |w ˆ (0, t)|2 .

(28)

(29)

(19) Proof. (24): By using Young’s Inequality, we obtain

By virtue of (9), (12), (13), (14), (15) and (16), the PDE for u˜ (x, t) is obtained as

τ u˜ t (x, t) = u˜ x (x, t) − u˜ (1, t) = 0.

τ˜ + τ τ˙ˆ (x − 1) w ˆ x (x, t), τˆ

(1 + x)(τ˜ + τ τ˙ˆ (x − 1))u˜ T (x, t)w ˆ x (x, t)dx 0

(20) (21)

The constructed Lyapunov functional used in the stability analysis to be carried out in Section 4 also contains w ˆ x (x, t) besides u˜ (x, t) and w ˆ (x, t). Thus, in view of (12), (14), (17) and (18), we derive the governing PDE for w ˆ x (x, t) as

τˆ w ˆ xt (x, t) = w ˆ xx (x, t)(1 + τ˙ˆ (x − 1)) + τ˙ˆ w ˆ x (x, t), T ∂P T w ˆ x (1, t) = −τˆ B γ˙ (t)X (t) − τˆ B P(γ (t)) ∂γ ( × (A − BBT P(γ (t)))X (t) ) + Bu˜ (0, t) + Bw ˆ (0, t) .

1

∫ −

(22)

(23)

) ∫ 1( ) ( 1 1 ϵ|˜u(x, t)|2 + |w ˆ x (x, t)|2 dx ≤ |τ˜ | + τ |τ˙ˆ | 2 ϵ 0 ( ) 1 ˙ )( 1 = |τ˜ | + τ |τˆ | ϵ∥˜u(t)∥2 + ∥w ˆ x (t)∥2 , 2 ϵ where ϵ is any positive constant. (25): It follows from w ˆ (1, t) = 0 and integration by parts that ∫ 1 (1 + x)w ˆ T (x, t)w ˆ x (x, t)dx 0 ∫ 1 T 1 = (1 + x)w ˆ (x, t)w ˆ (x, t)|0 − w ˆ T (x, t)w ˆ (x, t)dx 0 ∫ 1 − (1 + x)w ˆ T (x, t)w ˆ x (x, t)dx, 0

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

which implies that 1

∫ 0

implies that

1 ˆ (0, t)|2 + ∥w ˆ (t)∥2 ). (1 + x)w ˆ T (x, t)w ˆ x (x, t)dx = − (|w 2

(30)

1

(x2 − 1)w ˆ T (x, t)w ˆ x (x, t)dx = 0

1 2

|w ˆ (0, t)|2 −

1

∫

2

x|w ˆ (x, t)| dx

0

implies that

τ˙ˆ

1

∫

(x2 − 1)w ˆ T (x, t)w ˆ x (x, t)dx 0

1 ˆ (0, t)|2 + ∥w ˆ (t)∥2 ). ≤ |τ˙ˆ |( |w

(31)

2

Thus, (25) readily follows from adding (30) and (31). (26): With the help of (6) and Young’s Inequality, we have

∂P γ˙ (t)X (t)dx ∂γ ∫ 1 τ˙ˆ ∂P (1 + x)w ˆ T (x, t) hBT =− X (t)dx τˆ ∂γ 0 ⏐ ⏐ ∫ 1⏐ ⏐ τ˙ˆ 3 T ∂ P 1 ⏐ 14 T ⏐ ≤2 ˆ (x, t) h 4 B ˆ (t)∥2 X (t)⏐ dx ≤ h 2 ∥w ⏐h w ⏐ τˆ ∂γ 0 ⏐ ( τ˙ˆ )2 3 ∂P T ∂P + h 2 X T (t) BB X (t). τˆ ∂γ ∂γ (1 + x)w ˆ T (x, t)τˆ BT

0

(1 + x)w ˆ T (x, t)τˆ BT P(γ (t)) 0

)

× A − BB P(γ (t)) X (t) + Bu˜ (0, t) + Bw ˆ (0, t) dx ∫ 1 1 1 (1 + x)w ˆ T (x, t)BT P 2 (γ (t))P 2 (γ (t)) = τˆ (0 ) × (A − BBT P(γ (t)))X (t) + Bu˜ (0, t) + Bw ˆ (0, t) dx ∫ 1 w ˆ T (x, t)BT P(γ (t))Bw ˆ (x, t)dx ≤ τˆ 0 ∫ 1( ) + τˆ (A − BBT P(γ (t)))X (t) + Bu˜ (0, t) + Bw ˆ (0, t) T P(γ (t)) ( 0 ) × (A − BBT P(γ (t)))X (t) + Bu˜ (0, t) + Bw ˆ (0, t) dx ∫ 1( X T (t)(A − BBT P(γ (t)))T P(γ (t)) ≤ τˆ nγ (t)∥w ˆ (t)∥2 + 3τˆ T

)

Thus, (28) holds. (29): We evaluate w ˆ x (1, t) by using (23), Young’s Inequality and Lemma 1 as follows,

∂P ∂P X (t) |w ˆ x (1, t)|2 ≤ 2τˆ 2 γ˙ 2 (t)X T (t) BBT ∂γ ∂γ ( ) + 2τˆ 2 (A − BBT P(γ (t)))X (t) + Bu˜ (0, t) + Bw ˆ (0, t) T ( ×P(γ (t))BBT P(γ (t)) (A − BBT P(γ (t)))X (t) ) +Bu˜ (0, t) + Bw ˆ (0, t)

×(A − BBT P(γ (t)))X (t) + u˜ T (0, t)BT P(γ (t))Bu˜ (0, t) ) +w ˆ T (0, t)BT P(γ (t))Bw ˆ (0, t)

1

((

1 ˆ xx (x, t)dx ≤ |τ˙ˆ |( |w (x2 − 1)w ˆ x T (x, t)w ˆ x (0, t)|2 + ∥w ˆ x (t)∥2 ). 2

∂P ∂P X (t) + 6nγ (t)τˆ 2 ≤ 2τˆ 2 γ˙ 2 (t)X T (t) BBT ∂γ ∂γ ( × X T (t)(A − BBT P(γ (t)))T P(γ (t))

(27): By Lemma 1 and Young’s Inequality, we derive

∫

1

∫

∂P ∂P ≤ 2τˆ 2 γ˙ 2 (t)X T (t) BBT X (t) + 2nγ (t)τˆ 2 ∂γ ∂γ ( ) × (A − BBT P(γ (t)))X (t) + Bu˜ (0, t) + Bw ˆ (0, t) T ( ×P(γ (t)) (A − BBT P(γ (t)))X (t) ) + Bu˜ (0, t) + Bw ˆ (0, t)

1

∫

τ˙ˆ

0

On the other hand,

∫

101

∂P ∂P ≤ 2τˆ 2 γ˙ 2 (t)X T (t) BBT X (t) + 6τˆ 2 n2 γ 2 (t) ∂γ ∂γ (n + 1 γ (t)X T (t)P(γ (t))X (t) × 2 ) +|˜u(0, t)|2 + |w ˆ (0, t)|2 . □ 4. Direct stability analysis Based on the fact that τˆ (t) has a continuous second derivative, we can establish the differentiability of the feedback gain matrix K (γ (t)), the state X (t) and the input U(t). The information of differentiability of these signals is extensively involved in the construction of the Lyapunov functional and stability analysis to be carried out next.

0

× (A − BBT P(γ (t)))X (t)

Lemma 3. The time-varying feedback gain matrix K (γ (t)) in (5) is bounded and has a continuous second derivative on t ∈ [−τ , ∞).

) +˜uT (0, t)BT P(γ (t))Bu˜ (0, t) + w ˆ T (0, t)BT P(γ (t))Bw ˆ (0, t) dx 3

≤ τˆ nγ (t)∥w ˆ (t)∥2 + τˆ n(n + 1)γ 2 (t)X T (t)P(γ (t))X (t) 2 + 3τˆ nγ (t)(|˜u(0, t)|2 + |w ˆ (0, t)|2 ). (28): It can be verified that 1

∫

2

(1 + x)w ˆ x T (x, t)w ˆ xx (x, t)dx = |w ˆ x (1, t)| − 0

1 2

|w ˆ x (0, t)|2

1

− ∥w ˆ x (t)∥2 . 2

On the other hand, 1

∫

(x2 − 1)w ˆ x T (x, t)w ˆ xx (x, t)dx = 0

1 2

|w ˆ x (0, t)|2 −

1

∫

2

x|w ˆ x (x, t)| dx 0

Proof. By the time-varying low gain feedback design (7), τˆ (t) ∈ C 0 [−τ , ∞) is positive and has a finite limit τ , and hence τˆ (t) is bounded on t ∈ [−τ , ∞). Suppose that inft ≥−τ τˆ (t) = τmin and supt ≥−τ τˆ (t) = τmax , where 0 < τmin ≤ τmax . By the inversely proportional relationship between γ (t) and τˆ (t), γ (t) is also bounded with inft ≥−τ γ (t) = h/τmax and supt ≥−τ γ (t) = h/τmin . The increasing monotonicity of P(γ ) with respect to γ by Lemma 1 implies that P(γ (t)) is bounded with P(h/τmax ) ≤ P(γ (t)) ≤ P(h/τmin ). Therefore, the boundedness of K (γ (t)) follows readily from the construction of K (γ (t)) = −BT P(γ (t)). On the other hand, as noted in [22], P(γ ) is a rational matrix function of γ , which implies that P(γ ) is infinitely differentiable with respect to the feedback parameter. The first and second

102

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

derivatives of K (γ (t)) with respect to t are given as follows,

∂P h ˙ τˆ , ∂γ τˆ 2 h ( ∂ 2P h ˙ 2 ∂ P τ˙ˆ 2 ∂P ¨) K¨ (γ (t)) = BT 2 − τ ˆ − 2 + τˆ , τˆ ∂γ 2 τˆ 2 ∂γ τˆ ∂γ K˙ (γ (t)) = BT

(32)

Proof. Inspired by the Lyapunov functional for the stability analysis of linear systems under an adaptive predictor feedback law [18], where a term involving τ˜ 2 (t) is introduced to bound τ˜ (t), we consider V (t) = X T (t)P(γ (t))X (t) + b1 τ

from which it follows that K (γ (t)) has a continuous second derivative with respect to t since τˆ ∈ C 2 [−τ , ∞). □

1

∫

2

(1 + x)|˜u(x, t)| dx 0

+ b2 τˆ

1

∫

2

2

(1 + x)(|w ˆ (x, t)| + |w ˆ x (x, t)| )dx,

(35)

0

Lemma 4. With the initial condition X (θ ) = ψ (θ ) ∈ C 0 [−τ , 0], system (2) under the time-varying low gain feedback law (5) has a unique solution X (t) ∈ C 0 [−τ , ∞). Moreover, U(t) ∈ C 0 [−τ , ∞) and X (t), U(t) ∈ C 1 (0, ∞) ∩ C 2 (τ , ∞). Proof. The proof follows the general idea of the proof of Theorem 3.1 in [23]. Under the time-varying low gain feedback law (5), the open loop system (2) becomes X˙ (t) = AX (t) + BK (γ (t − τ ))X (t − τ ), t ≥ 0.

X (t) = e ψ (0) +

eA(t −s) BK (γ (s − τ ))ψ (s − τ )ds, 0

on t ∈ [0, τ ], which implies that X (t) ∈ C 0 [−τ , τ ]. Furthermore, the solution X (t) on t ∈ [τ , 2τ ] is obtained as follows, X (t) = e

A(t −τ )

X (τ ) +

t

∫ τ

eA(t −s) BK (γ (s −τ ))X (s −τ )ds, t ∈ [τ , 2τ ],

which implies that X (t) ∈ C 0 [−τ , 2τ ] due to the continuity of X (t) and K (γ (t)) on [0, τ ]. Similarly, the existence and uniqueness of X (t) can be established along the time axis toward positive infinity. The continuity of X (t) and its uniqueness on t ∈ [−τ , ∞) follow readily. From the right-hand side of (33), we obtain X˙ (t) ∈ C 0 (0, ∞) in view of the fact that X (t − τ ) is continuous on t ∈ (0, ∞). Taking the time derivative of both sides of (33) yields X¨ (t) = A2 X (t) + ABK (γ (t − τ ))X (t − τ ) + BK˙ (γ (t − τ ))X (t − τ )

+BK (γ (t − τ ))AX (t − τ ) +BK (γ (t − τ ))BK (γ (t − 2τ ))X (t − 2τ ), which implies that X (t) ∈ C 2 (τ , ∞) since X (t) ∈ C 0 [−τ , ∞) and K (γ (t)) is continuously differentiable for t ∈ [−τ , ∞), as proven in Lemma 3. Considering the time-varying low gain feedback law U(t) = K (γ (t))X (t), we know that U(t) ∈ C 0 [−τ , ∞) because both X (t) and K (γ (t)) are continuous on t ∈ [−τ , ∞). Also, the first and second derivatives of U(t) take the following form,

˙ U(t) = K˙ (γ (t))X (t) + K (γ (t))X˙ (t), ¨ = K¨ (γ (t))X (t) + 2K˙ (γ (t))X˙ (t) + K (γ (t))X¨ (t), U(t)

1

(

(1 + x)u˜ T (x, t) u˜ x (x, t)

+2b1 0

∈

t

∫

˙ γ (t))X (t) V˙ = 2X˙ T (t)P(γ (t))X (t) + X T (t)P(

∫ (33)

Considering the continuity of both K (γ (θ )) and ψ (θ ) on θ [−τ , 0], there exists a unique solution At

by discarding the term associated with τ˜ 2 (t) because τˆ (t) in our case is not an estimator of τ , but only provides information of the upper bound of delay to the time-varying low gain feedback law in order to achieve regulation. Here, b1 and b2 are two positive constants whose values are to be determined later. Taking the time derivative of V along the closed-loop trajectory gives,

(34)

which, in view of X (t) ∈ C 0 [−τ , ∞) ∩ C 1 (0, ∞) ∩ C 2 (τ , ∞) and the continuity of the second derivative of K (γ (t)), imply that U(t) ∈ C 1 (0, ∞) ∩ C 2 (τ , ∞). □ With the time-varying low gain feedback laws and the PDE representation of the closed-loop system in hand, we establish the global regulation of the system by a direct Lyapunov stability analysis as follows. Theorem 1. There exists a sufficiently small positive constant h⋆ such that, for each h ∈ (0, h⋆ ], the time-varying low gain feedback laws (5) globally regulate X (t) and U(t) of system (2).

) ∫ 1 w ˆ x (x, t) (1 + x)w ˆ T (x, t) dx + 2b2 −(τ˜ + τ τ˙ˆ (x − 1)) τˆ 0 ( × w ˆ x (x, t)(1 + τ˙ˆ (x − 1)) (( ) ∂P γ˙ (t)X (t) + τˆ BT P(γ (t)) A − BT P(γ (t)) X (t) ∂γ )) +Bu˜ (0, t) + Bw ˆ (0, t) dx ∫ 1 (1 + x)w ˆ x T (x, t) +2b2 0 ( ) × w ˆ xx (x, t)(1 + τ˙ˆ (x − 1)) + τ˙ˆ w ˆ x (x, t) dx ∫ 1 ˙ (1 + x)(|w ˆ (x, t)|2 + |w ˆ x (x, t)|2 )dx, +b2 τˆ

+τˆ BT

(36)

0

where (12), (17), (20) and (22) are used. To determine a time domain where V˙ is well defined, we first observe from the right-hand side of (36) that the highest order derivative is w ˆ xx (x, t), which can be computed as,

w ˆ xx (x, t) = uˆ xx (x, t) =

∂ 2 U 2 ⏐⏐ τˆ ⏐ s=t +τˆ (x−1) ∂ s2

(37)

by virtue of (14) and (11). Recall from Lemma 4 that U(t) has a continuous second derivative on t ∈ (τ , ∞). Then, w ˆ xx (x, t) is well defined if t > τ + τmax , where τmax is defined in the proof of Lemma 3 as the supremum of τˆ on t ∈ [−τ , ∞). This guarantees that s = t + τˆ (x − 1) > τ for any x ∈ [0, 1]. It then follows from X (t) ∈ C 1 (0, ∞), the continuous differentiability of P(γ (t)) with respect to t, which is equivalent to that of K (γ (t)) as indicated in Lemma 3, and τˆ (t), γ (t) ∈ C 1 [−τ , ∞), which is implied by the time-varying low gain feedback design (6) and (7), that V˙ ∈ C 0 [τ + τmax + 1, ∞) is well defined. Let ts = τ + τmax + 1 denote the starting point of the consideration of V˙ as a function of t. With the help of the parametric algebraic Riccati equation (4) and the closed-loop representation (19), we obtain from (36) that

(

)

V˙ = X T (t) −γ (t)P(γ (t)) − P(γ (t))BBT P(γ (t)) X (t)

+2X T (t)P(γ (t))Bu˜ (0, t)

∂P +2X T (t)P(γ (t))Bw ˆ (0, t) + X T (t) γ˙ (t)X (t) ∂γ ∫ 1 +2b1 (1 + x)u˜ T (x, t)u˜ x (x, t)dx 0

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

−

2b1

τˆ

∫

(1 + x)(τ˜ + τ τ˙ˆ (x − 1))u˜ T (x, t)w ˆ x (x, t)dx

+∥w ˆ x (t)∥2

0 1

(1 + x)w ˆ T (x, t)w ˆ x (x, t)(1 + τ˙ˆ (x − 1))dx

+2b2 0 1

∫

T

(

V˙ ≤ γ (t)X T (t)P(γ (t))X (t) − 1 + 6b2 n2 (n + 1)h2

) ( +3b2 n(n + 1)h + |˜u(0, t)|2 2 − b1 + 6b2 nh ) ( +12b2 n2 h2 + |w ˆ (0, t)|2 b2 (|τ˙ˆ | − 1) + 2 + 6b2 nh ) ( |τ˜ | + 12 τ |τ˙ˆ | ) +12b2 n2 h2 + ∥˜u(t)∥2 b1 −1 + 2ϵ τˆ ( τ˙ˆ )2 ∂ P ( h ∂P ∂ P 3 + 2b2 BBT h2 +X T (t) − 2 τ˙ˆ τˆ ∂γ τˆ ∂γ ∂γ ( τ˙ˆ )2 ∂ P ) ∂ P +4b2 BBT h2 X (t) + ∥w ˆ x (t)∥2 τˆ ∂γ ∂γ ( 2b |τ˜ | + 1 τ |τ˙ˆ | ) 1 2 × + 8b2 |τ˙ˆ | − b2 ϵ (τˆ ) 1 2 +∥w ˆ (t)∥ b2 −1 + 2nh + 2h 2 + 4|τ˙ˆ |

0

Next, we use 1

∫ 0

1 2 (1 + x)u˜ T (x, t)u˜ x (x, t)dx = − (|˜u(0, t)| + ∥˜u(t)∥2 ), 2

Young’s Inequality and the properties (24)–(28) in Lemma 2 to estimate V˙ as, 2

V˙ ≤ −γ (t)X T (t)P(γ (t))X (t) + 2|˜u(0, t)| + 2|w ˆ (0, t)| ∂P 2 T +X (t) γ˙ (t)X (t) − b1 (|˜u(0, t)| + ∥˜u(t)∥2 )

(38)

Substitution of |w ˆ x (1, t)| by its estimate in (29) of Lemma 2 and arranging terms in (38) give

(

V˙ ≤ X T (t)P(γ (t))X (t) −γ (t) + 6b2 τˆ 2 n2 (n + 1)γ 3 (t)

(

+3b2 τˆ n(n + 1)γ 2 (t) + |˜u(0, t)|2 2 − b1 ) +6b2 τˆ nγ (t) + 12b2 τˆ 2 n2 γ 2 (t) + |w ˆ (0, t)|2 ( ) × 2 + b2 (|τ˙ˆ | − 1) + 6b2 τˆ nγ (t) + 12b2 τˆ 2 n2 γ 2 (t) |τ˜ | + 21 τ |τ˙ˆ | ) +∥˜u(t)∥ −b1 + 2b1 ϵ τˆ ( ∂P ( τ˙ˆ )2 ∂ P ∂P 3 +X T (t) γ˙ (t) + 2b2 h 2 BBT ∂γ τˆ ∂γ ∂γ ∂P T ∂P ) 2 2 +4b2 τˆ γ˙ (t) BB X (t) ∂γ ∂γ 2

(

in (40), we consider

where Lemma 1 is used. It then follows that

2

2

∂P ∂γ

( ∂ P ) 12 ( ∂ P ) 12 ( ∂ P ) 12 ( ∂ P ) 12 ∂P T ∂P BB = BBT ∂γ ∂γ ∂γ ∂γ ∂γ ∂γ ( ∂P ) ∂P ∂ ∂P ∂P ≤ tr BT B = (tr(BT P(γ )B)) =n , ∂γ ∂γ ∂γ ∂γ ∂γ

2

+2b2 |τ˙ˆ |(∥w ˆ (t)∥2 + ∥w ˆ x (t)∥2 ).

(40)

To group terms that involve

( 3 +2b2 τˆ nγ (t)∥w ˆ (t)∥2 + τˆ n(n + 1)γ 2 (t)X T (t)P(γ (t))X (t) 2 ) 2 ˆ (0, t)|2 ) +3τˆ nγ (t)(|˜u(0, t)| + |w ( 1 ( τ˙ˆ )2 3 ) ∂P T ∂P +2b2 h 2 ∥w h 2 X T (t) ˆ (t)∥2 + BB X (t) τˆ ∂γ ∂γ ( 1 ˙ 2 2 +2b2 |w ˆ x (1, t)| + (|τˆ | − 1)|w ˆ x (0, t)| 2 ( ) ) 1 + |τ˙ˆ | − ∥w ˆ x (t)∥2 + 4b2 |τ˙ˆ | ∥w ˆ x (t)∥2

)

+|w ˆ x (0, t)|2 b2 (|τ˙ˆ | − 1).

2

∂γ ) |τ˜ | + 12 τ |τ˙ˆ | ( 1 +2b1 ϵ∥˜u(t)∥2 + ∥w ˆ x (t)∥2 τˆ ϵ ( ) (1 1) 2 ˙ ˆ (0, t)| + |τ˙ˆ | − ∥w ˆ (t)∥2 +2b2 (|τˆ | − 1)|w 2

(39)

Recall that γ (t) = h/τˆ (t). Then, substitution of γ (t)τˆ (t) by h simplifies the estimate of V˙ as follows,

+2b2 (1 + x)w ˆ (x, t)τˆ B P(γ (t)) ( 0 ) × (A − BBT P(γ (t)))X (t) + Bu˜ (0, t) + Bw ˆ (0, t) dx ∫ 1 ∂P (1 + x)w ˆ T (x, t)τˆ BT +2b2 γ˙ (t)X (t)dx ∂γ 0 ∫ 1 +2b2 (1 + x)w ˆ x T (x, t)w ˆ xx (x, t)(1 + τ˙ˆ (x − 1))dx 0 ∫ 1 +2b2 (1 + x)|w ˆ x (x, t)|2 τ˙ˆ dx 0 ∫ 1 ˙ +b2 τˆ (1 + x)(|w ˆ (x, t)|2 + |w ˆ x (x, t)|2 )dx. T

103

( 2b |τ˜ | + 1 τ |τ˙ˆ | ) 1 2 − b2 + 8b2 |τ˙ˆ | ϵ τˆ 1 +∥w ˆ (t)∥2 (−b2 + 2b2 τˆ nγ (t) + 2b2 h 2 + 4b2 |τ˙ˆ |) +|w ˆ x (0, t)|2 b2 (|τ˙ˆ | − 1).

1

∫

−

) ( τ˙ˆ )2 ∂ P h ˙ ∂P ∂P ( 3 τˆ + 2b2 BBT h 2 + 2h2 2

τˆ ∂P ≤h ∂γ

∂γ τˆ ∂γ ∂γ ˙ |τˆ | 1 (1 + 2b2 |τ˙ˆ |h 2 n + 4b2 |τ˙ˆ |hn) τˆ 2 ( ) ∂P τmax λmax maxγ ∈[h/τmax ,h/τmin ] { ∂γ } ≤ γ (t)P(γ (t)) h )) λmin (P( τmax |τˆ˙ | 1 × 2 (1 + 2b2 |τ˙ˆ |h 2 n + 4b2 |τ˙ˆ |hn), τmin

(41)

where λmin and λmax denote respectively the minimum and the maximum eigenvalue of a real symmetric matrix, τmin and τmax are respectively the infimum and the supremum of τˆ (t) over [−τ , ∞), and we have used the boundedness of τˆ (t), γ (t) and P(γ (t)) as shown in the proof of Lemma 3. By denoting

σ =

( ) ∂P τmax λmax maxγ ∈[h/τmax ,h/τmin ] { ∂γ } h 2 τmin λmin (P( τmax ))

,

and recalling from the design of τˆ (t) that limt →∞ τˆ (t) = τ and limt →∞ τ˙ˆ (t) = 0, we see that there exists a sufficiently large time constant t0 ≥ ts such that, for each t ≥ t0 ,

{ |τ˙ˆ | ≤ min

1

,

h

1 2b2 h 2 n + 4b2 hn σ

} .

Thus, (41) can be continued as follows,

−

( τ˙ˆ )2 ∂ P ) h ˙ ∂P ∂P ( 3 τˆ + 2b2 BBT h 2 + 2h2 ≤ 2hγ (t)P(γ (t)). 2

τˆ

∂γ

τˆ

∂γ

∂γ

104

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

on t ≥ t3 . It is worth mentioning here that h⋆ is independent of any information of the delay, including τ , but only depends on the dimension of the system n. To complete the proof, it remains to establish global regulation of both X (t) and U(t). We first demonstrate the square integrability of X (t) over t ≥ 0. By denoting

The estimate of V˙ then takes the form,

( V˙ ≤ γ (t)X T (t)P(γ (t))X (t) −1 + 2h + 6b2 n2 (n + 1)h2 ) ( +3b2 n(n + 1)h + |˜u(0, t)|2 2 − b1 + 6b2 nh ) ( +12b2 n2 h2 + |w ˆ (0, t)|2 b2 (|τ˙ˆ | − 1) + 2 + 6b2 nh

1 − 2h − 6b2 n2 (n + 1)h2 − 3b2 n(n + 1)h = η,

|τ˜ | + 12 τ |τ˙ˆ | ) +12b2 n2 h2 + ∥˜u(t)∥2 b1 −1 + 2ϵ τˆ ( ) |τ˜ | + 12 τ |τ˙ˆ | 2 2b1 +∥w ˆ x (t)∥ + 8b2 |τ˙ˆ | − b2 τˆ (ϵ ) 1 2 +∥w ˆ (t)∥ b2 −1 + 2nh + 2h 2 + 4|τ˙ˆ |

)

(

which is a positive constant as long as h is chosen small enough, we have from (42) that V˙ ≤ −ηγ (t)X T (t)P(γ (t))X (t), t ≥ t3 . Then, in view of the boundedness of γ (t) and P(γ (t)), as shown in the proof of Lemma 3, we get

+|w ˆ x (0, t)|2 b2 (|τ˙ˆ | − 1).

(42)

We next simplify those terms in (42) that contain |τ˙ˆ |. Again, in view of limt →∞ τˆ (t) = τ and limt →∞ τ˙ˆ (t) = 0, both of which are requirements of the time-varying parameter design, as given in (7), there exists a sufficiently large positive constant t1 ≥ t0 such that, for each t ≥ t1 ,

( |τ˜ | ) τmin τ ≤ 2 or ≤ 1 , |τ˙ˆ | ≤ , 0≤ τˆ τˆ τ |τ˜ | + 21 τ |τ˙ˆ | τmin 3 ≤1+ ≤ . τˆ 2τˆ 2 1 . 4

−1 + 2ϵ

which leads to ∞

∫

|X (t)|2 dt ≤ t3

∞

∫

|X (t)|2 dt < ∞,

(43) t3

Then,

which, together with the fact that X (t) ∈ C 0 [0, t3 ], as indicated in Lemma 3, imply that X (t) is square integrable on t ∈ [0, ∞) because

|τ˜ | + 12 τ |τ˙ˆ | < 0. τˆ

∞

∫

|X (t)|2 dt =

2

Then, the term associated with ∥˜u(t)∥ in (42) becomes negative. Also, there exists a sufficiently large positive constant t2 ≥ t1 such 1 that |τ˙ˆ | ≤ 16 , t ≥ t2 . Thus, with the help of (43), we obtain 2b1 |τ˜ | + τ |τ˙ˆ | 1 2

ϵ

τˆ

b2 + 8b2 |τ˙ˆ | − b2 ≤ 12b1 − .

(44)

2

To make the right-hand side of (44) non-positive, we let b1 and b2 satisfy 24b1 ≤ b2 . It then follows that the term corresponding to ∥w ˆ x (t)∥2 in (42) is non-positive. By choosing b2 > 4, we know that there exists a sufficiently large constant t3 ≥ t2 such that, for each t ≥ t3 , |τ˙ˆ | ≤ 21 − b2 , which is equivalent to 1 − |τ˙ˆ | −

2 b2

2

≥ 21 . Then, b2 (|τ˙ˆ | − 1) + 2 + 6b2 nh +

12b2 n2 h2 < 0, which leads to the negative definiteness of the term associated with |w ˆ (0, t)|2 in (42), is implied by 6nh + 12n2 h2 < 12 . Note that the definition of t3 naturally gives rise to the negativeness 1 of the term involving |w ˆ x (0, t)|2 in (42), and −1 + 2nh + 2h 2 + 1

4|τ˙ˆ | < 0 if 2nh + 2h 2 < 12 . In view of (42), we see that there exists a sufficiently large constant t3 such that, for each t ≥ t3 , terms associated with ∥˜u(t)∥2 , ∥w ˆ x (t)∥2 , ∥w ˆ (t)∥2 , |w ˆ (0, t)|2 and |w ˆ x (0, t)|2 in (42) are all non1 positive as long as b2 > 4, 24b1 ≤ b2 , nh(1 + 2nh) < 12 and 1

nh + h 2 <

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 . 4

τmax V (t3 ) τmax (V (t3 ) − V (∞)) ) ≤ ) . ( ( h h ) h ) h ηλmin P( τmax ηλmin P( τmax

Recall that V is continuously differentiable on t ∈ [ts , ∞). We see that V is bounded for t ∈ [ts , t3 ], where t3 ≥ ts according to the definition of t3 . Then,

which together guarantee that

Let ϵ =

( h ) h ) |X (t)|2 ≤ −V˙ , t ≥ t3 , ηλmin P( τmax τmax

It is then clear that V˙ < 0 on t ≥ t3 if

2h + 6b2 n2 (n + 1)h2 + 3b2 n(n + 1)h < 1, 6b2 nh(1 + 2nh) < b1 − 2, 1 1 1 nh(1 + 2nh) < , nh + h 2 < , 12 4 24b1 ≤ b2 , b1 > 2, b2 > 4

holds. We first choose b1 and b2 according to the last row of (45). Then, substitution of b1 and b2 in the rest of (45) shows that there exists a sufficiently small h⋆ such that, for each h ∈ (0, h⋆ ], V˙ < 0

|X (t)|2 dt +

∫

∞

|X (t)|2 dt t3

0

0

2

∫

∞

|X (t)|2 dt < ∞.

≤ t3 max {|X (t)| } + t ∈[0,t3 ]

t3

On the other hand, from the definition of V in (35) and the boundedness of V for t ∈ [ts , ∞), which can be justified by combining the boundedness of V on t ∈ [ts , t3 ] and V˙ < 0 on t > t3 , we obtain the boundedness of X (t) on t ∈ [ts , ∞). Then, X (t) < ∞ for t ∈ [−τ , ∞) readily follows from the continuity of X (t) on t ∈ [−τ , ts ]. Thus, (33) implies the boundedness of X˙ (t) on t > 0 since K (γ (t)) is bounded, as indicated in Lemma 3. With the square integrability of X (t) and the boundedness of X˙ (t), global regulation of X (t), which further implies that limt →∞ U(t) = 0 by (5) and the boundedness of K (γ (t)), follows from the Barbalat’s lemma. □ Remark 2. The requirement for τˆ (t) to have a continuous second derivative comes from the need for the well-definedness of the partial derivatives in the PDEs (12), (17), (20) and (22), where w ˆ xt and w ˆ xx are the highest order derivatives. From the definition of w ˆ (x, t) in (14) and the time-varying low gain feedback law (5), it follows that both the second order partial derivatives contain the same term τ¨ˆ (s)|s=t +τˆ (x−1) . Take w ˆ xx (x, t) for example,

w ˆ xx (x, t) = (45)

t3

∫

∂ 2P h ˙2 τˆ (s) τˆ ∂γ 2 (s) τˆ 2 (s) ∂ P τ˙ˆ 2 (s) ∂P ¨ ) −2 + τˆ (s) X (s) ∂γ (s) τˆ (s) ∂γ (s) ∂P h ˙ +2BT τˆ (s)X˙ (s) ∂γ (s) τˆ 2 (s) ⏐ ) ⏐ T 2 ¨ −B P(γ (s))X (s) τˆ (t)⏐⏐ ,

(

BT

h (

2 (s)

−

s=t +τˆ (x−1)

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

where we have used (32), (32), (34), (37) and the fact that K (γ (t)) = −BT P(γ (t)). Thus, the well-definedness of τ¨ˆ (t) is necessary for that of w ˆ xx (x, t). Without loss of generality, we assume that τˆ (t) ∈ C 2 [−τ , ∞) as in the design of the time-varying low gain feedback laws in Section 2. This requirement on τˆ (t) facilitates all our derivations in Sections 3 and 4, and in general, cannot be relaxed unless a new Lyapunov analysis on the closed-loop system is established involving at most the first derivative of τˆ (t). Theorem 1 reveals a group of time-varying low gain feedback laws under which global regulation of the closed-loop system is achieved. By the structure of the time-varying feedback parameter in (6) and Theorem 1, h and τˆ (t) need to be separately designed in order to construct γ (t). Thus, for the sake of simplicity, we consider directly designing γ (t) without involving h and τˆ (t). Corollary 1. There exists a sufficiently small positive constant γ ⋆ , which is inversely proportional to the delay bound τ , such that, the time-varying low gain feedback laws (5) globally regulate X (t) and U(t) of system (2) as long as

γ (t) ∈ C 2 [−τ , ∞), γ (t) > 0, lim γ (t) ∈ (0, γ ⋆ ], lim γ˙ (t) = 0. t →∞

(46)

t →∞

Proof. Given a γ (t) satisfying (46), we write γ (t) in the form of (6), where h and τˆ (t) are selected as h = τ limt →∞ γ (t) > 0 and τˆ (t) = h/γ (t). It can be readily verified that limt →∞ τˆ (t) = τ . Also, γ (t) ∈ C 2 [−τ , ∞), γ (t) > 0 and limt →∞ γ˙ (t) = 0 imply τˆ (t) ∈ C 2 [−τ , ∞), τˆ (t) > 0 and limt →∞ τ˙ˆ (t) = 0, respectively, because

−hγ˙ (t) ¨ h 2γ˙ (t) τ˙ˆ (t) = , τˆ (t) = 2 − γ¨ (t) . γ 2 (t) γ (t) γ (t)

(

2

)

105

{ ( ) β = min γ 1 − 120n2 (n + 1)τ 2 γ 2 − 60n(n + 1)τ γ , ( ) } 1 1 20 1 − 2nτ γ − 2τ 2 γ 2 , 1 , ζ = max{1, 40τ }, and V (t) is as given in (35) with γ (t) = γ and τˆ (t) = τ . Proof. The proof follows an analysis similar to that of the proof of Theorem 1, except the distinction between a constant feedback parameter and a time-varying parameter. With a Lyapunov functional V (t) given by (35), its estimated time derivative (39) along the trajectory of the closed-loop system (33) for the special case of the constant parameter design takes the form of

(

V˙ ≤ γ X T PX −1 + 6b2 n2 (n + 1)τ 2 γ 2 + 3b2 n(n + 1)τ γ

) ( +|˜u(0, t)|2 2 − b1 + 6b2 nτ γ + 12b2 n2 τ 2 γ 2 ( ) +|w ˆ (0, t)|2 −b2 + 2 + 6b2 nτ γ + 12b2 n2 τ 2 γ 2 ( 2b |τ˜ | ) ( |τ˜ | ) 1 + ∥w ˆ x (t)∥2 − b2 +∥˜u(t)∥2 b1 −1 + 2ϵ τˆ ϵ τˆ ( ) 1 1 2 +∥w ˆ (t)∥ b2 −1 + 2nτ γ + 2τ 2 γ 2 − |w ˆ x (0, t)|2 b2 ,

|τ˜ | τ −τ = <1 τˆ τ and taking ϵ =

1 , 3

in (48), we arrive at a further estimate of V˙ ,

V˙ ≤ γ X T PX −1 + 6b2 n2 (n + 1)τ 2 γ 2 + 3b2 n(n + 1)τ γ

)

) ( +|˜u(0, t)|2 2 − b1 + 6b2 nτ γ + 12b2 n2 τ 2 γ 2 ) ( +|w ˆ (0, t)|2 −b2 + 2 + 6b2 nτ γ + 12b2 n2 τ 2 γ 2 1

− ∥˜u(t)∥2 b1 + ∥w ˆ x (t)∥2 (6b1 − b2 ) 3 ( ) 1 1 +∥w ˆ (t)∥2 b2 −1 + 2nτ γ + 2τ 2 γ 2 − |w ˆ x (0, t)|2 b2 .

max 120n2 (n + 1)τ 2 γ 2 + 60n(n + 1)τ γ , 2nτ γ + 2τ 2 γ

5. Convergence rate analysis

where we note that

Theorem 2. The delay independent truncated predictor feedback law with the constant parameter design, U(t) = −BT P(γ )X (t), for a sufficiently small γ > 0, exponentially stabilizes system (2) with β

−ζ t 1 V (0), t ≥ 0, |X (t)|2 ≤ λ− min (P(γ ))e

(47)

(49)

With the selection of b1 > 2 and b2 > 6b1 , the terms involving ∥˜u(0, t)∥2 , |w ˆ (0, t)|2 or ∥w ˆ x (t)∥2 in (49) are negative if

Remark 3. The time-varying low gain feedback law is allowed to have constant feedback parameter γ (t) = γ > 0 according to (46). Thus, the group of traditional low gain feedback laws can be considered as a subset of the group of time-varying low gain feedback laws. In particular, Corollary 1 concludes that there exists a sufficiently small positive constant γ ⋆ , which is inversely proportional to the upper bound of delay τ such that, for each γ ∈ (0, γ ⋆ ], the low gain feedback law U(t) = −BT P(γ )X (t) globally regulates system (2). This observation is consistent with Theorem 2 of [14] and Theorem 1 of [16].

To illustrate the merits of the time-varying low gain feedback design over the constant parameter design with regard to the closed-loop performance, we compare convergence rates of the closed-loop system under a constant parameter feedback with different values of the feedback parameter within the range where exponential stability of the closed-loop system is ensured.

(48)

where we have replaced h with γ τ and all the terms involving γ˙ or τ˙ˆ disappear because of the constant parameter design. Noting that

(

Note from the selection of h = τ limt →∞ γ (t) that limt →∞ γ (t) ∈ (0, γ ⋆ ] is equivalent to h ∈ (0, τ γ ⋆ ]. Theorem 1 concludes that there exists a sufficiently small positive constant h⋆ , which is independent of any information of delay, such that, for any h ∈ (0, h⋆ ], the time-varying low gain feedback law U(t) = −BT P(γ (t))X (t) globally regulates the system. Thus, there exists a sufficiently small positive constant γ ⋆ = h⋆ /τ , which is inversely proportional to τ , such that the closed-loop regulation is guaranteed if limt →∞ γ (t) ∈ (0, γ ⋆ ]. □

)

6b2 nτ γ + 12b2 n2 τ 2 γ 2 < b1 − 2. For illustration, we choose b1 = 3 and b2 = 20. Then, a sufficiently small γ satisfying

{

1

1 2

}

<1

results in

(

V˙ ≤ −β X T PX + ∥˜u(t)∥2 + ∥w ˆ (t)∥2 + ∥w ˆ x (t)∥2

)

β ≤ − V, ζ

V (t) ≤ max{1, 2b1 τ , 2b2 τˆ }

( ) × X T PX + ∥˜u(t)∥2 + ∥w ˆ (t)∥2 + ∥w ˆ x (t)∥2 ( ) ≤ ζ X T PX + ∥˜u(t)∥2 + ∥w ˆ (t)∥2 + ∥w ˆ x (t)∥2 . Consequently, an estimate of V (t) readily follows from the comparison lemma, V (t) ≤ e

− βζ t

V (0), t ≥ 0,

which, by the definition of V (t) in (35), further implies the exponential convergence of X (t) expressed by (47). □

106

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

Fig. 1. The system evolution under γ (t) given by (50) and γ = 0.068.

Remark 4. Theorem 2 reveals a guaranteed convergence rate β of 2ζ for the state of the closed-loop system under the constant parameter design of the delay independent truncated predictor feedback law. Examination of this guaranteed convergence rate indicates that γ only appears in β , rendering

6. Simulation study To compare the closed-loop performance of system (2) under the traditional constant parameter low gain feedback and the timevarying parameter low gain feedback, we make the comparison from a theoretical point of view. Take system (2) with

lim β = lim β = 0,

γ →0

[

γ →γ

A=

where γ = min{γ 1 , γ 2 } with γ 1 and γ 2 being the unique positive solutions to the nonlinear equations, 1

120n2 (n + 1)τ 2 γ 2 + 60n(n + 1)τ γ = 1, 2nτ γ + 2τ 2 γ

1 2

= 1,

respectively. Thus, there exists a maximum β on the interval γ ∈ (0, γ ), corresponding to the fastest convergence rate implied by Theorem 2. Remark 5. Dealing with the conservativeness incurred in the Lyapunov stability analysis on the upper bound of feedback parameter γ under the constant parameter low gain feedback design, as given by (3), the time-varying parameter design proposed in this paper takes a proactive selection of γ at a relatively large value, which corresponds to a fast convergence rate, during the starting phase of the system evolution. After the system reaches a state near the equilibrium point zero, reducing γ to a sufficiently small constant, as required by Corollary 1, would not affect the closedloop transient performance but guarantee stability. Therefore, such a time-varying parameter design would manifest its merits in the closed-loop performance compared with that of the constant parameter design. Remark 6. Remark 5 provides a guideline to construct a timevarying γ (t) that outperforms a constant γ with respect to the closed-loop performance, as demonstrated by simulation. A theoretical proof of such an improvement is challenging due to the time-varying design of γ (t) and remains to be carried out.

]

0 0

[ ]

1 0 ,B = , τ = τ = 1 s, 0 1

[

]

1 ψ (θ ) = , θ ∈ [−1, 0], −1 as an example. The initial values of γ (t) are designed to decrease slowly from 0.3. On the other hand, limt →∞ γ (t) is computed as 7.6 × 10−4 by employing (45) with b1 = 3 and b2 = 72, Corollary 1 and the fact that τ = 1. Simulation shows that, regardless of how conservative the bound on limt →∞ γ (t) is, it does not prevent us from designing a time-varying low gain feedback law with better closed-loop performance than that under the traditional constant parameter design. This can be demonstrated by selecting

γ (t) = −0.0953 arctan(t − 20) + 0.1504, t ≥ −1,

(50)

which satisfies all the design requirements in (46). Evolution of the time-varying low gain parameter, the system state and the control input under the time-varying parameter design is illustrated in Fig. 1. The system evolution under the constant parameter design with γ = 0.068, which is the theoretical upper bound given by (3), is also given for comparison. Obviously, this choice of γ (t) achieves better closed-loop performance than with a constant γ . 7. Conclusions Global regulation of a linear system whose open loop poles are at the origin with an arbitrarily large unknown input delay was studied. With the knowledge of an upper bound on delay, a low gain feedback design with a time-varying low gain parameter was proposed to replace the traditional design with a constant

Y. Wei and Z. Lin / Systems & Control Letters 123 (2019) 98–107

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