Practical tracking control for a class of high-order switched nonlinear systems with quantized input

ISA Transactions xxx (xxxx) xxx

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Practical tracking control for a class of high-order switched nonlinear systems with quantized input Yan Jiang, Junyong Zhai

∗

Key Laboratory of Measurement and Control of CSE, Ministry of Education, School of Automation, Southeast University, Nanjing, Jiangsu, 210096, PR China

highlights • This paper focuses on the switched systems with switching powers and quantized input. • An improved practical tracking result is obtained under a weaker growth condition. • Construct a homogeneous reduced-order observer to estimate the unmeasurable states.

article

info

Article history: Received 3 February 2019 Received in revised form 21 June 2019 Accepted 21 June 2019 Available online xxxx Keywords: Homogeneous domination Practical tracking Switched nonlinear systems Quantized input

a b s t r a c t This paper concentrates on the global practical tracking problem for a class of high-order switched nonlinear systems under arbitrary switching, whose powers and nonlinearities count on the switching signal. The sector-bounded approach is utilized to dispose of the control input, which is quantized by a logarithmic quantizer. Firstly, through the medium of adding a power integrator approach, a homogeneous output feedback controller is designed for the nominal part of the switched systems. Then, by virtue of the homogeneous domination idea and a common change of coordinates, scaled homogeneous output feedback controllers are achieved to assure global boundedness of all the states in the whole system and ensure the tracking error to converge into an arbitrarily small neighborhood of origin in a finite time. Finally, two examples are provided to test the validity of the proposed method. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Switched systems are a subclass of hybrid systems defined by a aggregation of a finite number of subsystems and a law deciding which of them to be active. Due to the variable environment, numerous practical systems can be formulated as switched systems, for instance chemical processes, automotive engineering, and so on. Lots of researchers dedicate to the study of the performance of switched systems [1–7]. The control methods for switched system are cataloged as a common Lyapunov function (CLF) approach, the single Lyapunov function approach and multiple Lyapunov functions approach. If the switching mechanism cannot be changed by the controller, the systems operate under arbitrary switching. For this case, the CLF method can be adopted to design controllers and analyze the stability. It has been shown in [8] that a switched system is stable under arbitrary switching if and only if it admits a CLF. Recently, lots of attempt have been made to enhance the theory and application of high-order switched systems. The ∗ Corresponding author. E-mail addresses: [email protected] (Y. Jiang), [email protected] (J. Zhai).

work [9] utilized this method to investigate the high-order nominal switched systems with switching powers. Combined with the homogeneous domination approach, the work [10] successfully extended this control scheme to regulate all the states of the high-order switched nonlinear systems to the origin in finite time, in which the powers are unrelated to the switching signals. With switching dependent powers, the work [11] further addressed the same problem for the switched systems. In addition, many important results related with the practical tracking problems of switched nonlinear systems have been achieved in [12–18]. In the scheme of practical tracking, the output or state of the controlled plant can track the given reference into a small neighborhood around the origin. By using the state-feedback control paradigm, the work [17] considered the practical tracking problems for the switched nonlinear systems. In practical and engineering systems, it is difficult to measure all the system states. From this point of view, designing output-feedback controllers to ensure practical tracking is more meaningful. Immediately, an interesting problem arises: How to ensure global practical tracking via output feedback controllers for high-order switched systems for the case that both the powers and nonlinear terms rely on the switching signals?

https://doi.org/10.1016/j.isatra.2019.06.022 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

2

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx

On the other hand, quantization is an effective technique to keep communication unblocked since communication constraints including limited bits rate of communication channels and the limited bandwidth are unavoidable. Quantization is the process in which the continuous-valued signals are transformed into discrete-valued ones. In fact, it is the fundamental step for the transmission of signals through the communication network. The distinct features of quantization lie in sufficient precision and low communication rate. Apparently, quantization errors will reduce performance and result in instability. Therefore, some valuable results on quantized control and quantization error have been obtained in [19–25]. The work [20] considered the coarsest quantizer satisfying logarithmic law for linear system. Based on the sector bounded approach, the logarithmic quantizer was investigated for linear systems in [23]. Combined with backstepping technique, the work [24] solved the robust tracking issue for nonlinear systems with uncertainties. In the meantime, some results focused on the quantized feedback control for switched systems. For instance, the work [26] studied finite-level quantizers for switched linear systems. With quantized input, an adaptive fuzzy control idea was adopt to cope with output tracking issue for switched nonlinear systems in [27]. However, the quantized feedback control issues for switched nonlinear systems with high-order switched powers remain challenges. Inspired by the works [10,11], we concentrate on the practical tracking problem for the high-order switched nonlinear systems with quantized input. The main contributions of this article are concluded as: (i) Distinct from the work [10], this paper concentrates on the switched systems with switching powers and the quantized input, and achieves an improved globally practically tracking result under a weaker growth condition. (ii) A homogeneous reduced-order observer is presented to estimate the unmeasurable states. A special decomposition is introduced to deal with the quantized input. (iii) Different from the previous state-feedback results, such as [17,18], a series of switching related gains are incorporated in the observers and the output feedback controllers to conquer the difficulties caused by the switching powers and switchingdependent nonlinearities via the CLF method. (iv) With the assistance of the Lyapunov stability and the homogeneity theory, all the states of the closed-loop system are assured to be bounded and the tracking error is regulated into an arbitrarily small neighborhood of origin in a finite time. The framework of the rest part of this paper is: Section 2 reviews the idea of homogeneity, logarithmic quantizer and four necessary lemmas. Section 3 gives the problem formulate. The main result is shown in Section 4. Section 5 present two examples. Conclusion is provided in Section 6. Notation. Let m, n be integers. N = {1, 2, . . . , n}, N0 = {0} ∪ N , N − = N \ {n}, M = {1, 2, . . . , m}. Let R refer to the collection of real numbers, R+ indicate the collection of positive real numbers, Rn stand for the n-dimensional Euclidean space. 1 ∥ · ∥ is the Euclidean norm of a vector X . R≥ odd ≜ {θ ∈ R|θ ≥ 1 and θ is a ratio of two odd integers}. K stands for the set of all continuous and strictly increasing functions R+ → R+ , which are vanishing at zero; K∞ indicates the collection of all unbounded functions which are of class K. 2. Preliminaries In what follows, we review the ideas of homogeneity and logarithmic quantizer, and introduce four lemmas which occupy a significant position in the rest of this paper.

Fig. 1. The uniform quantizer.

Definition 1 ([28]). For coordinates (x1 , x2 , . . . , xn ) ∈ Rn , and ri ∈ R+ , i ∈ N , ∀ε > 0, we have (i) △ε (x) = (ε r1 x1 , ε r2 x2 , . . . , ε rn xn ) stands for the dilation, where ri , i ∈ N are the weights of the coordinates. The dilation weight is defined as △ = (r1 , r2 , . . . , rn ) for simplicity; (ii) a continuous function V : Rn → R is homogeneous of degree τ , if for ∀x ∈ Rn /{0}, τ ∈ R, V (△ε (x)) = ε τ V (x1 , x2 , . . . , xn ); (∑n ) 2/ri 1/2 (iii) ∥x∥△,2 = , ∀x ∈ Rn , stands for a homoi=1 |xi | geneous 2− norm, which is abbreviated as ∥x∥△ . Lemma 1 ([29]). Let V : Rn → R be homogeneous of degree τ . We have (i) ∂∂ xV is homogeneous of degree τ − ri ; i (ii) V (x) ≤ c1 ∥x∥τ△ with a constant c1 . Lemma 2 ([9]). For c , d ∈ R, and a constant κ , if κ > 1 then 1 1 1 κ−1 |c + d|κ ≤ 2κ−1 |c κ + dκ |, (|c | + |d|) κ ≤ |c | κ + |d| κ ≤ 2 κ (|c | + 1 1 1 1 1 1 κ − d κ | ≤ 21− κ |c − d| κ ; |c κ − dκ | ≤ |d|) κ ; if κ ∈ R≥ odd , then |c κ−1 κ−1 κ|c − d|(c − d ). Lemma 3 ([9]). Let a1 , b2 , θ be positive constants and χ1 , χ2 ∈ R, then a a1 a2 − a1 a1 a2 a1 +a2 a1 +a2

|χ1 | |χ2 |

≤

a1 + a2

θ|χ1 |

+

a1 + a2

θ

2

|χ2 |

.

Lemma 4 ([9]). Let x, y ∈ R and q ∈ Rodd , then ≥1

−(x − y)(xq − yq ) ≤ −

1 2q−1

(x − y)q+1 .

This paper considers the logarithmic quantizer proposed in [30], which is modeled as ⎧ ( si si ] ⎪ , ⎪ si , s ∈ ⎪ 1+δ 1−δ ⎨ [ ] q(s) = 0, s ∈ 0, s1 (1) ⎪ ⎪ 1 + δ ⎪ ( ) ⎩ − q(−s), s ∈ −∞, 0 where s is the information to be quantized, q(s) is the quantized value, si = ϱ(1−i) smin , i ∈ N , smin is a quantization parameter. The 1−ϱ parameters 0 < ϱ < 1, δ = 1+ϱ determine the quantization density of q(s). q(s) ∈ S := {0, ±si , i ∈ N }. The size of the dead zone of q(s) is determined by smin . The map of the logarithmic quantizer q(s) for s > 0 is shown in Fig. 1. Remark 1. As a measurement of quantization density, ϱ will influence the quantitative effect. The quantizer becomes coarser 1−ϱ as ϱ is smaller. From the relation δ = 1+ϱ , we can obtain that δ

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx

is close to 1 as ϱ approaches zero. In this case, the quantization levels of q(s) is fewer as s within this interval. To discover the influence of the quantization on the system performance, we define the quantization error as qe (s) = q(s) − s. Thus, the logarithmic quantizer is decomposed as q(s) = s + qe (s). Consider the fact that

{

(2)

where

{

q2 (t) =

0, |s| ≥ smin ,

θ2 (t)smin , |s| ≤ smin

with θi (t) ∈ [−1, 1], i = 1, 2 being time-varying parameters. 3. Problem formulate Consider the high-order switched nonlinear systems

⎧ pi,ϑ (t) − ⎪ ⎨ x˙ i = xi+1 + φ˜ i,ϑ (t) (x), i ∈ N , x˙ n = q(uϑ (t) ) + φ˜ n,ϑ (t) (x), ⎪ ⎩ y = x1 − y r

|φ˜ i,k (·)| ≤ c˜k

∑

| xj |

l =1

ri +τk rj

(p1,k ···pl,k )

+ c˜k

j=1

(4)

Assumption 2. There is a constant M > 0, so that the continuously differentiable reference signal yr satisfies the following inequality

|yr | + |˙yr | ≤ M , ∀t ∈ [0, +∞).

1 + θ1 (t)δ, |s| ≥ smin , 1, |s| ≤ smin ,

{

1+

i

r1 = 1, ri pi−1,k = ri−1 + τk , i = 2, 3, . . . , n.

which implies

q1 (t) =

Assumption 1. For every (i, k) ∈ N × M, there are constants c˜k > 0 and τk ∈ (− ∑n−1 1 , 0), such that

where

|qe (s)| ≤ δ|s|, |s| ≥ smin , |qe (s)| ≤ smin , |s| ≤ smin

q(s) = q1 (t)s + q2 (t)

3

(3)

Remark 4. Assumption 1 is a homogeneous growth condition, which is more general compared with linear growth conditions. Although the output feedback stabilization issues for switched systems with Assumption 1 were considered in [10], the result is only applicable to the case of common powers (i.e. pi,ϑ (t) = pi ) without quantized input. It is worth mentioning that it is interesting to design output feedback controllers under Assumption 1 for system (3) in which both the nominal part and nonlinear terms are dependent on switching signal. For simplicity, the numerators of τk , k ∈ M are assumed to be even, and the denominators of τk are assumed to be odd. Together with (4), both the denominator and numerator of ri , i ∈ N are odd. Remark 5. In the works about practical tracking problems, Assumption 2 is a standard condition for the reference signal, which has been used in [17,18].

where x = (x1 , x2 , . . . , xn )T ∈ Rn is the system state, y ∈ R is the output, and yr ∈ R is the reference signal. ϑ (t) : [0, +∞) → M represents the switching signal. We suppose ϑ (t) is a piecewise continuous (from the right) function with respect to t. For each k ∈ M, uk ∈ R is the control input, and q(uk ) represents the ≥1 quantized control input. For each (i, k) ∈ N − × M, pi,k ∈ Rodd . n ˜ For each (i, k) ∈ N × M, the nonlinear function φi,k (·) : R → R is continuous with φ˜ i,k (0) = 0. In this paper, the only measurable signal is assumed to be the error y between x1 and the reference signal yr for the two reasons. On the one hand, only the error signal can be directly measured in some engineering systems, such as the missile guidance system. On the other hand, exact information of the reference signals are not needed in some cases, which contributes the controller to adapt to different reference signals.

The homogeneous output feedback controllers of system (3) will be designed in this section. Based on the change of coordinates: z1 = y, z2 = x2 , . . . , zn = xn , system (3) becomes

Remark 2. Practical tracking has an extensive range of applications in practice, for instance, missile guidance systems and permanent magnet synchronous motors [31]. System (3) stands for a wide class of underactuated mechanical systems, which is unstable and weakly coupled [32].

|φ¯ i,k (·)| ≤ ck

Remark 3. Compared with the work [10], both the powers and nonlinearities in system (3) are dependent of the switching signal ϑ (t). This brings troublesome obstacle to the output feedback controller design under the CLF method. Therefore, control method in [10] cannot be applied directly. In addition, the control input is quantized by a logarithmic quantizer (1), which satisfies the decomposition (2). We will put forward a modified output feedback control scheme to cope with the practical tracking problem. The object of this paper is to design homogeneous output feedback controllers to guarantee the global boundedness of all the states and the practical tracking between x1 and yr . Before moving on, we make two assumptions for the nonlinear terms φ˜ i,k (·) and the reference signal yr .

4. Main results

⎧ pi,ϑ (t) − ⎪ ⎨ z˙i = zi+1 + φ¯ i,ϑ (t) (z), i ∈ N , z˙n = q(uϑ (t) ) + φ¯ n,ϑ (t) (z), ⎪ ⎩ y = z1

(5)

where φ¯ 1,ϑ (t) (z) = φ˜ 1,ϑ (t) (x) − y˙ r , φ¯ i,ϑ (t) (z) = φ˜ i,ϑ (t) (x), i = 2, 3, . . . , n. Under this coordinates change and Assumption 1, for each k ∈ M, one arrives at i ∑

|zj |

ri +τk rj

+ ck , i ∈ N

(6)

j=1

with ck > 0 is a constant. Before proceeding, a homogeneous output feedback controller is firstly designed for the nominal part of system (5), i.e. p ξ˙i = ξi+i,ϑ1 (t) , i ∈ N − , ξ˙n = q1 (t)vϑ (t) , y = ξ1

(7)

where ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ R denotes the system state, vϑ (t) ∈ R is the control input and y ∈ R is the output. In view of the design scheme presented in [9], the construction of homogeneous output feedback controllers for system (7) can be completed by three steps. T

n

4.1. State feedback controller design Step 1: Denoting ξ1∗ = 0, we consider a Lyapunov function candidate as V1 =

∫ ξ1 ξ∗ 1

1

∗ r1

(s r1 − ξ1

1

1 r

)2−r1 ds. Let η1 = ξ1 1 and

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

4

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx

calculate the time derivative of V1 along the trajectory of nominal system (7) rendering

where

p

∗p1,k

V˙ n ≤ − 2

∗p1,k

+ ξ1 (ξ2 1,k − ξ2

)

ξ2∗

is

controller

V˙1 = ξ1 ξ2

a

virtual

1 p1,k

r

ξ2∗ = −β1 η12 , β1 ≥ maxk∈M {n 2+τk 1

V˙1 ≤ −nη

p1,k 2−r1 ( 2 1

+ξ

with

the

form

− ξ2

V˙ i−1 ≤ −(n − i + 2)

η

2+τk j

+ξ

).

(8)

pi−1,k 2−ri−1 ( i i−1

∗pi−1,k

ξ

− ξi

)

(9)

1 r

r

∗ r1

j

, j = 2, . . . , i

(10)

where βj > 0, j = 1, . . . , i − 1. Next, we will prove that (9) still establishes at Step i. Consider a Lyapunov function candidate as Vi (ξ1 , . . . , ξi ) = Vi−1 +

V˙ i ≤ − (n − i + 2)

i−1 ∑

η

2+τk j

∗ r1

1

∫ ξi

Wi (ξ1 , . . . , ξi ), where Wi = is

i

(s ri −ξi

ξi∗

V˙ n ≤ −

ξn∗+1,k =−

∗p

+ξ

ξ

∗pi−1,k

− ξi

2−ri ∗pi,k i i+1

)+η

ξ

=−

βn

1−δ

η

ξ

− ξi

where Bi,1,k one has i−1 ∑ ∂ Wi

∑ ∂ Wi ∂ξj

ξ˙j

(11)

) ≤ |ηi−1 |

j=1

∂ξj

1

2+τk i−1

1 ri

|(ξi )

ri pi−1,k

∗ r1

− (ξ i

i

)

ri pi−1,k

|

2+τ Bi,1,k i k

η + η (12) 2 > 0 is a constant. From [9], with a constant Bi,2,k > 0,

ξ˙j ≤

i−1 1∑

2

η

2+τk j

+ q1 (t)ηn2−rn ξn∗+1,k + q1 (t)ηn2−rn (vk − ξn∗+1,k ) .

ηnrn +τk 1 1 1 1 1 ( 1 )r +τ r r r r βn ξnrn + βnr−n 1 (ξn−n−11 + · · · + β2 3 (ξ2 2 + β1 2 ξ1 ) · · ·) n k

+

n ∑

2+τk

ηj

+ q1 (t)ηn2−rn (vk − ξn∗+1,k ).

(17)

j=1

2−ri−1

≤

2+τk

ηj

(16)

V˙ n ≤ −

Note that 0 < ri pi−1,k < 1. By virtue of Lemmas 2 and 3, we have ∗pi−1,k

ξ˙j

with βn ≥ maxk∈M {1 + Bn,1,k + Bn,2,k }. Under which pi−1,k 2−ri−1 ( i i−1

j=1

pi−1,k 2−ri−1 ( i i−1

n−1 ∑

1−δ 1

i−1

p

∂ξj

j=1

)2−ri ds. Its time derivative

j=1

+ ηi2−ri (ξi+i,1k − ξi+1i,k ) +

n−1 ∑ ∂ Wn

From the definition of q1 (t), the virtual controllers ξn∗+1,k is designed as

1

ξj∗ = − βj−1 ηj−j 1 , ηj = ξj j − ξj

∗pn,k

) + q1 (t)ηn2−rn ξn+1,k

In line with (12) and (13), there exist constants Bn,j,k > 0, k ∈ M, j = 1, 2, so that

2+τk

∗ r1

ξ1 = 0, η1 = ξ1 − ξ1

∗pn−1,k

j=1

+ (Bn,1,k + Bn,2,k )ηn

with a series of virtual controllers ξ1∗ , ξ2∗ , . . . , ξi∗ 1 r1

p

2−r

+ ξn−1n−1 (ξi n−1,k − ξn

j=1

j=1

∗

2+τk

ηj

+ q1 (t)ηn2−rn (vk − ξn∗+1,k ) +

Inductive Step: For the hypothesis that there is a Lyapunov function candidate Vi−1 (ξ1 , . . . , ξi−1 ) at Step i − 1, and i−1 ∑

n−1 ∑

of

}, then

∗p1,k

ξ

whose time derivative is

2+τ Bi,2,k i k

η

.

(13)

4.2. Homogeneous output feedback controller design We design the following reduced-order observer r2

p ζ˙2 = − b1 ξˆ2 1,ϑ (t) , ξˆ2 = (ζ2 + b1 ξ1 ) r1 ri

p ζ˙i = − bi−1 ξˆi i−1,ϑ (t) , ξˆi = (ζi + bi−1 ξˆi−1 ) ri−1 , i = 3, . . . , n. (18)

Considering the estimated states ξˆ2 , . . . , ξˆn into (16), we design the output feedback controllers as

vk = −

1 1−δ

1 1 1 1 1 )r +τ ( 1 r r r r βn ξˆnrn + βnr−n 1 (ξˆn−n−11 + · · · + β2 3 (ξˆ2 2 + β1 2 ξ1 ) · · ·) n k .

(19)

j=1

Considering (12) and (13) into (11), we can achieve

For observer (18), we design a Lyapunov function candidate as

i−1

V˙i ≤ −(n − i + 1)

∑

2+τk

ηj

2−ri

+ ηi

∗p

ξi+1i,k + ηi

2−ri

pi,k

∗pi,k

(ξi+1 − ξi+1 )

∫

j=1

+ (Bi,1,k + Bi,2,k )η

Ui =

2+τk i

ξi

2−ri−1 ri

(ζi +bi−1 ξi−1 )

2−ri−1 ri−1

(

ri−1

s 2−ri−1 − (ζi + bi−1 ξi−1 ) ds, i = 2, . . . , n

)

in which ξi∗+1 is a virtual controller ri+1 i i

ξi∗+1 = −β η

(20)

, βi ≥ max{(n − i + 1 + Bi,1,k + Bi,2,k )

1 pi,k

and the time derivative is

}

k∈M

U˙ i =

then V˙i ≤ −(n − i + 1)

i ∑

η

2+τk j

+η

pi,k 2−ri ( i+1 i

ξ

−

∗p ξi+1i,k ).

n ∑ i=1

Wi , Wi =

∫

2−ri−1 −ri ri

(

ξi

ri−1 ri

) − (ζi + bi−1 ξi−1 ) ξ˜i+1

pi,k

This completes the inductive proof. Step n: Since the existence of quantizer (1), there is a tiny difference between the Step n and the preceding steps. We choose a Lyapunov function in the same form of the preceding steps Vn =

ri

ξi

2−ri−1 ) ( 2−rri−1 p p − bi−1 (ξi i−1,k − ξˆi i−1,k ) ξi i − (ζi + bi−1 ξi−1 ) ri−1

(14)

j=1

2 − ri−1

xi x∗ i

1

∗ r1

(s ri − xi

i

)2−ri ds

(15)

where ξ˜i+1 = ξi+1 , i ∈ N − ξ˜n+1 = q1 (t)vk . pi−1,k

Denoting ei = (ξi U˙ i =

2 − ri−1 ri

ξi

2−ri−1 −ri ri

r pi−1,k

− bi−1 ei i

(

p

1

− ξˆi i−1,k ) ri pi−1,k , i ∈ N , we have that

ξi

ri−1 ri

) − (ζi + bi−1 ξi−1 ) ξ˜i+1

2−ri−1 ( 2−rrii−1 ) r ξi − ξˆi i

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx r pi−1,k

− bi−1 ei i 2−ri−1 ri pi−1,k

Since

( ξˆi

2−ri−1 ri

− (ζi + bi−1 ξi−1 )

2−ri−1 ri−1

)

.

(21)

From (17), Proposition 4 and (23), it yields V˙ ≤ −

≥ 1, from Lemma 4, one has

2−ri−1 ( 2−rri−1 ) rp r − bi−1 ei i i−1,k ξi i − ξˆi i 2−ri−1 ( p ) 2−ri−1 1− rp p ≤ −2 ri pi−1,k bi−1 ei i i−1,k ξi i−1,k − ξˆi i−1,k ri pi−1,k

−

γ

(22)

where γi,1,k = 2 . We summarize the estimate of remaining terms in (21) as several propositions, whose proofs will be presented in Appendix.

Proposition 1. For every i = 2, 3, . . . , n − 1, k ∈ M, there is a constant γi,2,k > 0 and a continuous function hi,k (bi−1 ) ≥ 0, then 2 − ri−1 ri

≤

ξi

ξi

η

2+τk j

+γ

+

2 − rn−1 rn

≤

n 1∑

8

ξn

2+τk

ηj

+ γn,2,k (δ )

j=1

n ∑

2+τk

ej

r pi−1,k

1 16

2+τ

2+τk

2+τk

) + ei

1∑

−

−

i=1

n 1∑

4

2+τk

ei

.

(25)

i=2

From the construction of V , we can verify that V is positive definite and proper respect to X := (ξ1 , . . . , ξn , ζ2 , . . . , ζn )T . Then, the whole system (7)–(18)–(19) is transformed into

p

2+τ + h˜ i,k (bi−1 )ei−1 k .

(26)

Ui can be formulated as

2

4

2+τk

ηi

p

pi,ϑ (t) bi i+1

,i ∈ N . where fn+i,ϑ (t) = − ξˆ As a matter of fact, we choose ∆ = (r1 , . . . , rn , r1 , . . . , rn−1 ). −

    for ξ1 ,...,ξn

n

U˙ ≤

n 1∑

= (ξ2 1,ϑ (t) , . . . , ξn n−1,ϑ (t) , q1 (t)vϑ (t) , fn+1,ϑ (t) , . . . , f2n−1,ϑ (t) )T

∑nFrom (22) and Propositions 1–3, the time derivative of U = i=2

V˙ ≤ −

X˙ = Fϑ (t) (X )

2−ri−1 ) ( 2−rrii−1 ξˆi − (ζi + bi−1 ξi−1 ) ri−1

(ηi−1 k + ηi

2+τk

ηi

2+τ − (b1 γ2,1,k − γ2,2,k − γn,2,k (δ ) − h3,k (b2 ) − h˜ 3,k (b2 ))e2 k

i=1 n−1 ∑

2+τk

(bi−1 γi,1,k − 1 − γi,2,k − γn,2,k (δ ) − hi+1,k (bi ) − h˜ i+1,k (bi ))ei

i=3 2+τk

− (bn−1 γn,1,k − 1 − γn,2,k (δ ))en

.

(23)

In this subsection, the observer gains bi , i ∈ N − will be determined. Due to the redundant term in (17), a proposition is introduced whose proof will be given in Appendix. Proposition 4. For each k ∈ M, the following inequality holds n

1∑ 4

n 2+τk

ηi

+ γ3,k (δ )

i=1

where γ3,k (δ ) ≥ 0 is a continuous function.

∑ j=2

2+τk

ej



for ζ2 ,...,ζn



Thus, we can obtain from [10] that V is homogeneous of degree 2 about ∆. On account of Lemma 1, one can obtain V (X ) ≤ α∥X ∥2∆

(27)

with a positive constant α . Combined (27) with (25), it yields

∂V 2+τ Fk (X ) ≤ −αk ∥X ∥∆ k ∂X where αk > 0, k ∈ M are constants. V˙ =

4.3. Observer gains determination

q1 (t)ηn2−rn (vk − ξn∗+1 ) ≤

(24)

4

k∈M

4.4. Stability analysis

2+τ

+ hn,k (bn−1 , δ )en−1k .

j=2

− bi−1 ei i

+ γ2,2,k + γn,2,k (δ ) + γ3,k (δ ) + h3,k (b2 ) ) + h˜ 3,k (b2 ) /γ2,1,k }, (5 = max{ + γi,2,k + γn,2,k (δ ) + γ3,k (δ ) + hi+1,k (bi ) k∈M 4 ) + h˜ i+1,k (bi ) /γi,1,k }, i = 3, . . . , n − 1, ) (5 = max{ + γn,2,k (δ ) + γ3,k (δ ) /γn,1,k } 4

under which

( rnr−n 1 ) ξn − (ζn + bn−1 ξn−1 ) q1 (t)vk

Proposition 3. There are continuous functions h˜ i,k (bi−1 ) ≥ 0, i = 3, . . . , n, k ∈ M, then

≤

(1

bn−1

Proposition 2. There are two non-negative continuous functions γn,2,k (δ ) and hn,k (bi−1 , δ ), such that

bi−1 γi,1,k − 1 − γi,2,k − γn,2,k (δ ) − γ3,k (δ ) − hi+1,k (bi )

k∈M

2+τ hi,k (bi−1 )ei−1 k

with h2,k (b1 ) = 0.

n−1 ∑ (

b1 = max{

j=i−1

2−rn−1 −rn rn

i=1

where V = Vn + U. From the above inequality, we choose the observer gains bi , i ∈ N − as

) − (ζi + bi−1 ξi−1 ) ξ˜i+1

2+τk i,2,k ei

( − b1 γ2,1,k − γ2,2,k − γn,2,k (δ ) − γ3,k (δ )

) 2+τ − h˜ i+1,k (bi ) ei k ( ) 2+τ − bn−1 γn,1,k − 1 − γn,2,k (δ ) − γ3,k (δ ) en k

bi−1

i+1 1 ∑

12

(

ri−1 ri

4

2+τk

ηi

i=3

2 −r 1− r p i−1 i i−1,k

2−ri−1 −ri ri

n 1∑

) 2+τ − h3,k (b2 ) − h˜ 3,k (b2 ) e2 k

2+τ bi−1 i,1,k ei k

=−

5

(28)

To handle the problem caused by the switching signal in both system powers and nonlinearities, a common coordinates change is introduced for system (5):

ξi =

zi

ϖi

, i ∈ N , and vϑ (t) = qi,ϑ (t)

uϑ (t)

(29)

ϖn+1 qn,ϑ (t) +1

where ϖi = ϵi,ϑ (t) , i ∈ N , ϖn+1 = ϵn+1,ϑ (t) , and q1,ϑ (t) =

0, qi+1,ϑ (t) =

qi,ϑ (t) +1 pi,ϑ (t)

, i ∈ N − . 1 ≤ ϵ1,ϑ (t) ≤ ϵ2,ϑ (t) ≤ · · · ≤

ϵn,ϑ (t) ≤ ϵn+1,ϑ (t) are constants to be designed later.

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

6

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx

According to (2), system (5) becomes

⎧ pi,ϑ (t) − ⎪ ⎨ ξ˙i (t) = ϵ¯i,ϑ (t) ξi+1 (t) + φi,ϑ (t) (·), i ∈ N ξ˙n (t) = q1 (t)ϵ¯n,ϑ (t) vϑ (t) (t) + q¯ 2 (t) + φn,ϑ (t) (·), ⎪ ⎩ y(t) = ξ1 (t) qi,ϑ (t) +1

qi,ϑ (t)

where ϵ¯i,ϑ (t) = ϵi+1,ϑ (t) /ϵi,ϑ (t) , φi,ϑ (t) (·) = q2 (t)

(30)

φ¯ i,ϑ (t) (·) , ϖi

i ∈ N , q¯ 2 (t) =

. ϖn From (4), one has

qi,k ri

(1 +

∑i−2

j=1 p1,k · · · pj,k )τk + 1 1 1+

∑n−1

that q2,k r2

q3,k

≤

r3

l=1

(p1,k ···pl,k )

qn,k

≤ ··· ≤

, i = 2, 3, . . . , n.

i ck ∑

ϖi

ri +τk rj

, 0), one can deduce from (31) (32)

+

j=1

≤ ck Li,k

i −qi,k ∑

|ξj |

ri +τk rj

+

j=1

ck

ϖi

+

ck

ϖi

.

(33)

Proof. By virtue of (18), we design the homogeneous reducedorder observer with ϵ¯i,ϑ (t) , i ∈ N − as r2

p ζ˙2 = − ϵ¯1,ϑ (t) b1 ξˆ2 1,ϑ (t) , ξˆ2 = (ζ2 + b1 ξ1 ) r1

ri

p ζ˙i = − ϵ¯i−1,ϑ (t) bi−1 ξˆi i−1,ϑ (t) , ξˆi = (ζi + bi−1 ξˆi−1 ) ri−1 ,

i = 3, 4, . . . , n

(34)

with bi > 0, i ∈ N − are determined by (24). Combining (19) and (29), the homogeneous output feedback controllers can be designed as 1−δ

1 1 1 1 1 ( r1n )rn +τk qn,k +1 r3 r2 rn ˆ ˆ rn−1 ˆ r2 ϵn+ . 1,k βn ξn + βn−1 (ξn−1 + · · · + β2 (ξ2 + β1 ξ1 ) · · ·)

(35) In line with (26), (30)–(34)–(19) can be formulated as X˙ = diag(ϵ¯1,ϑ (t) , . . . , ϵ¯n,ϑ (t) , ϵ¯1,ϑ (t) , . . . , ϵ¯n−1,ϑ (t) )Fϑ (t) (X )

+ (0, . . . , 0, q¯ 2 (t), 0, . . . , 0)T + (φ1,ϑ (t) (·), . . . , φn,ϑ (t) (·), 0, . . . , 0)T .

(36)

Now, we calculate the time derivative of V as 2+τk

V˙ (X ) ≤ −ϵαk ∥X ∥∆

+

⏐ ⏐ ⏐ n ⏐ ∑ ⏐ ⏐ ∂V ⏐ ⏐ ∂V ⏐ ⏐+⏐ ⏐ ¯ φ q (t) i , k 2 ⏐ ∂ Xi ⏐ ⏐ ∂ Xn ⏐ i=1

where ϵ = mini∈N ,k∈M {¯ϵi,k }.

i

k

⏐ ⏐ ⏐ ⏐ 2+τk ⏐ ⏐ ∂V ⏐ ∂V ⏐ − ⏐ ⏐ ⏐ ⏐ ≤ ρ¯ 1,k ∥X (t)∥2+τk + ρ¯ 2,k u rn +τk ¯ q (t) ≤ u 2 min ∆ min ⏐ ⏐ ∂ Xn ⏐ ∂ Xn ⏐

(38)

(39)

with constants ρ¯ i,k > 0, i = 1, 2, k ∈ M. Substituting (38) and (39) into (37), it yields

ϵ > max

Theorem 1. Consider the quantized-input switched system (3) satisfying Assumptions 1–2, then there are homogeneous output feedback controllers such that for arbitrary switching, all the states of the closed-loop system are globally bounded and the tracking error converges into an arbitrarily small neighborhood of origin after a finite time.

1

2+τ

− r +τk

ρ˜ i,k ck

2+τk

)

2+τ − r +τk i k

+ ρ¯ 3,k .

(40)

2+τ − rn +τk k

where ρ¯ 3,k = ˜ i,k ck + ρ¯ 2,k umin . i=1 ρ We choose the parameter ϵ as follows

The main result of this paper is summarized as following.

uk = −

∑

∑n

(

ϖi

|ξj |

+

where ρk = i=1 ρi,k . In line with (38), we can obtain

∑n

j=1 i ∑

≤ (1 + ρk )∥X (t)∥∆

V˙ (X ) ≤ −ϵ αk − ϵ −1 (1 + ρk + ρ¯ 1,k ) ∥X ∥∆

ck

ri +τk rj

i=1 n

2+τk

i=1

.

rn

|ϖj ξj |

r −τ qi,k i r k i

≤ ck

i=1

(31)

Noting that 1 ≤ ϵ1,ϑ (t) ≤ ϵ2,ϑ (t) ≤ · · · ≤ ϵn,ϑ (t) ≤ ϵn+1,ϑ (t) . From (32) and Assumption 1, for i ∈ N , it yields

|φi,ϑ (t) (·)| ≤

⏐ ∑ ⏐ n ⏐ n n ⏐ ∑ ∑ ⏐ ⏐ ∂V ⏐ ∂ V ⏐ ck 2+τk ⏐ ⏐ ⏐ ⏐ ρi,k ∥X (t)∥∆ + ⏐ ∂ Xi φi,k (·)⏐ ≤ ⏐ ∂ Xi ⏐ ϖi i=1

j=1 p1,k · · · pj,k

Since τk ∈ (− 1<

that

∑i−2

1+

=

⏐ ∑n ⏐⏐ ∂ V ⏐ φ ( · ) ⏐ ⏐. i=1 ∂ Xi i,k From (33), it is easy to find that φi,k (·) is homogeneous of degree ri + τk in regard with ∆. From Lemma 1, one can achieve that ∂∂XV is homogeneous of degree 2 − ri in regard with ∆. Hence, i there are constants ρk > 0, ρi,k > 0 and ρ˜ i,k , i ∈ N , k ∈ M, so Now, we estimate the term

(37)

{

maxk∈M {1 + ρk + ρ¯ 1,k } mink∈M {αk } − λ

} ,1

(41)

where 0 < λ < mink∈M {αk } is a constant. Thus 2+τk

V˙ (X ) ≤ −λϵ∥X ∥∆

+ ρ¯ 3,k .

With (27) in mind, one has V˙ (X ) ≤ −λϵα −

2+τk 2

(V (X ))

2+τk 2

+ ρ¯ 3,k .

(42)

Then, there is a finite time T ∗ > 0 such that 1 2

|y|2 ≤ V (X ) ≤ α (

2 ρ¯ 3,k 2+τ ) k , ∀t ≥ T ∗ λϵ

It is clear that all the states of the closed-loop system are globally bounded and the tracking error y between x1 and the reference signal yr can converge to an arbitrarily small neighborhood of the origin with a sufficiently large ϵ in a finite time. ■ Remark 6. From the definitions of ϖi , i = 1, 2, . . . , n + 1 in the change of coordinates (29), ϖi are unrelated to the switching signal ϑ (t), which guarantee the feasibility of the CLF method in this paper. The homogeneous output feedback controllers (35) are dependent of the quantized parameter δ , a set of design parameters τk , ϵn+1,k , and the observer states ξˆi , i = 2, 3, . . . , n. Large control gain ϵn+1,k could benefit the control effect. However, large control gain ϵn+1,k enlarge the control input. Different from non-switching gain, ϵi,k , i = 1, 2, . . . , n + 1, k ∈ M can enlarge the freedom in the homogeneous output feedback controllers (35) to resolve the troubles caused by switching powers. Remark 7. Based on the inverted coordinates changes, the practical tracking problem of system (3) is converted into the practical stabilization problem for system (30). The output feedback controllers (35) for system (3) is constructed by four steps: Firstly, based on adding a power integrator method and a CLF, an output feedback controller (19) is designed for the nominal system (7). Secondly, introducing a series of switching-dependent gains ϵi,ϑ (t) , i ∈ N into a reduced-order observer (34) with quantized input to estimate ξ2 , ξ3 , . . . , ξn in system (30). Thirdly, the output

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx

7

feedback controller of system (30) is constructed by virtue of the homogeneous domination approach. Finally, based on the inverted coordinates changes, we obtain the output feedback controllers (35) for system (3), which regulates the tracking error into an arbitrarily small region around the origin. 5. Examples In what follows, a numerical example and a practical example are provided to test the validity of the above control approach. Example 1. Consider

⎧ p1,ϑ (t) ⎪ + φ˜ 1,ϑ (t) (·), ⎨ x˙ 1 = x2 x˙ 2 = q(uϑ (t) ) + φ˜ 2,ϑ (t) (·), ⎪ ⎩ y = x1 − y r

(43)

with M = {1, 2, 3}, p1,1 = 1, p1,2 = 3

x15 (t) sin(x1 (t)), φ˜ 1,2 (·) = 1

21

1 25 x (t), 2 1

7 , 5

p1,3 =

φ˜ 1,3 (·) = 11

13

13 , 9

φ˜ 1,1 (·) =

Fig. 2. The curves of the tracking effect in Example 1.

1 15 x (t) sin(x1 (t)), 3 1

φ˜ 2,1 (·) = x15 (t) sin(x1 (t)), φ˜ 2,2 (·) = x125 (t) sin(x2 (t)), φ˜ 2,3 (·) = 1

2

The reference signal is yr = 2 sin(2t) + 0.5 cos(π t). It is obvious that Assumptions 1 and 2 hold. Under z1 = y, z2 = x2 , system (43) becomes 1 3 x (t)x29 (t). 5 1

⎧ p1,ϑ (t) ⎪ + φ¯ 1,ϑ (t) (·), ⎨ z˙1 = z2 z˙2 = q(uϑ (t) ) + φ¯ 2,ϑ (t) (·), ⎪ ⎩ y = z1

(44)

where φ¯ 1,ϑ (t) (·) = φ˜ 1,ϑ (t) (·) − y˙ r , φ¯ 2,ϑ (t) (·) = φ˜ 2,ϑ (t) (·). 4 2 , τ3 = − 15 , r1 = 1, q1,1 = Selecting τ1 = − 52 , τ2 = − 25 3 9 q1,2 = q1,3 = 0, one has r2 = 5 , q2,1 = 1, q2,2 = 57 , q2,3 = 13 . With the help of the coordinates change (29), system (44) can be formulated as

⎧ p1,ϑ (t) ⎪ + φ1,ϑ (t) (·), ⎨ ξ˙1 = ϵ¯1,ϑ (t) ξ2 ξ˙2 = q1 (t)ϵ¯2,ϑ (t) vϑ (t) + q¯ 2 (t) + φ2,ϑ (t) (·), ⎪ ⎩ y = ξ1 qi,ϑ (t) +1

qi,ϑ (t)

where ϵ¯i,ϑ (t) = ϵi+1,ϑ (t) /ϵi,ϑ (t) , φi,ϑ (t) (·) = qi,ϑ (t) i,ϑ (t)

Fig. 3. The curves of the control inputs uk (t), k = 1, 2, 3 in Example 1.

(45)

φ¯ i,ϑ (t) (·) , ϖi

q¯ 2 (t) =

, ϖi = ϵ , i = 1, 2. By Theorem 1, the practical tracking issue of system (43) can be handled under arbitrary switching by the observers and controllers q2 (t)

ϖ2

r2

p ζ˙2 = − ϵ¯1,ϑ (t) b1 ξˆ2 1,ϑ (t) , ξˆ2 = (ζ2 + b1 ξ1 ) r1 , (46) 1 1 ( ) 1 q +1 r +τ r r uϑ (t) = − ϵ 2,ϑ (t) β2 ξˆ2 2 + β1 2 ξ1 2 ϑ (t) (47) 1 − δ 3,ϑ (t) with δ = 0.2, β1 = 4.5, β2 = 5.5, b1 = 8, ϵ1,1 = 1.6, ϵ1,2 = 2, ϵ1,3 = 1.1, ϵ2,1 = 1.65, ϵ2,2 = 2, ϵ2,3 = 1.15, ϵ3,1 = 1.7, ϵ3,2 = 2.05, ϵ3,3 = 1.5, x1 (0) = 1.5, x2 (0) = −1, ζ2 (0) =

0. Figs. 2–5 indicate the validity of the control method. Fig. 2 suggests that the output feedback controllers (47) can guarantee desired practical tracking performance. From Figs. 3–4, we can obtain that the quantization can save the bandwidth. Example 2. Some continuous stirred tank reactor (CSTR) with two-mode feed streams can be molded as the following switched system under a coordinate transformation and smooth feedback [33]:

{

x˙ 1 = x2 + f1,ϑ (t) (·), x˙ 2 = uϑ (t) ,

(48)

with ϑ (t) : [0, ∞) → M = {1, 2}, y = x1 , f1,1 = 0.5x1 , f1,2 = 2x1 and p1,1 = p1,2 = 1. Moreover, it is supposed that

Fig. 4. Quantized inputs q(uk (t)), k = 1, 2, 3 in Example 1.

there exist uncertainties in system (48), the control input u is quantized by a logarithmic quantizer (1) and the temperature tracks a given reference yr = 1.3 sin(π t)+0.7 cos(3t), which leads to the following system:

⎧ ⎪ ⎨ x˙ 1 = x2 + φ˜ 1,ϑ (t) (·), x˙ 2 = q(uϑ (t) ), ⎪ ⎩ y =x −y 1 r

(49)

3

where φ˜ 1,1 (·) = x15 (t) sin(x1 (t)), φ˜ 1,2 (·) = 0. It is obvious that Assumptions 1 and 2 hold.

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

8

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx

Fig. 8. Quantized inputs q(uk (t)), k = 1, 2 in Example 2.

Fig. 5. The switching signal ϑ (t) in Example 1.

Fig. 9. The switching signal ϑ (t) in Example 2.

Fig. 6. The curves of the tracking effect in Example 2.

system (44) can be formulated as

⎧ ⎪ ⎨ ξ˙1 = ϵ¯1,ϑ (t) ξ2 + φ1,ϑ (t) (·), ξ˙2 = q1 (t)ϵ¯2,ϑ (t) vϑ (t) + q¯ 2 (t), ⎪ ⎩ y = ξ1 qi,ϑ (t) +1

(51)

qi,ϑ (t)

where ϵ¯i,ϑ (t) = ϵi+1,ϑ (t) /ϵi,ϑ (t) , φ1,ϑ (t) (·) =

ϖi =

qi,ϑ (t) Li,ϑ (t)

, i = 1, 2.

φ¯ 1,ϑ (t) (·) , ϖ1

q¯ 2 (t) =

q2 (t)

ϖ2

,

By Theorem 1, the practical tracking issue of system (49) can be managed under arbitrary switching by the observers and controllers r2

(52) ζ˙2 = − ϵ¯1,ϑ (t) b1 ξˆ2 , ξˆ2 = (ζ2 + b1 ξ1 ) r1 , 1 ( r12 )r2 +τϑ (t) 1 q2,ϑ (t) +1 r2 uϑ (t) = − ϵ β2 ξˆ2 + β1 ξ1 (53) 1 − δ 3,ϑ (t) with δ = 0.2, β1 = 3, β2 = 7, b1 = 6, ϵ1,1 = 1.3, ϵ1,2 = 2.2, ϵ2,1 = 1.35, ϵ2,2 = 2.03, ϵ3,1 = 1.9, ϵ3,2 = 2.15, x1 (0) = 1.2, x2 (0) = −2, ζ2 (0) = 0. Figs. 6 and 9 indicate the validity of the control method.

Fig. 7. The curves of the control inputs uk (t), k = 1, 2 in Example 2.

Under z1 = y, z2 = x2 , system (49) becomes

⎧ ⎨ z˙1 = z2 + φ¯ 1,ϑ (t) (·), z˙ = q(uϑ (t) ), ⎩ 2 y = z1

Fig. 6 suggests that the output feedback controllers can guarantee desired practical tracking performance. (53). From Figs. 7 and 8, we can obtain that the quantization can save the bandwidth. (50)

where φ¯ 1,ϑ (t) (·) = φ˜ 1,ϑ (t) (x) − y˙ r . Selecting r1 = 1, τk = − 25 , q1,k = 0, k ∈ M, one has r2 = 35 , q2,k = 1, k ∈ M. Based on the coordinates change (29),

6. Conclusion This paper has solved the practical tracking issue for switched nonlinear systems with quantized input, whose powers and nonlinearities are related to the switching signal. Combined with the CLF method and the homogeneity theory, we have constructed

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx

homogeneous output feedback controllers to ensure the desired practical tracking performance. Two examples have been presented to test the effectiveness of the control approach. Some directions still can be further discussed, such as extending the analysis and design approaches to nonlinear switched systems with stochastic perturbation or unknown homogeneous growth rate [34]

9 2 ri

∑n

1

Proof of Proposition 2. Define ∥ξˆ ∥∆ = ( i=1 |ξˆi | ) 2 . From Definition 1 and the controller (19), one can obtain that vk (ξˆ ) ≤ r +τ αˆ k ∥ξˆ ∥∆n k . From Lemma 2 and (58), one has

|vk (ξˆ )| ≤ αˆ k (

n ∑

2

|ξˆi | ri )

rn +τk 2

i=1 n rn +τk rn +τk ∑ ) ( p rp ≤ αˆ k |ξ1 | r1 + |ξi i−1,k − ei i i−1,k | ri pi−1,k

Acknowledgments

i=2

This work is supported in part by National Natural Science Foundation of China [grant number 61873061], and Qing Lan Project, China.

≤ αˆ k

Declaration of competing interest

≤ αˆ k

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

rn

2 − ri−1 ri 2 − ri−1

=

ri

ξi ξi

2−ri−1 −ri ri

− bi−1 [(ξ

ξ

1

pi−2,k p i−2,k i−1 )

Noting that Lemma 2

ri−1 ri pi−1,k

ri−1

p

pi,k i+1

{

r pi−1,k r pi−1 i i−1,k ) i

(ξ

− (ξˆ

}

(54)

ri−1

|(ξi i−1,k ) ri pi−1,k − (ξˆi i−1,k ) ri pi−1,k | ≤ αˆ k |ei |ri−1 + αˆ 1,k |ξi | 1

pi−2,k p i−2,k i−1 )

|(ξ

1

pi−2,k p i−2,k i−1 )

− (ξˆ

|≤

1− 1 2 pi−2,k

ri−1 ri

,

|ei−1 |ri−1

(55) (56)

with a constant αˆ 1,k > 0. Taking (10) into account, one has pi,k i+1 ri−1

|ξ |ξi

ri

| ≤ αˆ k (|ηi+1 |

ri +τk

ri +τk

+ |ηi |

),

| ≤ αˆ (|ηi |ri−1 + |ηi−1 |ri−1 ).

i+1 ∑ ⏐ 2−ri−ri1 −ri pi,k ⏐ ⏐ξ ξi+1 ⏐ ≤ αˆ k |ηj |2+τk −ri−1 . i

(58)

ri

≤ αˆ k

i+1 ∑

(59)

|ηj |

( ri−ri 1 ) p ξi − (ζi + bi−1 ξi−1 ) ξi+i,1k

2+τk −ri−1

+ bi−1

8

|ei−1 |

2−rn−1 −rn rn

ξn

2+τk

ηj

( rnr−n 1 ) ξn − (ζn + bn−1 ξn−1 ) q1 (t)vk (ξˆ )

+ γn,2,k (δ )

j=1

n ∑

2+τk

ej

2+τ

+ hn,k (bn−1 , δ )en−1k

(62)

j=2

2−ri−1 ri

(63)

> 1, from (18), Lemma 2 and the definition of ξˆi ,

there is a continuous function h˜ i,k (bi−1 ) > 0, so that 2−ri−1 ) ( 2−rrii−1 ξˆi − (ζi + bi−1 ξi−1 ) ri−1 2−2ri−1 ⏐ ⏐⏐ ≤ αˆ bi−1 |ei |ri pi−1,k ⏐ζi + bi−1 ξˆi−1 − (ζi + bi−1 ξi−1 )⏐⏐(ζi + bi−1 ξˆi−1 ) ri−1 2−2ri−1 ⏐ + (bi−1 ξˆi−1 − bi−1 ξi−1 ) ri−1 ⏐

r pi−1,k

− bi−1 ei i

2−2ri−1 ( 2−2ri−1 ) r ≤ αˆ αˆ k b2i−1 |ei |ri pi−1,k |ei−1 |ri−1 |ξˆi | ri + bi−1i−1 |ei−1 |2−2ri−1

1 16

2+τ

2+τk

(ηi−1 k + ηi

2+τk

) + ei

+ h˜ i,k (bi−1 )e2i−+τ1 k .

(64)

Proof of Proposition 4. Utilizing Lemma 2 to (19), we can obtain that

1 1 1 1 1 ( 1 )r +τ { 1 r r r r βn ξˆnrn + βnr−n 1 (ξˆn−n−11 + · · · + β2 3 (ξˆ2 2 + β1 2 ξ1 ) · · ·) n k 1−δ 1 1 1 1 1 ( 1 )r +τ } 1 r r r r − βn ξnrn + βnr−n 1 (ξn−n−11 + · · · + β2 3 (ξ2 2 + β1 2 ξ1 ) · · ·) n k 1−δ n 1 1 (∑ αˆ k (1 + δ ) r r )r +τ ≤ |ηn |2−rn |ξi i − ξˆi i | n k . (65) 1−δ

≤ q1 (t)ηn2−rn

In line with (55) and (58), one has

|ξi | ≤ αˆ (|ηi | + |ηi−1 |), 1 r

) .

(61)

i=2

1 ri

( αˆ k |ei |ri−1 + αˆ 1,k αˆ (|ηi |ri−1 + |ηi−1 |ri−1 ) ri−1

)

i=2

j=i−1 1− 1 2 pi−2,k

|ei |rn +τk

q1 (t)ηn2−rn (vk − ξn∗+1 )

Considering (55), (56), (58) and (59) into (54) renders 2−ri−1 −ri ri

|ηi |rn +τk +

(57)

j=i−1

ξi

i=2 n ∑

)

where γn,2,k (δ ) and hn,k (bn−1 , δ ) are two non-negative continuous functions of bn−1 and δ .

≤

From Lemmas 2 and 3, it yields

2 − ri−1

n 1∑

Since

> 1, we can get the following estimate from

p

≤

− (ξˆ ]

i=1 n (∑

|ei |rn +τk

|ξˆi | ≤ αˆ k (|ei |ri + |ηi |ri + |ηi−1 |ri ).

r pi−1,k r pi−1 i i−1,k ) i

1

pi−2,k p i−2,k i−1 )

n ∑

Proof of Proposition 3. Recalling that definitions of ei , ξi∗ , i = 2, . . . , n, one has

) ( ri−ri 1 − (ζi + bi−1 ξi−1 ) ξ˜i+1 ξi

2−ri−1 −ri ri

+

In line with the proof of Proposition 1, we can obtain the estimate

Appendix

Proof of Proposition 1.

|ξi |

rn +τk ri

i=1

2 − rn−1

For the sake of simplification, we utilize two generic constants to stand for some finite positive constants which perhaps are not same in different places, i.e. αˆ which is independent of the switching signal and αˆ k , k ∈ M relate to the switching signal.

n (∑

(60)

Proposition 1 can be obtained by applying Lemma 3 to the above inequality with e1 = 0 and h2,k (b1 ) = 0.

1 r

p

1

p

1

1

|ξi i − ξˆi i | = |(ξi i−1,k ) ri pi−1,k − (ξˆi i−1,k ) ri pi−1,k | ≤ αˆ k |ei | + αˆ 2,k |ξi | ri where αˆ 2,k > 0 is a constant. Under which q1 (t)ηn2−rn (vk − ξn∗+1 )

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.

10

Y. Jiang and J. Zhai / ISA Transactions xxx (xxxx) xxx n

≤

∑( )r +τ αˆ k (1 + δ ) |ηn |2−rn αˆ k |ei | + αˆ 2,k αˆ (|ηi | + |ηi−1 |) n k 1−δ i=2

n ∑ ( rn +τ ) αˆ k (1 + δ ) ≤ |ηn |2−rn |ei | k + |ηi |rn +τk + |ηi−1 |rn +τk 1−δ i=2

≤

n 1∑

4

2+τk

ηi

i=1

+ γ3,k (δ )

n ∑

2+τk

ej

(66)

j=2

where γ3,k (δ ), k ∈ M are non-negative continuous functions of δ . References [1] Liberzon D, Morse A. Basic problems in stability and design of switched systems. IEEE Control Syst 1999;19(5):59–70. [2] Lin H, Antsaklis P. Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans Automat Control 2009;54(2):308–22. [3] Long L, Wang Z, Zhao J. Switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems. Automatica 2015;54:217–28. [4] Long L, Zhao J. Adaptive disturbance rejection for strict-feedback switched nonlinear systems using multiple Lyapunov functions. Internat J Robust Nonlinear Control 2014;24(13):1887–902. [5] Sakly A, Kermani M. Stability and stabilization studies for a class of switched nonlinear systems via vector norms approach. ISA Trans 2015;57:144–61. [6] Huang S, Xiang Z. Finite-time stabilisation of a class of switched nonlinear systems with state constraints. Internat J Control 2018;91(6):1300–13. [7] Fu J, Ma R, Chai T. Global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers. Automatica 2015;54:360–73. [8] Liberzon D. Switching in systems and control. Boston: Birkhauser; 2003. [9] Zhai J, Song Z, Fei S, Zhu Z. Global finite-time output feedback stabilization for a class of switched high-order nonlinear systems. Internat J Control 2018;91(1):170–80. [10] Zhai J, Song Z. Global finite-time stabilization for a class of switched nonlinear systems via output feedback. Int J Control Autom Syst 2017;15(5):1975–82. [11] Zhai J, Song Z, Karimi H. Global finite-time control for a class of switched nonlinear systems with different powers via output feedback. Int J Syst Sci 2018;49(13):2776–83. [12] Yu J, Shi P, Dong W, Yu H. Observer and command-filter-based adaptive fuzzy output feedback control of uncertain nonlinear systems. IEEE Trans Ind Electron 2015;62(9):5962–70. [13] Yu J, Shi P, Zhao L. Finite-time command filtered backstepping control for a class of nonlinear systems. Automatica 2018;92:173–80. [14] Kim H, Yoo S. Memoryless disturbance-observer-based adaptive tracking of uncertain pure-feedback nonlinear time-delay systems with unmatched disturbances. ISA Trans 2017;70:419–31.

[15] Niu B, Zhao J. Barrier Lyapunov functions for the output tracking control of constrained nonlinear switched systems. Syst Control Lett 2013;62(10):963–71. [16] Wang X, Li H, Zong G, Zhao X. Adaptive fuzzy tracking control for a class of high-order switched uncertain nonlinear systems. J Franklin Inst 2017;354(15):6567–87. [17] Song Z, Zhai J. Practical output tracking control for switched nonlinear systems: A dynamic gain based approach. Nonlinear Anal Hybrid Syst 2018;30:147–62. [18] Huang S, Xiang Z. Finite-time output tracking for a class of switched nonlinear systems. Internat J Robust Nonlinear Control 2017;27(6):1017– 38. [19] Wang H, Yang G. Dynamic output feedback controller design for affine T-S fuzzy systems with quantized measurements. ISA Trans 2016;64:202–15. [20] Elia N, Mitter S. Stabilization of linear systems with limited information. IEEE Trans Automat Control 2001;46(9):1384–400. [21] Liu J, Elia N. Quantized feedback stabilization of non-linear affine systems. Internat J Control 2004;77(3):239–49. [22] Ao W, Song Y, Wen C. Adaptive robust fault tolerant control design for a class of nonlinear uncertain MIMO systems with quantization. ISA Trans 2017;68:63–72. [23] Fu M, Xie L. The sector bound approach to quantized feedback control. IEEE Trans Automat Control 2005;50(11):1698–711. [24] Xing L, Wen C, Su H, Liu Z, Cai J. Robust control for a class of uncertain nonlinear systems with input quantization. Internat J Robust Nonlinear Control 2016;26(8):1585–96. [25] Zhou J, Wen C, Wang W. Adaptive control of uncertain nonlinear systems with quantized input signal. Automatica 2018;95:152–62. [26] Wakaiki M, Yamamoto Y. Stability analysis of sampled-data switched systems with quantization. Automatica 2016;69:157–68. [27] Sui S, Tong S. Fuzzy adaptive quantized output feedback tracking control for switched nonlinear systems with input quantization. Fuzzy Sets and Systems 2016;290:56–78. [28] Bacciotti A, Rosier L. Liapunov functions and stability in control theory. London: Springer; 2005. [29] Hermes H. Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. Differ Equations Stab Control 1991;109:249–60. [30] De Persis C, Mazenc F. Stability of quantized time-delay nonlinear systems: a Lyapunov–Krasowskii-functional approach. Math Control Signals Systems 2010;21(4):337–70. [31] Yu J, Shi P, Dong W, Lin C. Neural network-based adaptive dynamic surface control for permanent magnet synchronous motors. IEEE Trans Neural Netw Learn Syst 2015;26(3):640–5. [32] Rui C, Reyhanoglu M, Kolmanovsky I, Cho S, McClamroch N. Nonsmooth stabilization of an underactuated unstable two degrees of freedom mechanical system. In: Proceedings of the 36th IEEE conference on decision and control, San Diego, California, USA, 1997, p. 3998–4003. [33] Ma R, Zhao J. Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings. Automatica 2010;46:1819–23. [34] Zhai J, Karimi H. Global output feedback control for a class of nonlinear systems with unknown homogenous growth condition. Internat J Robust Nonlinear Control 2019;29(7):2082–95.

Please cite this article as: Y. Jiang and J. Zhai, Practical tracking control for a class of high-order switched nonlinear systems with quantized input. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.06.022.