State-constrained robust stabilisation for a class of high-order ...

IET Control Theory & Applications Brief Paper

State-constrained robust stabilisation for a class of high-order switched non-linear systems Ben Niu1

ISSN 1751-8644 Received on 13th August 2014 Revised on 19th March 2015 Accepted on 19th April 2015 doi: 10.1049/iet-cta.2014.0898 www.ietdl.org

, Zhengrong Xiang2

1

College of Mathematics and Physics and Automation Research Institute, Bohai University, Jinzhou, Liaoning Province 121013, People’s Republic of China 2 College of Automation, Nanjing University of Science andTechnology, Nanjing, Jiangsu Province 210094, People’s Republic of China E-mail: [email protected]

Abstract: This study deals with the problem of robust stabilisation for a class of state-constrained high-order uncertain switched non-linear systems in lower triangular form. Bounded state-feedback controllers are designed to assure that asymptotic stabilisation is achieved in a domain without violation of the state constraint, and all closed-loop signals are bounded, when an appropriate requirement on the initial condition is imposed. Simulation results are provided to show the efficiency of the developed method.

1

Introduction

During the past few decades, switched systems have received considerable attention [1–8]. The motivation for the study of such systems stems from two aspects. On the one hand, so many realworld systems can be modelled as switched systems, such as chemical systems, power systems and communication networks [9– 14]. On the other hand, the switching control strategy has been widely used in various advanced controls, such as in [15–20]. When the switching law has no given mode (or is arbitrary), one way to investigate the stability and stabilisation problems is to find a common Lyapunov function (CLF) for all the subsystems. However, finding a CLF for a family of subsystems is still an open problem, even though some progress has been made [21–24]. Backstepping design technique is a power tool for constructing stabilising controller for non-linear systems in lower triangular form, and many significant developments have been achieved based on this technique [25–30]. This design method is, of course, expected to be useful for stabilising a switched non-linear system. However, according to the standard backstepping procedure, backstepping cannot be directly extended to the switched non-linear system in lower triangular form because of the different coordinate transformations for different subsystems [31, 32]. Furthermore, the so-called adding a power integrator backstepping design method is proposed in [33–36], which is used in the controller design for high-order non-linear systems. Nevertheless, there exist only a few results on high-order switched non-linear system in lower triangular form [37, 38]. In addition, it is worth pointing out that many systems in practice have states/output constraints. Therefore the constrained control problem of switched systems is of great significance. Unfortunately, these constraints are not widespreadly taken into account in the aforementioned papers. A few exceptions are [39], in which tracking control for switched lower triangular systems with time-varying output constraints is achieved, and [40], which proposed a control design method for a class of switched non-linear systems with full state constraints. On the other hand, there exist several techniques to solve the constrained control problem for linear systems and non-linear systems. In the literature, most of the results are based on the notions of set invariance and admissible set control [41, 42]. Model predictive control that represents an effective control design methodology for handling both constraints and performance issues has been investigated in [43]. Besides, reference governors have also been proposed to tackle the problem of constraints for non-linear systems

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1901–1908 © The Institution of Engineering and Technology 2015

[44]. The approaches mentioned above are numerical in nature or depend heavily on computationally intensive algorithms to solve the control problems. Recently, a barrier Lyapunov function (BLF) is proposed in [45] to deal with the problem of asymptotical stabilisation in a domain for a non-switched feedback linearisable system, which can be transformed into a Brunovsky normal form with state constraints. However, a resulting problem is that the constructed asymmetric BLF is of a switching type, a C 1 function. Consequently, the subsequent stabilising functions must be of a high power. This is somewhat undesirable and could increase the control effort and decrease the robustness of the controlled system. Further, it is worth pointing out that p-times differentiable unbounded functions, firstly introduced in [46] to handle the output-constrained control problem, are proposed only for low-order non-linear systems. In this paper, we propose a novel robust controller design method to handle the state-constrained robust stabilisation (SCRS) problem for a class of high-order uncertain switched non-linear systems in lower triangular form. Bounded state-feedback controllers and an appropriate CLF for the switched system are explicitly constructed based on the adding a power integrator backstepping technique. In the proposed approach, the p-times differentiable unbounded functions are introduced to realise system conversion. Then, the backstepping technique is employed to design controllers for the conversional system. The contributions of this paper can be characterised by the following features: (i) The p-times differentiable unbounded functions are employed to solve the SCRS problem of high-order switched non-linear systems with uncertainties for the first time. It should be noted that the results obtained in this paper is not a simple parallel extension of [46]. Because in each step of the controller design procedure, we must choose a common virtual control law to counteract the influence of switched subsystems on the performance of closed-loop system, which is different with [46]. (ii) For the BLFs approach of switched non-linear systems with output/state constraints in [9, 10], a conservative assumption is imposed in the backstepping design procedure, which is called the simultaneous domination assumption. However, in the controller design procedure of this paper, the simultaneous domination assumption is removed via choosing a maximal common stabilising function for all subsystems at each step. (iii) When the state or output constraint is asymmetrical, the proposed asymmetric BLF in [9, 45] is of a switching type. Owing

1901

to the structural features of the employed p-times differentiable unbounded functions in this paper, no switchings are needed in our proposed controllers despite the asymmetric limit range, which avoids the defects in [9, 45]. The remainder of this paper is organised as follows. Section 2 offers problem statements. The SCRS control design method for high-order uncertain switched non-linear systems is given in Section 3. After that, in Section 4, two simulation examples are presented to show the effectiveness of the controllers. The paper is concluded in Section 5.

2

Problem statements

Consider a class of high-order uncertain switched non-linear systems

(3) All signals of the closed-loop system (1) with the designed controllers are bounded. In the following paragraph, we review the following definition that will be used in the development of the main results. Definition 1 [46]: A function (x, a, b) is said to be a p-times differentiable unbounded function if it holds the following properties: (1) x = 0 ⇔ (x, a, b) = 0, (2) limx→a− (x, a, b) = −∞, limx→b+ (x, a, b) = ∞, (3) (x, a, b) is p times differentiable with respect to x, for all x ∈ (a, b), (4)  (x, a, b) > 0, ∀x ∈ (a, b), (5)  (x, a, b) ≥ δ3 (ρ3 ) > 0, ∀x ∈ (a + ρ3 , b − ρ3 ), with 0 < ρ3 < [(b − a)/2],

p

x˙ 1 = f1,σ (t) (x1 , d(t)) + g1,σ (t) (x1 )x21 x˙ 2 =

p f2,σ (t) (¯x2 , d(t)) + g2,σ (t) (¯x2 )x32

··· p

x˙ n = fn,σ (t) (¯xn , d(t)) + gn,σ (t) (¯xn )uσn(t)

(1)

where (x1 , x2 , . . . , xn )T ∈ Rn and u ∈ R are the state and control inputs of the system, respectively, and x¯ i = (x1 , x2 , . . . , xi )T , i = 1, 2, . . . , n. d(t) is an unknown piecewise continuous disturbance or parameter belonging to a known compact set  ⊂ Rs , pi , i = 1, 2, . . . , n are positive integers. σ (t) : [0, ∞) → Im = {1, 2, . . . , m} denotes the ‘switching function’, which is assumed to be a piecewise constant function continuous from the right. All functions are smooth with fi,k (0, d(t)) = 0 and 0 < g ≤ gi,k (¯xi ) ≤ gi (¯xi ), g, gi (¯xi ) are a positive constant and a known function, i = 1, 2, . . . , n, k ∈ Im . It is assumed that the state x(t) is continuous at each switching instants. In addition, the Zeno behaviour is not considered in this paper. In the following section, we suppose that the following growth condition is satisfied.

Remark 1: In Definition 1, if a = −b, then it is easy to get many p-times differentiable unbounded functions. An example is the function tan(−[π/2a]x). If a = −b, it is more difficult to find a p-times differentiable unbounded function. However, we can construct a p-times differentiable unbounded function by using Lemma 2 in [46]. Assumption 4: The p-times differentiable unbounded function in Definition 1 is strictly monotonic and satisfies (x, a, b) = x[1 + χ(x, a, b)]

(5)

where χ(x) is a non-negative smooth function. Lemma 1 [33]: For any positive real numbers a, d and any realvalued function ρ(x, y) > 0,

Assumption 1: For i = 1, 2, . . . , n, |fi,k (¯xi , d(t))| ≤ (|x1 |pi + |x2 |pi + · · · + |xi |pi )μi,k (¯xi )

where p is a positive integer, a and b are constants such that a < 0 < b,  (x, a, b) = [{∂(x, a, b)}/∂x], and δ3 (ρ3 ) is a positive constant depending on the positive constant ρ3 . Moreover, if p = ∞, then the function (x, a, b) is said to be a smooth unbounded function.

(2)

where μi,k (¯xi ), k ∈ Im are a set of known non-negative smooth functions.

|x|a |y|d ≤

a d ρ(x, y)|x|a+d + ρ −a/d (x, y)|y|a+d a+d a+d

(6)

Assumption 2: p1 ≥ p2 ≥ · · · ≥ pn ≥ 1 are odd integers.

Lemma 2 ([47] Barbalat’ s lemma): Consider a differentiable ˙ is uniformly function h(t). If limt→∞ h(t) is finite and h(t) ˙ = 0. continuous, then limt→∞ h(t)

Assumption 3: At the initial time t0 , there exist strictly positive constants L1 < L and L1 < L such that

3

−L1 ≤ x1 (t0 ) ≤ L1

(3)

where x1 (t0 ) is the initial value of the state x1 . In this paper, we solve the following SCRS problem for system (1). 2.1

SCRS problem

For system (1) satisfying Assumptions 1–3, under arbitrary switchings, design state feedback controllers to ensure that: (1) The closed-loop system (1) with the designed controllers is asymptotically stable in a domain. (2) The state x1 is within a pre-specified limit range, that is −L ≤ x1 (t) ≤ L

(4)

for all t ≥ t0 ≥ 0, where L and L are strictly positive constants.

1902

Controllers design and analysis

In this section, we present a controller design approach for system (1) by the adding a power integrator backstepping technique, which solves the SCRS problem. The state-feedback controllers will be constructed explicitly through the following several steps. First, we choose a change of coordinates z1 = (x1 , a, b)

(7)

where (x1 , a, b) is a p-times differentiable unbounded function with p ≥ n − 1, and the constants a and b are chosen such that −L ≤ a < −L1 ,

L1 < b ≤ L¯

(8)

On the basis of the properties of (x1 , a, b) shown in Definition 1, we conclude that if we are able to design a control input u such that limt→∞ z1 (t) = 0 and keep all signals of the corresponding closed-loop system bounded for a bounded z1 (t0 ), then the SCRS problem of system (1) is solved. It is seen that z1 (t0 ) is bounded under the choice of the constants a and b in (8), the assumption

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1901–1908 © The Institution of Engineering and Technology 2015

on the initial state of x1 in (3), and the properties of the function (x1d , a, b) listed in Definition 1. Differentiating both sides of (7) in conjunction with system (1), one can rewrite them in the form

q −1

p

x˙ 2 = f2,k (¯x2 , d(t)) + g2,k (¯x2 )x32

+ z2 2

···

V˙ 1 (z1 ) = 

k ∈ Im

(9)

p (x1 , a, b)z1 [f1,k (z1 , d(t)) + g1,k (z1 )x21 ]



p

+ g1,k (z1 )z1 (x21 − φ1 1 (z1 ))

|f1,k (z1 , d(t))| ≤ |z1 | μˆ 1,k (z1 ),

p +1

ϕ˜1,k (z1 ),

∀k ∈ Im

(11)

p

q −1

+ |z2 2

p

2,k (¯z2 , d(t))| + g2,k (¯z2 )φ2 2 (¯z2 ) p

p

+ g2,k (¯z2 )(x32 − φ2 2 (¯z2 ))

(16)

∀k ∈ Im

 p V˙ 1 ≤  (x1 , a, b) |z1 1,k (z1 , d(t))| + g1,k (z1 )z1 φ1 1 (z1 )  p p + g1,k (z1 )z1 (x21 − φ1 1 (z1 ))  p +1 p ≤  (x1 , a, b) z1 1 ϕ˜1,max (z1 ) + g1,k (z1 )z1 φ1 1 (z1 )  p p (13) + g1,k (z1 )z1 (x21 − φ1 1 (z1 ))

(14) Substituting (14) into (13), one has  g1,k (z1 ) p1 +1 p +1 V˙ 1 (z1 ) ≤  (x1 , a, b) z1 1 ϕ˜1,max (z1 ) − z1 ϕ˜1,max (z1 ) g g1,k (z1 ) −1 g1 (z1 ) p +1 p +1 g1,k (z1 )z1 1 − n (x1 , a, b)z1 1 g g  p p + g1,k (z1 )z1 (x21 − φ1 1 (z1 ))

−

p +1

−  (x1 , a, b)g1 (z1 )z1 1

(18)

p +1

+  (x1 , a, b)g1 (z1 )z1 1

p +1

+  (x1 , a, b)g1 (z1 )z2 1 p +1

+ z2 1

p +1

ϕ˜2 (¯z2 ) + z1 1

q −1 p2 φ2 (¯z2 )

ϕˆ2,k (¯z2 ) + g2,k (¯z2 )z2 2 q −1

+ g2,k (¯z2 )z2 2

p +1

≤ −(n − 1)z1 1

p

+ g2,k (¯z2 )z2 2

p +1

≤ −(n − 1)z1 1

p

(x32 − φ2 2 (¯z2 ))

+ ( (x1 , a, b)g1 (z1 )ϕ˜2 (¯z2 )

q −1

1 φ1 (z1 ) = z1 − (ϕ˜1,max (z1 ) + g1 (z1 ) + n/ (x1 , a, b)) g

ϕ˜2 (¯z2 )

p +1 + z2 1 ϕˆ2,k (¯z2 )

p +1 p +1 V˙ 2 (¯z2 ) ≤ −nz1 1 −  (x1 , a, b)g1 (z1 )z1 1

+ ϕˆ2,k (¯z2 ))z2 1

(1/p1 )

≤

p +1 z1 1

p +1

+ z2 1

where ϕ˜2 (¯z2 ) ≥ 0, ϕˆ2,k (¯z2 ) ≥ 0, k ∈ Im are some smooth functions. Thus, we obtain

p +1

where ϕ˜1,max (z1 ) ≥ ϕ˜1k (z1 ) ≥ 0, ∀k ∈ Im is a smooth function. Design the common stabilising function 

p +1

q −1 |z2 2 2,k (¯z2 , d(t))|

(12)

(17)

where μˆ 2,k (¯z2 ) are a set of non-negative smooth functions. Furthermore, according to Lemma 1 and (17), one can infer that p



p +  (x1 , a, b)g1,k (z1 )z1 (x21

+  (x1 , a, b)g1 (z1 )|z1 ((z2 + φ1 (z1 ))p1 − φ1 1 (z1 ))|

|z1 ((z2 + φ1 (z1 ))p1 − φ1 1 (z1 ))| ≤ z1 1

where 1,k (z1 , d(t)) = f1,k (z1 , d(t)) and ϕ˜1,k (z1 ) = μˆ 1,k (z1 ), k ∈ Im are non-negative smooth functions. Then, it can be seen that

p +1

p +1

−  (x1 , a, b)g1 (z1 )z1 1

|2,k (¯z2 , d(t))| ≤ (|z1 |p2 + |z2 |p2 )μˆ 2,k (¯z2 )

where μˆ 1,k (z1 ) are a set of non-negative smooth functions. Then, we can obtain |z1 1,k (z1 , d(t))| ≤ z1 1

p +1

≤ −nz1 1

(10)

Using Assumptions 4 and 1, it can be found that p1

p

where 2,k (¯z2 , d(t)) = f2,k (¯z2 , d(t)) − [{∂φ1 (z1 )}/∂z1 ] 1,k (¯z2 ), 1,k (¯z2 ) =  (x1 , a, b)(f1,k (z1 , d(t)) + g1,k (z1 )(z2 + φ1 (z1 ))), k ∈ Im . Using Assumption 5 gives that

 p ≤  (x1 , a, b) |z1 f1,k (z1 , d(t))| + g1,k (z1 )z1 φ1 1 (z1 )

≤ −nz1 1

p

(2,k (¯z2 , d(t)) + g2,k (¯z2 )φ2 2 (¯z2 ) p

p

p

+ g2,k (¯z2 )(x32 − φ2 2 (¯z2 )))

Step 1: Choose V1 (z1 ) = 12 z12 and let z2 = x2 − φ1 (z1 ), where φ1 (z1 ) is the common stabilising function to be designed. The derivative of V1 (z1 ) is given by

p +1 p +1 V˙ 2 (¯z2 ) = −nz1 1 −  (x1 , a, b)g1 (z1 )z1 1 p

p

p fn,k (¯xn , d(t)) + gn,k (¯xn )uk n ,

and

+  (x1 , a, b)g1,k (z1 )z1 (x21 − φ1 1 (z1 ))

z˙1 =  (x1 , a, b)(f1,k (x1 , d(t)) + g1,k (x1 )x21 )

x˙ n =

q

Choose V 2 (¯z2 ) = V1 (z1 ) + (1/q2 )z2 2 , q2 = p1 − p2 + 2, then the time derivative of V 2 (¯z2 ) is given by

q −1 p2 φ2 (¯z2 )

+ g2,k (¯z2 )z2 2 p

p

(x32 − φ2 2 (¯z2 )) p +1

+ z2 1

ϕ2,max (¯z2 )

q −1 p + g2,k (¯z2 )z2 2 φ2 2 (¯z2 ) q −1 p p + g2,k (¯z2 )z2 2 (x32 − φ2 2 (¯z2 ))

(19)

where ϕ2,max ≥  (x1d , a, b)g1 (z1 )ϕ˜2 (¯z2 ) + ϕˆ2,k (¯z2 ) is a smooth function. Design the common stabilising function (1/p2 )  1 φ2 (¯z2 ) = z2 − (ϕ2,max (¯z2 ) + (n − 1)) g

(20)

Substituting (20) into (19) yields

p − φ1 1 (z1 ))

(15)

where the coupling term  (x1 , a, b)g1,k (z1 )z1 (x21 − φ1 1 (z1 )) will be canceled in the subsequent step. Step 2: Let z3 = x3 − φ2 (¯z2 ), where φ2 (¯z2 ) is the common stabilising function to be designed. p

p

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1901–1908 © The Institution of Engineering and Technology 2015

p +1 p +1 V˙ 2 (¯z2 ) ≤ −(n − 1)(z1 1 + z2 1 ) q −1

+ g2,k (¯z2 )z2 2

q −1

where the coupling term g2,k (¯z2 )z2 2 celed in the subsequent step.

p

p

p

p

(x32 − φ2 2 (¯z2 ))

(21)

(x32 − φ2 2 (¯z2 )) will be can-

1903

Step i: Let zi+1 = xi+1 − φi (¯zi ), where φi (¯zi ) is the common stabilising function to be designed. Assume that we have completed the first i − 1(2 ≤ i ≤ n) steps, that is, for the following collection of auxiliary (z1 , . . . , zi−1 )equations

where ϕ˜i (¯zi ) ≥ 0, ϕˆi,k (¯zi ) ≥ 0 are some smooth functions. Thus, one can obtain p +1 p1 +1 p +1 ) + z1 1 V˙ i (¯zi ) ≤ −(n − i + 2)(z1 1 + · · · + zi−1 p +1

p +1

1 + · · · + zi−1 + ϕˆi,k (¯zi )zi 1

p

j , z˙j = j,k (¯zj , d(t)) + gj,k (¯zj )xj+1

j = 1, . . . , i − 1

p +1

+ gi−1,k (¯zi−1 )ϕ˜i (¯zi )zi 1

(22)

q −1

+ gi,k (¯zi )zi i

where

p

p

i (xi+1 − φi (¯zi ))

p +1

≤ −(n − i + 2)(z1 1

j,k (¯zj , d(t)) = fj,k (¯zj , d(t)) −

p +1 p +1

1 + · · · + zi−1 + (ϕˆi,k (¯zi ) + gi−1,k (¯zi−1 )ϕ˜i (¯zi ))zi 1

q −1 pi p q −1 pi − φi (¯zi )) φi (¯zi ) + gi,k (¯zi )zi i (xi+1 p +1 p1 +1 p +1 ) + z1 1 ≤ −(n − i + 2)(z1 1 + · · · + zi−1 p1 +1 p +1 q −1 p + · · · + zi−1 + ϕi,max (¯zi )zi 1 + gi,k (¯zi )zi i φi i (¯zi ) q −1 pi p (26) − φi (¯zi )) + gi,k (¯zi )zi i (xi+1

+ gi,k (¯zi )zi i

l=1

we have a set of common stabilising functions (14), (20) and  φj (¯zj ) = zj

p +1

1 + · · · + zi−1 ) + z1 1

p +1

j−1  ∂φj−1 (¯zj−1 )

l,k (¯zl−1 ) ∂zl

(1/pj ) 1 − (ϕj,max (¯zj ) + (n − j + 1)) g

q −1 pi φi (¯zi )

+ gi,k (¯zi )zi i

(23)

where 3 ≤ j ≤ i − 1, such that there exists a CLF for the transform system (22)

where ϕi,max (¯zi ) ≥ gi−1,k (¯zi−1 )ϕ˜i (¯zi ) + ϕˆi,k (¯zi ) is a smooth functions. Design the common stabilising function 

V i−1 (¯zi−1 ) = V1 (z1 ) +

i−1  1 ql z ql l

φi (¯zi ) = zi

(24)

1 − (ϕi,max (¯zi ) + (n − i + 1)) g

(1/pi ) (27)

l=2

Then, substituting (27) into (26) yields and the time derivative of V i−1 (¯zi−1 ) satisfies V˙ i−1 (¯zi−1 ) ≤ − p1 +1 qi−1 −1 pi−1 pi−1 p +1 ) + gi−1,k (¯zi−1 )zi−1 (xi − φi−1 (n − i + 2)(z1 1 + · · · + zi−1 (¯zi−1 )). q Choose V i (¯zi ) = V i−1 (¯zi−1 ) + q1i zi i , qi = p1 − pi + 2, then one conclude

p +1 p1 +1 ) V˙ i (¯zi ) ≤ −(n − i + 1)(z1 1 + · · · + zi−1 p +1

+ ϕi,max (¯zi )zi 1 −

p +1 p1 +1 V˙ i (¯zi ) ≤ −(n − i + 2)(z1 1 + · · · + zi−1 ) q

i−1 + gi−1,k (¯zi−1 )zi−1

q −1

+ zi i

−1

p

q

i−1 × |zi−1

−1

p +1

q −1 pi + gi,k (¯zi )zi i (xi+1

p

q −1

+ |zi i

p

p

q −1 pi φi (¯zi )

p

p

i (xi+1 − φi i (¯zi ))

)

p − φi i (¯zi )) p

(28) p

i − φi i (¯zi )) will be canwhere the coupling term gi,k (¯zi )zi i (xi+1 celled in the subsequent step. Step n: By repeating the inductive argument above, it is straightforward to see that at the final step, there exists a CLF of system (1)

i,k (¯zi , d(t))| + gi,k (¯zi )zi i q −1

p +1

+ · · · + zi 1

q −1

p +1

1 + · · · + zi−1 ) + gi−1,k (¯zi−1 )

i−1 (xi i−1 − φi−1 (¯zi−1 ))|

+ gi,k (¯zi )zi i

p

≤ −(n − i + 1)(z1 1

i + gi,k (¯zi )(xi+1 − φi,ki (¯zi )))

p +1

gi,k (¯zi ) q −1 p +1 (n − i + 1)zi 1 + gi,k (¯zi )zi i g p

p

≤ −(n − i + 2)(z1 1

gi,k (¯zi ) p +1 ϕi,max (¯zi )zi 1 g

i × (xi+1 − φi (¯zi ))

p

i−1 (xi i−1 − φi−1 (¯zi−1 ))

(i,k (¯zi , d(t)) + gi,k (¯zi )φi,ki (¯zi ) p

−

1 2 zl 2 n

(25)

V n (¯zn ) = V1 (z1 ) +

(29)

l=2

where

Then, we can explicitly design individual controller for each subsystem

i,k (¯zi , d(t)) = fi,k (¯zi , d(t)) −

i−1  ∂φl−1 (¯zl−1 )

∂zi

l=1

l,k (¯zl−1 )

Similarly to Step 2, one has q

i−1 |zi−1

−1

p

p +1

1 zi−1 z1 1 p +1 + zi 1 ϕ˜i (¯zi ), + ··· + 2gi−1 (¯zi−1 ) 2gi−1 (¯zi−1 )

q −1

× |zi i

i (¯zi , d(t))|

1 p1 +1 1 p +1 p +1 + zi 1 ϕˆi,k (¯zi ) ≤ z1 1 + · · · + zi−1 2 2

1904

∀k ∈ Im

(30)

such that

i−1 ((zi + φi−1 (¯zi−1 ))pi−1 − φi−1 (¯zi−1 ))|

p +1

≤

(1/pn )  1 (ϕn,max (¯zn ) + 1) , uk (¯zn ) = zn − gi,k

p +1 p +1 V˙ n (¯zn ) ≤ −(z1 1 + · · · + zn 1 ),

∀¯zn = 0

(31)

Under the discussion of the above section, we are now in a position to show the main result of this paper. Theorem 1: Suppose that Assumptions 1–4 hold, then the SCRS problem of system (1) is solved by the controllers designed as in (30) under arbitrary switchings.

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1901–1908 © The Institution of Engineering and Technology 2015

Proof: (i) Forward completeness. From (31) and  (x1 , a, b) > 0 for all x1 (t) ∈ (a, b), see Property (4) of the function (x1 , a, b) in Definition 1, one obtains V˙ n ≤ 0 ⇒ V n (t) ≤ V (t0 ),

∀t ≥ t0 ≥ 0

(32)

This implies that n  i=1

zi (t) ≤

n 

zi (t0 )

(33)

i=1

for all t ≥ t0 ≥ 0. Under the initial condition specified in (3), and the choice of the constants a and b in (8), the right-hand side of (33) is bounded. This means that the left-hand side of (33) must be bounded. Boundedness of the left-hand side of (33) implies that all zi , i = 1, 2, . . . , n are bounded. Since |z1 (t)| is bounded for all t ≥ t0 ≥ 0, the state x1 (t) never reaches its boundary values a and b, that is, x1 (t) ∈ (a, b) for all t ≥ t0 ≥ 0. This in turn implies from (8) and L1 < L and L1 < L (Assumption 3) that x1 (t) is always in its constraint range, that is, −L < x1 (t) < L for all t ≥ t0 ≥ 0. Boundedness of all xi , i = 1, 2, . . . , n follows from the boundedness of all zi , and smooth property of all functions fi,k (¯xi , d(t)), gi,k (¯xi ), k ∈ Im and (x1 , a, b). Boundedness of all xi , i = 1, 2, . . . , n also implies that the closed-loop system of (1) is forward complete. (ii) Asymptotic convergence. From the fact that xi (t), zi (t), i = 1, 2, . . . , n are bounded, it is easy to deduce that V¨ n (¯zn ) is bounded, which means that V˙¯ n (¯zn ) is uniformly continuous. Then, by Lemma 2, we obtain limt→∞ zi (t) = 0, i = 1, 2, . . . , n. Thus, limt→∞ x1 (t) = 0 by Property (1) of the function (x1 , a, b). When Im = {1}, system (1) degrades into the following class of non-switched high-order uncertain non-linear systems in lower triangular form p

x˙ 1 = f1,1 (x1 , d(t)) + g1,1 (x1 )x21 p

x˙ 2 = f2,1 (¯x2 , d(t)) + g2,1 (¯x2 )x32 ··· x˙ n = fn,1 (¯xn , d(t)) + gn,1 (¯xn )u

pn

For system (34), we can obtain the following corollary.

(34) 

Corollary 1: Suppose that Assumptions 1–4 hold with k ≡ 1, then a state-feedback controller can be designed similar to the one in (30), such that the following properties hold: (1) The closed-loop system (34) with the designed controller is asymptotically stable in a domain. (2) The state x1 is within a pre-specified limit range, that is −L ≤ x1 (t) ≤ L

(35)

for all t ≥ t0 ≥ 0, where L and L are strictly positive constants. (3) All signals of the closed-loop system (34) with the designed controller are bounded. Remark 2: In this paper, the powers of systems (1) are high order. In fact, it is easy to know that, when pi ≡ 1, i = 1, 2, . . . , n in (1), the proposed approach in Section 3 is still useful. Remark 3: This paper considers only a scalar state constraint, the reason is that it is quite hard to find a vector p-times differentiable unbounded function. How to expand the developed method in this paper to deal with partial state constraints or full state constraints is a valuable issue for us to research further.

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Fig. 1

4

Switched RLC circuit

Illustrative examples

In this section, we present simulation studies to demonstrate the effectiveness of the main results. Example 1: Consider a switched RLC circuit [48] that is widely employed in order to perform low-frequency signal processing in integrated circuits. As shown in Fig. 1, the circuit consists of an input power source, a resistance, an inductance and two capacitors that could be switched between each other. The two measurable state variables are the charge in the capacitor and the flux in the inductance x = [qc , φL ]. The input u is the voltage. The dynamic equations are given by ⎧ 1 ⎪ ⎨x˙ 1 = x2 L 1 ⎪ ⎩x˙ 2 = − x1 − RL x2 + u, Ci

i = 1, 2

(36)

where the system parameters are L = 0.1 H, C1 = 50 μF, C2 = 100 μF and R = 1 . It is assumed that the state x(t) = (x1 (t), x2 (t))T in (36) is continuous at each switching instants. Our control objective is to design a controller to ensure that the output x1 of the system converge to 0, while it does not destroy a symmetric constraint L = L = 0.2. Owing to the symmetric constraint L = L = 0.2, choose z1 = (x1 , −0.2, 0.2) = tan([5π/2]x1 ) and V1 (z1 ) = [1/2]z12 . Under Remark 2 and using the design method presented in Section 3, one can obtain the common stabilising function φ1 (z1 ) for each subsystem at the initial step  φ1 (z1 ) = −z1 1 +

2 25π sec2 ( 52 x1 )

 (37)

Let z2 = x2 − φ1 (z1 ) and the CLF for system (36) is given by V 2 (z 2 ) = 12 z12 + 12 z22 . On the basis of (30), one can design the statefeedback controller as follows 

 2 u(¯z2 ) = −z2 1+ 25π sec2 ( 52 π x1 ) ⎞ 2  2 7⎠ (38) + 2.2 × 10 + 5 1+ 25πsec2 ( 52 π x1 ) Let the initial values be x1 (0) = 0.19, x2 (0) = −1.2. Fig. 2 shows that x1 converge to 0 while keeping in the set(−0.2, 0.2), under a random switching signal depicted in Fig. 3. Then, the state response of the designed controller u in (38) is provided in Fig. 4. Moreover,

1905

Fig. 2

State response of x1 Fig. 5

State response of the p-times differentiable unbounded function z1

where σ : [0, +∞) → {1, 2}, f1,1 (¯x1 , d(t)) = 0, g1,1 (x1 ) = 1, f2,1 (¯x2 , d(t)) = x13 sin(x12 x2 ), g2,1 (¯x2 ) = 1, f3,1 (¯x3 , d(t)) = 0, f1,2 (¯x1 , d(t)) = 0, g1,2 (x1 ) = 1, f2,2 (¯x2 , d(t)) = θ x23 , g2,2 (¯x2 ) = 1, f3,2 (¯x3 , d(t)) = 0, θ ∈ [0, 1]. It is assumed that the state x(t) = (x1 (t), x2 (t), x3 (t))T in (39) is continuous at each switching instants. Our control objective is to design a controller to stabilise system (39), while x1 does not destroy a symmetric constraint L = L = 0.3. Owing to the symmetric constraint L = L = 0.3, one can choose 1 2 z1 = (x1 , −0.3, 0.3) = tan( 5π 3 x1 ) and V1 (z1 ) = 2 z1 . Using the design method presented in Section 3, one can obtain the common stabilising function φ1 (z1 ) for each subsystem in (39) at the initial step ⎞1

⎛ φ1 (z1 ) = −z1 ⎝1 + Fig. 3

Given switching signal for system (36)

5π 2 3 sec

3 

3

5π 3 x1

⎠

(40)

Then, let z2 = x2 − φ1 (z1 ) and V 2 (z 2 ) = 12 z12 + 12 z22 , one can further obtain the following common stabilising function φ2 (¯z2 ) for each subsystem in (39)     27c2 81c4 5π 5π 9 sec2 x1 + + φ (¯z2 ) = −z2 3 3 16 4 16  +

  13 3a2 +a +2 16

(41)

where ⎞1

⎛ c = ⎝1 +

5π 2 3 sec

3 

3

5π 3 x1

⎠ ,

a = 1 + 4c + 6c2 + 4c3 + c4

Finally, let z3 = x3 − φ2 (¯z2 ) and V 3 (z 3 ) = V 2 (z 2 ) + 12 z32 , one can design the following state feedback controller for system (39) Fig. 4

u(¯z3 ) = −((1 + γ )z33 + c0 )

State response of the controller u for system (36)

with the state response of the p-times differentiable unbounded function z1 = tan([5π/2]x1 ) is shown in Fig. 5. Example 2: Consider the following switched non-linear system x˙ 1 = f1,σ (t) (¯x1 , d(t)) + g1,σ (t) (x1 )x23 x˙ 2 = f2,σ (t) (¯x2 , d(t))+g2,σ (t) (¯x2 )x33 x˙ 3 = f3,σ (t) (¯x3 , d(t)) + u(¯x3 )

1906

(39)

(42)

 27c02 81c04 9 γ = + + , 16 4 16     27c2 81c4 5π 5π 9 c0 = sec2 x1 + + 3 3 16 4 16  2   13 3a + +a +2 16

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1901–1908 © The Institution of Engineering and Technology 2015

In the simulation, Fig. 6 shows that x1 does not transgress its constraint with various initial values. The given switching signal for system (39) is shown in Fig. 7. Then, the state response of the controller u in (42) is demonstrated in Fig. 8. Finally, the state response of the p-times differentiable unbounded function z1 = tan( 5π 3 x1 ) is shown in Fig. 9 verifying the validity of the designed controller (42).

Fig. 9 State response of the p-times differentiable unbounded function z1 = tan( 5π3 x1 )

5

Fig. 6

State response of x1

Conclusions

This paper has dealt with the SCRS problem for a class of highorder uncertain switched non-linear systems in lower triangular form under arbitrary switchings. The designed state-feedback controllers ensure that the equilibrium at the origin of the closed-loop system is asymptotically stable in a domain without transgression of the constraint. Two simulation examples are used to demonstrate the effectiveness of the developed results. Some issues under current investigation are how to generalise the results in this paper to more general class of high-order switched non-linear systems; how to design an output-feedback controller for system (1).

6

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos: 61304054, 61273120 and 61403041), the Program for Liaoning Provincial Excellent Talents in University, China (no. LJQ2014122) and the Innovation Project of Jiangsu Province Postgraduate Training Project (no: CXZZ13-0208).

7 Fig. 7

Given switching signal for system (39)

1 2 3

4 5

6

7

8

9

10

11

Fig. 8

State response of the controller u for system (39)

IET Control Theory Appl., 2015, Vol. 9, Iss. 12, pp. 1901–1908 © The Institution of Engineering and Technology 2015

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