Adaptive stabilization of a class of uncertain switched nonlinear systems with backstepping control

Automatica 50 (2014) 2128–2135

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Brief paper

Adaptive stabilization of a class of uncertain switched nonlinear systems with backstepping control✩ Ming-Li Chiang, Li-Chen Fu 1 Department of Electrical Engineering, National Taiwan University, 10617 Taipei, Taiwan

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Article history: Received 1 March 2013 Received in revised form 13 November 2013 Accepted 16 April 2014 Available online 17 June 2014 Keywords: Switched systems Adaptive control Nonlinear control systems Backstepping control

abstract In this paper, we focus on the problem of adaptive stabilization for a class of uncertain switched nonlinear systems, whose non-switching part consists of feedback linearizable dynamics. The main result is that we propose adaptive controllers such that the considered switched systems with unknown parameters can be stabilized under arbitrary switching signals. First, we design the adaptive state feedback controller based on tuning the estimations of the bounds on switching parameters in the transformed system, instead of estimating the switching parameters directly. Next, by incorporating some augmented design parameters, the adaptive output feedback controller is designed. The proposed approach allows us to construct a common Lyapunov function and thus the closed-loop system can be stabilized without the restriction on dwell-time, which is needed in most of the existing results considering output feedback control. A numerical example and computer simulations are provided to validate the proposed controllers. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Adaptive stabilization of nonlinear systems has been extensively discussed in the last two decades, e.g., Krstić, Kanellakopoulos, and Kokotović (1995), Lin and Qian (2002) and Marino and Tomei (1995). Recently, many results about analysis and control of switched systems have been developed, see e.g., Branicky (1998), Geromel and Colaneri (2006), Hespanha, Liberzon, and Teel (2008) and Lin and Antsaklis (2009) for stability analysis, and Cheng, Guo, and Wang (2005), Han, Ge, and Lee (2010), Sang and Tao (2012) and Vu and Liberzon (2011) for controller design. Despite these ample results, there are very few in the literature which try to tackle the problem of adaptively stabilizing the switched nonlinear systems with unknown parameters and unknown switching information. The present paper exactly attempts to achieve this goal. Generally speaking, control of fast time-varying systems is a challenging task in adaptive control field since when parameters change rapidly or abruptly, stability and convergence properties

✩ This paper was supported by NSF-100-2221-E-002-082-MY3, AE00-00-06, and NTU-CESRP-103R7617. The material in this paper was partially presented at the 52nd IEEE Conference on Decision and Control (CDC), December 10–13, 2013, Florence, Italy. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected] (M.-L. Chiang), [email protected] (L.-C. Fu). 1 Tel.: +886 2 2362220; fax: +886 2 2 23654267.

http://dx.doi.org/10.1016/j.automatica.2014.05.029 0005-1098/© 2014 Elsevier Ltd. All rights reserved.

will be affected or destroyed. As stated in Tao (2003): ‘‘For a general linear system with non-small parameter variations including unknown jumping parameters at unknown time instants, new adaptive control designs are yet to be developed.’’ Nonetheless, adaptive control of unknown switched systems remains an attractive problem though relatively fewer results are available to date. Some alternative adaptive control schemes for fast time varying systems using multiple models are proposed in Han and Narendra (2012) and Hespanha et al. (2001), where the indirect adaptive control method which estimates the plant parameters using multiple models are discussed. The design philosophy of multiple model adaptive control (MMAC) may be more suitable than the traditional adaptive control for plants with rapid change parameters; however, the stability analysis for unknown switched nonlinear systems is still missing in the existing literature. Instead of estimating the switching parameters and then switching the controller to the corresponding one from the multiple models, in this work, we directly estimate the controller parameters which are some transformed bounds on the switching parameters whereby the system can be stabilized without explicitly considering the changes of plant parameters. Thus, stabilization under arbitrary switching signals can be achieved without the restriction on dwell-time. However, the price is that the plants that can be dealt with are subjected to the constraint that the non-switching part of the switched system should be feedback linearizable. For the choice of controller structure, state feedback backstepping controllers are designed in Ma and Zhao (2010) and Wu (2009)

M.-L. Chiang, L.-C. Fu / Automatica 50 (2014) 2128–2135

to stabilize the completely known switched nonlinear systems subjected to lower-triangular structure under arbitrary switching signals. For a more general switched nonlinear system with impulses, in Han, Ge, and Lee (2009) the authors proposed a state feedback discontinuous adaptive neural backstepping controller to track the desired output, however, average dwell-time condition is needed to ensure system stability. Among the existing results in the literature, state feedback design is discussed in most of the works and there are relatively fewer results of output feedback control of switched systems. Owing to the lack of information of full states, system parameters, and switching signals, it is challenging to design controllers for the uncertain switched systems using only output information. A new output feedback controller using adaptive high gain observer is designed in Han et al. (2010) to stabilize a general class of nonlinearly parameterized switched systems. However, the persistent dwell-time is needed for stability. In the present paper, we design adaptive state feedback and adaptive output feedback controllers for a class of uncertain switched nonlinear systems. Design of controllers for uncertain systems using the bound on the parameters or on the norm of the vector functions have been discussed in robust adaptive control such as Lin and Qian (2002) and Marino and Tomei (1993a). Based on the idea of estimating the bound on switching parameters, we design an adaptive state feedback controller, whereas in Ma and Zhao (2010) and Wu (2009), all system parameters are assumed to be known. Then, by incorporating some augmented design parameters, an adaptive output feedback controller is proposed for all relative degree cases. The main contribution of this paper is that we propose the new adaptive controllers such that the considered uncertain switched systems can be stabilized under arbitrary switching signals; whereas in the existing results, dwell-time condition is usually needed. Moreover, as a special case, a general class of switched linear systems can also be stabilized by the proposed controller. It is worth mentioning that there are relatively few results about adaptive output feedback stabilization of unknown switched systems under arbitrary switching signals. By the proposed approach, a common Lyapunov function can be constructed with output feedback. This paper is organized as follows. Problem formulation is given in Section 2. In Sections 3 and 4, adaptive state feedback and output feedback controllers are proposed respectively for the considered system, and stability analysis under arbitrary switching signals is given. In Section 5, some simulations are provided to validate the design, and the conclusions are given in Section 6. 2. Problem statement Consider the SISO switched nonlinear system which can be linσ (t ) early parameterized by the switching parameters θ σ (t ) and θb : σ (t )

x˙ = Fσ (t ) (x) + θb p

= f 0 ( x) +



g (x)u

θiσ (t ) fi (x) + θbσ (t ) g (x)u

i=1

y = h(x),

(1) σ (t )

where x = [x1 , . . . , xn ]T ∈ Rn , u ∈ R, θ σ (t ) = [θ1 σ (t )

, . . . , θpσ (t ) ]T

∈ R and θb ∈ R are the unknown piecewise constant switching parameters, and σ (t ) ∈ P = {1, 2, . . . , ps } is a non-Zeno switching signal which is right continuous. The output function h is smooth with h(0) = 0, Fk and g are smooth vector fields with Fk (0) = 0, ∀k ∈ P , and g (x) ̸= 0 ∀x ∈ Rn . In this paper, the parameters θ k and θbk , k ∈ P , the switching time instants Tk , k = 1, 2, . . . , and the switching index σ (t ), are all unknown. p

2129

The control objective is to design the adaptive state feedback and output feedback control to stabilize the switched system under arbitrary switching signals. The system considered here is assumed to be feedback linearizable in the non-switching part of the dynamics. In the following, | · | denotes the absolute value of a scalar and ∥ · ∥ denotes the induced 2-norm of a vector or a matrix. 3. Adaptive state feedback stabilization of switched nonlinear systems If the considered system (1) satisfies the involutive condition in the non-switching part (that is, f0 and g are feedback linearizable) and the strict triangularity condition (Marino & Tomei, 1995), then, there exists a function hs (x) and a parameter independent diffeomorphic transformation z = Ts (x) = (hs (x), . . . , Lnf0−1 hs (x)), Ts (0) = 0, such that system (1) can be transformed into the parametric strict-feedback form: z˙j = zj+1 +

p 

ψji (Zj )θiσ ,

1 ≤ j ≤ n − 1,

i =1

z˙n = S1 (Zn ) + θbσ S2 (Zn )u +

p 

ψni (Zn )θiσ ,

(2)

i=1

where Zk := [z1 , . . . , zk ]T , for 1 ≤ k ≤ n, ψji , 1 ≤ j ≤ n, 1 ≤ i ≤ p, are smooth functions generated from fi after coordinate transformation, and S1 (Zn ) = Lnf0 hs (x) = Lnf0 hs (Ts−1 (Zn )), S2 (Zn ) =

Lg Lnf0−1 hs (x) = Lg Lfn0−1 hs (Ts−1 (Zn )). Note that S1 (0) = 0 and S2 (Zn ) ̸= 0, ∀Zn ∈ Rn , since {f0 , g } is feedback linearizable. Details about the transformation, the notations of Lie derivatives, and relevant differential geometric tools can be found in Isidori (1995) and Marino and Tomei (1995). We will design the adaptive state feedback controller from (2). For an unknown switched system, control performance through adaptively estimating the switching parameters may not be satisfactory since the system is persistently switching. Our design strategy is to find controller parameters, which are some transformed bounds on the switching parameters, to adaptively stabilize the system. A backstepping design for non-switched systems with unknown time-varying parameters is proposed in Marino and Tomei (1993b). Based on the robust adaptive control approach, we design a new state feedback controller for (2). In order to make the problem here more tractable, a common but practical assumption is made for the adaptive state feedback controller: (A1) θbσ for all σ ’s have the same sign and whose common lower bound is known, that is, 0 < θ b ≤ |θbσ |, ∀t ≥ 0. Without loss of generality, we assume that θbσ , σ ∈ P , are positive. The adaptive state feedback backstepping control is designed by the following cases: p For the case that n = 1, z˙1 = S1 (z1 ) + θbσ S2 (z1 )u + i =1 ψ1i (z1 )θiσ . Since ψ1i , 1 ≤ i ≤ p, are smooth functions and ψ1i (0) = 0, there exist continuous functions ψ¯ 1i (z1 ) such that ψ1i (z1 ) = ψ¯ 1i (z1 )z1 . Moreover, since the parameters lie in a compact set, we can find a smooth function α1 (z1 ) and an unknown positive constant l1 such that

  p     ψ¯ 1i (z1 )θiσ  ≤ l1 α1 (z1 ).   i =1  Similarly, since S1 (0) = 0 and S1 is smooth, we have |S1 (z1 )| ≤ |z1 |W1 (z1 ) for a smooth function W1 (z1 ) ≥ 0. Let ˆl1 denotes the estimate of l1 , and design u such that u = u1 = S (1Z ) (u∗11 + u∗12 ), 2 1 where u∗11 , u∗12 will be determined based on a Lyapunov design process as shown below. Consider a Lyapunov function candidate V1 = 21 (z12 +˜l21 ), where ˜l1 = l1 −ˆl1 , and evaluate the time derivative

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M.-L. Chiang, L.-C. Fu / Automatica 50 (2014) 2128–2135

˙

and then design u∗21 , u∗22 , and adaptation law ˆl2 as follows:

of V1 with the following design: u∗11 = −

1

θb

z1 W1 ,

˙ˆl = z 2 α , 1 1 1

u∗12 =

1 

 −k1 z1 − ˆl1 α1 z1 ,

θb

u∗21 = −

ˆl1 (0) > 0,

(3)

where k1 is a designed positive gain. Then, by Young’s inequality θσ

and the fact that θb ≥ 1, we obtain a promising analysis on V˙ 1 as b follows:



 σ ∗

σ ∗

V˙ 1 = z1 (S1 (z1 ) + θb u11 ) +

θb u12 +

p 

 σ

ψ1i θi

+ ˜l1˙˜l1



˙

1

1



2



c2 z˜i2 ≤ 0,

(6)

i=1

z˙˜ 2 = S1 (Z2 ) + θbσ S2 (Z2 )u +

p 

ψ2i (Z2 )θiσ

p 



ψ1i (z1 )θiσ (t ) +

i =1

for some positive constant c2 . Hereafter, we define ck , k = 2, 3, . . . , n, to be some appropriately chosen positive constants. Note k θσ that the relations ( j=1 |˜zj |)/k ≤ ∥Z˜k ∥, θb ≥ 1, and Young’s b

inequality are applied in the above derivations. From the fact that V˙ 2 ≤ 0 in (6), signal boundedness can be guaranteed, and by Barbalat’s lemma, z˜1 , and z˜2 converge to zero, which in turn implies that x1 and x2 converge to zero. For the general case where n > 2, define the virtual control that p stabilizes the system z˙i = zi+1 + j=1 ψij θjσ , i = 1, 2, . . . , n − 1, as zi∗+1 (Zi , ˆl1 , . . . , ˆli−1 )

i=1

z2 +

1

1



≤ −(k1 − 1)z12 − (k2 − 1)˜z22 ≤ −

where ˆl1 is designed by (3). Note that z2 = z2∗ can stabilize z1 in p σ ˜2 = z˙1 = z2 + i=1 ψ1i (z1 )θi . We design u = u2 such that z ∗ z2 − z2 converges to zero, and hence z1 and z2 will converge to zero, respectively. Note that

∂ z2 ∂ z1

∂ z∗ ˙

∂ z∗

where W2 = S1 (Z2 ) − ∂ z2 z2 − ˆ2 ˆl1 , which shows that V˙ 2 is indeed ∂ l1 1 negative semidefinite as shown below:

2

For the case where n = 2, let z˜1 = z1 and z2∗ = −k1 z1 − ˆl1 α1 z1 ,

−

2

(5)

2

which implies that z1 and all signals are bounded. By Barbalat’s lemma, z1 will converge to zero, which in turn implies that x = x1 will converge to zero.



θb

 1 −k2 z˜2 − ˆl2 α22 z˜2 ,

z12 − k2 − − z˜22 ≤ − k1 − − 2 2 2 2    1 1  + − ∥Z˜2 ∥2 + l2 |α2 |˜z2 ∥Z˜2 ∥ − ( l2 α2 z˜2 )2

≤ 0,

∗



z˜22 α22 ,



 θσ ≤ |z1 |2 W1 (z1 ) − b z12 W1 (z1 ) θb   σ     θ + z1 b −k1 z1 − ˆl1 α1 z1 + l1 α1 z1 + ˜l1 −α1 z12 θb



1

u∗22 =





≤−

2

θb

sgn(˜z2 W2 )W2 ,

V˙ 2 ≤ −k1 z12 + z1 z˜2 + z˜2 θbσ u∗22 + |˜z2 | l2 |α2 | ∥Z˜2 ∥ + ˜l2˜l2

i=1

k1 z12

˙ˆl = 2

1

1

  i−1  ∂ zi∗ ˙ˆ 1 ∂ zi∗ zj+1 + lj − ˆli z˜i αi2 , = −˜zi−1 − ki z˜i + ˆ ∂ z 2 j ∂ l j j =1



∂ z2 ˙ˆ l1 . ∂ ˆl1 ∗

Since ψji are smooth functions and ψji (0) = 0, 1 ≤ i ≤ p, 1 ≤ ¯ ji such j ≤ 2, by the same token there exist continuous functions ψ

˙ˆl = 1 z˜ 2 α 2 , i i i 2

i = 2, 3, . . . , n − 1

(7)

ψ¯ ki (Zj )zk . Moreover, since 2 ∗ ˆ ˜ ˜ ˆ zk for z˜2 = z2 − z2 (z1 , l1 ), we can write ψ2i (Z2 ) = k=1 ψki (Z2 , l1 )˜ T ˜ ˜ some function ψki where Zj := [˜z1 , . . . , z˜j ] , 1 ≤ j ≤ n. Hence, we

where ki are properly chosen positive constants. Let z˜i = zi − zi∗ , i = 2, . . . , n, then using similar design we can obtain the adaptive state feedback controller

have

u = un =

that for all Z2 ∈ R2 , ψji (Zj ) =

p  i =1

ψ2i (Z2 )θiσ −

p  ∂ z∗ 2

i =1

∂ z1

j

k=1

ψ1i (z1 )θiσ :=

p  2 

˜ ki (Z˜2 , ˆl1 )θiσ . z˜k ψ

1 (u∗n1 + u∗n2 ) S2 (Zn ) 1 u∗n1 = − sgn(˜zn Wn )Wn ,

i=1 k=1

Besides, we can find a smooth function α2 and a positive constant l2 such that for j = 1, 2,

˙ˆl = n

  √ p   l2   ψ˜ ji (Z˜2 , ˆl1 )θiσ  ≤ |α2 (Z˜2 , ˆl1 )|.   i=1  2

(4)

Let ˆl2 be the estimate of l2 , ˜l2 = l2 − ˆl2 , and V2 = V1 + 12 z˜22 + 12 ˜l22 , then we will devise the control u = u2 =

˙

1 S2 (Z2 )

(u∗21 + u∗22 ) and

the adaptive law ˆl2 so that the control u2 will fulfill the Lyapunov analysis design as shown below. First, differentiate V2 as follows:

˙ 

u∗n2 =

V˙ 2 ≤ −k1 z12 + z1 z˜2 + z˜2 z˙˜ 2 + ˜l2˜l2

 ≤ −k1 z12 + z1 z˜2 + ˜l2˙˜l2 + z˜2 S1 (Z2 ) + θbσ (u∗21 + u∗22 )   2 √  l2 ∂ z∗ ∂ z∗ + |˜z2 | |α2 (Z˜2 , ˆl1 )| |˜zj | + z˜2 − 2 z2 − 2 ˙ˆl1 2 ∂ z1 ∂ ˆl1 j =1

1 2

1

θb 

θb

 1 −kn z˜n − ˆln αn2 z˜n , 2

z˜n2 αn2

(8)

where ˆln is the estimation of the parameter ln and αn is a smooth function, both are  defined by the  relation similar to (4), and Wn =

S1 (Zn ) −

n−1 j=1

∂ zn∗ z ∂ zj j+1

+

∂ zn∗ ˙ˆ l ∂ ˆlj j

.

The result of the proposed adaptive state feedback controller can be summarized as follows: Theorem 1. Consider the switched nonlinear system (1). If the system can be transformed into (2) and assumption (A1) is satisfied, then the system can be stabilized under arbitrary switching signals by the adaptive state feedback controller (8). Proof. For the transformed switched system (2), signal boundedness of the whole system can be proved by showing that the

M.-L. Chiang, L.-C. Fu / Automatica 50 (2014) 2128–2135

time derivative of a common Lyapunov function along the system is negative semidefinite. Let the Lyapunov function candidate n 1 2 zi + 12 ˜l2i ), where z˜i and ˜li , i = 1, 2, . . . , n, are deVn = i=1 2 (˜ fined in the above analysis. Then, by similar derivations, the time derivative of this function along the closed-loop system consisting of controller (8) with properly chosen gains ki , i = 1, 2, . . . , n, satisfies V˙ n ≤ −

n−1 



cn−1 z˜i2 + z˜n−1 z˜n + z˜n θbσ u∗n2 +





˙

ln |αn | ∥Z˜n ∥ + ˜ln˜ln

i=1

≤−

n 

cn z˜i2 ≤ 0

i=1

for some positive constant cn . Hence, all signals are bounded. Furthermore, by Barbalat’s lemma, Z˜n converges to zero, which in turn implies that z and x converge to zero, since the transformation is diffeomorphic and parameter independent. Example 1. Consider the following switched system parameterized by the switching parameters θ σ = [θ1σ , θ2σ , θ3σ ]T and θbσ under the unknown switching signal σ (t ) ∈ {1, 2}: x˙ = f0 (x) +

3 

θiσ fi (x) + θbσ g (x)u

i=1









x2 −x21 = x21 x2 + x3  + θ1σ  x21  + θ2σ 0 0

 σ

+ θ3

0 0 sin x1



0 0 x1 ex1





0 0 u 1

  σ

+ θb

(9)

where θ 1 = [a1 , a2 , 0]T , θ 2 = [0, 0, b1 ]T , θb1 = a3 , and θb2 = b2 . The parameters a1 , a2 , a3 , b1 , b2 are unknown and a3 , b2 are non-zero constants. It can be verified that this system can be transformed into (2). Let hs (x) = x1 , then the parameter independent transformation z = Ts (x) = [hs , Lf0 hs , L2f0 hs ]T will transform the system into z12 1σ z12 1σ

z˙1 = z2 + (− )θ

σ

= z2 + ψ1 (z1 )θ1 , σ

z˙2 = z3 + ( )θ

= z3 + ψ2 (z1 )θ1 p  z˙3 = Lnf0 hs + θbσ (Lg Lfn0−1 hs )u + ψ3i (Z3 )θiσ i =1

= 2z1 z22 + z12 z3 + θbσ u + θ1σ (−2z13 z2 + z14 ) + θ2σ (z1 ez1 ) + θ3σ (sin z1 ). According to previous discussions, let α1 (z1 ) = (1 + z12 ) and

z2∗ = −k1 z1 − ˆl1 (z1 + z13 ). By computation and (7), we can choose

α2 = (1 + k1 + ˆl1 )z1 + ˆl1 z13 and z3∗ = −z1 − k2 z˜2 + [−k1 − ˆl1 (1 + 3z12 )]z2 + z13 + 2z15 + z17 − 21 ˆl2 z˜2 α22 . Note that α1 and α2 contain z1

only, and then we can represent p  i=1

 ψ3i (Z3 ) −

2  ∂ z∗ 3

j =1

∂ zj

 ψji (Zj ) θiσ =

p 

˜ i (Z˜3 , ˆl1 , ˆl2 )θiσ . z˜1 ψ

i =1

Thus, we can choose



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Remark 3.1. In Marino and Tomei (1993b), a state feedback controller is designed for a nonlinear system with piecewise continuous time-varying uncertain parameters and θbσ ≡ 1. For state feedback design, if θbσ ≡ 1, the term S1 (·) in the transformed dynamics can be eliminated completely. For the case of switching θbσ in this paper, our design of u∗n1 can help to compensate the dynamics of S1 (·), which cannot be eliminated directly with the switching uncertainty θbσ . Moreover, note that there is a discontinuous sign function in the proposed state feedback controller. For practical implementation, we can use the smooth function tanh(·) to approximate the sign function. By the relation 0 ≤ |u| − u tanh( uϵ ) ≤ δϵ , where ϵ > 0 and δ satisfies δ = e−(δ+1) (Polycarpous & Ioannou, 1996), we can make the convergence error arbitrarily small by adjusting the design parameter ϵ . 4. Adaptive output feedback stabilization of switched nonlinear systems For output feedback design, it is difficult to design a switching observer or a single observer for the considered system such that the states can be estimated for the whole time interval under arbitrary switching signals, in addition to the unknown parameters and unknown switching signals. Thus, we consider the conditions that the switched nonlinear systems can be transformed into output feedback form by a parameter independent transformation. It has been shown in Marino and Tomei (1993a) (for nonswitching case) that for a linearly parameterized system, if certain conditions related to observability are satisfied, then, there exists a diffeomorphism z = [z1 , . . . , zn ]T = T (x), which is independent of θ σ and θbσ , that transforms system (1) into the following form: z˙ = A0 z + θbσ bα(y)u + φ0 (y) +

p 

φi (y)θiσ

i =1

y = c T z = [1, 0, . . . , 0]z = z1 ,

(10)

where α(y) ∈ R is a smooth function and α(y) ̸= 0 for all y since g0 (x) ̸= 0. φi (y) = [φi1 , . . . , φin ]T ∈ Rn , i = 0, 1, 2, . . . , p, are vector functions that depend on the system output y only, and



0



A0 =  .. 0

In − 1 

.

,

b = 0, . . . , bm , . . . , b0



T

,

(11)

···0

where 0 ≤ m ≤ n − 1, and the relative degree ρ := n − m. To simplify the notation, we let θ0σ ≡ 1, ∀t > 0, and write

φ0 (y) +

p

φi (z1 )θiσ (t ) =

p

σ (t )

. More details about i=0 φi (z1 )θi the transformation conditions can be found in Marino and Tomei (1995). Based on this result, the considered system can be transformed into a canonical form with some perturbations. The following assumptions are made for the output feedback controller: i =1

(A1′ ) The common lower and upper bounds of θbσ , σ ∈ P , are known, that is, 0 < θ b ≤ θbσ ≤ θ¯b . (A2) bm sm + · · · + b1 s + b0 is a Hurwitz polynomial. A filtered transformation will be employed to design the output feedback controller for (10). Define the stable filters λ 1(s) , i = i

1, 2, . . . , ρ − 1, where λi (s) = (s + λc )ρ−i and λc > 0 is a design parameter. Let the filtered input be ηi = λ 1(s) α(y)u, i = i





α3 = (1 + z12 ) 1 + ˆl1 z12 z22 + z12 z22 + (1 + z14 + z16 )     2 2 2 2 ˆ ˆ ˆ × 1 + l2 (α2 + z˜2 α2 ) L(z1 , l1 ) + k2 + l2 α2 + ez1 where L(z1 , ˆl1 ) := 1 + k1 + ˆl1 + 3ˆl1 z12 . Then, the controller can be designed by (8).

1, 2, . . . , ρ − 1, and let u = α(y)−1 ηρ . Define the vectors d(i) = [d1 (i), . . . , dn (i)]T ∈ Rn , 1 ≤ i ≤ ρ , where the components of d(i) are defined by the following relation

(bm sm + · · · + b1 s + b0 )λi (s) = d1 (i)sn−1 + · · · + dn (i), (12) with 1 ≤ i ≤ ρ − 1. Note that d(ρ) = b and d(i) = [0, . . . , 0, di (i), . . . , dn (i)]T . From (12), we can observe that d(i − 1) = A0 d(i) + λc d(i), i = 2, 3, . . . , ρ . Then, by the filtered

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M.-L. Chiang, L.-C. Fu / Automatica 50 (2014) 2128–2135

ρ

transformation zp = z − i=2 d(i)ηi−1 which involves the filtered input signals (Marino & Tomei, 1993a), one can rewrite (10) as p 

z˙p = A0 zp + θbσ dη1 +

φi (y)θiσ ,

i =0

y = c T zp = zp1 = z1 ,

(13)

where d := d(1) := [d1 , . . . , dn ] is derived from (bm s + · · · + b0 )λ1 (s) = d1 sn−1 +· · ·+ dn . Note that zp1 = z1 = y, d1 = bm > 0, and d1 sn−1 + · · · + dn−1 s + dn is a Hurwitz polynomial. Define ζ = [ζ1 , . . . , ζn−1 ]T with ζ1 = zp2 − dd2 z1 , ζ2 = zp3 − dd3 z1 , . . . , ζn−1 = T

m

1

1

zpn − ddn z1 , then the system can be transformed into the following 1 switching dynamics: z˙1 = ζ1 +

d2 d1

z1 + θbσ d1 η1 +

p 

φi1 (y)θiσ

(14)

ζ˙ = Λζ + Ψ z1 +

  p  σ ˙V1 = z1 z˙1 − q∥ζ ∥2 + 2ζ T P Ψ z1 + Φi (z1 )θi i=0

+ ˜l11˙˜l11 + ˜l12˙˜l12 , and from this we design η1 as

η1 = η1∗ =



1 d1 θ b

−

Φi (z1 )θiσ ,

(15)

σ

V˙ 1 ≤ −q∥ζ ∥ + z1 θb d1

i=0

Λ = d¯



In−1 , 0



d¯ = −

d2 d1

,−

d3 d1

,...,−

dn d1

T ∈R

n −1



,

(17)



1

v1

∗

d1 θ b



 + z1

p 

 σ

φ¯ i1 θi

i=0

    p  1 T σ ¯ i θi + 2∥ζ ∥|z1 | ∥P ∥ ∥Ψ ∥ + + 2ζ Pz1 Φ

,

2

(d2 )2  − (d1 )2   d1   d2 d3   d4  −   d1 (d1 )2      Ψ = ... ,     d2 dn−1   dn  −   d1 (d1 )2    d2 dn − (d1 )2   d2 φi2 − φi1   d1 p p       σ . σ .. Φi (z1 )θi =   θi .   i =0 i =0   dn φin − φi1  d3

i =0

(18) + ˜l11˙˜l11 + ˜l12˙˜l12 , where |ζ1 | ≤ ∥ζ ∥ is used in the above inequality. Now we are ready to specify v1∗ , and to design adaptation laws for ˆl11 , ˆl12 as v1∗ = −(q − 1)z1 − β1ˆl11 z1 − Wo2 z1 − (∥P ∥β2 )2ˆl12 z1

˙ˆl = z 2 β , 11 1 1

˙ˆl = z 2 β 2 ∥P ∥2 , 12 1 2

(19)

where Wo = (∥P ∥ ∥Ψ ∥ + 12 ), based on which, time derivative of the Lyapunov function candidate V1 can be further assessed as V˙ 1 ≤ −q∥ζ ∥2 − (q − 1)|z1 |2 − ˆl11 β1 |z1 |2 − Wo2 |z1 |2

− (∥P ∥β2 )2ˆl12 |z1 |2 + |z1 |2 l11 β1 + 2∥ζ ∥|z1 |Wo  + 2∥ζ ∥ ∥P ∥|z1 | l12 β2 − ˜l11 |z1 |2 β1 − ˜l12 |z1 |2 β22 ∥P ∥2 ≤ −(q − 2)(∥ζ ∥2 + |z1 |2 ) := −q1 (∥ζ ∥2 + |z1 |2 ) ≤ 0.

d1

4.1. Design of adaptive output feedback control As in the state feedback case, the design strategy is to adaptively tune the controller parameters instead of directly estimating the values of switching parameters. The proposed controller is designed with some augmented parameters ˆli ’s to compensate for the effects of switching dynamics when the relative degree ρ is greater than one. First, consider the case where ρ = 1. In this case, d = b = [bn−1 , . . . , b0 ]T , and we design η1 = η1∗ , u = α −1 η1 to stabilize the system. Since z = T (x) used in (10) is a smooth transformation, we have φi (0) = 0 from fk (0) = 0, ∀k ∈ P . Thus, there exist smooth functions φ¯ ij such that φij (y) = yφ¯ ij (y), for i = 0, 1, 2, . . . , p, j = ¯ i (y)y for i = 0, 1, . . . , p. 1, 2, . . . , n. Similarly, we have Φi (y) = Φ Given the above facts, we can see that there exist positive constants l11 , l12 , and smooth functions β1 (z1 ), β2 (z1 ) such that

  p    σ ¯ φi1 (z1 )θi  ≤ l11 β1 (z1 ),   i=0    p     ¯ i (z1 )θiσ  ≤ l12 β2 (z1 ). Φ   i=0 

d1

z1 + v1∗

 2

where



d2

where v1∗ will be determined later. Substitute (14) and (17) into V˙ 1 and we have

i=0 p 

Let ˆl11 , ˆl12 be the estimates of l11 and l12 , respectively, and let ˜l1i = l1i − ˆl1i , i = 1, 2. Define V1 = 12 (z1 )2 + ζ T P ζ + 21 (˜l11 )2 + 12 (˜l12 )2 , where P = P T > 0 satisfies the Lyapunov equation ΛT P + P Λ = −qI < 0, and q is a properly chosen positive constant. Note that the existence of P is guaranteed since the eigenvalues of Λ are the roots of d1 sn−1 + · · · + dn . To proceed with the Lyapunov analysis, we derive the time derivative of V1 as

This implies that all signals are bounded, and z1 and ∥ζ ∥ are concluded to converge to zero by Barbalat’s lemma. Since in this case zp = z, we further conclude that z → 0 by the fact of convergence of z1 and ∥ζ ∥, which in turn implies that x → 0 as t → ∞. When relative degree ρ = 2, since η1 cannot be assigned directly and η˙ 1 = −λc η1 + η2 , we will design η2∗ = η2 such that η1 − η1∗ → 0, which results in u = α(y)−1 η2 . Let η˜ 1 = η1 − η1∗ , then we have

η˙˜ 1 = −λc η˜ 1 − λc η1∗ − η˙ 1∗ + η2∗ .

(20)

Moreover, since η1 is a function of z1 , ˆl11 , and ˆl12 , it should be ∗

2 ∂η1∗ ˙ ∂η∗ ˆ straightforward that η˙ 1∗ = ∂ z 1 z˙1 + i=1 ∂ ˆl l1i . To stabilize the 1 1i

system, an additional parameter ˆl2 will be added into the controller. Let ˜l2 = (l11 )2 − ˆl2 and the Lyapunov function candidate V2 = V1 + 21 (η˜ 1 )2 + 21 (˜l2 )2 . To assess the time rate of change of V2 , we differentiate V2 as follows:

˙

V˙ 2 ≤ −q1 (∥ζ ∥2 + |z1 |2 ) + z1 (θbσ d1 η˜ 1 ) + η˜ 1 η˙˜ 1 + ˜l2˜l2 (16)

= −q1 (∥ζ ∥2 + |z1 |2 ) + z1 (θbσ d1 η˜ 1 ) + ˜l2˙˜l2     2  ∂η1∗ ∂η1∗ ˙ˆ ∗ ∗ z˙1 + l1i + η2 (21) + η˜ 1 −λc η˜ 1 − λc η1 − ˆ ∂ z1 i=1 ∂ l1i

M.-L. Chiang, L.-C. Fu / Automatica 50 (2014) 2128–2135

which implies the following design:

η2∗ = λc η1∗ − (k1 + q1 )η˜ 1 +

2  ∂η1∗ ˙ˆ l1i + v2∗ , ˆ ∂ l 1i i=1

(22)

(θ¯b d1 )2 and v2∗ will be determined later. This yields an assessment of V˙ 2 as where k1 ≥

1 2

˙

V˙ 2 ≤ −q1 (∥ζ ∥2 + |z1 |2 + (η˜ 1 )2 ) + z1 (θbσ d1 η˜ 1 ) + ˜l2˜l2



− λc (η˜ 1 )2 − k1 (η˜ 1 )2 + η˜ 1 −

∂η1∗ z˙1 + v2∗ . ∂ z1 

(23)

In particular, note that

∂η1∗ ∂η1∗ z˙1 = ∂ z1 ∂ z1

 ζ1 +

d2 d1

z1 + θbσ d1 η1∗ +

p 

 φi1 θiσ + θbσ d1 η˜ 1 ,

i=0

v2 = −η˜ 1



∂η1∗ ∂ z1

2  2 1 2

 +

2d2

2

d1

Barbalat’s lemma. Thus, zp → 0 and from the definition of the filtered transformation, we can conclude that z → 0, which in turn implies that x → 0 as t → ∞. Similarly, for the case where ρ = 3, we can design η3∗ such that η˜ 2 = η2 − η2∗ → 0. Instead of going to higher ρ incrementally, we now discuss general ρ subjected to 2 ≤ ρ ≤ n, where the most important step is to design ηρ∗ such that η˜ i = ηi − ηi∗ → 0, i = 1, 2, . . . , ρ − 1, so that the exogenous control input is designed as follows: u = α(y)−1 ηρ∗ .

1 2

(˜lρ )2 , where ˜lρ = (l11 )2 − ˆlρ and ˆlρ is an augmented controller

parameter. Then, by going through similar design and derivation process, we can come to the following state of V˙ ρ :



j =1

+ (2θ¯b d1 K1 (t ))2

+ η˜ ρ−1 η˙˜ ρ−1 + ˜lρ (−ˆ˙lρ ). Given the following relation:

 2  ˆl2 β1   ∗ 2  ∂η1  (θ¯b d1 )2  − η˜ 1 +  ,  2λc  ∂ z1 2 

 (24)

∗ η˙˜ ρ−1 = −λc η˜ ρ−1 − λc ηρ− 1 −

 where we use a simplified notation η1∗ = K1 (t )z1 from (17), and  | pi=0 φi1 θiσ | ≤ |l11 β1 (z1 )z1 |. We set aside the following fact for calculation of V˙ 2 : 1

1

4

4



− ∥ζ ∥2 − |z1 |2 − |z1 |2 − λc η˜ 12 + η˜ 1 − ∂η∗ ≤ 0 − η˜ 1 1 ∂ z1 

2

 ( ˆl2 β1 )2 2

∂η1 z˙1 + v2∗ ∂ z1 ∗



   ∂η1∗   + η˜ 1 z1 l11 β1  , ∂ z1

(25)

which leads to the following more concise result: V˙ 2 ≤ −(q1 − 1)(∥ζ ∥ + |z1 | ) − 2

2

ˆl2  ∂η1∗ 2 η˜ − η˜ 1 (β1 )2 2 ∂ z1

q1 12

−

which finally leads to the desirable result making V˙ 2 negative semidefinite as shown below:



ˆl2  ∂η1∗ 2 η˜ 1 (β1 )2 2 ∂ z1  

 1 ∂η∗ 2 ∂η∗ + η˜ 1 1 z1 l11 β1 + ˜l2 − η˜ 1 1 (β1 )2 ∂ z1 2 ∂ z1   3 ≤ −(q1 − 1)(∥ζ ∥2 + (η˜ 1 )2 ) − q1 − |z1 |2 2

≤ −q2 (∥ζ ∥2 + |z1 |2 + (η˜ 1 )2 ) ≤ 0, for some positive constant q2 . As a result, signal boundedness are guaranteed and convergence of ∥ζ ∥, z1 , η˜ 1 can be concluded by

∂ z1

z˙1 +

ρ−2 ∗  ∂ηρ− 1 i=1

∂ηi

 η˙ i

 ρ−1 ∗ ∗ 2   ∂ηρ− ∂ηρ− 1˙ 1˙ ˆl1i + ˆli + ηρ∗ , ˆ ˆ ∂ l1i ∂ li i =1 i =2

(28)

∗ ∗ ηρ∗ = −qρ−1 η˜ ρ−1 + λc ηρ− 1 + vρ   ρ−1 ρ−2 ∗ ∗ ∗ 2    ∂ηρ− ∂ηρ− ∂ηρ− 1˙ 1˙ 1 ˆ ˆ l1i + li , η˙ i + + ∂ηi ∂ ˆl1i ∂ ˆli i=2 i =1 i=1

(29)

where vρ∗ will be determined later, so that the evaluation of V˙ ρ can go one step further as:

 V˙ ρ ≤ −qρ−1

z12

2

+ ∥ζ ∥ +

ρ−1 

 η˜

+ η˜ ρ−2 η˜ ρ−1

2 j

j =1

  ∗ ∂ηρ− 1 z˙1 . + η˜ ρ−1 vρ∗ − ∂ z1

1

 where we have used the relation |z1 (θbσ d1 η˜ 1 )| − 12 |z1 |2 +  (θ¯b d1 η˜ 1 )2 ≤ 0. Now, the last step is to design the adaptive law for ˆl2 as  ∗ 2 ˙ˆl = 1 η˜ ∂η1 (β )2 , (26) 1 2 1 2 ∂ z1

∗ ∂ηρ− 1

we design

   ∂η∗  + η˜ 1 1 z1 l11 β1  + ˜l2 (−˙ˆl2 ), ∂z

V˙ 2 ≤ −(q1 − 1)(∥ζ ∥2 + |z1 |2 ) − q1 η˜ 12 −

(27)

By the same token, we establish the Lyapunov function candidate Vρ in a recursively manner, namely, let Vρ = Vρ−1 + 12 (η˜ ρ−1 )2 +

  ρ−2  2 ˙Vρ ≤ −qρ−1 z12 + ∥ζ ∥2 + η˜ j + η˜ ρ−2 η˜ ρ−1

and this inspires us to design v2∗ as ∗

2133

(30)

As in the previous derivations, now vρ∗ is designed as

vρ∗



∗ ∂ηρ− 1

2 

d2

2



2 + (2θ¯b d1 K1 (t )) ∂ z1 d1     ∗ 2  2 ( ˆlρ β1 )2 ¯ b d1 ) 2 ∂ηρ−1 1 ( θ   − η˜ ρ−1 + + ∂ z1 2 2λc 2

= −η˜ ρ−2 − η˜ ρ−1

2

(31)

and the adaptive law is designed as

 2 ∗ ∂ηρ− 1 ˙ˆl = 1 η˜ β12 . ρ ρ−1 2 ∂ z1

(32)

By the proposed design, we end up with the desirable analysis result on V˙ ρ , namely, V˙ ρ is negative semidefinite as shown below:

 V˙ ρ ≤ −(qρ−1 − 1) z12 + ∥ζ ∥2 +

ρ−2  j =1

 −

η˜ ρ−1

∗ ∂ηρ− 1

∂ z1

2

(β1 )2

ˆlρ 2

 η˜ j2

2134

M.-L. Chiang, L.-C. Fu / Automatica 50 (2014) 2128–2135

   2 ∗ ∗   ˜lρ ∂ηρ− ∂ηρ− 1 1 + η˜ ρ−1 z1 l11 β1  − η˜ ρ−1 (β1 )2 ∂ z1 2 ∂ z1   ρ−1  2 2 2 ≤ −(qρ−1 − 2) z1 + ∥ζ ∥ + η˜ j j =1

 := −qρ z12 + ∥ζ ∥2 +

ρ−1 

 η˜ j2

≤ 0.

(33)

j =1

The result of the proposed adaptive output feedback backstepping controller can be summarized as follows: Theorem 2. Consider the switched nonlinear system (1). If the system can be transformed into (10), and assumptions (A1′ ) and (A2) are satisfied, then the system can be stabilized under arbitrary switching signals by the adaptive output feedback controller (27), (29), (31), and (32). Proof. Consider the common Lyapunov function candidate Vρ and it is shown in (33) that the time derivative of this function along the transformed closed-loop system is negative semidefinite. Thus, signal boundedness are guaranteed. Moreover, convergence of z1 , ∥ζ ∥, η˜ i to zero for i = 1, 2, . . . , ρ − 1, can be concluded by Barbalat’s Lemma. Then, from the definition of ζ and η, ˜ x will converge to zero under arbitrary switching signals. Example 2. Consider the system (9) in Example 1. It can be verified that the system can be transformed into (10) by the transformation x3

z = T (x) with z1 = x1 = y, z2 = − 31 + x2 , z3 = x3 , which transforms the system into (10) with α(y) = 1, b =

[0, 0, 1]T , φ0 (y) = 3 



y3 3

T , 0, 0 , and

 T φi (y)θiσ = −θ1σ y2 , θ1σ y2 + θ1σ y4 , θ2σ yey + θ3σ sin y .

i =1

By derivations we can choose β1 (z1 ) = 2z12 + 1 and β2 (z1 ) = 2(z14 + z12 + ez1 + 1). Then, the controller can be designed as in (27), (29), (31) and (32). Remark 4.1. In Hespanha (2004), a detailed discussion about different types of dwell-time switching signals are provided. Based on the related stability theories and multiple Lyapunov functions, kinds of dwell-time conditions are usually needed for the stabilization of switched systems. Generally speaking, handling of systems with more general switching signals usually demands for more restrictive conditions. Some existing results discussed about the control of more general switched systems, e.g., Chiang and Fu (2014), Han et al. (2010) and Ma and Zhao (2010); however, the controllers there are either state feedback or dwell-time dependent. To stabilize switched systems with arbitrary switching signals, we usually look for the design such that a common Lyapunov function can be constructed. However, the existence conditions of common Lyapunov is still an open problem. The system we considered here is restricted to the subsystems that are linearly parameterized with switching parameters, and satisfied the conditions such that it can be transformed into (2) or (10). Due to the conditions of the subsystems, we can construct a common Lyapunov function by the proposed control and thus stabilize the system under arbitrary switching signals. It is not always available to stabilize the switched system with arbitrarily fast switching by output feedback. The system considered here can also be regarded as some criteria of the class of switched systems that can be stabilized under arbitrary switching signals by output feedback.

Fig. 1. State feedback: x1 , x2 , x3 , u, σ (t ), and estimated parameters of Example 1.

Remark 4.2. Consider the switched linear system represented by the following transfer function, y=

sn

θbσ (bm sm + · · · + b1 s + b0 ) u, + θ1σ sn−1 + · · · + θnσ−1 s + θnσ

(34)

which can be rewritten as (10), where p = n and φ0 (y) = 0, φ1 (y) = [y, 0, . . . , 0]T , φ2 (y) = [0, y, 0, . . . , 0]T , . . . , φn (y) = [0, . . . , 0, y]T . If bm sm + · · · + b1 s + b0 is Hurwitz, the proposed adaptive controller can be applied directly. It is worth mentioning that in Wulff, Wirth, and Shorten (2009), a pole placement based switching controller is designed for (34) with m = 0. In that work, all system parameters and the switching signal are assumed to be known, and then, individual pole placement linear controllers are designed and synchronously switching with the system. For stabilization purpose, our proposed output feedback controller uses less information and assumptions with the price of a more complex adaptive backstepping design. To the best of our knowledge, there is no similar result in the literature for the stabilization of (34) under arbitrary switching signals. 5. Simulation results In this section, we provide some simulations to validate the controllers proposed in Sections 3 and 4. Let (a1 , a2 , a3 ) = (1, 1, 2), (b1 , b2 ) = (1, 0.5), and assume that θ b = 0.4, θ¯b = 3. For state feedback design in Example 1, we choose k1 = k2 = k3 = 5. A simulation with x(0) = [0.5, −1, 2]T , ˆl1 (0) = ˆl2 (0) = ˆl3 (0) = 0.2, and a randomly given switching signal which switches every 0.2 s in average is given in Fig. 1. For output feedback design in Example 2, the system has relative degree ρ = n = 3. We choose λc = 2 and the vector d = [1, 4, 4]T . Let q = 4, then P can be obtained and we can derive the controller as in Example 2. Fig. 2 shows the state trajectories and control input of the output feedback adaptive control with the same switching signal of Fig. 1. 6. Conclusions Adaptive stabilization of switched nonlinear systems with unknown switching signals and unknown parameters is investigated in this paper. By the proposed controller, the considered system can be stabilized under arbitrary switching signals without the restriction on dwell-time. Existing results in the literature show that some systems with special structures can be stabilized without dwell-time condition but with the knowledge of full states and/or

M.-L. Chiang, L.-C. Fu / Automatica 50 (2014) 2128–2135

Fig. 2. Output feedback: x1 , x2 , x3 , u, and estimated parameters of Example 2.

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Ming-Li Chiang received his Ph.D. degree in Control Engineering from National Taiwan University in 2010. From 2010 to 2013, he was a postdoctoral researcher in the Department of Electrical Engineering, National Taiwan University. He received the Best Paper Award in the International Automatic Control Conference in Taiwan, 2012. He is currently a visiting postdoctoral researcher in the Department of Computer, Control, and Management Engineering, University of Rome, Italy. His research interests include switched systems, adaptive control, nonlinear control systems and its applications.

Li-Chen Fu received B.S. degree from National Taiwan University, Taiwan, R.O.C., in 1981, and M.S. and Ph.D. degrees from University of California, Berkeley, USA in 1985 and 1987, respectively. Since 1987, he has been a member of the faculty, and is currently a full professor in the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, Taiwan, R.O.C. He was awarded Lifetime Distinguished Professorship from his university in 2007. He has received numerous academic recognitions, such as Distinguished Research Awards from National Science Council, Taiwan, R.O.C., the Irving T. Ho Chair Professorship, Macronix Chair Professorship, and IEEE Fellow in 2004. He currently serves as Editor-in-Chief of the Asian Journal of Control, President of Asian Control Association, and Distinguished Lecturer of the IEEE Control Systems Society during 2013–2015. His research interests include robotics, smart home, visual detection and tracking, intelligent vehicle, production scheduling, virtual reality, and control theory & applications.