Global Stabilization of Feedforward Nonlinear System ...

Vol. 36, No. 4

ACTA AUTOMATICA SINICA

April, 2010

Global Stabilization of Feedforward Nonlinear System Based on Nested Saturated Control WANG Yong1

MA Ru-Ning2

Abstract In this paper, we discuss global stabilization procedure for convergence of a more general feedforward nonlinear systems. Our stabilizer consists of a nested saturation function, which is a nonlinear combination of saturation functions. We extend the existing stabilization results and prove that our stabilizer is exponential convergent. Key words Feedforward nonlinear system, nested saturated control, global stabilization, exponential convergence DOI 10.3724/SP.J.1004.2010.00528

The problem of stabilizing feedforward system has been studied extensively[1−4] . In 1992, Teel[5−6] presented some bounded control algorithms for feedforward linear system  x˙ 1 = x2     .. .   x˙ = xn   n−1 x˙ n = u where the control u is the combination of following nested saturation functions[6] : u = −σn (xn + σn−1 (xn−1 + · · · + σ1 (x1 ))) After Teel0 s work, there appeared many results on global stabilization of feedforward systems, such as Jankovic et al.[1, 7] , Mazenc et al.[8] , Teel[9] , Liu et al.[10] , Zhong et al.[11] and so on. In [12], the author has studied the following feedforward nonlinear system:  x˙ 1 = x2 + ϕ1 (x3 , · · · , xn , u)     x   ˙ 2 = x3 + ϕ2 (x4 , · · · , xn , u) .. .     x˙ n−1 = xn + ϕn−1 (u)   x˙ n = u where ϕi is vanishing and locally Lipschitz continuous at zero for i = 1, · · · , n − 1. In this paper, we study a more general class of feedforward nonlinear systems  x˙ 1 = f1 (x2 , x3 , · · · , xn , u)     x   ˙ 2 = f2 (x3 , x4 , · · · , xn , u) .. (1) .     x˙ = fn−1 (xn , u)   n−1 x˙ n = fn (u) where x = [x1 , · · · , xn ]T ∈ Rn is the state, fi is continuously differentiable for its variable, vanishing at zero. In this work, we used nested saturation stabilizer to make above-mentioned nonlinear system globally stable at the equilibrium x = 0. The stabilizer we used is basically similar to that proposed in [6]. The saturation levels are determined by properties of fi . Manuscript received December 30, 2008; accepted June 18, 2009 Supported by National Natural Science Foundation of China (10771101) and Science Research and Development Foundation of Changchun University of Technology (2008A29) 1. Changchun University of Technology, Changchun 130012, P. R. China 2. Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China

The rest of this paper is organized as follows. In Section 1, we give our main result about global stability and exponential convergence of nonlinear system (1). An extended result is given in Section 2. In Section 3, we give a simulation example. The paper is concluded in Section 4.

1

Global stability and exponential convergence of saturated control In this section, we always suppose that (1) satisfies ¯ ¯ ∂fn ¯¯ ∂fi ¯¯ = c = 6 0 (i = 1, · · · , n − 1), = cn 6= 0 i ∂xi+1 ¯0 ∂u ¯0

Then, we have fi (xi+1 , · · · , xn , u) = ci xi+1 + ϕi (xi+1 , · · · , xn , u), i = 1, · · · , n − 1 (2) and fn (u) = cn u + ϕn (u)

(3)

At the same time, there exist ai and ri such that |ϕi (xi+1 , · · · , xn , u)| ≤ ri (|xi+1 |2 + |xi+2 | + · · · + |xn | + |u|), i = 1, · · · , n − 1 (4) and

|ϕn (u)| ≤ rn |u|2

(5)

whenever max{|xi+1 |, |xi+2 |, · · · , |xn |, |u|} ≤ ai . Using (2) and (3), system (1) can be written as  x˙ 1 = c1 x2 + ϕ1 (x2 , x3 , · · · , xn , u)    x˙ = c x + ϕ (x , x , · · · , x , u)  2 3 2 3 4 n   2 .. .     x˙ n−1 = cn−1 xn + ϕn−1 (xn , u)   x˙ n = cn u + ϕn (u)

(6)

Next, we will use the linear coordinate transformation yn−i =

i X j=0

where Cij =

j

Cij ε n xn−j , cn−j · · · cn

i! j!(i−j)!

i = 1, · · · , n

and ε ≤ 1 is a scaling factor. Not-

i n

ing that ε xn−i depends only on yn−i , · · · , yn , it can be concluded that there exist positive numbers Kn−i ≥ 1 (independent of ε) such that i

ε n |xn−i | ≤ Kn−i (|yn−i | + · · · + |yn |), i = 0, · · · , n − 1 (7)

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WANG Yong and MA Ru-Ning: Global Stabilization for Feedforward Nonlinear System · · ·

529

(8), we can get

Since

yn−i+1 + · · · + yn =

i−1 P j=0

i−1 P j=0

j+1

i−1 P k=j

cn−j ···cn

j

Ci

ε n xn−j = cn−j ···cn

¯ ¯ 0 ¯ ¯¯ 1 1 ¯ ¯ Ci ¯ ¯ Ci ε n ¯ |Φn−i | ≤ ¯¯ ϕn (u)¯¯ + ¯ ϕn−1 (xn , u)¯ + · · · + ¯ cn−1 cn ¯ cn ¯ ¯ i ¯ ¯ i n Ci ε ¯ ¯ ϕn−i (xn−i+1 , · · · , xn , u)¯ ≤ ¯ ¯ ¯ cn−i+1 · · · cn

j

Ck

i P j=1

j

ε n xn−j = j

j−1 Ci ε n xn−j+1 cn−j+1 ···cn

1

Ci0 C 1 ε n rn−1 rn |u|2 + i (|xn |2 + |u|) + · · · + cn cn−1 cn i Cii ε n rn−i (|xn−i+1 |2 + |xn−i+2 | + · · · + |xn | + |u|) = cn−i+1 · · · cn 1 (i) (i) Nn |u|2 + Nn−1 ε n (|xn |2 + |u|) + · · · +

we have y˙ n−i =

i P j=0

1 n

j

j

Ci ε n x˙ cn−j ···cn n−j

=

ε (yn−i+1 + · · · + yn ) + u + Φn−i where Φn−i =

i

(i)

Nn−i ε n (|xn−i+1 |2 + |xn−i+2 | · · · + |xn | + |u|)

1 Ci0 C1 ϕ (u) + cn−1i cn ε n ϕn−1 (xn , u) + · · · + cn n i Cii ε n ϕn−i (xn−i+1 , xn−i+2 , · · · , xn , u) cn−i ···cn

u = −ε σn (yn + σn−1 (yn−1 + · · · + σ1 (y1 ))) where

½ σi (x) =

x, εi sgn(x),

if |x| ≤ εi if |x| > εi

(8)

constants an−i depend only on the functions fi (independent of ε). Moreover, by (7) and (12), we have

(9)

(10)

|xn−j | ≤

j

ε− n Kn−j (|yn | + · · · + |yn−j |) ≤ µ ¶ j n−j 3 3 1 −n ε Kn−j ε + ··· + ε ≤ ε n Kn−j 4 4 4j

Thus, 2

(i)

(i)

|Φn−i | ≤ ε1+ n [Nn + Nn−1 (Kn2 + 1) + · · · +

and 1 ε1 = ε2 = · · · = 4

µ ¶n−1 1 εn , 4

2 (i) 2 + kn−i+2 + · · · + Kn + 1)] = ε1+ n A˜i Nn−i (Kn−i+1

εn = ε

=

1

ε n [y2 + · · · + yn − σn (yn + σn−1 × (yn−1 + · · · + σ1 (y1 )))] + Φ1

=

where

1

ε n [yn − σn (yn + σn−1 (yn−1 + · · · +

(11)

(i)

(i)

µ δi =

¶n

1 4i+1 A˜i

,

i = 0, · · · , n − 1

then in finite time, yn−i ∈ Pn−i = {yn−i : |yn−i | < 43 εn−i }, where i = 0, · · · , n − 1. Proof. First, consider the evolution of the state yn

σ1 (y1 )))] + Φn−1 =

2 + where A˜i = Nn + Nn−1 (Kn2 + 1) + · · · + Nn−i (Kn−i+1 Kn−i+2 + · · · + Kn + 1). ¤ Lemma 2. If the scaling factor ε satisfies ½ µ ¶n ¾ min ai ε ≤ min 1, δn , δn−1 · · · , δ1 , (13) max Ki (i)

By using (9), the system is transformed to the following closed loop system  y˙ 1           ..    .   y˙ n−1            y˙ n    

j = 1, · · · , i. Obviously, the

(i) Nn−i ,

Define the nested saturation control 1 n

j

Ci rn−j , cn−j+1 ···cn

(i)

where Nn−j =

1

−ε n σn (yn + σn−1 (yn−1 + · · · +

1 dyn = −ε n σn (yn + σn−1 (yn−1 + · · · + σ1 (y1 ))) + Φn (14) dt

σ1 (y1 ))) + Φn

where Φ1 , · · · , Φn are defined by (8). In the following, we tried to choose ε ≤ 1 to see that the system (11) is globally stable and exponentially convergent. Lemma 1. For i = 0, · · · , n − 1, assume that max{|xn−i+1 |, · · · , |xn |, |u|} ≤ an−i and for j = 0, · · · , i − 1 yn−j ∈ Pn−j = {yn−j : |yn−j |
εi

µ ¶n−1 1 εn , 4

εn = ε

System (21) is then transformed to the following closed loop system  y˙ 1 = z2 + · · · + zn − σn (zn + · · · + σ1 (z1 )) + Φ1     y˙ 2 = z3 + · · · + zn − σn (zn + · · · + σ1 (z1 )) + Φ2   .. .     y ˙ = zn − σn (zn + · · · + σ1 (z1 ))] + Φn−1 n−1   y˙ n = −σn (zn + · · · + σ1 (z1 )) + Φn Now, we will prove that system (21) is globally stable with small enough ε. Theorem 2. Choose ε to be small enough, then in finite time, zn−i ∈ Pn−i = {zn−i : |zn−i | < 34 εn−i }, where i = 0, · · · , n − 1. Proof. Since all xi → 0 if and only if all zi → 0, hence this theorem implies that the state of control system (1) is globally stable. We begin by considering the evolution of the state yn dyn = −σn (zn + σn−1 (zn−1 + · · · + σ1 (z1 ))) + Φn dt

(22)

¯ Pn = {zn : |zn | < 34 εn }, then |zn | ≥ 34 εn , and If zn ∈ |zn + σn−1 (zn−1 + · · · + σ1 (z1 ))| ≥ |zn | − εn−1 ≥

1 εn 2

532

ACTA AUTOMATICA SINICA

¯ Pn , then Now, it can be seen from (22) that if zn ∈ ¯ ¯ ¯ dyn ¯ ¯ ¯ ¯ dt ¯ ≥ |σn (zn + σn−1 (zn−1 + · · · + σ1 (z1 )))| − |Φn | ≥ 1 1 1 εn − rn ε2n ≥ εn , ε ≤ 2 4 4rn and

µ sgn

dyn dt

Vol. 36

Simulation results are described in Fig. 1. The initial point x 0 are (2, −3, 1), (−2, 1, −1), (1.5, 1, −1), and (−1, 1.5, 1), respectively, and the parameter ε is 0.65.

¶ = −sgn(zn )

Note that all pi are odd numbers and all yi → 0 if and only if all zi → 0. Therefore, when zn ≥ 34 εn , yn decreases at the rate 14 εn until zn < 34 εn . Similarly, when zn ≤ − 34 εn , yn increases at the rate 41 εn until zn > − 34 εn . Consequently, zn enters Pn in finite time and remains in Pn thereafter. Now, we proceed by induction. Suppose zn−j ∈ Pn−j = {zn−j : |zn−j | < 34 εn−j }, j = 0, · · · , i − 1, we consider the evolution of the state zn−i . The argument of σn−j , j = 0, · · · , i − 1, is bounded as |zn−j + σn−j−1 (zn−j−1 + · · · + σ1 (z1 ))| ≤ 3 εn−j + εn−j−1 = εn−j 4 By the definition of σi (see (10)), the evolution of zn−i is given by dyn−i = −σn−i (zn−i + · · · + σ1 (z1 )) + Φn−i dt ¯ Pn−i = {zn−i : |zn−i | < Similarly, if zn−i ∈ |zn−i | ≥ 34 εn−i , and

3 ε }, 4 n−i

|σn−i (zn−i + σn−i−1 (zn−i−1 + · · · + σ1 (z1 )))| ≥

then

1 εn−i 2

Hence, for small enough ε, we have ¯ ¯ ¯ dyn−i ¯ ¯ ¯ ¯ dt ¯ ≥ |σn−i (zn−i + · · · + σ1 (z1 ))| − |Φn−i | ≥ µ pm +1 ¶ 1 1 pm εn−i − ri εn−i + ε2 ≥ εn−i 2 4 where pm = max{p2 , · · · , pn } and µ ¶ dyn−i sgn = −sgn(zn−i ) dt Therefore, when zn−i ≥ 34 εn−i , yn−i decreases at the rate 1 ε until zn−i < 34 εn−i . Similarly, when zn−i ≤ − 34 εn−i , 4 n−i yn−i increases at the rate 14 εn−i until zn−i > − 34 εn−i . Consequently, zn−i enters Pn−i in finite time and remains in Pn−i thereafter. ¤

3

Numerical test

Consider the feed forward nonlinear system:  x˙ = − sin(x2 + x3 + u) + cos(x3 + u) + tan(u) − 1   1 x˙ 2 = tan(x3 ) − sin(u2 x3 ) 2   x˙ = − sin(u − u2 ) + u 3 4 √ 3 Let u = − ε(σ3 (y3 + σ2 (y2 + σ1 (y1 )))), where      2 ε 3 x1 y1 1 −2 −1   y2  =  0 −1 −1   ε 13 x  2  y3 0 0 −1 x3

Fig. 1

Simulation results

From the simulation test, we observe that the solution trajectory x3 (t), x2 (t), and x1 (t) converge to zero one by one, that is, x3 (t) converges to zero firstly, then x2 (t) and x1 (t) converges to zero finally (especially from Fig. 1 (d)). In fact, from the proof of Lemma 2, the control (9) makes the solution trajectories of nonlinear system (1) come into the small interval Pn−i by the order of yn , yn−1 , · · · , y1 . The numerical simulation verifies that exactly.

4

Conclusion

In this paper, we study the saturated control of a general class of feedforward nonlinear systems. By linear coordinate transformation, we transfer this feedforward nonlinear system to a closed loop system. We used nested saturation

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WANG Yong and MA Ru-Ning: Global Stabilization for Feedforward Nonlinear System · · ·

stabilizer[6] to make the nonlinear system globally stable and exponentially convergent at the equilibrium x = 0. The results of this paper clearly enhance the class of feedforward nonlinear systems investigated before. Numerical example is given, by which efficiency of the result given above is verified. References 1 Jankovic M, Sepulchre R, Kokotovic P V. Global adaptive stabilization of cascade nonlinear system. Automatica, 1997, 33(2): 263−268 2 Jia X C, Zhang D W, Zheng L H, Zheng N N. Modeling and stabilization for a class of nonlinear networked control systems: a T-S fuzzy approach. Progress in Natural Science, 2008, 18(8): 1031−1037 3 Liberzon D. Output-input stability implies feedback stabilization. Systems and Control Letters, 2004, 53(3-4): 237−248 4 Stankovic S S, Siljak D D. Robust stabilization of nonlinear interconnected systems by decentralized dynamic output feedback. Systems and Control Letters, 2009, 58(4): 271−275

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9 Teel A R. A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Transactions on Automatic Control, 1996, 41(9): 1256−1270 10 Liu X, Marquez J, Lin Y P. Input-to-state stabilization for nonlinear dual-rate sampled-data systems via approximate discrete-time model. Automatica, 2008, 44(12): 3157−3161 11 Zhong J H, Cheng D Z, Hu X M. Constructive stabilization for quadratic input nonlinear systems. Automatica, 2008, 44(8): 1996−2005 12 Ye X D. Universal stabilization of feedforward nonlinear systems. Automatica, 2003, 39(1): 141−147 13 Bhatia R. A note on the Lyapunov equation. Linear Algebra and Its Applications, 1997, 259(1): 71−76

WANG Yong Received his master degree from Jilin University in 2007. He is currently a lecturer at Changchun University of Technology. His research interest covers control theory and optimization. E-mail: wangyong [email protected]

5 Teel A R. Using saturation to stabilize a class of single-input partially linear composite systems. In: Proceedings of the 2nd IFAC Symposium on Nonlinear Control Systems. Bordeaux, France: Springer, 1992. 379−384 6 Teel A R. Global stabilization and restricted tracking for multiple integrators with bounded controls. Systems and Control Letters, 1992, 18(3): 165−171 7 Jankovic M, Sepulchre R, Kokotovic P V. Constructive Lyapunov stabilization of nonlinear cascade systems. IEEE Transactions on Automatic Control, 1996, 41(12): 1723−1735 8 Mazenc F, Praly L. Adding integrations, saturated controls, and stabilization for feedforward systems. IEEE Transactions on Automatic Control, 1996, 41(11): 1559−1578

MA Ru-Ning Received his Ph. D. degree from Fudan University in 2003. He is currently an associate professor at Nanjing University of Aeronautics and Astronautics. His research interest covers image processing, machine learning, control theory and optimization. Corresponding author of this paper. E-mail: [email protected]