Control of State-Constrained Nonlinear Systems Using ... - IEEE Xplore

2 downloads 0 Views 361KB Size Report
Abstract— This paper presents a control design for nonlinear systems with state constraints, based on the use of our newly introduced Integral Barrier Lyapunov ...
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA

Control of State-Constrained Nonlinear Systems Using Integral Barrier Lyapunov Functionals Keng Peng Tee and Shuzhi Sam Ge Abstract— This paper presents a control design for nonlinear systems with state constraints, based on the use of our newly introduced Integral Barrier Lyapunov Functionals (iBLF). The integral functional allow the mixing of the original state constraints with the errors in a form amenable to stable backstepping control design. This reduces some of the conservatism associated with the use of purely error-based functions with transformed error constraints. We show that, under the proposed iBLF-based control, output tracking error is bounded by an exponentially decreasing function of time, all states always remain in the constrained state space, and that the stabilizing functions and control input are bounded, subject to significantly relaxed feasibility conditions. A numerical example illustrates the performance of the proposed control.

I. I NTRODUCTION Constraints are commonly encountered in physical systems (e.g. mechanical stoppages, saturation, safety specifications), and the violation of constraints during operation can be detrimental in terms of performance degradation and/or hazards. Driven by these practical needs, as well as theoretical challenges, the rigorous handling of constraints in the control design stage has gained attention as an increasingly important topic of research. Well known methods for constrained nonlinear control include model predictive control (MPC) and reference governors (RG). The MPC method, widely adopted in industry, tackles the problem within an optimization paradigm inherently suitable for handling constraints, by solving an on-line finite horizon open-loop optimal control problem, subject to the system dynamics and constraints [1]. On the other hand, RG-based control works by modulating the reference signal, using online optimization algorithms, to avoid any violation of system constraints [2]. Other notable methods include extremum seeking control [3], nonovershooting control [4], and error transformation [5]. For linear systems, methods to handle constraints are usually based on notions of set invariance using Lyapunov analysis [6], [7], exploiting the fact that positive invariant sets can be obtained constructively in linear systems. Similar notions of invariance control has been extended to the nonlinear setting by switching between a nominal controller in the interior of the admissible set and an intervention K.P. Tee is with the Institute for Infocomm Research, A*STAR, Singapore 138632. [email protected] S. S. Ge is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, and The Robotics Institute, and School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611813, China. [email protected]

978-1-4673-2066-5/12/$31.00 978-1-4673-2064-1/12/$31.00 ©2012 ©2012 IEEE IEEE

control at the boundary [8], [9], using the idea of barrier certificates [10] to ensure invariance. In the spirit of shaping a control Lyapunov function to simultaneously achieve tracking and satisfy constraints, Barrier Lyapunov Functions (BLF) have been proposed. The method uses barrier functions that grow to infinity at some finite limits, and designs a control to keep the BLF bounded in the closed loop system and thus guarantee that the barriers are not transgressed. To date, BLFs have been used to design controls for systems in Brunovsky form [11], outputconstrained systems in strict feedback form [12], [13], [14], output-constrained systems in output feedback form [15], as well as state-constrained systems [16]. Besides these theoretical contributions, BLF-based controls have been applied to practical problems, such as the control of electromagnetic oscillators [17], electrostatic parallel plate microactuators [18], and electrostatic torsional micromirrors [19]. A limitation of existing BLF-based controls for stateconstrained systems is that the feasibility conditions that ensure constraint satisfaction tend to be conservative, since the original state constraints are enforced indirectly by imposing transformed constraints on the errors. Inspired by the Integral Lyapunov Function proposed [20], we propose, in this paper, novel Integral Barrier Lyapunov Functionals (iBLFs) for designing a control for a class of nonlinear systems with state constraints. The new integral Lyapunov functionals allow the original state constraints to be mixed with the error terms, in contrast to our previous BLF method [16] which used logarithmic Lyapunov functions composed solely of errors. Thus, the proposed iBLF approach enables a control to be designed that directly enforces the original state constraints. By avoiding the formulation of error constraints that indirectly enforces the state constraints, a process that introduces some degree of conservatism, we overcome the aforementioned limitation and achieve significant simplification and relaxation of the feasibility conditions. The rest of this paper is organized as follows. Section II formulates the state constraint problem for strict feedback systems. Section III provides an exposition of iBLFs, followed by backstepping control design. In Section IV, we briefly touch on a constrained optimization algorithm for offline feasibility checking and determination of the control gains. The simulation study in Section V illustrates the performance of the iBLF-based control, before concluding remarks for this paper are finally made.

3239

II. P ROBLEM F ORMULATION Throughout this paper, we denote by R+ the set of nonnegative real numbers, and k • k the Euclidean vector norm in Rm . We also denote x ¯i = [x1 , x2 , ..., xi ]T , z¯i = (1) (2) (i) T [z1 , z2 , ..., zi ] , and y¯di = [yd , yd , yd , ..., yd ]T . Consider the following nonlinear system in strict feedback form: x˙ i x˙ n y

= = =

fi (¯ xi ) + gi (¯ xi )xi+1 , fn (¯ xn ) + gn (¯ xn )u x1

i = 1, 2, ..., n − 1

Vi ≤

|xi | ≤ kci

(2)

with kci as a positive constant, for i = 1, ..., n. We denote the set X := {x ∈ Rn : |xi | < kci , i = 1, ..., n} ⊂ Rn . The control objective is to track desired trajectory yd while ensuring that all closed loop signals are bounded and that state constraints are not violated. Note that the state constraints are not necessarily physical constraints but can also be performance requirements. The following assumptions are in order. Assumption 1: For any kc1 > 0, there exist positive constants A0 , Yi , i = 1, ..., n, such that the desired trajectory yd (t) and its time derivatives satisfy (i)

|yd (t)| < Yi

(3)

for all t ≥ 0 and i = 1, ..., n. Assumption 2: The functions gi (¯ xi ), i = 1, 2, ..., n, are known, and there exists a positive constant g0 such that 0 < g0 ≤ |gi (¯ xi )| for |xj | < kcj , j = 1, 2, ..., i. Without loss of generality, we further assume that the gi (¯ xi ), i = 1, 2, ..., n, are all positive for |xj | < kcj , j = 1, 2, ..., i.

Vi (zi , αi−1 )

=

n X

Vi (zi , αi−1 )

i=1 Z zi 0

σkc2i dσ, kc2i − (σ + αi−1 )2

0

for |zi + αi−1 | < kci , which leads to (7) after substituting for pi . We use the backstepping procedure [21] to design the controller. It consists of n steps for our plant (1). Step 1 In the first step, we design the stabilizing function α1 . Consider the functional (5) for i = 1. The time-derivative is given by V˙ 1 =

kc21 z1 (f1 2 kc1 − x21

which is useful for establishing uniform stability.

∂V1 y˙ d ∂yd

(10)

where ̺1 (z1 , yd ) = =

=

1

kc21 dβ 2 2 0 kc1 − (βz1 + yd ) µ · ¶ kc1 z1 + yd tanh−1 z1 kc1 ¶¸ µ yd − tanh−1 kc1 kc1 (kc1 + z1 + yd )(kc1 − yd ) (12) ln 2z1 (kc1 − z1 − yd )(kc1 + yd )

Z

It can be shown, using L’Hˆopital’s rule, that

(5)

where zi = xi − αi−1 , α0 := yd and α1 , ..., αn−1 are continuously differentiable functions satisfying |αi | ≤ Ai < kci+1 for positive constants Ai , i = 0, 1, .., n − 1. Then, Vi is positive definite, continuously differentiable, and satisfies the decrescent condition in the set |xi | < kci for i = 1, ..., n: Z 1 βkc2i zi2 dβ (6) ≤ Vi ≤ zi2 2 2 2 0 kci − (βzi + sgn(zi )Ai−1 )

+ g1 z2 + g1 α1 − y˙ d ) +

where it is straightforward to show, using integration by parts and the substitution σ = βz1 [22], that ¶ µ kc21 ∂V1 − ̺1 (z1 , yd ) (11) = z1 ∂yd kc21 − x21

(4) i = 1, ..., n

(8)

which is positive in the set |σ + αi−1 | < kci . Since pi (0, αi−1 ) = 0 for |αi−1 | < kci , and pi (σ, αi−1 ) is monotonically increasing with σ in the set |σ + αi−1 | < kci , we can easily see that Z zi pi (σ, αi−1 )dσ ≤ zi pi (zi , αi−1 ) (9)

For system (1), consider the Integral Barrier Lyapunov Functional (iBLF) candidate =

(7)

2 k 2 + σ 2 − αi−1 ∂pi = 2ci ∂σ kci − (σ + αi−1 )2

III. C ONTROL S YNTHESIS U SING AN I NTEGRAL BARRIER LYAPUNOV F UNCTIONAL

V (z, α)

kc2i zi2 kc2i − x2i

for |xi | < kci . Proof: Denote pi (σ, αi−1 ) = σkc2i /(kc2i − (σ + αi−1 )2 ). We can show that

(1)

where f1 , ..., fn , g1 , ..., gn are smooth functions, x1 , ..., xn are the states, u and y are the input and output respectively. Every state xi is required to remain in the set

|yd (t)| ≤ A0 < kc1 ,

Lemma 1: The functional Vi , i = 1, ..., n, described in (5), satisfies

lim ̺1 (z1 , yd )

z1 →0

= =

lim

z1 →0

(kc21

kc21 kc21 − yd2

kc21 − (z1 + yd )2 ) (13)

Since |yd | < kc1 by Assumption 1, ̺1 (z1 , yd ) is well-defined in a neighborhood of z1 = 0. By designing the stabilizing function as µ ¶ (kc21 − x21 )y˙ d ̺1 1 α1 = −f1 − κ1 z1 + (14) g1 kc21

3240

where κ1 is a positive control gain, we obtain V˙ 1

kc21 g1 z2 k 2 κ1 z12 + − 2c1 kc1 − x21 kc21 − x21

=

limit exists. To this end, we have (15)

Step i (i = 2, ..., n) In the ith step, we design the stabilizing function αi . Consider integral-type functions described by (5), whose time-derivatives are given by V˙ i =

kc2i zi ∂Vi (fi + gi zi+1 + gi αi − α˙ i−1 ) + α˙ i−1 kc2i − x2i ∂αi−1

where ¶ kc2i − ̺ (z , α ) (16) i i i−1 kc2i − x2i Z 1 kc2i = dβ 2 2 0 kci − (βzi + αi−1 ) (kci + zi + αi−1 )(kci − αi−1 ) kci ln = 2zi (kci − zi − αi−1 )(kci + αi−1 ) (17)

∂Vi ∂αi−1

µ

= zi

̺i (zi , αi−1 )

The partial derivatives of ̺i (zi , αi−1 ), i = 1, ..., n, are given by: µ ¶ kc2i ∂̺i 1 (18) − ̺ = i ∂zi zi kc2i − (zi + αi−1 )2 kc2i (zi + 2αi−1 ) ∂̺i = 2 ) (19) ∂αi−1 (kc2i − (zi + αi−1 )2 )(kc2i − αi−1 where α0 := yd . Using L’Hˆopital’s rule, we have lim ̺i (zi , αi−1 ) =

zi →0

lim

zi →0

∂̺i ∂zi

=

kc2i 2 kc2i − αi−1 kc2i αi−1 2 )2 2 (kci − αi−1

Thus, ̺i , ∂̺i /∂zj and ∂̺i /∂αj−1 are well-defined in a neighborhood of zi = 0, in the set |αi−1 | < kci . In fact, we can generalize this result for higher order derivatives of ̺i , as shown in the following Lemma. Lemma 2: The function ̺i (zi , αi−1 ) is C n−i in the set Ψ = {zi ∈ R, αi−1 ∈ R : |αi−1 | < kci , |zi + αi−1 | < kci }. Proof: To facilitate analysis, let us define ψi = zi ̺i . We can express the jth partial derivative of ̺ with respect to zi as ∂ j ̺i ∂zij

=

η , d

j = 1, ..., n − i

(20)

where the numerator η and denominator d are defined by à ! ∂ j ψi ∂ j−1 ̺i j η = zi − ∂zij ∂zij−1 d

=

zij+1

∂η ∂zi

jzij−1

=

∂ j ψi ∂zij

+ zij

∂ j+1 ψi

∂zij+1 ∂ j ̺i ∂ j−1 ̺i −j 2 zij−1 j−1 − jzij j ∂zi ∂zi

(22)

The first term on the right hand side can be rewritten as µ ¶ j−1 ∂ j ψi ∂̺i j−1 ∂ jzij−1 = jz z + ̺ i i i ∂zi ∂zij ∂zij−1 " ¶# µ j−1 ∂̺i ̺i ∂ j−2 ∂ 2 ̺i j−1 ∂ = jzi + j−2 zi 2 + ∂zi ∂zi ∂zij−1 ∂zi à ! ∂ j−1 ̺i ∂ j ̺i = jzij−1 j j−1 + zi j (23) ∂zi ∂zi Substituting (23) into (22), it is easy to see all but the second term of (22) are eliminated, yielding: ∂η ∂zi

=

zij

∂ j+1 ψi ∂zij+1

(24)

As a result, we know that lim

zi →0

∂ j ̺i ∂zij

=

1 ∂ j+1 ψi lim j + 1 zi →0 ∂zij+1

(25)

Note that the limit always exists, since the function ψi =

kci (kci + zi + αi−1 )(kci − αi−1 ) ln 2 (kci − zi − αi−1 )(kci + αi−1 )

(26)

is C ∞ in the set (zi , αi−1 ) ∈ Ψ. Hence, ̺i (zi , αi−1 ) is at least n − i times continuously differentiable with respect to zi in the set (zi , αi−1 ) ∈ Ψ. The analysis for the partial derivatives of ̺i with respect to αi−1 is simpler. By inspection of (19), we see that ∂̺i /∂αi−1 is C ∞ in the set (zi , αi−1 ) ∈ Ψ. Therefore, j ∂ j ̺i /∂αi−1 , for j = 1, ..., n−i, exists in the set (zi , αi−1 ) ∈ Ψ. Now, we consider the mixed partial derivatives of ̺i (zi , αi−1 ). First, we note that ! Ã ∂ j+k ̺i ∂k ∂ j ̺i lim = (27) lim k zi →0 ∂αk ∂z j zi →0 ∂z j ∂αi−1 i−1 i i which exists and is continuous in the set (zi , αi−1 ) ∈ Ψ by virtue of (25)-(26), for any (j + k) ∈ {2, ..., n − i} and j, k ≥ 1. Then, using Clairaut’s Theorem [23], we can obtain any mixed partial derivative of ̺i of order j + k regardless of the order of differentiation, and show that it is continuous in the set (zi , αi−1 ) ∈ Ψ based on (27). Since the pure and mixed partial derivatives of ̺i up to order n − i exist and are continuous, ̺i is C n−i in the set (zi , αi−1 ) ∈ Ψ.

(21)

Based on L’Hˆopital’s rule, we can obtain limzi →0 ∂ j ̺i /∂zij = limzi →0 (∂η/∂zi )/(∂d/∂zi ) if the 3241

Design the stabilizing functions as µ (k 2 − x2i )α˙ i−1 ̺i 1 αi = −fi − κi zi + ci gi kc2i ! 2 2 2 kci−1 (kci − xi )gi−1 zi−1 − kc2i (kc2i−1 − x2i−1 )

(28)

where κi is a positive control gain, for i = 2, ..., n, and the control law as u = αn (29) Then, we obtain the time-derivative of the Lyapunov function candidate (4) as V˙

=



n X kc2i κi zi2 k 2 − x2i i=1 ci

(30)

Using Lemma 1, we obtain the inequality: V˙ ≤ −ρV

(31)

where ρ = 2 mini {κi }, in the set x ∈ X . The closed loop system is given by: µ 2 ¶ (kc1 − x21 )̺1 z˙1 = −κ1 z1 + g1 z2 + − 1 y˙ d kc21 ¶ µ 2 (kci − x2i )̺i − 1 α˙ i−1 z˙i = −κi zi + gi zi+1 + kc2i kc2 (kc2 − x2i )gi−1 zi−1 − i−12 i2 kci (kci−1 − x2i−1 ) µ 2 ¶ (kcn − x2n )̺n z˙n = −κn zn + − 1 α˙ n−1 kc2n kc2 (kc2 − x2n )gn−1 zn−1 − n−12 n2 (32) kcn (kcn−1 − x2n−1 ) which can be rewritten in the form z˙ = h(t, z)

(33)

where h(t, z) is piecewise continuous in t and locally Lipschitz in z, uniformly in t, in the set (zi (t), αi−1 (t)) ∈ Ψ, where Ψ is defined in Lemma 2. Thus, (31) and (33) allow us to establish the existence and uniqueness of the solution z(t) ∀t ∈ [0, ∞). In the following result, we establish the fact that x(t) remains in the constrained set X ∀t > 0, as well as the convergence properties of z(t). Theorem 1: Consider known system (1) under Assumptions 1-2, control law (14),(28),(29), and initial condition x(0) ∈ X := {x ∈ Rn : |xi | < kci , i = 1, ..., n}. Let Ai =

max

(¯ zn ,¯ ydn )∈Ω

|αi (¯ zi , y¯di )|,

i = 1, ..., n − 1

(34)

where Ω =

n p z¯n ∈ Rn , y¯dn ∈ Rn+1 : |zj | ≤ 2V |t=0 , o (j) |yd (t)| ≤ A0 , |yd | ≤ Yj , j = 1, ..., n (35)

If there exist positive constants κ1 , ..., κn−1 that satisfy the feasibility condition: kci

> Ai−1 (κ1 , ..., κi−1 ),

i = 1, ..., n

(36)

where A0 satisfies |yd (t)| ≤ A0 < kc1 , then the following properties hold. 3242

i) The error signals zi (t), i = 1, 2, ..., n, have an exponentially decreasing bound: p ρt (37) |zi (t)| ≤ 2V |t=0 e− 2 ∀t > 0

ii) The state x(t) remains, for all t > 0, in the constrained set X . iii) The stabilizing functions αi (t), i = 1, ..., n − 1, and control input u(t) are bounded ∀t > 0. Proof:

i) From (31), we obtain that V |t ≤ V |t=0 e−ρt in the set x ∈ X . Since initial condition x(0) ∈ X , we have V |t ≤ P V |t=0 e−ρt ∀t > 0. Then, using the fact that n V |t ≥ i=1 zi2 (t)/2, as shown in (6), we thus arrive at zi2 (t) ≤ 2V |t=0 e−ρt , which leads to (37). ii) Using proof by contradiction, we first assume that there exists some t = T and some i ∈ {1, ..., n} such that |xi (T )| = kci , starting from the initial condition x(0) ∈ X . Then, from (5) and (31), we have V |t=T

= =

n X

Vi |t=T i=1 n Z zi (T ) X i=1



0

V |t=0

σkc2i dσ kc2i − (σ + αi−1 (T ))2 (38)

Integrating Vi |t=T by parts,we arrive at: ¸z · σ + αi−1 i Vi |t=T = kci σ tanh−1 kci 0 Z zi −1 σ + αi−1 tanh −kci dσ kci 0 (1 + αi−1 (T ))(1 − xi (T )) = kci αi−1 (T ) ln (1 − αi−1 (T ))(1 + xi (T )) 2 k2 (T ) k 2 − αi−1 + ci ln ci2 2 kci − x2i (T ) ≤ V |t=0 (39) for all i = 1, ..., n. Now, substituting |xi (T )| = kci , it is clear that Vi |t=T becomes unbounded, contradicting the boundedness result V |t=T ≤ V |t=0 ∀t > 0. As such, |xi (t)| 6= kci ∀t > 0, i = 1, ..., n. Then, since x(0) ∈ X , it is clear that x(t) ∈ X ∀t > 0. iii) Given that the feasibility condition (36) is satisfied, we have |αi−1 (t)| < kci ∀t > 0. From item(ii), |xi (t)| < kci ∀t > 0. Thus, (zi (t), αi−1 (t)) ∈ Ψ ∀t > 0, i = 1, ..., n, where Ψ is defined in Lemma 2. Then, Lemma 2 yields that ̺i (zi , αi−1 ) is C n−i . Furthermore, ̺i (zi (t), αi−1 (t)) and its partial derivatives are all bounded ∀t > 0. Consequently, by inspection of the stabilizing functions αi and control input u , it is clear that they are bounded, by virtue of the boundedness of y¯dn (t), ̺i and its derivatives, and the fact that |xi (t)| < kci , ∀t > 0, ∀i = 1, ..., n.

IV. F EASIBILITY C HECK

3

yd = −0.5

The validity of the proposed control is contingent on feasibility condition (36) satisfied. In this section, we check if there exists a solution to an offline optimization problem maximizing the sum of the control gains to seek better tracking performance, similar to our previous work [16]. Denote the solution

2

x

2

1

ξ = [κ1 , ..., κn−1 ]T

0

−1

for the optimization problem: −2

maximize P (ξ) =

n−1 X

κi

(40)

−3 −1

i=1

subject to: kci > Ai−1 (ξ), i = 2, ..., n κi > 0, i = 1, ..., n − 1

1

yd = 0

(41) 2

x

2

1

0

−1

−2

−3 −1

(42)

where θ1 = 0.1, θ2 = 0.1, and θ3 = −0.2. The objective is for x1 to track desired trajectory yd , subject to full state constraint |x1 | < kc1 = 0.8 and |x2 | < kc2 = 2.5. First, we consider a stabilization task at 3 different set points yd = −0.5, yd = 0 and yd = 0.5. Figure 1 shows that the states, starting from various initial points spanning the constrained state space, converge to the respective set point. Even though the set point is shifted from the origin towards

−0.5

0 x

0.5

1

0.5

1

1

3

yd = 0.5 2

x

2

1

0

−1

−2

−3 −1

We present a simulation study on the full state constraint problem for a nonlinear system: θ1 x21 + x2 θ2 x1 x2 + θ3 x1 + (1 + x21 )u

0.5

3

V. N UMERICAL E XAMPLE

= =

0 x

1

where Ai (ξ) is defined in (34). If a solution ξ ∗ exists, then the proposed control (14),(28),(29) with ξ = ξ ∗ is feasible in ensuring output tracking for system (1) with state constraint. The use of the iBLF has advantageously resulted in the number of constraints of the optimization greatly reduced, and the constraints themselves relaxed. In contrast to [16] which required 5 sets of feasibility conditions, this paper requires only 2 sets, as shown in (41). The design in [16] involved an intermediate bound kbi on the errors zi , i = 1, ..., n, which led to one of the feasibility conditions kci > Ai−1 (ξ) + kbi , where kbi > 0. This feasibility condition is clearly more conservative than (41). Furthermore, the feasibility conditions in [16] specified the requirement for the initial state to belong to a subset of the constrained state space, i.e. −kci < ai ≤ xi (0) ≤ bi < kci , i = 1, ..., n, while (41) specifies no such requirement. Indeed, as stated in Theorem 1, the initial state can be any point in the constrained state space X . Remark 1: The proposed design can deal with partial state constraints by assigning iBLFs to constrained states and quadratic functions to free states, and to deal with model uncertainty by using adaptive backstepping design, similar to the treatment given in [16].

x˙ 1 x˙ 2

−0.5

−0.5

0 x

1

Fig. 1. Closed-loop state trajectories on the state space resulting from iBLF-based control. Initial states are marked by ‘+’.

the sides of the constrained state space, there is no shrinkage of the attraction basin. Next, we consider a tracking task, where the desired trajectory is described by yd (t) = 0.7 sin(2.5t) As shown in Figure 2, x1 (t), starting from various initial positions, converges to yd (t) and tracks it closely thereafter. Even though 2 of the initial positions are very close to the

3243

constraint and yielded initial trajectories that approached the constraints, the iBLF-based control aggressively pulls back the trajectories and keeps them in the constrained state space. A similar behavior is observed for x2 (t), which converges to and tracks the stabilizing function α1 (t) while avoiding any transgressions of the constraints. 0.8 0.6 0.4

x

1

0.2 0 −0.2 −0.4 −0.6 −0.8 0

1

2

3

4

5

time

2.5 2 1.5 1

x

2

0.5 0 −0.5 −1 −1.5 −2 −2.5 0

1

2

3

4

5

time

Fig. 2. States starting from different initial conditions in the feasible set converge to zero while confined in constrained region. The dotted line in the top figure represents yd while that for the bottom figure represents α1 .

VI. C ONCLUSIONS We have newly introduced Integral Barrier Lyapunov Functionals for control design of nonlinear systems with state constraints. A key advantage of using iBLFs is that the original state constraints are mixed with the error terms, unlike existing methods using BLFs composed solely of errors. This results in the relaxation of feasibility conditions for constraint satisfaction, since conservative error constraint transformations are no longer necessary. Provided that the feasibility conditions are satisfied, the closed loop tracking error has an exponentially decreasing bound, the state is guaranteed to remain in the constrained region, and the control input is always bounded.

R EFERENCES [1] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, pp. 789–814, 2000. [2] A. Bemporad, “Reference governor for constrained nonlinear systems,” IEEE Trans. Automatic Control, vol. 43, no. 3, pp. 415–419, 1998. [3] D. DeHaan and M. Guay, “Extremum-seeking control of stateconstrained nonlinear systems,” Automatica, vol. 41, pp. 1567–1574, 2005. [4] M. Krstic and M. Bement, “Nonovershooting control of strict-feedback nonlinear systems,” IEEE Trans. Automatic Control, vol. 51, no. 12, pp. 1938–1943, 2006. [5] K. D. Do, “Control of nonlinear systems with output tracking error constraints and its application to magnetic bearings,” International Journal of Control, vol. 83, no. 6, pp. 1199–1216, 2010. [6] D. Liu and A. N. Michel, Dynamical Systems with Saturation Nonlinearities. London, U.K.: Springer-Verlag, 1994. [7] T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis and Design. Boston, MA: Birkhuser, 2001. [8] J. Wolff, C. Weber, and M. Buss, “Continuous control mode transitions for invariance control of constrained nonlinear systems,” in Proc. 48th IEEE Conference on Decision & Control, pp. 542–547, 2007. [9] M. B¨urger and M. Guay, “Robust constraint satisfaction for continuous-time nonlinear systems in strict feedback form,” IEEE Trans. Automatic Control, vol. 55, no. 11, pp. 2597–2601, 2010. [10] S. Prajna, Optimization-based methods for nonlinear and hybrid systems verification. Dissertation (Ph.D.), California Institute of Technology, 2005. [11] K. B. Ngo, R. Mahony, and Z. P. Jiang, “Integrator backstepping using barrier functions for systems with multiple state constraints,” in Proc. 44th IEEE Conf. Decision & Control, (Seville, Spain), pp. 8306–8312, December 2005. [12] K. P. Tee, S. S. Ge, and E. H. Tay, “Barrier Lyapunov Functions for the control of output-constrained nonlinear systems,” Automatica, vol. 45, no. 4, pp. 918–927, 2009. [13] K. P. Tee, B. Ren, and S. S. Ge, “Control of nonlinear systems with time-varying output constraints,” Automatica, vol. 47, no. 11, pp. 2511–2516, 2011. [14] F. Yan and J. Wang, “Non-equilibrium transient trajectory shaping control via multiple barrier lyapunov functions for a class of nonlinear systems,” in Proc. American Control Conference, pp. 1695–1700, 2010. [15] B. Ren, S. S. Ge, K. P. Tee, and T. H. Lee, “Adaptive neural control for output feedback nonlinear systems using a Barrier Lyapunov Function,” IEEE Trans. Neural Networks, vol. 21, no. 8, pp. 1339– 1345, 2010. [16] K. P. Tee and S. S. Ge, “Control of nonlinear systems with partial state constraints using a Barrier Lyapunov Function,” International Journal of Control, vol. 84, pp. 2008–2023, 2011. [17] H. S. Sane and D. S. Bernstein, “Robust nonlinear control of the electromagnetically controlled oscillator,” in Proc. American Control Conference, (Anchorage, AK), pp. 809–814, May 2002. [18] K. P. Tee, S. S. Ge, and E. H. Tay, “Adaptive control of electrostatic microactuators with bidirectional drive,” IEEE Trans. Control Systems Technology, vol. 17, no. 2, pp. 340–352, 2009. [19] G. Zhu, C. G. Agudelo, L. Saydy, and M. Packirisamy, “Torque multiplication and singularity avoidance in the control of electrostatic torsional micro-mirrors,” in Proc. 17th IFAC World Congress, (Seoul, Korea), pp. 1189–1194, 2008. [20] S. S. Ge, C. C. Hang, and T. Zhang, “A direct adaptive controller for dynamic systems with a class of nonlinear parameterizations,” Automatica, vol. 35, pp. 741–747, 1999. [21] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley and Sons, 1995. [22] S. S. Ge, C. C. Hang, and T. Zhang, “Stable adaptive control of nonlinear multivariable systems with triangular control structure,” IEEE Trans. Automatic Control, vol. 45, pp. 1221–1225, 2000. [23] T. M. Apostol, Mathematical Analysis. Reading, Mass: AddisonWesley, 2nd ed., 1974.

ACKNOWLEDGMENTS This work was partially supported by the National Basic Research Program of China (973 Program) under Grant 2011CB707005. 3244