Control of nonlinear systems with partial state ...

International Journal of Control 2011, 1–16, iFirst

Control of nonlinear systems with partial state constraints using a barrier Lyapunov function Keng Peng Teea* and Shuzhi Sam Gebc a

Institute for Infocomm Research, ASTAR, 138632 Singapore; bDepartment of Electrical and Computer Engineering, National University of Singapore, 117576 Singapore; cInstitute of Intelligent Systems and Information Technology (ISIT), and School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China

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(Received 20 October 2010; final version received 9 October 2011) This article addresses the problem of control design for strict-feedback systems with constraints on the states. To prevent the states from violating the constraints, we employ a barrier Lyapunov function (BLF), which grows to infinity whenever its arguments approaches some finite limits. Based on BLF-based backstepping, we show that asymptotic output tracking is achieved without violation of any constraint, provided that the initial states and control parameters are feasible. We also establish sufficient conditions to ensure feasibility, which can be checked offline without precise knowledge of the initial states. The feasibility conditions are relaxed when handling the partial state constraint problem as compared to the full state constraint problem. In the presence of parametric uncertainties, BLF-based adaptive backstepping is useful in preventing the states from transgressing the constrained region during the transient stages of online parameter adaptation. To relax the feasibility conditions, asymmetric error bounds are considered and asymmetric barrier functions are used for control design. The performance of the BLF-based control is illustrated with two simulated examples. Keywords: constrained systems; adaptive control; backstepping; barrier functions

1. Introduction Constraints are ubiquitous in physical systems, and manifest themselves as physical stoppages, saturation, as well as performance and safety specifications, among others. Violation of the constraints during operation may result in performance degradation, hazards or system damage. In some cases, it is possible to neglect constraints in control design, and circumvent the problem through ad-hoc engineering fixes, although such solutions are highly context specific, require substantial human intervention and do not provide any guarantee of success. Driven by practical needs and theoretical challenges, the rigorous handling of constraints in the control design stage has become an important research topic in recent decades. For linear systems, methods to handle constraints are usually based on notions of set invariance using Lyapunov analysis (Liu and Michel 1994; Hu and Lin 2001), exploiting the fact that positive invariant sets can be obtained constructively in linear systems. 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 control at the boundary (Wolff, Weber, and Buss 2007; Bu¨rger and Guay 2010), using the idea of barrier certificates (Prajna 2005) to ensure invariance. Other *Corresponding author. Email: [email protected] ISSN 0020–7179 print/ISSN 1366–5820 online  2011 Taylor & Francis http://dx.doi.org/10.1080/00207179.2011.631192 http://www.tandfonline.com

well-known methods for the control of constrained nonlinear systems include model predictive control (MPC) and reference governors (RG). MPC considers the problem within an optimisation framework inherently suitable for handling constraints, by solving an on-line finite horizon open-loop optimal control problem, subject to the system dynamics and constraints (see, e.g. Mayne, Rawlings, Rao, and Scokaert 2000; Allgo¨wer, Findeisen, and Ebenbauer 2003). On the other hand, RG-based control works by modulating the reference signal, using online optimisation algorithms, to avoid any violation of system constraints (Bemporad 1998; Gilbert and Kolmanovsky 2002; Gilbert and Ong 2009). Other notable methods include extremum seeking control (DeHaan and Guay 2005), nonovershooting control (Krstic and Bement 2006), adaptive variable structure control (Su, Stepanenko, and Leung 1995) and error transformation (Do 2010). As the literature on constrained control is rather rich, it is out of this article’s scope to provide an exhaustive review. Recently, the use of barrier Lyapunov function (BLF) for the control of nonlinear systems with output and state constraints has been proposed. It involves the construction of a control Lyapunov function that grows to infinity whenever its arguments approaches some limits. Then, by keeping the BLF bounded in the

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2

K.P. Tee and S.S. Ge

closed-loop system, it is thus guaranteed that the limits are never transgressed. BLFs have been used to control systems in Brunovsky form (Ngo, Mahony, and Jiang 2005), output-constrained systems in strict feedback form (Tee, Ge, and Tay 2009b; Tee, Ren, and Ge 2011; Yan and Wang 2010), output-constrained systems in output feedback form (Ren, Ge, Tee, and Lee 2010), as well as state-constrained strict feedback systems (Tee and Ge 2009). In addition, BLF-based control has been applied to practical problems, such as the control of electromagnetic oscillators (Sane and Bernstein 2002), electrostatic parallel plate microactuators (Tee, Ge, and Tay 2009a) and electrostatic torsional micromirrors (Zhu, Agudelo, Saydy, and Packirisamy 2008). In this article, we extend the work of Tee et al. (2009b), on output-constrained systems, to a more difficult problem wherein constraints are present in some or even all of the states. The additional challenge of state-constrained systems comes from the fact that the intermediate stabilising functions, which ensure that certain states are constrained, need to satisfy constraints themselves. This necessitates feasibility analysis of the designed control, and thus the results provided here are different from that of Tee et al. (2009b). A preliminary version of this work, which deals only with full state constraint for a known system, has been presented in Tee and Ge (2009). However, a significant improvement of this article over Tee and Ge (2009) is that now we do not require precise knowledge of the initial state in establishing the feasibility conditions and determining the design parameters. The new feasibility conditions are formulated in terms of a box region of initial states, such that all points in the region share the same design parameters and achieve output tracking without constraint violation. This provides robustness to uncertainty in sensing the initial state. We provide a new algorithm to check the feasibility conditions and determine suitable design parameters prior to actual operation, by solving a constrained optimisation problem. For the first time, we propose asymmetric barrier functions to design control and feasibility conditions for the state constraint problem. Asymmetric barrier functions accommodate asymmetric error bounds that reduce conservatism of the conditions in Tee and Ge (2009). Furthermore, to deal with parametric uncertainty, we present an adaptive control that ensures constraint satisfaction and asymptotic output tracking, despite perturbations induced by transient online parameter adaptation. A further contribution is a generalised design and analysis to deal with partial state constraint, which includes full state constraint and output constraint as special cases. For known cases

considered, it is shown that locally exponential output tracking is achieved without violation of any constraint, whereas for the uncertain case, asymptotic output tracking is achieved. The remainder of this article is organised as follows. Section 2 formulates the partial state constraint problem for strict feedback systems and provides an exposition of barrier functions in Lyapunov synthesis. Section 3.1 presents the control design for a known system, Section 3.2 the adaptive control the design to handle model uncertainty and Section 3.3 control design using asymmetric barrier functions. In Section 4, we outline constrained optimisation algorithms for offline checking of the feasibility of the proposed control and determination of the design parameters. The simulation study in Section 5 illustrates the performance of the BLF-based control in comparison with one based on a quadratic Lyapunov function (QLF), and concluding remarks are made in Section 6.

2. Problem formulation and preliminaries Notation: Throughout this article, we denote by Rþ the set of nonnegative real numbers, k.k the Euclidean vector norm in Rm and @(.) the boundary of set (.). The symbols max(.) and min(.) denote the maximum and minimum eigenvalues of ., respectively. We also  :¼ ½ðÞ , ðÞ , . . . , ðÞ T for positive intedenote ðÞ i:j i iþ1 j  :¼ ½ðÞ , ðÞ , . . . , ðÞ T , where (.) can gers i < j, and ðÞ i 1 2 i be any variable except for yd, in which case ð2Þ ðiÞ T y di :¼ ½ yd , yð1Þ d , yd , . . . , yd  . Consider the following nonlinear system in strict feedback form: x_ i ¼ fi ðx i Þ þ gi ðx i Þxiþ1 , x_ n ¼ fn ðx n Þ þ gn ðx n Þu

i ¼ 1, 2, . . . , n  1 ð1Þ

y ¼ x1 where f1, . . . , fn, g1, . . . , gn are smooth functions, x1, . . . , xn are the states, u and y are the input and output, respectively. The nonlinear functions fi ðx i Þ may be uncertain, in which case they satisfy the following linear-in-the-parameters (LIP) condition: fi ðx i Þ ¼ T i ðx i Þ,

i ¼ 1, . . . , n

ð2Þ l

where 1, . . . , n are smooth functions, and  2 R is a vector of uncertain parameters satisfying  2  with known compact set . Due to smoothness property, there exist positive constants i such that k i ðx i Þk  i for jxi j  kci , i ¼ 1, 2, . . . , n. Consider the partition of the full state x ¼ [x1, . . . , xn]T into free states xr ¼ ½xr1 , xr2 , . . . , xrnr T and constrained states xs ¼ ½xs1 , xs2 , . . . , xsns T , where nr þ ns ¼ n, and the number sequences, fr1 , r2 , . . . , rnr g

3

International Journal of Control

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and fs1 , s2 , . . . , sns g, are both ascending. The states xi(t), i ¼ s1 , . . . , sns , are required to remain, 8t  0, in the set jxi j  kci , with kci positive constants. The generalised representation can be noted from the fact that xs ¼ x, xr ¼ ; yields the full state constraint problem in Tee and Ge (2009), while xs ¼ x1, xr ¼ [x2, . . . , xn]T yields the output constraint problem tackled in Tee et al. (2009b). The control objective is to track a desired trajectory yd while ensuring that all closed-loop signals are bounded, and that the partial state constraint is not violated. Note that the state constraints can represent not only physical constraints but also performance requirements. Assumption 1: For any kc1 4 0, there exist positive constants A0, Y1, Y2, . . . , Yn such that the desired trajectory yd(t) and its time derivatives satisfy j yd ðtÞj  A0 5 kc1 ,

jyðiÞ d ðtÞj 5 Yi ,

i ¼ 1, . . . , n ð3Þ

for all t  0.

where  :¼ [w, z]T 2 N is the state, and the function h : Rþ N ! Rnþl is piecewise continuous in t and locally Lipschitz in , uniformly in t, on Rþ N . Let Z i :¼ fzi 2 R : jzi j 5 kbi g  R. Suppose that there exist positive definite functions U : Rl ! Rþ and Vi : Z i ! Rþ (i ¼ 1, . . . , n), both of which are also continuously differentiable on Rl and Z i, respectively, such that Vi ðzi Þ ! 1 Let VðÞ :¼ inequality

Pn

i¼1

Remark 1: An example of a practical application is an electrostatic microactuator, which can be modelled by a second-order differential equation in the strict feedback form (1) where the states x1 and x2 represent the position and velocity, respectively, of the movable electrode (Tee et al. 2009a). It is desired to control the movable electrode within an airgap without coming into contact with the fixed electrode, and also limit the speed of the movable electrode. This amounts to constraints on x1 and x2 in the form of jx1 j 5 kc1 and jx2 j 5 kc2 , where kc1 and kc2 are positive constants specified according to problem requirements. A BLF candidate is a continuously differentiable, positive definite function V(z) that is defined on an open region D containing the origin and is proper on D, i.e. limz!@D VðzÞ ¼ þ1. The following lemma formalises a result on the use of a BLF candidate for constraint satisfaction. Lemma 1 (Tee and Ge 2009): For any positive let Z :¼ fz 2 Rn : jzi j 5 kbi , constants kbi , n i ¼ 1, 2, . . . , ng  R and N :¼ Rl Z  Rnþl be open sets. Consider the system _ ¼ hðt, Þ

ð4Þ

ð5Þ

Vi ðzi Þ þ UðwÞ and z(0) 2 Z. If the @V V_ ¼ h  0 @

ð6Þ

holds in the set z 2 Z, then z(t) 2 Z 8t 2 [0, 1). Lemma 2 (Ren et al. 2010): For any positive constant kb, positive integer p and any z 2 R satisfying jzj < kb, we have log

Assumption 2: The functions gi ðx i Þ, i ¼ 1, 2, . . . , n, are known, and there exists a positive constant g0 such that 0 5 g0  j gi ðx i Þj for jxj j 5 kcj , j 2 fs1 , . . . , sns g. Without loss of generality, we further assume that the gi ðx i Þ, i ¼ 1, 2, . . . , n, are all positive for jxj j 5 kcj , j 2 fs1 , . . . , sns g.

as zi ! kbi

k2p b k2p b

 z2p

5

z2p k2p b

 z2p

ð7Þ

3. Control design for partial state constraints For the problem of output constraint tackled in Tee et al. (2009b), only the first step of the backstepping design involves the use of a barrier function. By enforcing a constraint on the output tracking error z1 ¼ y  yd, the output y is constrained within a specified zone, provided that the desired trajectory yd is also within the same zone. In Tee and Ge (2009), full state constraint has been considered, and the use of barrier functions extended to every step, in order to keep each error signal zi ¼ xi  i1 (i ¼ 2, . . . , n) constrained. In this section, we present a generalised design to deal with partial state constraint, where only some of the states have constraints. In fact, the design for full state constraint and output constraint are special cases of this design. The design procedure is modified such that barrier functions are only used up to the step with the highest order state under constraint, and the feasibility conditions can be relaxed. Let F :¼ fr1 , r2 , . . . , rnr g \ f1, 2, . . . , sns g

ð8Þ

be a set of nF positive integers used to denote fkci gi2F , a set of artificial constraints that we impose on the free states as part of the design. In order to ensure that xi never transgresses the constrained region, feasibility conditions related to

4

K.P. Tee and S.S. Ge

the design parameters and an initial state region are formulated. Different from Tee and Ge (2009), the new feasibility conditions in this article do not require precise knowledge of the initial state x(0). These conditions can be checked offline to determine if the given problem can be solved under this approach. In the following, we will consider the case when the functions fi ðx i Þ in system (1) are known, and also the case when they contain uncertain parameters.

where i, i ¼ 1, . . . , n, are positive constants. This yields the closed-loop system z_1 ¼ 1 z1 þ g1 z2 z_i ¼ i zi 

z_sns þ1 ¼ sns þ1 zsns þ1 

gsns zsns 2 kbsn  z2sns s

i ¼ 2, ... ,sns

þ gsns þ1 zsns þ2

z_j ¼ j zj  gj1 zj1 þ gj zjþ1 , j ¼ sns þ 2, ... ,n  1 z_n ¼ n zn  gn1 zn1 ð12Þ The derivative of V along the closed-loop trajectories can be written as

3.1 Known case Denote z1 ¼ x1  yd and zi ¼ xi  i1, i ¼ 2, . . . , n. Consider the BLF candidate:

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k2bi  z2i gi1 zi1 þ gi ziþ1 , 2 kbi1  z2i1

V_ ¼ 

sns X j z2j j¼1

V¼

sns X

Vi þ U

ð9Þ

 2

k2bj  zj

n X

j z2j

ð13Þ

j¼sns þ1

Let the closed-loop system (12) be written as

i¼1

z_ ¼ hðt, zÞ

ð14Þ

where n X 1 2 z, U¼ 2 j j¼s þ1 ns

k2bi

1 Vi ¼ log 2 , 2 kbi  z2i

i ¼ 1, 2, . . . , sns

The positive constants kbi , i ¼ 1, . . . , n, are to be determined from subsequent feasibility analysis. Note that V is positive definite and continuously differentiable in the set jzi j 5 kbi for all i ¼ 1, 2, . . . , n. According to the backstepping methodology, we employ barrier functions Vi to design stabilising functions from step 1 to step sns : 1 ¼ i ¼

1 ðf1  1 z1 þ y_d Þ g1

j ¼

gs z s fsns þ1 þ _ sns  sns þ1 zsns þ1  2 ns ns2 kbsn  zsns

!

s

Theorem 1: Consider known system (1) under Assumptions 1 and 2, stabilising functions and control law (10)–(11). Let ðx sns , zsns , yds ns Þ2

where   :¼ x sns 2 Rsns , zsns 2 Rsns , y ds ns 2 Rsns þ1 : jxj j  Dzj þ Aj1 , jzj j  Dzj , o j yðdj Þ j  Yj , j ¼ 1, . . . , sns vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u Pn s ns 2 2 u  Z2 Y kbk  Zk t k¼sns þ1 k Dzj :¼ kbj 1  e k2bk k¼1 x0 :¼ fx 2 Rn : aj  xj  bj , j ¼ 1, . . . , ng

1 ðfj þ _ j1  j zj  gj1 zj1 Þ, gj j ¼ sns þ 2, sns þ 3, ...,n

u ¼ n

Then, together with the fact that V_  0 in the set zsns 2 Z sns and zsns ð0Þ 2 Z sns , we invoke Lemma 1 to obtain that zsns ðtÞ 2 Z sns for all t > 0.

x2x0

From step ðsns þ 1Þ onwards until the final step, the quadratic function U is used for the design of the remaining stabilising functions and final control law:

gsns þ1

ð15Þ

max ji ðx i , zi , y di Þj, i ¼ 1, . . . , sns  1   Zi ¼ max zi ðx i , ydi ð0ÞÞ, i ¼ 1, . . . , n ð16Þ

!

ð10Þ

sns þ1 ¼

Z sns :¼ fzsns 2 Rsns : jzi j 5 kbi , i ¼ 1, 2, . . . , sns g

Ai ¼

k2  z2i 1 fi þ _ i1  i zi  2 bi gi1 zi1 gi kbi1  z2i1

1

where h(t, z) is piecewise continuous in t and locally Lipschitz in z, uniformly in t, in the set z 2 Z sns , defined by

ð11Þ

ð17Þ

Given the constraints fkcs2 , kcs3 , . . . , kcsns g, suppose that the initial state xð0Þ 2 x0 , where bi 5 kci and ai 4 kci . If there exist positive constants that satisfy the  ¼ ½an , bn ,  n1 , kbsns , fkci gi2F T

International Journal of Control conditions kbi 4 Zi ðÞ,

i ¼ 1, . . . , sns

kci 4 Ai1 ðÞ þ kbi ,

i ¼ 1, . . . , sns

ð18Þ

where A0 satisfies j yd ðtÞj  A0 5 kc1 , then the following properties hold. (i) The signals zi(t), i ¼ 1, 2, . . . , n, remain, for all t > 0, in the compact set defined by z ¼ fz 2 Rn : jzi j  Dzi , i ¼ 1, 2, . . . , sns , pffiffiffiffiffiffiffiffiffiffiffiffi kzsns þ1:n k  2Vð0Þg

ð19Þ

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where zsns þ1:n :¼ ½zsns þ1 , zsns þ2 , . . . , zn T . (ii) The partial state xs(t) remains, for all t > 0, in the set xs :¼ fxs 2 Rns : jxi j  Dzi þ Ai1 5kci , i ¼ s1 , s2 ,...,sns g ð20Þ i.e. the partial state constraint is never violated. (iii) All closed-loop signals are bounded. (iv) The origin z ¼ 0 is locally exponentially stable. Proof: (i) Since the satisfaction of kbi 4 Zi in (18) implies zsns ð0Þ 2 Z sns , Lemma 1 yields zsns ðtÞ 2 Z sns 8t > 0. Then, from (13), we have VðtÞ 

sns X 1 i¼1



2

sns X 1 i¼1

log

k2bi k2bi  z2i ð0Þ

þ

n X 1 2 z ð0Þ 2 i i¼s þ1 ns

n X 1 2 log 2 Z þ 2 kbi  Z2i i¼sn þ1 2 i

k2bi

s

Using the identity log a þ log b ¼ log ab, we rewrite the above inequality into the following: ! Pn sns k2bk k2bi Z2 Y k¼sns þ1 k e , i ¼ 1, 2, . ..,sns k2bi  z2i ðtÞ k2  Z2k k¼1 bk ð21Þ The above inequality can be rearranged into the form jzi ðtÞj  Dzi 8t > 0. Then, based on the fact that n 1 X z2 ðtÞ  Vð0Þ ð22Þ 2 i¼s þ1 i ns pffiffiffiffiffiffiffiffiffiffiffiffi it is easy to see that kzsns þ1:n ðtÞk  2Vð0Þ. Hence, zi(t) remains in the compact set z 8t > 0. (ii) Since jz1 ðtÞj  Dz1 5 kc1  A0 , we can show that jx1 ðtÞj  Dz1 þ j yd ðtÞj 5 kc1  A0 þ j yd ðtÞj. Noting that jyd(t)j  A0 from Assumption 1, we therefore conclude that jx1 ðtÞj  Dz1 þ A0 5 kc1 , 8t > 0.

5

To show that jx2 ðtÞj  kc2 , we need to first verify that there exists a positive constant A1 such that j1(t)j  A1, 8t > 0. Since jx1 ðtÞj  Dz1 þ A0 , jz1 ðtÞj  Dz1 and jy_d ðtÞj  Y1 , it is clear that ðx1 ðtÞ, z1 ðtÞ, y d1 ðtÞÞ 2 1 , and thus, the stabilising function 1 ðx1 , z1 , yd1 Þ in (10) is bounded since it is a continuous function. As a result, A1 exists. Then, from jz2 ðtÞj  Dz2 5 kb2 , we can show that jx2 ðtÞj  Dz2 þ j1 ðtÞj 5 kb2 þ j1 ðtÞj. Since j1(t)j  A1, we conclude that jx2 ðtÞj  Dz2 þ A1 5 kb2 þ A1 5 kc2 , 8t > 0. We can progressively show that jxiþ1 ðtÞj  kciþ1 , i ¼ 2, . . . , sns  1, after verifying that there exist positive constants Ai such that ji(t)j  Ai, 8t > 0. Since jxi ðtÞj  Dzi þ Ai1 , jzi ðtÞj  Dzi and j yðiÞ d ðtÞj  Yi , it is clear that ðx i ðtÞ, zi ðtÞ, ydi ðtÞÞ 2 i , and thus, the stabilising function i ðx i , zi , ydi Þ in (10) is bounded since it is a smooth function. As a result, we have that Ai exists. Then, from jziþ1 ðtÞj  Dziþ1 5 kbiþ1 , we can show that jxiþ1 ðtÞj  Dziþ1 þ ji ðtÞj 5 kbiþ1 þ ji ðtÞj. As a result of ji(t)j  Ai, we have jxiþ1 ðtÞj  Dziþ1 þ Ai 5 kbiþ1 þ Ai 5 kciþ1 , 8t > 0. Finally, since the sequence fs1 , s2 , . . . , sns g  f1, 2, . . . , sns g, we conclude that xs ðtÞ 2 xs 8t > 0. (iii) By inspection of the stabilising functions i ðx i , zi , y di Þ and control uðx n , zn , y dn Þ, it is clear that they are bounded, by virtue of the boundedness of x n ðtÞ, zn ðtÞ, y dn ðtÞ, and, in particular, by jzi ðtÞj  Dzi 5 kbi , which prevents any term comprising ðk2bi  z2i ðtÞÞ in the denominator from becoming unbounded. Since jzi ðtÞj  Dzi 5 kbi , jxi ðtÞj 5 kp i(t)ffi  Ai ci ,ffiffiffiffiffiffiffiffiffiffiffi for i ¼ 1, . . . , sns and kzsns þ1:n k  2Vð0Þ, we can progressively show, via signal chasing, that the remaining i(t) and xi(t) are also bounded ði ¼ sns þ 1, . . . , nÞ. Then, it is straightforward to show that the control uðx n , zn , y dn Þ is bounded. Thus, all closed-loop signals are bounded. (iv) From (13) and Lemma 2, we have V_  V in the set zsns 2 Z sns , where  ¼ min{21, . . . , 2n}. This yields V(t)  V(0)et, and thus, 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < kbi 1  e2Vð0Þet i ¼ 1, . . . , sns jzi ðtÞj  pffiffiffiffiffiffiffiffiffiffiffiffi t : i ¼ sns þ 1, . . . , n 2Vð0Þe 2 To show exponential convergence of zi, i ¼ 1, . . . , sns , to 0, it suffices to show that there exist constants a, b > 0 and ts  0 such t that 1  e2Vð0Þe  aebt for t  ts. From the identity w þ ew  1 for any w  0,

6

K.P. Tee and S.S. Ge we substitute w for 2V(0)et and rearrange to obtain 1  e2Vð0Þe

t

 2Vð0Þet

It follows that ( pffiffiffiffiffiffiffiffiffiffiffiffi t kbi 2Vð0Þe 2 jzi ðtÞj  pffiffiffiffiffiffiffiffiffiffiffi ffi t 2Vð0Þe 2

i ¼ 1, . . . , sns i ¼ sns þ 1, . . . , n

Kokotovic 1995) for stable design of an adaptation law. Denote z1 ¼ x1  yd and zi ¼ xi  i1, i ¼ 2, . . . , n. We employ the same BLF candidate (9), except that U is augmented with a quadratic term of the parameter estimation error: n X 1 2 1 ~T 1 ~ z þ    U¼ ð23Þ 2 i 2 s þ1 ns

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which implies local exponential convergence œ of every zi, i ¼ 1, . . . , n, to zero. Remark 2: For i 2 fs1 , s2 , . . . , sns g, the given constraints kci need to satisfy feasibility conditions (18). However, for i 2 fr1 , r2 , . . . , rns g, where rns 5 sns , the constraints kci are not explicitly specified as problem requirements, but rather, they are artificially imposed as part of the design procedure. As such, they can be chosen as design parameters, thus relaxing the feasibility conditions. Remark 3: pIt is ffi noted from (19) that the bound ffiffiffiffiffiffiffiffiffiffiffi kzsns þ1:n k  2Vð0Þ, for the free states zsns þ1 , . . . , zn , depends on the initial state x(0), while that for the constrained states depends on a region of initial states. This is acceptable since the bound for the free states is for analytical purpose only and does not need to be computed during feasibility check.

where  ¼ diag( 1, . . . , l) > 0, ~ :¼ ^  , and kbi , i ¼ 1, . . . , n, are positive constants to be determined from subsequent feasibility analysis. From step 1 to step sns , adaptive backstepping with barrier functions yields the stabilising functions: 1 1 ¼ ð^T w1  1 z1 þ y_ d Þ g1 k2  z22 1 2 ¼ ^T w2  2 z2  b22 g1 z1 g2 kb1  z21 ! 1 X @1 @1 ð jþ1Þ @1 þ  2 g1 x2 þ y þ ðjÞ d @x1 @^ j¼0 @y i ¼

1 ^T wi  i zi  gi þ

i1 X @i1 j¼1

Remark 4: Computation of the bound Dzi for zi, i.e. jzi j  Dzi 5 kbi , is required so that the bound Ai for the stabilising function i can be computed in (16), and the feasibility conditions (18) checked. Remark 5: Unlike the approach of Tee and Ge (2009), the new feasibility conditions (18) in this article do not require precise knowledge of the initial state x(0), but are formulated in terms of a region of feasible initial states about the initial desired trajectory and its derivatives, providing robustness to the control scheme against effects of imprecision in state sensing. Specifically, Dzj is expressed in terms of Zk, k ¼ 1, . . . , n, instead of zk(0), k ¼ 1, . . . , n, since the computation of Ai for (18), which requires Dzj , j ¼ 1, . . . , i, must not require exact knowledge of zk(0).

When the nonlinearities fi ðx i Þ are uncertain, but can be linearly parameterised according to (2), the foregoing design methodology can be modified, based on the certainty equivalence approach, i.e. replacing instances of T i ðx i Þ in the controls with their estimates ^T i ðx i Þ, followed by the design of the adaptation law for ^ that guarantees closed-loop stability. We adopt the tuning functions approach (Krstic, Kanellakopoulos, and

gj xjþ1 þ

gi1 zi1

i1 X @i1 ðjÞ j¼0 @yd

yðdjþ1Þ

! i1 X @i1 zj @j1  i þ wi , þ k2  z2j @^ @^ j¼2 bj i ¼ 3, 4, . . . , sns

ð24Þ

From step ðsns þ 1Þ onwards until the final step, the quadratic function U is used to design the stabilising functions: sns þ1 ¼

1 gsns þ1

þ

j¼1

þ

^T wsns þ1  sns þ1 zsns þ1 

sns X @sn

sns X j¼2

3.2 Uncertain case

xj

d 2 kbi  z2i kbi1  z2i1

xj

s

gj xjþ1 þ

s ns X @sn

s

ðjÞ j¼0 @yd

zj @j1 wsns þ1 2 kbj  z2j @^

gsns zsns 2 kbsn  z2sns s

yðdjþ1Þ þ

!

@sns  sns þ1 @^

" i1 X 1 @i1 ^T wi  i zi  gi1 zi1 þ gi xiþ1 i ¼ gi xi j¼1 þ

þ

i1 X @i1 ðiÞ j¼0 @yd sns X j¼2

yðiþ1Þ þ d

@i1  i @^

# ! i1 X zj @j1 @j1 þ zj wi k2bj  z2j @^ @^ j¼sn þ1

i ¼ sns þ 2, . . . , n

s

ð25Þ

7

International Journal of Control The control and adaptation laws are u ¼ n _ ^ ¼  n

The derivative of V along the closed-loop trajectories can be written as: ð26Þ V_ ¼ 

where the intermediate functions wi, tuning functions i and adaptation law are given by w1 ¼

1 ðx1 Þ,

iÞ  i ðx

i1 X @i1

 j Þ, i ¼ 2, .. .,n j ðx @xj 8 < i1 þ wi zi , i ¼ 2, ...,sns w1 z1 k2bi  z2i

1 ¼ 2 , i ¼ : kb1  z21

i1 þ wi zi , i ¼ sns þ 1, . ..,n ð27Þ

wi ¼

j¼1

Downloaded by [Keng Peng Tee] at 11:41 09 November 2011

z_2 ¼ 2 z2  þ

k2b2  z22 k2b1



z21

1 ðx1 Þ

i1 X j¼2

j¼2

k2bi1  z2i1

i ¼ 3, . . . , sns

gsns zsns 2 kbsn  z2sns s

þ gsns þ1 zsns þ2

@sns zj @j1 wsns þ1 þ wsns þ1 zj 2 2 kbj  zj @^ @^

@i1 _^ ð i  Þ z_i ¼ i zi  gi1 zi1 þ gi ziþ1  ~T wi þ ^ @ sns X @sns zj @j1 þ wi þ wi zj , k2  z2j @^ @^ j¼2 b j

i ¼ sns þ 2, . . . , n  1 @n1 _^ z_n ¼ n zn  gn1 zn1  ~T wn þ ð n  Þ ^ @ sns n1 X X zj @j1 @j1 þ wn þ wn zj 2  z2 ^ ^ k @  j j¼2 b j¼s þ1 @ j

_ ~ ¼  n

N :¼ fz 2 Rn , ~ 2 Rl : jzi j 5 kbi , i ¼ 1, 2, . . . , ng ð30Þ Since V_  0 in the set zsns 2 Z sns , and zsns ð0Þ 2 Z sns , we invoke Lemma 1 to obtain that zsns ðtÞ 2 Z sns for all t > 0.

where 0 is defined by (17) and  by n  :¼ x sns 2 Rsns , zsns 2 Rsns , y ds ns 2 Rsns , ^ 2 Rl : jxj j

gi1 zi1 þ gi ziþ1

zj @j1 wi , k2bj  z2j @^

s ns X

Let the closed-loop system (28) be written as _ ¼ hðt, Þ, where  ¼ ½zT , ~T T . By inspection, h(t, ) is piecewise continuous in t and locally Lipschitz in , uniformly in t, in the set  2 N , defined by

ð31Þ

@sns _^ ð sns þ1  Þ  ~T wsns þ1 þ ^ @ þ

ð29Þ

j¼sns þ1

x2x0

k2bi  z2i

z_sns þ1 ¼ sns þ1 zsns þ1 

j z2j

   ^ , i ¼ 1, . ..,n Zi ¼ max zi ðx i , ydi ð0Þ, ð0ÞÞ

@i1 _^ ð i  Þ  ~T wi þ @^ þ

k2bj  zj

n X

^ ðx sns , zsns , y dsns , Þ2

g1 z1 þ g2 z3  ~T w2

@1 _^ ð 2  Þ @^

z_i ¼ i zi 

j¼1

 2

Theorem 2: Consider the closed-loop system (1) and (24)–(26), under Assumptions 1 and 2. Let    ^ , i ¼ 1, ...,sns  1 Ai ¼ max i ðx i , zi , y di , Þ

which yields the closed-loop system z_1 ¼ 1 z1 þ g1 z2  ~T

sns X j z2j

 Dzj þ Aj1 , jzj j  Dzj , j yðdj Þ j  Yj , ^  D ^, j ¼ 1, . . . , sns g kk  vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sns 2 Y u kbk  Z2k Dzj :¼ kbj t1  e2V  k2bk k¼1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2V D^ :¼ M þ min ð1 Þ n 1 X 2 1 ^  k2 V  :¼ Z þ max ð1 Þ max kð0Þ 2@ 2 k¼s þ1 k 2 ns

V :¼

sns 1X

2 k¼1

log

k2bk k2bk  Z2k

þ V  ð32Þ

Given the constraints fkcs2 , kcs3 , . . . , kcsns g, suppose that the initial state xð0Þ 2 x0 , where bi 5 kci and ai 4 kci . If there exist positive constants  ¼ ½an , bn ,  n1 , kbsns , l , fkci gi2F T that satisfy the conditions kbi 4 Zi ðÞ,

i ¼ 1, . . . , sns

kci 4 Ai1 ðÞ þ kbi ,

ns

ð28Þ

i ¼ 1, . . . , sns

ð33Þ

where A0 satisfies j yd ðtÞj  A0 5 kc1 , then the following properties hold.

8

K.P. Tee and S.S. Ge ^ remain, (i) The signals zi(t), i ¼ 1, 2, . . . , n, and ðtÞ for all t > 0, in the compact sets defined by  z :¼ z 2 Rn : jzi j  Dzi , i ¼ 1, 2, . . . , sns ,  pffiffiffiffiffiffiffiffiffiffiffiffio  zsns þ1:n k  2Vð0Þ ð34Þ n o l ^  D^ ^ :¼ ^ 2 R : kk  (ii) The partial state xs(t) remains, for all t > 0, in the set

xs :¼ fxs 2 Rns : jxi j  Dzi þ Ai1 5kci , i ¼ s1 , s2 ,...,sns g ð35Þ i.e. the partial state constraint is never violated. (iii) All closed loop signals are bounded. (iv) The origin z ¼ 0 is asymptotically stable.

Downloaded by [Keng Peng Tee] at 11:41 09 November 2011

Proof:

Remark 6: In the absence of uncertainty, exponential stability of the point z ¼ 0 is achieved by Theorem 1. However, when uncertainty is involved, the analysis is ~ and we can slightly complicated by the presence of , ~ remains only conclude that limt!1 z(t) ¼ 0 while ðtÞ bounded for all t > 0, as established by Theorem 2. Remark 7: If all constraints are lifted, this becomes a special case whereby sns ¼ 0 under the partial constraints framework in Section 3. The design reduces to standard (adaptive) backstepping with Quadratic Lyapunov Functions (QLF ). The simplification to a QLF-based design makes sense because BLFs are no longer needed in the absence of constraints.

3.3 Using asymmetric barrier functions

(i) From the fact that kk  M, and jzi(0)j  Zi,  Further, with we know that Vð0Þ  V. kbi 4 Zi , it follows that zsns ð0Þ 2 Z sns , and thus Lemma 1 yields zsns ðtÞ 2 Z sns 8t > 0. Then, from (29), it is clear that  and hence ^ remains in the VðtÞ  Vð0Þ  V, compact set ^ 8t > 0. Furthermore, we have k2bi k2bi  z2i ðtÞ

 log 

 s ns Y k2bk e2V

k2 k¼1 bk sns Y

k2bk

k2 k¼1 bk

 Z2k

 e2V

 z2k ð0Þ ,

i ¼ 1, 2, . . . , sns

ð36Þ

which yields jzi ðtÞjP Dzi 8t > 0. Then, based  it is easy on the fact that 12 ni¼sns þ1 z2i ðtÞ  V, pffiffiffiffiffiffi to see that kzsns þ1:n ðtÞk  2V . Hence, zi remains in the compact set z 8t > 0. (ii) Similar to the proof of Theorem 1(ii), we can show that jxi j  kci for i ¼ 1, 2, . . . , sns . Since the sequence fs1 , s2 , . . . , sns g  f1, 2, . . . , sns g, we can conclude that xs ðtÞ 2 xs 8t > 0. (iii) We have already established the boundedness ^  D ^, jzi ðtÞj  Dzi 5 kbi , jxi ðtÞj 5 results kk  1, . . . , sns . Together with kci and i  Ai for i ¼ pffiffiffiffiffiffi the fact kzsns þ1:n k  2V , we can progressively show, along the lines of the proof of Theorem 1(iii), that the remaining i and xi, for and the control i ¼ sns þ 1, . . . , n, ^ are all bounded. uðx n , zn , ydn , Þ, (iv) The proof is straightforward by using the LaSalle–Yoshizawa theorem and (29) to conclude that sns n X X j z2j ðtÞ j z2 ðtÞ ¼ 0 þ lim ð37Þ t!1 k2  z2j ðtÞ j¼sn þ1 j j¼1 bj s

and thus, limt!1 z(t) ¼ 0.

œ

While symmetric error bounds jzi j 5 kbi are mathematically easier to handle, they may lead to conservative feasibility conditions. To reduce conservatism, we formulate asymmetric error bounds kai 5 zi 5 kbi and ensure that they hold by using asymmetric barrier functions in the control design. This asymmetric formulation is general and can handle symmetric error bounds, but the control law and feasibility conditions are more complex. Consider asymmetric barrier Lyapunov function P candidate V ¼ ni¼1 Vi , where 8 2pi > > > qi log kbi > > 2pi i > 2pi > k2p > b i  zi > < i k2p 1  qi Vi ¼ ai þ log , i ¼ 1, . . . , sns > 2pi > i 2pi > k2p a i  zi > > > > > 1 > : z2i , i ¼ sns þ 1, . . . , n 2  1, zi 4 0 qi ¼ , i ¼ 1, . . . , sns 0, zi  0 ð38Þ and 2pi  n  i þ 1 to ensure that i is at least n  i times differentiable in the set Z as ¼ fzsns 2 Rsns : kai 5 zi 5 kbi , i ¼ 1, . . . , sns g ð39Þ The BLF candidate (38) is positive definite and continuously differentiable in the set z 2 Z as. We design the stabilising functions and control as 1 ðfi þ _ i1  i zi Þ þ i0 , gi u ¼ n

i ¼

i ¼ 1, . . . , n

ð40Þ

9

International Journal of Control 0 1 sns k 2pk X k2p k 1 b a V :¼ max@log 2p k 2pk , log 2pk k 2pk A k 2p k k  Zi k  Z a k k¼1 iþ b

where 0 :¼ yd, and 8 2p  1 i > > >  2p zi , > i > > > > 2p  1 > i > > zi  > > 2p i > < i1 gi1 2pi1 2pi þ1 i0 ¼  zi , > 2p > i1 i gi > > > i1 gi1 2pi1 1 > >  z , > > 2pi1 gi i > > > > > :  gi1 zi1 , gi

i¼1

k

n

1 X 2 þ max Zk , Z2k 2 k¼s þ1

i ¼ 2, . . . , sns i ¼ sns þ 1 i ¼ sns þ 2, . . . , n

Given the constraints fkcs2 , kcs3 , . . . , kcsns g, suppose that the initial state xð0Þ 2 x0 , where bi 5 kci and ai 4 kci . If there exist positive constants T      ¼ ½an , bn ,  n1 , kbsns , kasns , fkci gi2F  that satisfy the conditions

kai 4 Zi ðÞ,

Downloaded by [Keng Peng Tee] at 11:41 09 November 2011

and qi i k2p bi



i z2p i

þ

1  qi i k2p ai

i  z2p i

V_  

s ns X

i i i z2p i 

n X

i z2i

ð43Þ

Remark 8: The design of (40)–(41) is different from that in Sections 3.1 and 3.2 in that Young’s inequality _ is used to split the coupling term i gi zi2pi 1 ziþ1 in V, arising from the i-th step, into terms that can be jointly cancelled by i0 and (iþ1)0. Theorem 3: Consider the closed-loop system (1) and (24)–(26), under Assumptions 1 and 2. Let Ai ¼ Ai ¼

max

i ðx i , zi , y di Þ,

i ¼ 1, . . . , sns  1

min

i ðx i , zi , y di Þ,

i ¼ 1, . . . , sns  1

ðx sns , zsns , ydsns Þ2 ðx sns , zsns , ydsns Þ2

Zi ¼ max zi ðx i , y di ð0ÞÞ,

i ¼ 1, . . . , n

Zi ¼ min zi ðx i , y di ð0ÞÞ,

i ¼ 1, . . . , n

x2x0 x2x0

Zi

^ remain, (i) The signals zi(t), i ¼ 1, 2, . . . , n, and ðtÞ for all t > 0, in the compact sets defined by  z :¼ z 2 Rn : Dzi  zi  Dzi , i ¼ 1, 2, . . . , sns ,  pffiffiffiffiffiffiffiffiffiffiffiffio  ð47Þ zsns þ1:n k  2Vð0Þ (ii) The partial state xs(t) remains, for all t > 0, in the set xs :¼ fxs 2 Rns : kci 5 Dzi þ Ai1  xi  Dzi þ Ai1 5 kci , i ¼ s1 , s2 , . . . , sns g

i.e. the partial state constraint is never violated. (iii) All closed-loop signals are bounded. (iv) The origin z ¼ 0 is locally exponentially stable.

(i) The derivative of V along (1) and (40) satisfies V_  

where 0 is defined by (17) and  by n  :¼ x sns 2 Rsns , zsns 2 Rsns , y ds ns 2 Rsns : Dzj þ Aj1  xj  Dzj þ Aj1 , Dzj  zj  Dzj ,  o  ðjÞ  yd j  Yj , j ¼ 1, . . . , sns 1



1

Dzj :¼ kaj ð1  e2pj V Þ2pj

sns X

þ

n X

i i i z2p i 

i¼1

ð44Þ



ð48Þ

Proof: We only present the proofs of parts (i) and (iv), as parts (ii)–(iii) can be proved along the lines of Theorem 1 and are omitted.

  ¼ max 0, Zi , i ¼ 1, . . . , sns   ¼ min 0, Zi , i ¼ 1, . . . , sns

Dzj :¼ kbj ð1  e2pj V Þ2pj

ð46Þ

where A0 satisfies j yd ðtÞj  A0 5 kc1 , then the following properties hold.

i¼sns þ1

i¼1

i ¼ 1, . . . , sns

kci 4 Ai1 ðÞ þ kbi , i ¼ 1, . . . , sns kci 4 Ai1 ðÞ þ kai , i ¼ 1, . . . , sns

ð42Þ

Substituting (40)–(41) into V_ yields

Ziþ

i ¼ 1, . . . , sns

kbi 4 Zi ðÞ,

ð41Þ

i ¼

ð45Þ

ns

i z2i

i¼sns þ1

n1 X

ziþ1 ð gi zi þ giþ1 ðiþ1Þ0 Þ

i¼sns þ1

þ

sns X

i gi zi2pi 1 ðziþ1 þ i0 Þ þ gsns þ1 zsns þ1 ðsns þ1Þ0

i¼1

ð49Þ By using Young’s inequality as follows zi2pi 1 ziþ1 

2pi  1 2pi 1 2pi zi þ z 2pi 2pi iþ1

ð50Þ

10

K.P. Tee and S.S. Ge

and noting that i > 0 and gi > 0, i ¼ 1, . . . , sns , in the set zsns 2 Z as , and (gizi þ giþ1(iþ1)0) ¼ 0, we have V_  

sns X

i i i z2p i



i¼1

þ

sns X

 i gi

i¼1

n X

i z2i

i¼sns þ1

2pi  1 2pi 1 i zi þ z2p þ zi2pi 1 i0 2pi 2pi iþ1



Downloaded by [Keng Peng Tee] at 11:41 09 November 2011

i 1 i gi z2p i



Substituting i0 from (41) into the above expressoin yields V_  

s ns X

n X

i i i z2p i 

i¼1

i z2i

i k2p bi i k2p bi



i z2p i

þ ð1  qi Þ log

i  z2p i

 2pi Vð0Þ, i ¼ 1, . . . , sns n X z2i  2Vð0Þ

ð53Þ

i¼sns þ1

and thus Dzi  zi ðtÞ  Dzi , i ¼ 1, . . . , sns , and pffiffiffiffiffiffiffiffiffiffiffiffi kzsns þ1:n k  2Vð0Þ, 8t > 0. (ii) From (43) and Lemma 2, we have V_  V in the set zsns 2 Z as , where  ¼ min{2p11, . . . , 2psns sns , 2sns þ1 , . . . , 2n g. This yields V(t)  V(0)et, and thus t

 kai ð1  e2pi Vð0Þe Þ1=ð2pi Þ  zi ðtÞ t

 kbi ð1  e2pi Vð0Þe Þ1=ð2pi Þ , i ¼ 1, . . . , sns pffiffiffiffiffiffiffiffiffiffiffiffi jzi ðtÞj  2Vð0Þet=2 , i ¼ sns þ 1, . . . , n As shown in the proof of Theorem 1(iv), we have 1  e2pi Vð0Þe

t

 2pi Vð0Þet

 t

Remark 9: Large control action may result when the states approach the boundary of the constrained region, and can be viewed as a drawback of the proposed method. However, seen in another light, provision for a potentially large control effort to pull away from the boundary can be useful as an insurance against any tendency for constraint transgression caused by disturbances and uncertainties. If large control effort is not possible, then careful selection of the control parameters may suffice to limit the growth of the control signal within a desirable operating range. In fact, a straightforward extension will be to add an additional feasibility condition involving the input and its constraint: kcnþ1 4 An ¼

i k2p ai i k2p ai

 zi ðtÞ

ð52Þ

i¼sns þ1

From kbi 4 Zi , it follows that zsns ð0Þ 2 Z as . Together with the fact that V_  0 in the set zsns 2 Z as , Lemma 1 yields zsns ðtÞ 2 Z as 8t > 0. Since V(t)  V(0) 8t > 0, it follows that qi log

i

Thus, we obtain local exponential convergence of œ every zi, i ¼ 1, . . . , n, to zero.

2pi  1 i1 gi1 2pi1 2pi þ1 zi þ zi þ i0 2pi1 i gi 2pi i¼2  sns gsns 2psns 1 þ gsns þ1 zsns þ1 z þ ðsns þ1Þ0 2psns gsns þ1 sns þ1  2pi  1 1 1 zi þ 10 ð51Þ þ 1 g1 z2p 1 2pi þ

t 2p

 kbi ð2Vð0ÞÞ e 2pi , i ¼ 1, . . . , sns pffiffiffiffiffiffiffiffiffiffiffiffi t jzi ðtÞj  2Vð0Þe 2 , i ¼ sns þ 1, . . . , n

i¼sns þ1

i¼1

1

 kai ð2Vð0ÞÞ2pi e

1 2pi

þ gsns þ1 zsns þ1 ðsns þ1Þ0 sns n X X i  i i z2p  i z2i i sns X

which implies that

max

ðx sns , zsns , y ds ns Þ2

juðx, z, y dn Þj

swhere kcnþ1 is the input constraint. The new feasibility conditions, albeit conservative, can be checked in a similar manner as presented. For known systems, as long as neither the initial state set nor the desired trajectory is extremely close to the constraint boundary, the input signals are unlikely to grow large, thanks to the exponentially convergent errors. Remark 10: Asymmetric barrier functions can also be used for the treatment of uncertain plants, along the lines of tuning functions-based design (24)–(26). The control design is omitted in this article.

4. Feasibility check In this section, we address the issue of feasibility pertaining to the existence of a set of design parameters for the control such that output tracking is achieved without violating any of the state constraints, given a region of initial condition. If the states are to be constrained in a small set, such a control may not exist. The feasibility conditions are formulated as sufficient conditions (18), (33) and (46) in Theorems 1–3, respectively. They depend on the state constraints, the initial conditions and the design parameters  n1 , kbn , l . As such, if we are able to find a set of design parameters that satisfies the conditions, then the control using the design parameters is feasible.

11

Downloaded by [Keng Peng Tee] at 11:41 09 November 2011

International Journal of Control The feasibility conditions are formulated in terms of a box region of initial states ai  xi(0)  bi, i ¼ 1, . . . , n, such that all points in the region share the same design parameters, and are feasible in achieving output tracking without constraint violation. On the one hand, we want to maximise the box region to increase robustness, but note that increasing the box region increases the bounds Ai and Zi, making it more difficult to satisfy the feasibility conditions. On the other hand, we also aim to achieve good tracking performance with sufficiently large control gains i, but increasing i also increases Ai and Zi. The tradeoff between robustness and performance is formulated as a nonlinear constrained optimisation problem that is solved offline, prior to actual implementation. Recall that the full state is partitioned into free states xr ¼ ½xr1 , xr2 , . . . , xrnr T and constrained states xs ¼ ½xs1 , xs2 , . . . , xsns T . Then, the parameters kci , for i 2 F , where F is defined in (8), are no longer hard constraints imposed by the problem, but are now design constants at our disposal. Additionally, there are less conditions to satisfy, except for the special case when sns ¼ n, i.e. xn needs to be constrained. When the plant is known, we check if there exists a solution

for the optimisation problem: n n1 X X X ðbi  ai Þ þ i  kci i¼1

i¼1

i2F

ð54Þ subject to: kbi 4 Zi ðÞ, i ¼ 1, . . . , sns kci  Ai1 ðÞ þ kbi , i ¼ 1, . . . , sns  kci 5 ai 5 bi 5 kci , i ¼ 1, . . . , sns ai 5 bi , i ¼ sns þ 1, . . . , n i 4 0,

ð55Þ

i ¼ 1, . . . , n  1

where Ai() is defined in (16), and , are positive weighting constants. Note that the penalty term P  i2F kci is appended in the above objective function to limit the growth of the design constants kci , i 2 F , during the optimisation. If a solution  exists, then the proposed control (11) with  ¼  is feasible in ensuring output tracking for the system (1) with partial state constraint. When the plant is uncertain, we check if there is a solution  ¼ ½a n , bn ,  n1 , kbsns , l , fkci gi2F T

PðÞ ¼

n n1 l X X X X ðbi  ai Þ þ i þ % i  kci i¼1

i¼1

ð56Þ

i2F

i¼1

subject to the optimisation constraints described in (55), where Ai() is defined in (31). If we find a solution , then the proposed adaptive control (26) with  ¼  achieves output tracking for the system (1) with guaranteed satisfaction of the partial state constraint. When the asymmetric barrier function is used, we check if there exists a solution  ¼ ½a n , bn ,  n1 , kb , ka , fkc gi2F T s ns

sns

i

to the optimisation problem maximising cost function (54) subject to the constraints: kbi 4 Zi ðÞ, i ¼ 1, . . . , sns kai 4 Zi ðÞ, i ¼ 1, . . . , sns kci 4 Ai1 ðÞ þ kbi ,

i ¼ 1, . . . , sns

kci 4 Ai1 ðÞ þ kai , i ¼ 1, . . . , sns  kci 5 ai 5 bi 5 kci , i ¼ 1, . . . , sns ai 5 bi ,

i ¼ sns þ 1, . . . , n

i 4 0,

i ¼ 1, . . . , n  1

ð57Þ

Existence of a solution  implies that the control (40) with  ¼  is feasible in ensuring output tracking for the system (1) with partial state constraint.

 ¼ ½an , bn ,  n1 , kbsns , fkci gi2F T

maximise PðÞ ¼

to an optimisation problem that maximises the objective function

5. Simulation To illustrate the workings of the proposed BLF-based control, we present two simulation studies, one based on a 2-dimensional nonlinear system with full state constraint, and another a 3-dimensional manipulator system with partial state constraint. The 2-dimensional example is useful for illustrating phase plane trajectories and level curves of the BLF, while the 3-dimensional example is practically motivated. Furthermore, for illustration purpose, symmetric barrier functions are used for the 2-dimensional system, and an asymmetric one for the manipulator system.

5.1 2-Dimensional system Consider the system: x_ 1 ¼ 1 x21 þ x2 x_ 2 ¼ 2 x1 x2 þ 3 x1 þ ð1 þ x21 Þu

ð58Þ

where 1 ¼ 0.1, 2 ¼ 0.1 and 3 ¼ 0.2. The objective is for x1 to track desired trajectory yd, subject to full state constraint jx1 j 5 kc1 ¼ 0:8 and jx2 j 5 kc2 ¼ 2:5. For

12

K.P. Tee and S.S. Ge

2

1

1 x2

x2

3

2

0

0

−1

−1

−2

−2

−3 −1

Downloaded by [Keng Peng Tee] at 11:41 09 November 2011

QLF

BLF

3

−0.5

0 x1

0.5

1

−3 −1

−0.5

0 x1

0.5

1

Figure 1. Closed-loop state trajectories on the state space resulting from BLF-based (a) and QLF-based (b) control. Initial states are marked by ‘þ’.

simplicity, we consider the system to be known for a stabilisation task, i.e. yd  0. Comparison with a conventional QLF-based control is presented. First, for the stabilisation task, we provide a step by step description for the design of the BLF-based control given by (11), as follows: Step 1: Determine A0, Y1, upper bounds for jyd(t)j and jy_ d ðtÞj, respectively, according to Assumption 1. Since yd  0, we have A0 ¼ Y1 ¼ 0. Step 2: Compute, in terms of  ¼ ½a n , bn ,  n1 , kbn T , the following: Z21 ¼ max x21 ¼ maxða21 , b21 Þ a1 x1 b1

Z22 ¼ max ðx2 þ 0:1x21 þ 1 x1 Þ2 x2@x0

where the latter follows from the fact that z22 ðx 2 , y d2 ð0ÞÞ has a global minimum for x 2 R2, and thus the maximum value in the region x 2 x0 must be found at the boundary @x0 . Step 3: Obtain Dz1 ðÞ from (17) as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u Y k2bk  Z2k Dz1 ¼ kb1 t1  k2bk k¼1 Step 4: From (16), compute A1() as A1 ðÞ ¼ 0:1ðDz1 þ A0 Þ2 þ 1 Dz1 Step 5: Find a solution  ¼ ½a1 , b1 , a2 , b2 , 1 , kb1 , kb2 T , if it exists, for the optimisation problem described in (55). Using the Matlab routine

fmincon.m, we obtain a1 ¼ 0:798, b1 ¼ 0:798, a2 ¼ 2:417, b2 ¼ 2:341, 1 ¼ 0:0165, kb1 ¼ 0:799 and kb2 ¼ 2:420. Step 6: Select any 2 > 0 and implement the control law (10)–(11) with design parameters 1 , kb1 and kb2 , which are valid for all initial states x(0) in ai  xi  bi . i ¼ 1, 2. In this study, we let 2 ¼ 4. For comparison purpose, we design a QLF as Vq ¼ d1 z21 þ d2 z22

ð59Þ

where d1 and d2 are positive constants, and design the QLF-based control as 1 ¼ x21  1 z1 1 d1 u¼ ð2 x1 x2  3 x1  2 z2  z1  _ 1 Þ d2 1 þ x21 ð60Þ where 1, 2, d1, d2 are positive constants, chosen in this study as 1 ¼ 2, 2 ¼ 4, and d1 ¼ 0.5, d2 ¼ 0.055. Figure 1 shows the closed-loop state trajectories corresponding to different initial conditions, which are indicated by ‘þ’. The BLF-based control ensures that the state trajectories remain in the interior of the constraint region for all time, even if the trajectories start near the boundary of the constraint region. Trajectories that approach the boundary are repelled away from it. In contrast, the QLF-based control allows the state trajectories starting near the corners to violate the constraint.

13

International Journal of Control BLF

2

2

1

1

0

−1

−2

−2

−0.5

0 x1

0.5

−3 −1

1

−0.5

0 x1

0.5

1

Figure 2. Level curves of a BLF (a) and a QLF (b). The state moves along a trajectory that decreases the value of the Lyapunov function.

Stabilisation

0.8

Stabilisation

2.5 2

0.6

1.5 0.4 1 0.2

0.5 x2

x1

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0

−1

−3 −1

QLF

3

x2

x2

3

0

0 −0.5

−0.2

−1 −0.4 −1.5 −0.6

−2

−0.8

−2.5 0

0.5

1

1.5

2 Time

2.5

3

3.5

4

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Figure 3. States starting from different initial conditions in the feasible set converge to zero while confined in constrained region.

From Figure 2, we see the level curves of the BLF and QLF . The BLF is well-aligned with the state constraints and tapers steeply to infinity when approaching the boundary of the constraint region. Since the state moves along a trajectory that decreases the value of the Lyapunov function, it never escapes the constraint region. On the other hand, the QLF extends radially across the boundary, so although the Lyapunov function decreases, the state may still transgress the constraint. Figure 3 shows that asymptotic stabilisation is achieved. From 25 different initial states in the feasible set xi(0) 2 [ai, bi], i ¼ 1, 2, the states x1(t) and x2(t) converge to zero without transgressing the constraints. Figure 4 shows, for an initial time window, the control

input trajectories corresponding to the 25 initial conditions. Most of the input trajectories are modest in magnitude, but two peaked with high values as a result of the state x1 initialised very close to a constraint boundary.

5.2 Manipulator We consider a 1-link manipulator actuated by a brush dc motor. The dynamics are modelled by Mq€ þ Dq_ þ mgl sin q ¼ kI I km1 I_ þ km2 I þ km3 q_ ¼ V

ð61Þ

14

K.P. Tee and S.S. Ge Stabilisation 300 200 100

u

0 −100 −200 −300

−500

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time

Figure 4. Control input trajectories for different initial conditions. Peaks are due to the state x1 approaching very closely to the constraint boundaries.

Tracking

1.5

Tracking

1.5

1

1

0.5 x2

0.5 x1

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−400

0

0 −0.5 −0.5

−1 −1.5

−1 0

5

10 Time

15

20

0

5

10 Time

15

20

Figure 5. State trajectories x1 and x2 converge the desired trajectory and 1, respectively, and all trajectories satisfy the state constraints.

where q is the manipulator joint angle, I the motor current, V the input voltage, M the inertia, D damping coefficient, m mass, l the distance of the center of mass of the link from the joint and kmi , i ¼ 1, 2, 3, positive constants in the electrical subsystem. To be consistent with the notation of this article, the above dynamics

are rewritten in the form x_ 1 ¼ x2 x_ 2 ¼ 1 sin x1  2 x2 þ 3 x3 x_ 3 ¼ 4 x2  5 x3 þ 6 u

ð62Þ

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International Journal of Control Tracking

1.5

Tracking

1.5 1

1

0.5

0.5 z2

z1

0 0

−0.5 −0.5

−1

−1 −1.5

−1.5

0

5

10

15

−2 0

5

Time

10

15

Time

Tracking

2

Tracking

15

1.5 10

1 0.5

5

0 u

z3

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Figure 6. Error trajectories z1 and z2 converge to zero and satisfy error constraints.

−0.5

0

−1 −1.5

−5 −2 −2.5

0

5

10

15

Time

−10 0

5

10 Time

15

20

Figure 7. Error trajectories z3 converge to zero. Figure 8. Input trajectories for different initial conditions.

_ x3 ¼ I, 1 ¼ mgl/M, 2 ¼ D/M, where x1 ¼ q, x2 ¼ q, 3 ¼ kI/M, 4 ¼ km3 =km1 , 5 ¼ km2 =km1 and 6 ¼ 1=km1 . In our simulation, the parameters are m ¼ 1, l ¼ 0.15, M ¼ 1, D ¼ 1, kI ¼ 1, km1 ¼ 0:05, km2 ¼ 0:5 and km3 ¼ 10. The objective is to track qd ¼ 0.2(1  cos(t)) while satisfying the partial state _ 5  8t > 0. constraints jq(t)j < /2 and jqðtÞj Unlike the previous example, we use asymmetric barrier function-based control (40)–(41) with the design parameters determined from (54) and (57). Using the Matlab routine fmincon.m, we obtain a1 ¼ 1:5688, b1 ¼ 1:1688, a2 ¼ 0:87665, a3 ¼ 1:0011, b3 ¼ 0:99928, b2 ¼ 0:87665, 1 ¼ 0:00172, 2 ¼ 0:5006, ka1 ¼ 1:5698, kb1 ¼ 1:1698, ka2 ¼ 2:0591 and kb2 ¼ 1:7584. Figure 5 shows the trajectories of x1 and x2 starting from 125 different initial conditions. State x1 converges to the desired trajectory while state x2 to 1, and all

trajectories satisfy the constraints. Tracking performance can be clearly seen in Figures 6–7, where all error trajectories z1, z2, z3, corresponding to 125 initial conditions, converge to zero, and z1, z2 satisfy the error constraints kai 5 zi 5 kbi , i ¼ 1, 2. The control input trajectories for various initial conditions are bounded with modest peaks, as shown in Figure 8.

6. Conclusions In this article, we have employed a BLF to design a control for strict feedback systems with partial state constraint. Besides the nominal case where the plant is fully known, the presence of parametric uncertainties has also been handled. When dealing with known systems, local exponential convergence of tracking

16

K.P. Tee and S.S. Ge

errors to the origin is achieved without violation of constraint, and all closed-loop signals remain bounded, under some feasibility conditions which involve the initial states and selection of control parameters. These feasibility conditions for partial state constraints can be checked offline without precise knowledge of initial states, and are more relaxed than those involving full state constraints. To reduce conservatism of the feasibility conditions, asymmetric error bounds have been formulated and asymmetric barrier functions used to satisfy them. For uncertain systems, only asymptotic convergence has been proved.

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Acknowledgement We acknowledge partial financial support from the Basic Research Program of China (973 Program) under Grant 2011CB707005.

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