- Email: [email protected]
Tracking Control of Nonlinear Systems with Box-Constrained States
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Tracking Control of Nonlinear Systems with Box-Constrained States Keng Peng Tee ∗ Shuzhi Sam Ge ∗∗ ∗
Institute for Infocomm Research, A*STAR, Singapore 138632 (e-mail: [email protected]). ∗∗ Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117576 (e-mail: [email protected]) Abstract: We present control design for strict-feedback systems with box constraints on all of the states. A Barrier Lyapunov Function (BLF) is used to ensure that the states do not transgress the box constraints. 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, without the need for precise knowledge of the initial states, and provide an algorithm to check them offline. In the presence of parametric uncertainties, BLF-based adaptive backstepping prevents the states from transgressing the constrained region during the transient stages of online parameter adaptation. The performance of the BLF-based control is illustrated with a numerical example. Keywords: Constraints, adaptive control, Lyapunov function, barrier, nonlinear control 1. INTRODUCTION
Driven by practical needs and theoretical challenges, control of constrained systems has become an important research topic in recent decades. Violation of the constraints, which are ubiquitous in physical systems, may result in performance degradation, hazards or system damage. To handle constraints, many techniques have been developed, including set invariance methods (e.g. Liu and Michel (1994); Hu and Lin (2001)), model predictive control (e.g. Mayne et al. (2000); Allg¨ower et al. (2003)), reference governors (e.g. Gilbert and Kolmanovsky (2002); Gilbert and Ong (2009)), nonovershooting control by Krstic and Bement (2006), and coordinate transformation by Do (2010). As the literature on constrained control is rather rich, it is out of this paper’s scope to provide an exhaustive review. Recently, the use of Barrier Lyapunov Function (BLF) for 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 closed loop system, it is thus guaranteed that the limits are never transgressed. BLFs have been used to control systems in Brunovsky form by Ngo et al. (2005), output-constrained systems in strict feedback form by Tee et al. (2009b), output-constrained systems in output feedback form by Ren et al. (2009, 2010), as well as systems with box-constrained states by Tee and Ge (2009). In addition, BLF-based control has been applied to practical problems, such as the control of electromagnetic oscillators by Sane and Bernstein (2002), and electrostatic parallel plate microactuators by Tee et al. (2009a). 978-3-902661-93-7/11/$20.00 © 2011 IFAC
In this paper, we tackle the problem of control design for systems with box-constrained states. Similar to Tee and Ge (2009), we employ a barrier function for each step of backstepping design to deal with the constraints, and formulate sufficient conditions of feasibility. However, different from Tee and Ge (2009), we do not require precise knowledge of the initial state in establishing the feasibility conditions and determining the design parameters. The 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 are feasible in achieving output tracking without constraint violation. This provides robustness to uncertainty in sensing the initial state. It is shown that asymptotic output tracking is achieved without violation of any constraint when these conditions are satisfied. We provide a means of checking the conditions and determining feasible design parameters, by solving a constrained optimization problem offline, prior to actual operation of the control. 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.
2. PROBLEM FORMULATION AND PRELIMINARIES Throughout this paper, we denote by R+ the set of nonnegative real numbers, k • k the Euclidean vector norm in Rm . The symbols λmax (•) and λmin (•) denote the maximum and minimum eigenvalues of •, respectively. We also denote x ¯i = [x1 , x2 , ..., xi ]T , z¯i = [z1 , z2 , ..., zi ]T , and (1) (2) (i) y¯di = [yd , yd , yd , ..., yd ]T for positive integers i, j.
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10.3182/20110828-6-IT-1002.03439
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Consider the following nonlinear plant in strict feedback form: x˙ i = fi (¯ xi ) + gi (¯ xi )xi+1 ,
i = 1, 2, ..., n − 1
x˙ n = fn (¯ xn ) + gn (¯ xn )u y = x1 (1) where f1 , ..., fn , g1 , ..., gn are smooth functions, x1 , ..., xn are the states, u and y are the input and output respectively. The plant is subjected to box constraints on all states, that is, every state xi is required to remain in the set |xi | ≤ kci , with kci as a positive constant, for i = 1, ..., n. The nonlinear functions fi (¯ xi ) may be uncertain, in which case they satisfy the following linear-in-the-parameters (LIP) condition: fi (¯ xi ) = θT ψi (¯ xi ), i = 1, ..., n (2) l where ψ1 , ..., ψn are smooth functions, and θ ∈ R is a vector of uncertain parameters satisfying kθk ≤ θM with known positive constant θM . Due to smoothness property, there exist positive constants Ψi such that kψi (¯ xi )k ≤ Ψi for |xi | ≤ kci , i = 1, 2, ..., n. 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 , Y1 , Y2 ,..., Yn such that the desired trajectory yd (t) and its time derivatives satisfy
Let V (η) := inequality
Pn
i=1
Vi (zi ) + U (w), and z(0) ∈ Z. If the
∂V V˙ = h ≤ 0 ∂η holds in the set z ∈ Z, then z(t) ∈ Z ∀t ∈ [0, ∞). 3. BLF-BASED CONTROL DESIGN
In this section, we consider the case when the functions fi (¯ xi ) in the plant are known. We use a barrier function in each step of backstepping design in order to keep each error signal zi = xi − αi−1 (i = 2, ..., n) constrained. In order to ensure that xi never transgresses the constrained region, feasibility conditions related to the design parameters and an initial state region are formulated. Different from Tee and Ge (2009), the new feasibility conditions in this paper do not require precise knowledge of the initial state x(0). Denote z1 = x1 − yd and zi = xi − αi−1 , i = 2, ..., n. Consider the BLF candidate: V =
n X
Vi ,
Vi =
i=1
k2 1 log 2 bi 2 , 2 kbi − z i
i = 1, ..., n
< Yi , i = 1, ..., n (3) |yd (t)| ≤ A0 < kc1 , for all t ≥ 0. 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. A Barrier Lyapunov Function (BLF) candidate is continuously differentiable, positive definite function V (z), defined on an open region D containing the origin, and is proper on D, that is, limz→∂D− V (z) = +∞. The following lemma formalizes a result on the use of a BLF candidate for constraint satisfaction. Lemma 1. Tee and Ge (2009) For any positive constant kb1 , let Z := {z ∈ Rn : |zi | < kbi , i = 1, 2, ..., n} ⊂ Rn and N := Rl × Z ⊂ Rn+l be open sets. Consider the system η˙ = h(t, η) (4) T where η := [w, z] ∈ N is the state, and the function h : R+ × N → Rn+l is piecewise continuous in t and locally Lipschitz in η, uniformly in t, on R+ × N . Let Zi := {zi ∈ R : |zi | < kbi } ⊂ R. Suppose that there exist positive definite functions U : Rl → R+ and Vi : Zi → R+ (i = 1, ..., n), both of which are also continuously differentiable on Rl and Zi respectively, such that Vi (zi ) → ∞ as zi → ±kbi
(5)
(7)
where kbi , i = 1, ..., n, are positive constants to be determined from subsequent feasibility analysis. Design the stabilizing functions and control law as α1 =
1 (−f1 − κ1 z1 + y˙ d ) g1
αi =
k 2 − zi2 1 (−fi + α˙ i−1 − κi zi − 2 bi 2 gi−1 zi−1 ) gi kbi−1 − zi−1
(8)
i = 2, ..., n (i) |yd (t)|
(6)
(9)
u = αn (10) where κi , i = 1, ..., n, are positive constants. This yields the closed loop system z˙1 = −κ1 z1 + g1 z2 k 2 − zi2 z˙i = −κi zi − 2 bi 2 gi−1 zi−1 + gi zi+1 , kbi−1 − zi−1 i = 2, ..., n − 1 z˙n = −κn zn −
kb2n kb2n−1
− zn2 gn−1 zn−1 2 − zn−1
(11)
The time derivative of V along (11) can be rewritten as V˙ = −
n X
κj zj2 − zj2
k2 j=1 bj
(12)
Let the closed loop system (11) be written as z˙ = h(t, z) (13) where h(t, z) is piecewise continuous in t and locally Lipschitz in z, uniformly in t, in the set z ∈ Z, defined by (14) Z := {z ∈ Rn : |zi | < kbi , i = 1, 2, ..., n} ˙ Then, together with the fact that V ≤ 0 in the set z ∈ Z, and that z(0) ∈ Z, we invoke Lemma 1 to obtain that z(t) ∈ Z for all t > 0.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
To show that |x2 (t)| ≤ kc2 , we need to first verify that there exists a positive constant A1 such that |α1 (t)| ≤ A1 , ∀ t > 0. Since |x1 (t)| ≤ Dz1 + A0 , |z1 (t)| ≤ Dz1 , and |y˙ d (t)| ≤ Y1 , it is clear that (x1 (t), z1 (t), y¯d1 (t)) ∈ Ω1 , and thus, the stabilizing function α1 (x1 , z1 , y¯d1 ) in (8) is bounded since it is a continuous function. As a result, A1 exists. Then, from |z2 (t)| ≤ Dz2 < kb2 , we can show that |x2 (t)| ≤ Dz2 + |α1 (t)| < kb2 + |α1 (t)|. Since |α1 (t)| ≤ A1 , we conclude that |x2 (t)| ≤ Dz2 + A1 < kb2 + A1 < kc2 , ∀ t > 0. We can progressively show that |xi+1 (t)| ≤ kci+1 , i = 2, ..., n−1, after verifying that there exist positive constants Ai such that |αi (t)| ≤ Ai , ∀ t > 0. Since (i) |xi (t)| ≤ Dzi + Ai−1 , |zi (t)| ≤ Dzi , and |yd (t)| ≤ Yi , it is clear that (¯ xi (t), z¯i (t), y¯di (t)) ∈ Ωi , and thus, the stabilizing function αi (¯ xi , z¯i , y¯di ) in (8) is bounded since it is a smooth function. As a result, we have that Ai exists. Then, from |zi+1 (t)| ≤ Dzi+1 < kbi+1 , we can show that |xi+1 (t)| ≤ Dzi+1 + |αi (t)| < kbi+1 + |αi (t)|. Since |αi (t)| ≤ Ai , we conclude that |xi+1 (t)| ≤ Dzi+1 + Ai < kbi+1 + Ai < kci+1 , ∀ t > 0. iii) By inspection of the stabilizing functions αi (¯ xi , z¯i , y¯di ) and control u(¯ xn , z¯n , y¯dn ) , it is clear that they are bounded, by virtue of the boundedness of x ¯n (t), z¯n (t), y¯dn (t), and, in particular, by |zi (t)| ≤ Dzi < kbi , which prevents any term comprising (kb2i − zi2 (t)) in the denominator from becoming unbounded. iv) Since V˙ (z) in (12) is negative definite, and V is proper in the set z ∈ Z, then, by the LaSalle-Yoshizawa Theorem, we have, for z(0) ∈ Z, that z(t) → 0 as t → ∞.
Theorem 1. Consider the closed loop system (1) and (10) under Assumptions 1-2. Let Ai =
max
(x,z,¯ ydn )∈Ω
i = 1, ..., n − 1
|αi (¯ xi , z¯i , y¯di )|,
xi , y¯di (0))| Zi = max |zi (¯
(15)
x∈Ωx0
where Ω is a compact set defined by: Ω := {x ∈ Rn , z ∈ Rn , y¯dn ∈ Rn+1 : (j)
|xj | ≤ Dzj + Aj−1 , |zj | ≤ Dzj , |yd | ≤ Yj ,
Dzj
j = 1, ..., n} v u n Y u kb2k − Zk2 t := kbj 1 − kb2k
(16) (17)
k=1
Ωx0 := {x ∈ Rn : aj ≤ xj ≤ bj , j = 1, ..., n}
(18)
Given the constraints kci > 0, i = 1, ..., n, suppose that the initial state x(0) ∈ Ωx0 , where bi < kci and ai > −kci . If there exist positive constants ξ = [a1 , b1 , ..., an , bn , κ1 , ..., κn−1 , kb1 , ..., kbn ]T that satisfy the conditions: kbi > Zi (ξ),
i = 1, ..., n i = 1, ..., n
kci > Ai−1 (ξ) + kbi ,
(19)
where A0 satisfies |yd (t)| ≤ A0 < kc1 , then the following properties hold. i) The signals zi (t), i = 1, 2, ..., n, remain in the compact set defined by Ωz = {¯ zn ∈ Rn : |zi | ≤ Dzi , i = 1, 2, ..., n}. ii) Every state xi (t) remains in the set Ωx := {x ∈ Rn : |xi | ≤ Dzi + Ai−1 < kci , i = 1, ..., n} ∀t > 0, i.e. the full state constraint is never violated. iii) All closed loop signals are bounded. iv) The origin z = 0 is asymptotically stable. Proof: i) From (19), kbi > Zi implies that z(0) ∈ Z, since |zj (0)| ≤ Zj for x(0) ∈ Ω0 . Thus, Lemma 1 yields z(t) ∈ Z ∀ t > 0. Then, from (12), we have V (t) ≤ V (0), which implies that n X kb2j k2 1 1 log 2 bi 2 ≤ log 2 2 kbi − zi 2 kbj − zj2 (0) j=1
≤
n X 1 j=1
2
log
kb2j
kb2j − Zj2
for i = 1, ..., n. Using the identity log a+log b = log ab, we rewrite (20) into the following log
kb2i kb2i − zi2
≤ log
n Y
j=1
kb2j
Remark 1. Unlike the approach of Tee and Ge (2009), the new feasibility conditions (19) in this paper 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. 4. ADAPTIVE CONTROL DESIGN When the nonlinearities fi (¯ xi ) are uncertain, but can be linearly parameterized according to (2), the foregoing design methodology can be modified, based on the certainty equivalence approach, i.e. replacing instances of θT ψi (¯ xi ) in the controls with their estimates θˆT ψi (¯ xi ), followed by the design of the adaptation law for θˆ that guarantees closed loop stability. We adopt the tuning functions approach Krstic et al. (1995) for stable design of an adaptation law. Denote z1 = x1 − yd and zi = xi − αi−1 , i = 2, ..., n. Consider the BLF candidate:
kb2j − Zj2
for i = 1, ..., n. Then, the above can be rearranged to yield |zi (t)| ≤ Dzi ∀ t > 0. ii) Since |z1 (t)| ≤ Dz1 < kc1 − A0 , we can show that |x1 (t)| ≤ Dz1 + |yd (t)| < kc1 − A0 + |yd (t)|. Noting that |yd (t)| ≤ A0 from Assumption 1, we therefore conclude that |x1 (t)| ≤ Dz1 + A0 < kc1 , ∀ t > 0. 6717
V =
n X
Vi + U
i=1
1 U = θ˜T Γ−1 θ˜ 2 k2 1 Vi = log 2 bi 2 , 2 kbi − zi
i = 1, ..., n
(20)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where Γ is a positive diagonal matrix, θ˜ := θˆ − θ, and , and kbi , i = 1, ..., n, are positive constants to be determined from subsequent feasibility analysis. The adaptive backstepping control is designed as:
Let the closed loop system (22) be written as η˙ = h(t, η), where η = [z T , θ˜T ]T . By inspection, h(t, η) is piecewise continuous in t and locally Lipschitz in η, uniformly in t, in the set η ∈ N , defined by
α1 = −θˆT w1 − κ1 z1 + y˙ d à k 2 − z22 ∂α1 1 g1 x2 g1 z1 + −θˆT w2 − κ2 z2 − b22 α2 = 2 g2 kb1 − z1 ∂x1 1 X ∂α1 (j+1) ∂α1 + y + Γτ2 (j) d ∂ θˆ ∂y
N := {z ∈ Rn , θ˜ ∈ Rl : |zi | < kbi , i = 1, 2, ..., n} (24) Then, together with the fact that V˙ ≤ 0 in the set z ∈ Z, and that z(0) ∈ Z, we invoke Lemma 1 to obtain that z(t) ∈ Z for all t > 0. Theorem 2. Consider the closed loop system (1) and (21) under Assumptions 1 and 2. Let
j=0
αi =
1 gi
Ã
d
−θˆT wi − κi zi −
i−1 X ∂αi−1
+
xj
j=1
+
i−1 X j=2
kb2i 2 kbi−1
gj xj+1 +
ˆ (x,z,¯ ydn ,θ)∈Ω
− zi2 2 gi−1 zi−1 − zi−1
i−1 X ∂αi−1 (j)
j=0
Ai = max
∂yd
(j+1)
yd
+
¯ ¯ ¯ ¯ ˆ xi , y¯di (0), θ(0)) Zi = max ¯zi (¯ ¯, x∈Ωx0
ˆ ≤ D ˆ, |xj | ≤ Dzj + Aj−1 , |zj | ≤ Dzj , kθk θ o (j) |yd | ≤ Yj , j = 1, ..., n v u n Y u kb2k − Zk2 Dzj := kbj t1 − ¯ kb2k e2Vθˆ k=1 s 2V¯ Dθˆ := θM + λmin (Γ−1 ) 1 ˆ + θM )2 V¯θˆ := λmax (Γ−1 )(kθ(0)k 2 n k2 1X ¯ V := log 2 bi 2 + V¯θˆ 2 i=1 kbi − Zi
w1 = ψ1 (x1 )
wi = ψi (¯ xi ) −
j=1
∂xj
ψj (¯ xj ),
i = 2, ..., n
wi zi w1 z1 , τi = τi−1 + 2 τ1 = 2 2 kb1 − z1 kbi − zi2 u = αn ˙ θˆ = Γτn which yields the closed loop system
(21)
z˙1 = −κ1 z1 + g1 z2 − θ˜T ψ1 (x1 ) k 2 − z22 z˙2 = −κ2 z2 − b22 g1 z1 + g2 z3 − θ˜T w2 kb1 − z12 ∂α1 ˆ˙ + (Γτ2 − θ) ∂ θˆ k 2 − zi2 ˜T z˙i = −κi zi − 2 bi 2 gi−1 zi−1 + gi zi+1 − θ wi kbi−1 − zi−1 +
∂αi−1 ˆ˙ + (Γτi − θ) ˆ ∂θ
z˙n = −κn zn − +
i−1 X
k2 j=2 bj
∂αn−1 ˆ˙ + (Γτn − θ) ˆ ∂θ
˙ θ˜ = Γτn
j=2
kbi > Zi (ξ),
∂αj−1 zj Γwi , 2 − zj ∂ θˆ
j=1
(28) (29) (30)
i = 1, ..., n
Ωz = {z ∈ Rn : |zi | ≤ Dzi , i = 1, 2, ..., n} n o ˆ ≤ Dˆ Ωθˆ = θˆ ∈ Rl : kθk θ
(32)
ii) Every state xi (t) remains in the set Ωx := {x ∈ Rn : |xi | ≤ Dzi + Ai−1 < kci , i = 1, ..., n} ∀t > 0, i.e. the full state constraint is never violated. iii) All closed loop signals are bounded. iv) The origin z = 0 is asymptotically stable.
zj ∂αj−1 Γwn kb2j − zj2 ∂ θˆ
The derivative of V along (22) can be written as: V˙ = −
(27)
ˆ i) The signals zi (t) and θ(t), i = 1, 2, ..., n, remain, for all t > 0, in the compact sets defined by
i = 3, ..., n − 1
κj zj2 kb2j − zj2
(26)
(31) kci > Ai−1 (ξ) + kbi , i = 1, ..., n where A0 satisfies |yd (t)| ≤ A0 < kc1 , then the following properties hold.
(22)
n X
(25)
Given the constraints kci > 0, i = 1, ..., n, suppose that the initial state x(0) ∈ Ωx0 , where bi < kci and ai > −kci . If there exist positive constants ξ = [a1 , b1 , ..., an , bn , κ1 , ..., κn−1 , kb1 , ..., kbn , γ1 , ..., γl ]T that satisfy the conditions:
kb2n − zn2 gn−1 zn−1 − θ˜T wn 2 2 kbn−1 − zn−1 n−1 X
i = 1, ..., n
where Ω is a compact set defined by: n Ω := x ∈ Rn , z ∈ Rn , y¯dn ∈ Rn+1 , θˆ ∈ Rl :
∂αi−1 Γτi ∂ θˆ
∂αj−1 zj Γwi , i = 3, ..., n kb2j − zj2 ∂ θˆ i−1 X ∂αi−1
ˆ i = 1, ..., n − 1 |αi (¯ xi , z¯i , y¯di , θ)|,
Proof: (23)
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i) Since kθk ≤ θM and |zi (0)| ≤ Zi , i = 1, ..., n, it can be shown that
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
V (0) ≤ V¯ (33) Since Zi < kbi , i = 1, ..., n, Lemma 1 yields z(t) ∈ Z ∀ t > 0. Then, from (23), we have V (t) ≤ V (0) ≤ V¯ , which yields s 2V¯ ˆ kθ(t)k ≤ θM + (34) λmin (Γ−1 )
with sufficiently large control gains κi , but increasing κi also increases Ai and Zi . The tradeoff between robustness and performance is formulated as a nonlinear constrained optimization problem that is solved offline, prior to actual implementation.
Therefore, θˆ remains in the compact set Ωθˆ ∀ t > 0. Furthermore, we have
ξ := [a1 , b1 , ..., an , bn , kb1 , ..., kbn , κ1 , ..., κn−1 ]T
For known plant, we check if there exists a solution
for the optimization problem:
2V¯θˆ
log
n Y kb2j e kb2i ≤ log kb2i − zi2 (t) k 2 − zj2 (0) j=1 bj
≤ log
¯ n Y kb2j e2Vθˆ
k2 j=1 bj
−
, Zj2
maximize P (ξ) =
n X
(bi − ai ) + ρ
i=1
subject to:
i = 1, ..., n
kbi > Zi (ξ),
which can be written as |zi (t)| ≤ Dzi < kbi , ∀ t > 0. ii) The proof follows the a similar line of argument as that in Theorem 1, and is omitted. iii) By inspection of the stabilizing functions αi (¯ xi , z¯i , ˆ , it is clear that ˆ and control u(¯ xn , z¯n , y¯dn , θ) y¯di , θ) they are bounded, by virtue of the boundedness ˆ and, in particular, by of x ¯n (t), z¯n (t), y¯dn (t), θ(t), |zi (t)| < kbi , which prevents any term comprising (kb2i − zi2 ) in the denominator from becoming unbounded. iv) Since V˙ (z) in (23) is negative semidefinite, and V is proper in the set z ∈ Z, then, by the LaSalleYoshizawa Theorem, we have, for z(0) ∈ Z, that lim
n X
κj zj (t)2 =0 − zj (t)2
κi
i=1
i = 1, ..., n
kci ≥ Ai−1 (ξ) + kbi , −kci < ai < bi < kci , κi > 0,
n−1 X
i = 1, ..., n i = 1, ..., n
i = 1, ..., n − 1
(35)
where ρ is a positive weighting constant, Ai , Zi are defined in (15). If a solution ξ ∗ to the above optimization problem exists, then the feasibility conditions (19) in Theorem 1 are satisfied, and the proposed control (10) with ξ = ξ ∗ is feasible in ensuring output tracking for (1) while respecting the full state constraint for all time. When the plant is uncertain, we check if there exists a solution ξ := [a1 , b1 , ..., an , bn , κ1 , ..., κn−1 , kb1 , ..., kbn , γ1 , ..., γl ]T
t→∞ k2 j=1 bj
for an optimization problem that maximizes the objective function
and thus, z(t) → 0 as t → ∞.
P (ξ) =
5. FEASIBILITY CHECK AND PARAMETER SELECTION
n n−1 l X X X (bi − ai ) + ρ κi + ̺ λi i=1
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. The feasibility conditions are formulated as sufficient conditions (19) and (31), which depend on the state constraints, the initial conditions and the design parameters κ1 , ..., κn−1 , kb1 , ..., kbn , γ1 , ..., γn , where γ1 , ..., γl are the elements of the diagonal matrix Γ. As such, if we are able to find a set of design parameters that satisfies (19) for known plant, and (31) for uncertain plant, then the control using the design parameters is feasible.
i=1
(36)
i=1
subject to (35), where ρ, ̺ are positive weighting constants, and Ai , Zi are defined in (25). If a solution ξ ∗ is found, then the proposed adaptive control (21)-(21), with ξ = ξ ∗ is feasible in ensuring output tracking with full state constraint, according to Theorem 2. 6. SIMULATION We present a simulation study on the full state constraint problem for a nonlinear plant: x˙ 1 = θ1 x21 + x2 x˙ 2 = θ2 x1 x2 + θ3 x1 + (1 + x21 )u
Unlike the approach of Tee and Ge (2009), the feasibility conditions are formulated in terms of a box region of initial states ai ≤ xi (0) ≤ bi , i = 1, ..., n, such that all points in the region share the same design parameters, and are feasible in achieving output tracking without constraint violation.
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. For simplicity, consider the plant to be known, and that yd ≡ 0, i.e., a stabilization task.
On the one hand, we want to maximize 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
Denote ξ = [a1 , b1 , a2 , b2 , κ1 , kb1 , kb2 ]T . We need to find a solution ξ ∗ = [a∗1 , b∗1 , a∗2 , b∗2 , κ∗1 , kb∗1 , kb∗2 ]T , if it exists, for the optimization problem (35). Using the Matlab routine fmincon.m, we obtain a∗1 = −0.798, b∗1 = 0.798, a∗2 = −2.417, b∗2 = 2.341, κ∗1 = 0.0165, kb∗1 = 0.799, and
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
kb∗2 = 2.420. Next, we implement the control law (8)-(10) with design parameters κ1 = κ∗1 , kb1 = kb∗1 , kb2 = kb∗2 , and κ2 = 4, which are valid for all initial states x(0) in a∗i ≤ xi ≤ b∗i , i = 1, 2.
BLF 3
2
Figure 1 shows the closed loop state trajectories corresponding to different initial conditions, which are indicated by ‘+’. The BLF-based control ensures that the state trajectories remain in the interior of the constraint region for all time, and converge to the origin, even if the trajectories start near the boundary of the constraint region.
x
2
1
−1
Figure 2 shows the level curves of the BLF. 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. 7. CONCLUSIONS
0
−2
−3 −1
−0.5
0 x1
0.5
1
Fig. 2. Level curves of a BLF. The state moves along a trajectory that decreases the value of the Lyapunov function.
In this paper, we have employed a Barrier Lyapunov Function to design a control for strict feedback systems with state constraint. Besides the nominal case where the plant is fully known, the presence of parametric uncertainties has also been handled. Asymptotic tracking 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 can be checked offline without precise knowledge of initial states. BLF 3
2
x
2
1
0
−1
−2
−3 −1
−0.5
0 x
0.5
1
1
Fig. 1. Phase plot of closed loop system resulting from BLF-based control.
REFERENCES Allg¨ ower, F., Findeisen, R., and Ebenbauer, C. (2003). Nonlinear model predictive control. Encyclopedia of Life Support Systems (EOLSS) article contribution 6.43.16.2. Do, K.D. (2010). Control of nonlinear systems with output tracking error constraints and its application to magnetic bearings. International Journal of Control, 83(6), 1199–1216. Gilbert, E.G. and Kolmanovsky, I. (2002). Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor. Automatica, 38, 2063–2073.
Gilbert, E.G. and Ong, C.J. (2009). An extended command governor for constrained linear systems with disturbances. In Proc. 48th IEEE Conf. Decision & Control, 6929–6934. Shanghai, China. Hu, T. and Lin, Z. (2001). Control Systems With Actuator Saturation: Analysis and Design. Birkhuser, Boston, MA. Krstic, M. and Bement, M. (2006). Nonovershooting control of strictfeedback nonlinear systems. IEEE Trans. Automatic Control, 51(12), 1938–1943. Krstic, M., Kanellakopoulos, I., and Kokotovic, P.V. (1995). Nonlinear and Adaptive Control Design. New York: Wiley and Sons. Liu, D. and Michel, A.N. (1994). Dynamical Systems with Saturation Nonlinearities. Springer-Verlag, London, U.K. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O.M. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36, 789–814. Ngo, K.B., Mahony, R., and Jiang, Z.P. (2005). Integrator backstepping using barrier functions for systems with multiple state constraints. In Proc. 44th IEEE Conf. Decision & Control, 8306– 8312. Seville, Spain. Ren, B., Ge, S.S., Tee, K.P., and Lee, T.H. (2009). Adaptive control for parametric output feedback systems with output constraint. In Proc. 48th IEEE Conf. Decision & Control, 6650–6655. Shanghai, China. Ren, B., Ge, S.S., Tee, K.P., and Lee, T.H. (2010). Adaptive neural control for output feedback nonlinear systems using a barrier lyapunov function. IEEE Trans. Neural Networks, 21(8), 1339– 1345. Sane, H.S. and Bernstein, D.S. (2002). Robust nonlinear control of the electromagnetically controlled oscillator. In Proc. American Control Conference, 809–814. Anchorage, AK. Tee, K.P. and Ge, S.S. (2009). Control of nonlinear systems with full state constraint using a barrier Lyapunov function. In Proc. 48th IEEE Conference on Decision & Control, 8618–8623. Shanghai, China. Tee, K.P., Ge, S.S., and Tay, E.H. (2009a). Adaptive control of electrostatic microactuators with bidirectional drive. IEEE Trans. Control Systems Technology, 17(2), 340–352. Tee, K.P., Ge, S.S., and Tay, E.H. (2009b). Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica, 45(4), 918–927.
6720
Tracking Control of Nonlinear Systems with Box-Constrained States Keng Peng Tee ∗ Shuzhi Sam Ge ∗∗ ∗
Institute for Infocomm Research, A*STAR, Singapore 138632 (e-mail: [email protected]). ∗∗ Department of Electrical & Computer Engineering, National University of Singapore, Singapore 117576 (e-mail: [email protected]) Abstract: We present control design for strict-feedback systems with box constraints on all of the states. A Barrier Lyapunov Function (BLF) is used to ensure that the states do not transgress the box constraints. 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, without the need for precise knowledge of the initial states, and provide an algorithm to check them offline. In the presence of parametric uncertainties, BLF-based adaptive backstepping prevents the states from transgressing the constrained region during the transient stages of online parameter adaptation. The performance of the BLF-based control is illustrated with a numerical example. Keywords: Constraints, adaptive control, Lyapunov function, barrier, nonlinear control 1. INTRODUCTION
Driven by practical needs and theoretical challenges, control of constrained systems has become an important research topic in recent decades. Violation of the constraints, which are ubiquitous in physical systems, may result in performance degradation, hazards or system damage. To handle constraints, many techniques have been developed, including set invariance methods (e.g. Liu and Michel (1994); Hu and Lin (2001)), model predictive control (e.g. Mayne et al. (2000); Allg¨ower et al. (2003)), reference governors (e.g. Gilbert and Kolmanovsky (2002); Gilbert and Ong (2009)), nonovershooting control by Krstic and Bement (2006), and coordinate transformation by Do (2010). As the literature on constrained control is rather rich, it is out of this paper’s scope to provide an exhaustive review. Recently, the use of Barrier Lyapunov Function (BLF) for 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 closed loop system, it is thus guaranteed that the limits are never transgressed. BLFs have been used to control systems in Brunovsky form by Ngo et al. (2005), output-constrained systems in strict feedback form by Tee et al. (2009b), output-constrained systems in output feedback form by Ren et al. (2009, 2010), as well as systems with box-constrained states by Tee and Ge (2009). In addition, BLF-based control has been applied to practical problems, such as the control of electromagnetic oscillators by Sane and Bernstein (2002), and electrostatic parallel plate microactuators by Tee et al. (2009a). 978-3-902661-93-7/11/$20.00 © 2011 IFAC
In this paper, we tackle the problem of control design for systems with box-constrained states. Similar to Tee and Ge (2009), we employ a barrier function for each step of backstepping design to deal with the constraints, and formulate sufficient conditions of feasibility. However, different from Tee and Ge (2009), we do not require precise knowledge of the initial state in establishing the feasibility conditions and determining the design parameters. The 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 are feasible in achieving output tracking without constraint violation. This provides robustness to uncertainty in sensing the initial state. It is shown that asymptotic output tracking is achieved without violation of any constraint when these conditions are satisfied. We provide a means of checking the conditions and determining feasible design parameters, by solving a constrained optimization problem offline, prior to actual operation of the control. 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.
2. PROBLEM FORMULATION AND PRELIMINARIES Throughout this paper, we denote by R+ the set of nonnegative real numbers, k • k the Euclidean vector norm in Rm . The symbols λmax (•) and λmin (•) denote the maximum and minimum eigenvalues of •, respectively. We also denote x ¯i = [x1 , x2 , ..., xi ]T , z¯i = [z1 , z2 , ..., zi ]T , and (1) (2) (i) y¯di = [yd , yd , yd , ..., yd ]T for positive integers i, j.
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10.3182/20110828-6-IT-1002.03439
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Consider the following nonlinear plant in strict feedback form: x˙ i = fi (¯ xi ) + gi (¯ xi )xi+1 ,
i = 1, 2, ..., n − 1
x˙ n = fn (¯ xn ) + gn (¯ xn )u y = x1 (1) where f1 , ..., fn , g1 , ..., gn are smooth functions, x1 , ..., xn are the states, u and y are the input and output respectively. The plant is subjected to box constraints on all states, that is, every state xi is required to remain in the set |xi | ≤ kci , with kci as a positive constant, for i = 1, ..., n. The nonlinear functions fi (¯ xi ) may be uncertain, in which case they satisfy the following linear-in-the-parameters (LIP) condition: fi (¯ xi ) = θT ψi (¯ xi ), i = 1, ..., n (2) l where ψ1 , ..., ψn are smooth functions, and θ ∈ R is a vector of uncertain parameters satisfying kθk ≤ θM with known positive constant θM . Due to smoothness property, there exist positive constants Ψi such that kψi (¯ xi )k ≤ Ψi for |xi | ≤ kci , i = 1, 2, ..., n. 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 , Y1 , Y2 ,..., Yn such that the desired trajectory yd (t) and its time derivatives satisfy
Let V (η) := inequality
Pn
i=1
Vi (zi ) + U (w), and z(0) ∈ Z. If the
∂V V˙ = h ≤ 0 ∂η holds in the set z ∈ Z, then z(t) ∈ Z ∀t ∈ [0, ∞). 3. BLF-BASED CONTROL DESIGN
In this section, we consider the case when the functions fi (¯ xi ) in the plant are known. We use a barrier function in each step of backstepping design in order to keep each error signal zi = xi − αi−1 (i = 2, ..., n) constrained. In order to ensure that xi never transgresses the constrained region, feasibility conditions related to the design parameters and an initial state region are formulated. Different from Tee and Ge (2009), the new feasibility conditions in this paper do not require precise knowledge of the initial state x(0). Denote z1 = x1 − yd and zi = xi − αi−1 , i = 2, ..., n. Consider the BLF candidate: V =
n X
Vi ,
Vi =
i=1
k2 1 log 2 bi 2 , 2 kbi − z i
i = 1, ..., n
< Yi , i = 1, ..., n (3) |yd (t)| ≤ A0 < kc1 , for all t ≥ 0. 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. A Barrier Lyapunov Function (BLF) candidate is continuously differentiable, positive definite function V (z), defined on an open region D containing the origin, and is proper on D, that is, limz→∂D− V (z) = +∞. The following lemma formalizes a result on the use of a BLF candidate for constraint satisfaction. Lemma 1. Tee and Ge (2009) For any positive constant kb1 , let Z := {z ∈ Rn : |zi | < kbi , i = 1, 2, ..., n} ⊂ Rn and N := Rl × Z ⊂ Rn+l be open sets. Consider the system η˙ = h(t, η) (4) T where η := [w, z] ∈ N is the state, and the function h : R+ × N → Rn+l is piecewise continuous in t and locally Lipschitz in η, uniformly in t, on R+ × N . Let Zi := {zi ∈ R : |zi | < kbi } ⊂ R. Suppose that there exist positive definite functions U : Rl → R+ and Vi : Zi → R+ (i = 1, ..., n), both of which are also continuously differentiable on Rl and Zi respectively, such that Vi (zi ) → ∞ as zi → ±kbi
(5)
(7)
where kbi , i = 1, ..., n, are positive constants to be determined from subsequent feasibility analysis. Design the stabilizing functions and control law as α1 =
1 (−f1 − κ1 z1 + y˙ d ) g1
αi =
k 2 − zi2 1 (−fi + α˙ i−1 − κi zi − 2 bi 2 gi−1 zi−1 ) gi kbi−1 − zi−1
(8)
i = 2, ..., n (i) |yd (t)|
(6)
(9)
u = αn (10) where κi , i = 1, ..., n, are positive constants. This yields the closed loop system z˙1 = −κ1 z1 + g1 z2 k 2 − zi2 z˙i = −κi zi − 2 bi 2 gi−1 zi−1 + gi zi+1 , kbi−1 − zi−1 i = 2, ..., n − 1 z˙n = −κn zn −
kb2n kb2n−1
− zn2 gn−1 zn−1 2 − zn−1
(11)
The time derivative of V along (11) can be rewritten as V˙ = −
n X
κj zj2 − zj2
k2 j=1 bj
(12)
Let the closed loop system (11) be written as z˙ = h(t, z) (13) where h(t, z) is piecewise continuous in t and locally Lipschitz in z, uniformly in t, in the set z ∈ Z, defined by (14) Z := {z ∈ Rn : |zi | < kbi , i = 1, 2, ..., n} ˙ Then, together with the fact that V ≤ 0 in the set z ∈ Z, and that z(0) ∈ Z, we invoke Lemma 1 to obtain that z(t) ∈ Z for all t > 0.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
To show that |x2 (t)| ≤ kc2 , we need to first verify that there exists a positive constant A1 such that |α1 (t)| ≤ A1 , ∀ t > 0. Since |x1 (t)| ≤ Dz1 + A0 , |z1 (t)| ≤ Dz1 , and |y˙ d (t)| ≤ Y1 , it is clear that (x1 (t), z1 (t), y¯d1 (t)) ∈ Ω1 , and thus, the stabilizing function α1 (x1 , z1 , y¯d1 ) in (8) is bounded since it is a continuous function. As a result, A1 exists. Then, from |z2 (t)| ≤ Dz2 < kb2 , we can show that |x2 (t)| ≤ Dz2 + |α1 (t)| < kb2 + |α1 (t)|. Since |α1 (t)| ≤ A1 , we conclude that |x2 (t)| ≤ Dz2 + A1 < kb2 + A1 < kc2 , ∀ t > 0. We can progressively show that |xi+1 (t)| ≤ kci+1 , i = 2, ..., n−1, after verifying that there exist positive constants Ai such that |αi (t)| ≤ Ai , ∀ t > 0. Since (i) |xi (t)| ≤ Dzi + Ai−1 , |zi (t)| ≤ Dzi , and |yd (t)| ≤ Yi , it is clear that (¯ xi (t), z¯i (t), y¯di (t)) ∈ Ωi , and thus, the stabilizing function αi (¯ xi , z¯i , y¯di ) in (8) is bounded since it is a smooth function. As a result, we have that Ai exists. Then, from |zi+1 (t)| ≤ Dzi+1 < kbi+1 , we can show that |xi+1 (t)| ≤ Dzi+1 + |αi (t)| < kbi+1 + |αi (t)|. Since |αi (t)| ≤ Ai , we conclude that |xi+1 (t)| ≤ Dzi+1 + Ai < kbi+1 + Ai < kci+1 , ∀ t > 0. iii) By inspection of the stabilizing functions αi (¯ xi , z¯i , y¯di ) and control u(¯ xn , z¯n , y¯dn ) , it is clear that they are bounded, by virtue of the boundedness of x ¯n (t), z¯n (t), y¯dn (t), and, in particular, by |zi (t)| ≤ Dzi < kbi , which prevents any term comprising (kb2i − zi2 (t)) in the denominator from becoming unbounded. iv) Since V˙ (z) in (12) is negative definite, and V is proper in the set z ∈ Z, then, by the LaSalle-Yoshizawa Theorem, we have, for z(0) ∈ Z, that z(t) → 0 as t → ∞.
Theorem 1. Consider the closed loop system (1) and (10) under Assumptions 1-2. Let Ai =
max
(x,z,¯ ydn )∈Ω
i = 1, ..., n − 1
|αi (¯ xi , z¯i , y¯di )|,
xi , y¯di (0))| Zi = max |zi (¯
(15)
x∈Ωx0
where Ω is a compact set defined by: Ω := {x ∈ Rn , z ∈ Rn , y¯dn ∈ Rn+1 : (j)
|xj | ≤ Dzj + Aj−1 , |zj | ≤ Dzj , |yd | ≤ Yj ,
Dzj
j = 1, ..., n} v u n Y u kb2k − Zk2 t := kbj 1 − kb2k
(16) (17)
k=1
Ωx0 := {x ∈ Rn : aj ≤ xj ≤ bj , j = 1, ..., n}
(18)
Given the constraints kci > 0, i = 1, ..., n, suppose that the initial state x(0) ∈ Ωx0 , where bi < kci and ai > −kci . If there exist positive constants ξ = [a1 , b1 , ..., an , bn , κ1 , ..., κn−1 , kb1 , ..., kbn ]T that satisfy the conditions: kbi > Zi (ξ),
i = 1, ..., n i = 1, ..., n
kci > Ai−1 (ξ) + kbi ,
(19)
where A0 satisfies |yd (t)| ≤ A0 < kc1 , then the following properties hold. i) The signals zi (t), i = 1, 2, ..., n, remain in the compact set defined by Ωz = {¯ zn ∈ Rn : |zi | ≤ Dzi , i = 1, 2, ..., n}. ii) Every state xi (t) remains in the set Ωx := {x ∈ Rn : |xi | ≤ Dzi + Ai−1 < kci , i = 1, ..., n} ∀t > 0, i.e. the full state constraint is never violated. iii) All closed loop signals are bounded. iv) The origin z = 0 is asymptotically stable. Proof: i) From (19), kbi > Zi implies that z(0) ∈ Z, since |zj (0)| ≤ Zj for x(0) ∈ Ω0 . Thus, Lemma 1 yields z(t) ∈ Z ∀ t > 0. Then, from (12), we have V (t) ≤ V (0), which implies that n X kb2j k2 1 1 log 2 bi 2 ≤ log 2 2 kbi − zi 2 kbj − zj2 (0) j=1
≤
n X 1 j=1
2
log
kb2j
kb2j − Zj2
for i = 1, ..., n. Using the identity log a+log b = log ab, we rewrite (20) into the following log
kb2i kb2i − zi2
≤ log
n Y
j=1
kb2j
Remark 1. Unlike the approach of Tee and Ge (2009), the new feasibility conditions (19) in this paper 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. 4. ADAPTIVE CONTROL DESIGN When the nonlinearities fi (¯ xi ) are uncertain, but can be linearly parameterized according to (2), the foregoing design methodology can be modified, based on the certainty equivalence approach, i.e. replacing instances of θT ψi (¯ xi ) in the controls with their estimates θˆT ψi (¯ xi ), followed by the design of the adaptation law for θˆ that guarantees closed loop stability. We adopt the tuning functions approach Krstic et al. (1995) for stable design of an adaptation law. Denote z1 = x1 − yd and zi = xi − αi−1 , i = 2, ..., n. Consider the BLF candidate:
kb2j − Zj2
for i = 1, ..., n. Then, the above can be rearranged to yield |zi (t)| ≤ Dzi ∀ t > 0. ii) Since |z1 (t)| ≤ Dz1 < kc1 − A0 , we can show that |x1 (t)| ≤ Dz1 + |yd (t)| < kc1 − A0 + |yd (t)|. Noting that |yd (t)| ≤ A0 from Assumption 1, we therefore conclude that |x1 (t)| ≤ Dz1 + A0 < kc1 , ∀ t > 0. 6717
V =
n X
Vi + U
i=1
1 U = θ˜T Γ−1 θ˜ 2 k2 1 Vi = log 2 bi 2 , 2 kbi − zi
i = 1, ..., n
(20)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where Γ is a positive diagonal matrix, θ˜ := θˆ − θ, and , and kbi , i = 1, ..., n, are positive constants to be determined from subsequent feasibility analysis. The adaptive backstepping control is designed as:
Let the closed loop system (22) be written as η˙ = h(t, η), where η = [z T , θ˜T ]T . By inspection, h(t, η) is piecewise continuous in t and locally Lipschitz in η, uniformly in t, in the set η ∈ N , defined by
α1 = −θˆT w1 − κ1 z1 + y˙ d à k 2 − z22 ∂α1 1 g1 x2 g1 z1 + −θˆT w2 − κ2 z2 − b22 α2 = 2 g2 kb1 − z1 ∂x1 1 X ∂α1 (j+1) ∂α1 + y + Γτ2 (j) d ∂ θˆ ∂y
N := {z ∈ Rn , θ˜ ∈ Rl : |zi | < kbi , i = 1, 2, ..., n} (24) Then, together with the fact that V˙ ≤ 0 in the set z ∈ Z, and that z(0) ∈ Z, we invoke Lemma 1 to obtain that z(t) ∈ Z for all t > 0. Theorem 2. Consider the closed loop system (1) and (21) under Assumptions 1 and 2. Let
j=0
αi =
1 gi
Ã
d
−θˆT wi − κi zi −
i−1 X ∂αi−1
+
xj
j=1
+
i−1 X j=2
kb2i 2 kbi−1
gj xj+1 +
ˆ (x,z,¯ ydn ,θ)∈Ω
− zi2 2 gi−1 zi−1 − zi−1
i−1 X ∂αi−1 (j)
j=0
Ai = max
∂yd
(j+1)
yd
+
¯ ¯ ¯ ¯ ˆ xi , y¯di (0), θ(0)) Zi = max ¯zi (¯ ¯, x∈Ωx0
ˆ ≤ D ˆ, |xj | ≤ Dzj + Aj−1 , |zj | ≤ Dzj , kθk θ o (j) |yd | ≤ Yj , j = 1, ..., n v u n Y u kb2k − Zk2 Dzj := kbj t1 − ¯ kb2k e2Vθˆ k=1 s 2V¯ Dθˆ := θM + λmin (Γ−1 ) 1 ˆ + θM )2 V¯θˆ := λmax (Γ−1 )(kθ(0)k 2 n k2 1X ¯ V := log 2 bi 2 + V¯θˆ 2 i=1 kbi − Zi
w1 = ψ1 (x1 )
wi = ψi (¯ xi ) −
j=1
∂xj
ψj (¯ xj ),
i = 2, ..., n
wi zi w1 z1 , τi = τi−1 + 2 τ1 = 2 2 kb1 − z1 kbi − zi2 u = αn ˙ θˆ = Γτn which yields the closed loop system
(21)
z˙1 = −κ1 z1 + g1 z2 − θ˜T ψ1 (x1 ) k 2 − z22 z˙2 = −κ2 z2 − b22 g1 z1 + g2 z3 − θ˜T w2 kb1 − z12 ∂α1 ˆ˙ + (Γτ2 − θ) ∂ θˆ k 2 − zi2 ˜T z˙i = −κi zi − 2 bi 2 gi−1 zi−1 + gi zi+1 − θ wi kbi−1 − zi−1 +
∂αi−1 ˆ˙ + (Γτi − θ) ˆ ∂θ
z˙n = −κn zn − +
i−1 X
k2 j=2 bj
∂αn−1 ˆ˙ + (Γτn − θ) ˆ ∂θ
˙ θ˜ = Γτn
j=2
kbi > Zi (ξ),
∂αj−1 zj Γwi , 2 − zj ∂ θˆ
j=1
(28) (29) (30)
i = 1, ..., n
Ωz = {z ∈ Rn : |zi | ≤ Dzi , i = 1, 2, ..., n} n o ˆ ≤ Dˆ Ωθˆ = θˆ ∈ Rl : kθk θ
(32)
ii) Every state xi (t) remains in the set Ωx := {x ∈ Rn : |xi | ≤ Dzi + Ai−1 < kci , i = 1, ..., n} ∀t > 0, i.e. the full state constraint is never violated. iii) All closed loop signals are bounded. iv) The origin z = 0 is asymptotically stable.
zj ∂αj−1 Γwn kb2j − zj2 ∂ θˆ
The derivative of V along (22) can be written as: V˙ = −
(27)
ˆ i) The signals zi (t) and θ(t), i = 1, 2, ..., n, remain, for all t > 0, in the compact sets defined by
i = 3, ..., n − 1
κj zj2 kb2j − zj2
(26)
(31) kci > Ai−1 (ξ) + kbi , i = 1, ..., n where A0 satisfies |yd (t)| ≤ A0 < kc1 , then the following properties hold.
(22)
n X
(25)
Given the constraints kci > 0, i = 1, ..., n, suppose that the initial state x(0) ∈ Ωx0 , where bi < kci and ai > −kci . If there exist positive constants ξ = [a1 , b1 , ..., an , bn , κ1 , ..., κn−1 , kb1 , ..., kbn , γ1 , ..., γl ]T that satisfy the conditions:
kb2n − zn2 gn−1 zn−1 − θ˜T wn 2 2 kbn−1 − zn−1 n−1 X
i = 1, ..., n
where Ω is a compact set defined by: n Ω := x ∈ Rn , z ∈ Rn , y¯dn ∈ Rn+1 , θˆ ∈ Rl :
∂αi−1 Γτi ∂ θˆ
∂αj−1 zj Γwi , i = 3, ..., n kb2j − zj2 ∂ θˆ i−1 X ∂αi−1
ˆ i = 1, ..., n − 1 |αi (¯ xi , z¯i , y¯di , θ)|,
Proof: (23)
6718
i) Since kθk ≤ θM and |zi (0)| ≤ Zi , i = 1, ..., n, it can be shown that
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
V (0) ≤ V¯ (33) Since Zi < kbi , i = 1, ..., n, Lemma 1 yields z(t) ∈ Z ∀ t > 0. Then, from (23), we have V (t) ≤ V (0) ≤ V¯ , which yields s 2V¯ ˆ kθ(t)k ≤ θM + (34) λmin (Γ−1 )
with sufficiently large control gains κi , but increasing κi also increases Ai and Zi . The tradeoff between robustness and performance is formulated as a nonlinear constrained optimization problem that is solved offline, prior to actual implementation.
Therefore, θˆ remains in the compact set Ωθˆ ∀ t > 0. Furthermore, we have
ξ := [a1 , b1 , ..., an , bn , kb1 , ..., kbn , κ1 , ..., κn−1 ]T
For known plant, we check if there exists a solution
for the optimization problem:
2V¯θˆ
log
n Y kb2j e kb2i ≤ log kb2i − zi2 (t) k 2 − zj2 (0) j=1 bj
≤ log
¯ n Y kb2j e2Vθˆ
k2 j=1 bj
−
, Zj2
maximize P (ξ) =
n X
(bi − ai ) + ρ
i=1
subject to:
i = 1, ..., n
kbi > Zi (ξ),
which can be written as |zi (t)| ≤ Dzi < kbi , ∀ t > 0. ii) The proof follows the a similar line of argument as that in Theorem 1, and is omitted. iii) By inspection of the stabilizing functions αi (¯ xi , z¯i , ˆ , it is clear that ˆ and control u(¯ xn , z¯n , y¯dn , θ) y¯di , θ) they are bounded, by virtue of the boundedness ˆ and, in particular, by of x ¯n (t), z¯n (t), y¯dn (t), θ(t), |zi (t)| < kbi , which prevents any term comprising (kb2i − zi2 ) in the denominator from becoming unbounded. iv) Since V˙ (z) in (23) is negative semidefinite, and V is proper in the set z ∈ Z, then, by the LaSalleYoshizawa Theorem, we have, for z(0) ∈ Z, that lim
n X
κj zj (t)2 =0 − zj (t)2
κi
i=1
i = 1, ..., n
kci ≥ Ai−1 (ξ) + kbi , −kci < ai < bi < kci , κi > 0,
n−1 X
i = 1, ..., n i = 1, ..., n
i = 1, ..., n − 1
(35)
where ρ is a positive weighting constant, Ai , Zi are defined in (15). If a solution ξ ∗ to the above optimization problem exists, then the feasibility conditions (19) in Theorem 1 are satisfied, and the proposed control (10) with ξ = ξ ∗ is feasible in ensuring output tracking for (1) while respecting the full state constraint for all time. When the plant is uncertain, we check if there exists a solution ξ := [a1 , b1 , ..., an , bn , κ1 , ..., κn−1 , kb1 , ..., kbn , γ1 , ..., γl ]T
t→∞ k2 j=1 bj
for an optimization problem that maximizes the objective function
and thus, z(t) → 0 as t → ∞.
P (ξ) =
5. FEASIBILITY CHECK AND PARAMETER SELECTION
n n−1 l X X X (bi − ai ) + ρ κi + ̺ λi i=1
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. The feasibility conditions are formulated as sufficient conditions (19) and (31), which depend on the state constraints, the initial conditions and the design parameters κ1 , ..., κn−1 , kb1 , ..., kbn , γ1 , ..., γn , where γ1 , ..., γl are the elements of the diagonal matrix Γ. As such, if we are able to find a set of design parameters that satisfies (19) for known plant, and (31) for uncertain plant, then the control using the design parameters is feasible.
i=1
(36)
i=1
subject to (35), where ρ, ̺ are positive weighting constants, and Ai , Zi are defined in (25). If a solution ξ ∗ is found, then the proposed adaptive control (21)-(21), with ξ = ξ ∗ is feasible in ensuring output tracking with full state constraint, according to Theorem 2. 6. SIMULATION We present a simulation study on the full state constraint problem for a nonlinear plant: x˙ 1 = θ1 x21 + x2 x˙ 2 = θ2 x1 x2 + θ3 x1 + (1 + x21 )u
Unlike the approach of Tee and Ge (2009), the feasibility conditions are formulated in terms of a box region of initial states ai ≤ xi (0) ≤ bi , i = 1, ..., n, such that all points in the region share the same design parameters, and are feasible in achieving output tracking without constraint violation.
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. For simplicity, consider the plant to be known, and that yd ≡ 0, i.e., a stabilization task.
On the one hand, we want to maximize 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
Denote ξ = [a1 , b1 , a2 , b2 , κ1 , kb1 , kb2 ]T . We need to find a solution ξ ∗ = [a∗1 , b∗1 , a∗2 , b∗2 , κ∗1 , kb∗1 , kb∗2 ]T , if it exists, for the optimization problem (35). Using the Matlab routine fmincon.m, we obtain a∗1 = −0.798, b∗1 = 0.798, a∗2 = −2.417, b∗2 = 2.341, κ∗1 = 0.0165, kb∗1 = 0.799, and
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
kb∗2 = 2.420. Next, we implement the control law (8)-(10) with design parameters κ1 = κ∗1 , kb1 = kb∗1 , kb2 = kb∗2 , and κ2 = 4, which are valid for all initial states x(0) in a∗i ≤ xi ≤ b∗i , i = 1, 2.
BLF 3
2
Figure 1 shows the closed loop state trajectories corresponding to different initial conditions, which are indicated by ‘+’. The BLF-based control ensures that the state trajectories remain in the interior of the constraint region for all time, and converge to the origin, even if the trajectories start near the boundary of the constraint region.
x
2
1
−1
Figure 2 shows the level curves of the BLF. 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. 7. CONCLUSIONS
0
−2
−3 −1
−0.5
0 x1
0.5
1
Fig. 2. Level curves of a BLF. The state moves along a trajectory that decreases the value of the Lyapunov function.
In this paper, we have employed a Barrier Lyapunov Function to design a control for strict feedback systems with state constraint. Besides the nominal case where the plant is fully known, the presence of parametric uncertainties has also been handled. Asymptotic tracking 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 can be checked offline without precise knowledge of initial states. BLF 3
2
x
2
1
0
−1
−2
−3 −1
−0.5
0 x
0.5
1
1
Fig. 1. Phase plot of closed loop system resulting from BLF-based control.
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