Backstepping-Forwarding Boundary Control Design for First-Order ...
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Backstepping-Forwarding Boundary Control Design for First-Order Hyperbolic Systems With Fredholm Integrals Federico Bribiesca-Argomedo and Miroslav Krstic Abstract— In this article, a backstepping-forwarding transform is developed for the boundary stabilization of a class of non-strict feedback first-order hyperbolic systems. Sufficient conditions for the well-posedness of the resulting kernel equations are given, both in terms of the spectrum of some integral operators and, more conservatively, in terms of the magnitude of the coefficients of the system equation. The results presented in this paper reduce to known backstepping results when the non-strict feedback terms are zero.
I. I NTRODUCTION Backstepping design has proven to be an effective tool for constructing boundary controllers and observers based on boundary measurements for a large class of Partial Differential Equations (PDEs) with non-local terms, see for instance [1], [2], [3], [4]. Its wide-ranging applications include the control of turbulent flows modeled by coupled parabolic PDEs, see [5], and delay compensation for finite-dimensional systems that can be modeled by coupled ODEs coupled with hyperbolic (transport) PDEs, see [6], However, its reliance on a Volterra integral transform reduces its applicability to systems that have a strict-feedback structure. Non-strict-feedback terms appear, for example, when analyzing some coupled PDE systems using singular perturbations. Existing stabilization approaches for such systems (developed for parabolic PDEs) rely either on the use of an additional control input to cancel the non-strict-feedback components and reduce the problem to one that can be tackled with traditional backstepping, see for instance [5], or the solution of operator Ricatti equations. In this article, we present an integral transform that allows the computation of stabilizing boundary control laws for a class of systems with non-strict-feedback terms. This transform is closely related to that used in [7], [8] yet it allows for a discontinuity along the diagonal of the integral kernels. It is also related to the transform used in [9] from which it differs by avoiding the existence of multiple equivalent representations for the gain kernels (due to the overlap between the two domains of integration used in that transform). Following a similar approach to that used in the standard backstepping design, we set to find a bounded (and boundedly invertible) transform that maps the original system to a stable target system. Instead of the Volterra transform used in backstepping, a Fredholm transform has to be considered in this case. This formulation introduces difficulties in the
analysis of the invertibility of the transform and the wellposedness of the kernel PDEs. In this article, sufficient conditions for the existence of the direct and inverse transforms are given in terms of the spectrum of some integral operators (more general, but hard to verify in some cases) and then in terms of the magnitude of the coefficients in the original PDE (much more restrictive but simple to check). Although considering general coefficients makes the conditions given in this article conservative, considering specific structures can lead to less restrictive conditions. In particular, it should be noted that the problem considered in this article is a generalization of that in [3] and, when applied to that problem, the same kernel equations are obtained (which were shown to be well-posed for arbitrarily large coefficients). In Section II the class of systems considered in this article is presented and the control objective is stated. Section III presents sufficient conditions for the existence of the direct and inverse transforms and then states the main result of this article. Finally, Section IV applies the method developed in this paper to a system consisting of a first-order hyperbolic PDE interconnected with second order ODE in the spatial variable. II. P ROBLEM S TATEMENT The class of systems under consideration in this article are of the form: u ¯t (x, t) = u ¯x (x, t) + e¯(x)¯ u(x, t) + f¯(x)¯ u(0, t) Z x Z 1 ¯ y)¯ + g¯(x, y)¯ u(y, t)dy + h(x, u(y, t)dy, 0
(1) with boundary condition: ¯ (t), ∀t ∈ (0, T ] u ¯(1, t) = U
(2)
and initial condition u ¯(x, 0) = u0 (x) ∈ L2 ([0, 1]; R). The ¯ ¯ are assumed to be continuous realcoefficients e¯, f , g¯ and h valued functions in their respective domains. . the same change of variables from [3] u(x, t) = R Using x e 0 e¯(ξ)dξ u ¯(x, t) that eliminates the reaction term, we consider in the rest of this article the alternate formulation: Z x ut (x, t) = ux (x, t) + f (x)u(0, t) + g(x, y)u(y, t)dy 0 Z 1 (3) + h(x, y)u(y, t)dy,
The authors are with the Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA.
978-1-4799-3271-9/$31.00 ©2014 AACC
x
∀(x, t) ∈ (0, 1) × (0, T ]
5428
x
∀(x, t) ∈ (0, 1) × (0, T ] u(1, t) = U (t), ∀t ∈ (0, T ]
(4)
. with initial condition u(x, 0) = u0 (x) ∈ L2 ([0, 1]; R). The coefficients f , g and h are assumed to be continuous, realvalued functions in their domains (this assumption will be formally stated once these domains are defined). U (t) is the control input. The problem under consideration in this article is to find a feedback gain γ ∈ C([0, 1]; R) such that, under the control law: Z 1 . U (t) = γ(y)u(y, t)dy (5) 0
the origin of the system (3)-(4) is finite-time stable in the topology of the L2 norm. Remark that well-posedness of system (3)-(5) follows readily from [4] (with the addition of a perturbation by a bounded linear operator). To tackle this problem we set to find continuous realvalued kernels p, q, k and l such that the (bounded) transform: Z
equipped with these norms. Furthermore, the chosen k·k∞ norm is equivalent to the usual Euclidean norm. Definition 2: We now define the following spaces: . Xl = C(Tl ; R) . Xu = C(Tu ; R)
(6)
z∈Tl
kskXu
. = sup |s(z)|,
or, equivalently (7)
with IL2 the identity operator on L2 ([0, 1]; R) and Πp,q : L2 ([0, 1]; R) → L2 ([0, 1]; R), has a bounded inverse: Z x u(x, t) = w(x, t) + k(x, y)w(y, t)dy 0 Z 1 + l(x, y)w(y, t)dy
(16)
Note that (Xl , k·kXl ) and (Xu , k·kXu ) are Banach spaces. Assumption 1: The coefficients in (3) satisfy: f ∈ C([0, 1]; R), g ∈ Xl and h ∈ Xu . Definition 3: Define now the space: . X = Xl × Xu
(17)
equipped with the norm:
(18) As defined, (X, k·kX ) is a Banach space. Definition 4: Let us now define the integral operator T : X → X (for A1,1 : Xl → Xl , A1,2 : Xu → Xl , A2,1 : Xl → Xu , A2,2 : Xu → Xu , F1 ∈ Xl F2 ∈ Xu ), for all p ∈ Xl , q ∈ Xu : T
(8)
p q
x
and maps the evolution of system (3)-(4) into the (finite-time stable) target system: wt (x, t) = wx (x, t), ∀(x, t) ∈ (0, 1) × (0, T ] w(1, t) = 0, ∀t ∈ (0, T ]
∀s ∈ Xu
z∈Tu
. . kϕkX = max{kϕ1 kXl , kϕ2 kXu }, ∀ϕ = (ϕ1 , ϕ2 ) ∈ X
x
w(x, t) = (IL2 − Πp,q )[u(·, t)](x)
(14)
equipped with the norm k·kXl (respectively k·kXu ) defined as: . kskXl = sup |s(z)|, ∀s ∈ Xl (15)
x
w(x, t) = u(x, t) − p(x, y)u(y, t)dy 0 Z 1 − q(x, y)u(y, t)dy
(13)
(9) (10)
III. BACKSTEPPING -F ORWARDING CONTROL DESIGN A. Preliminary Definitions
(19)
where: A1,1 [p](x, y) Z x−y . = f (s)p(x − y, s)ds 0 Z yZ σ + h(s, σ)p(σ + x − y, s)ds dσ
Definition 1: Let us define two (closed, bounded) subsets of R2 as follows: . Tl = {(x, y) ∈ [0, 1] × [0, 1], y ≤ x} (11) . Tu = {(x, y) ∈ [0, 1] × [0, 1], x ≤ y} (12)
0
Z
0 y
Z
x−y
g(s + σ, σ)p(σ + x − y, σ + s)ds dσ
+ 0
0
(20) . A1,2 [q](x, y) =
equipped with the norm: . . kzk∞ = max{|z1 |, |z2 |}, ∀z = (z1 , z2 ) ∈ R2
p . =A +F q A1,1 A1,2 p F1 . = + A2,1 A2,2 q F2
Z
1−x+y
f (x − y + s)q(x − y, x − y + s)ds 0
Z
where |·| denotes the absolute value of an element of R (N.B. whenever necessary, we consider R to be equipped with the topology induced by the absolute value metric, or the euclidean norm in R). We should note that (Tl , k·k∞ ) and (Tu , k·k∞ ) are compact in the topology induced by their norms. Hereafter, unless otherwise explicitly stated, we assume Tl and Tu to be
y
Z
1−σ−x+y
g(σ + x − y + s, σ)
+ 0
0
× q(σ + x − y, σ + x − y + s)ds dσ (21) . A2,1 [p](x, y) = −
5429
Z
1−y
Z
σ+x
h(s, σ + y)p(σ + x, s)ds dσ 0
0
(22)
1−y
Z
. A2,2 [q](x, y) = −
y−x
Z
h(s + σ + x, σ + y)
0
0
× q(σ + x, σ + y + s)ds dσ
. A¯2,1 [k](x, y) =
(23) . F1 (x, y) = −f (x − y) −
Z
(24)
. A¯2,2 [l](x, y) =
0 1−y
Z
h(σ + x, σ + y)dσ
(25)
0
in their respective domains. Definition 5: Given functions φ ∈ Xl , ψ ∈ Xu , we define an operator Rφ,ψ : X → X as: Rφ,ψ
k l
k φ . = S φ,ψ + l ψ " # φ,ψ φ,ψ S1,1 S1,2 k φ . = + φ,ψ φ,ψ l ψ S2,1 S2,2
Z
0 0 1−y Z 1−y−σ
h(σ + x, s + σ + y) 0
0
y
g(σ + x − y, σ)dσ
y
f (σ + x − y)l(0, σ)dσ (33) Z0 y Z σ g(σ + x − y, s)l(s, σ)ds dσ −
× q(σ + x, σ + x + s)ds dσ Z 1−y Z 1−σ−y g(s + σ + y, σ + y) −
. F2 (x, y) =
Z
. A¯1,2 [l](x, y) = −
0
0
(34)
× k(s + σ + y, σ + y)ds dσ Z 1−y f (σ + x)l(0, σ + y)dσ 0 Z 1−y Z σ+x g(σ + x, s)l(s, σ + y)ds dσ + 0 0 Z 1−y Z y−x h(σ + x, s + σ + x) + 0
0
× l(s + σ + x, σ + y)ds dσ (35) Z y . ¯ F1 (x, y) = −f (x − y) − g(σ + x − y, σ)dσ 0 Z 1−y . F¯2 (x, y) = h(σ + x, σ + y)dσ
(26)
(36) (37)
0
where:
in their respective domains. y
. φ,ψ S1,1 [k](x, y) =
Z
. φ,ψ S1,2 [l](x, y) =
Z
x
Z ψ(s, y)k(x, s)ds +
0
. φ,ψ S2,1 [k](x, y) =
φ(s, y)k(x, s)ds (27)
y 1
φ(s, y)l(x, s)ds
(28)
ψ(s, y)k(x, s)ds
(29)
x Z x 0
. φ,ψ S2,2 [l](x, y) =
Z
y
Z ψ(s, y)l(x, s)ds +
x
1
φ(s, y)l(x, s)ds y
(30) in their respective domains. Definition 6: Let us now define a third integral operator T¯ : X → X (for A¯1,1 : Xl → Xl , A¯1,2 : Xu → Xl , A¯2,1 : Xl → Xu , A¯2,2 : Xu → Xu , F¯1 ∈ Xl F¯2 ∈ Xu ), for all k ∈ Xl , l ∈ Xu :
T¯
k l
. ¯ k =A + F¯ l A¯1,1 A¯1,2 k F¯1 . = + ¯ A¯2,1 A¯2,2 l F2
(31)
B. Direct Transform Lemma 1: If the operator T , as defined in (19) has a unique fixed point in X (i.e. there exists a unique ζ ∈ X s.t. T ζ = ζ), then transform (6) with kernels p . =ζ (38) q . maps system (3)-(5), with γ(y) = p(1, y), into (9)-(10). . Proof: (Sketch) Condition γ(y) = p(1, y) follows directly from evaluating transform (6) at the boundary x = 1 and using the boundary conditions (4) and (10). Following a similar procedure to that used in standard backstepping, we plug (6) into (9) and (10) and, after integrating by parts some terms and changing the order of integration in some double integrals, we obtain the following set of hyperbolic PIDEs the kernels must verify: Z y px (x, y) + py (x, y) = −g(x, y) + h(s, y)p(x, s)ds 0 Z x + g(s, y)p(x, s)ds (39) y Z 1 + g(s, y)q(x, s)ds, x
where:
∀x, y ∈ [0, 1] s.t. y ≤ x, y 6= 0 Z y qx (x, y) + qy (x, y) = −h(x, y) + h(s, y)q(x, s)ds x Z 1 + g(s, y)q(x, s)ds (40) Zy x + h(s, y)p(x, s)ds,
A¯1,1 [k](x, y) Z y Z x−y . =− g(σ + x − y, s + σ)k(s + σ, σ)ds dσ 0
Z
0 y
Z
−
1−σ−x+y
h(σ + x − y, s + σ + x − y) 0
0
× k(s + σ + x − y, σ)ds dσ
0
(32) 5430
∀x, y ∈ [0, 1] s.t. x ≤ y, y 6= 1
which in turn implies
with boundary conditions: Z
x
p(x, 0) = −f (x) + p(x, y)f (y)dy 0 Z 1 + q(x, y)f (y)dy, ∀x ∈ [0, 1]
kA(ϕ − ϕ)k ˜ X ≤ 2cK (41)
x
q(x, 1) = 0,
∀x ∈ [0, 1]
then the operator T is a contraction. We start by noting that: kT ϕ − T ϕk ˜ X = kAϕ − Aϕk ˜ X = kA(ϕ − ϕ)k ˜ X (46) . Let us denote K = kϕ − ϕk ˜ X , and c defined as in the theorem statement then, after some computations we obtain the following norm estimate: kA(ϕ − ϕ)k ˜ X ≤ max{cK sup (1 + y), cK sup (1 − y)} y∈[0,1]
And therefore, if 2c < 1, T defines a contraction mapping. Corollary 2 completes the first part of the proof. The norm estimate comes from rewriting:
(42)
Integrating (39) and (40) along the level curves of x − y up to the boundary conditions (41) and (42) we obtain a set of integral equations to which the condition of the theorem guarantees a unique solution. Therefore, a suitable direct transform exists. Note that Lemma 1 can be alternatively formulated in terms of the spectrum of the operator A. As a first step toward finding concrete sufficient conditions in terms of the magnitude of the coefficients in equation (3) under which Lemma 1 can be applied we use Banach’s contraction mapping principle, see for instance [10, Theorem 3.1] to obtain the following Corollary. Corollary 2: If the operator T as defined in (19) is a contraction, then transform (6) with kernels p . = lim T n ϑ0 (43) q n→∞ . for any ϑ0 ∈ X, maps system (3)-(5), with γ(y) = p(1, y), into (9)-(10). In particular, if the operator T is a contraction, the spectral radius of A is less than 1 and {1} belongs to its resolvent set. The contraction property depends on the choice of norm for the Banach space and, even if an operator is not a contraction for a given norm, it may be a contraction for an equivalent norm. A similar result holds if there exists n ∈ N such that T n is a contraction. Using the supremum norm already defined in X, we obtain the following sufficient condition for the direct transform to exist: . Proposition 3: If the coefficients in equation (3) verify c = max{sups∈[0,1] |f (s)|, kgkXl , khkXu } < 12 , then transform (6) with kernels p . . = ζ = lim T n ϑ0 (44) q n→∞ . for any ϑ0 ∈ X, maps system (3)-(5), with γ(y) = p(1, y), into (9)-(10). Furthermore kζkX ≤ kF kX /(1 − 2c). Proof: If we can show that there exists C ∈ [0, 1) such that: kT ϕ − T ϕk ˜ X ≤ Ckϕ − ϕk ˜ X , ∀ϕ, ϕ˜ ∈ X (45)
y∈[0,1]
(47)
(48)
ζ=
∞ X
An F
(49)
n=0
and noting that it implies, using (48): kζkX ≤ kF kX
∞ X
(2c)n
(50)
n=0
this, and the condition c
0 and boundary conditions: u(1, t) = U (t)
(60)
vx (0, t) = 0
(61)
v(1, t) = 0
(62)
This system closely resembles the Korteweg-de Vries-like equation presented in [3]. The only two differences (other than notation) are the extra (destabilizing) term au(0, t) and the use of only one boundary to control the full interconnected system (instead of using one boundary of each subsystem). Solving (59) with boundary conditions (61)-(62), and plugging the solution into (58) we obtain a system of the form (3)-(4). We present simulation results for a = 1.25, b = 0.1, c = 0.01, d = 7.5. For these coefficients, a solution can still be found for both systems of integral equations (even though they are larger than the sufficient condition presented in Proposition 9) and therefore the direct and inverse transforms exist and are bounded. Figures 1 (a) and 1 (b) show the obtained direct (respectively inverse) transform kernels for this system. Figure 1 (c) shows the obtained control gain. Figure 2 shows the evolution of the state in open-loop (unstable) and closed-loop (finite-time stable).
[1] A. Smyshlyaev and M. Krstic, “Closed-form boundary state feedback for a class of 1-D partial integro-differential equations,” IEEE Transactions on Automatic Control, vol. 49, no. 12, pp. 2185–2201, 2004. [2] ——, “Backstepping observers for a class of parabolic PDEs,” Systems & Control Letters, vol. 54, pp. 613–625, 2005. [3] M. Krstic and A. Smyshlyaev, “Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actutator and sensor delays,” Systems & Control Letters, vol. 57, no. 9, pp. 750–758, 2008. [4] S. Nakagiri, “Deformation formulas and boundary control problems of first-order Volterra integro-differential equations with nonlocal boundary conditions,” IMA Journal of Mathematical Control and Information, vol. 30, no. 3, pp. 345–377, 2013. [5] R. Vazquez and M. Krstic, Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation, ser. Systems & Control: Foundations & Applications. Birkhäuser, 2008. [6] M. Krstic, Delay Compensation for Nonlinear, Adaptive and PDE Systems, ser. Systems & Control: Foundations & Applications. Birkhäuser, 2009. [7] N. Bekiaris-Liberis and M. Krstic, “Compensating the distributed effect of a wave PDE in the actuation or sensing path of MIMO LTI systems,” Systems & Control Letters, vol. 59, no. 11, pp. 713–719, November 2010. [8] ——, “Lyapunov stability of linear predictor feedback for distributed input delays,” IEEE Transactions on Automatic Control, vol. 56, no. 3, pp. 655–660, March 2011. [9] C. Guo, C. Xie, and C. Zhou, “Stabilization of a spatially noncausal reaction-diffusion equation by boundary control,” International Journal of Robust and Nonlinear Control, 2012. [10] M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, ser. Pure and Applied Mathematics. John Wiley & Sons, 2001. [11] W. A. Kirk and B. Sims, Eds., Handbook of Metric Fixed Point Theory. Kluwer Academic Publishers, 2001.
5433
Backstepping-Forwarding Boundary Control Design for First-Order Hyperbolic Systems With Fredholm Integrals Federico Bribiesca-Argomedo and Miroslav Krstic Abstract— In this article, a backstepping-forwarding transform is developed for the boundary stabilization of a class of non-strict feedback first-order hyperbolic systems. Sufficient conditions for the well-posedness of the resulting kernel equations are given, both in terms of the spectrum of some integral operators and, more conservatively, in terms of the magnitude of the coefficients of the system equation. The results presented in this paper reduce to known backstepping results when the non-strict feedback terms are zero.
I. I NTRODUCTION Backstepping design has proven to be an effective tool for constructing boundary controllers and observers based on boundary measurements for a large class of Partial Differential Equations (PDEs) with non-local terms, see for instance [1], [2], [3], [4]. Its wide-ranging applications include the control of turbulent flows modeled by coupled parabolic PDEs, see [5], and delay compensation for finite-dimensional systems that can be modeled by coupled ODEs coupled with hyperbolic (transport) PDEs, see [6], However, its reliance on a Volterra integral transform reduces its applicability to systems that have a strict-feedback structure. Non-strict-feedback terms appear, for example, when analyzing some coupled PDE systems using singular perturbations. Existing stabilization approaches for such systems (developed for parabolic PDEs) rely either on the use of an additional control input to cancel the non-strict-feedback components and reduce the problem to one that can be tackled with traditional backstepping, see for instance [5], or the solution of operator Ricatti equations. In this article, we present an integral transform that allows the computation of stabilizing boundary control laws for a class of systems with non-strict-feedback terms. This transform is closely related to that used in [7], [8] yet it allows for a discontinuity along the diagonal of the integral kernels. It is also related to the transform used in [9] from which it differs by avoiding the existence of multiple equivalent representations for the gain kernels (due to the overlap between the two domains of integration used in that transform). Following a similar approach to that used in the standard backstepping design, we set to find a bounded (and boundedly invertible) transform that maps the original system to a stable target system. Instead of the Volterra transform used in backstepping, a Fredholm transform has to be considered in this case. This formulation introduces difficulties in the
analysis of the invertibility of the transform and the wellposedness of the kernel PDEs. In this article, sufficient conditions for the existence of the direct and inverse transforms are given in terms of the spectrum of some integral operators (more general, but hard to verify in some cases) and then in terms of the magnitude of the coefficients in the original PDE (much more restrictive but simple to check). Although considering general coefficients makes the conditions given in this article conservative, considering specific structures can lead to less restrictive conditions. In particular, it should be noted that the problem considered in this article is a generalization of that in [3] and, when applied to that problem, the same kernel equations are obtained (which were shown to be well-posed for arbitrarily large coefficients). In Section II the class of systems considered in this article is presented and the control objective is stated. Section III presents sufficient conditions for the existence of the direct and inverse transforms and then states the main result of this article. Finally, Section IV applies the method developed in this paper to a system consisting of a first-order hyperbolic PDE interconnected with second order ODE in the spatial variable. II. P ROBLEM S TATEMENT The class of systems under consideration in this article are of the form: u ¯t (x, t) = u ¯x (x, t) + e¯(x)¯ u(x, t) + f¯(x)¯ u(0, t) Z x Z 1 ¯ y)¯ + g¯(x, y)¯ u(y, t)dy + h(x, u(y, t)dy, 0
(1) with boundary condition: ¯ (t), ∀t ∈ (0, T ] u ¯(1, t) = U
(2)
and initial condition u ¯(x, 0) = u0 (x) ∈ L2 ([0, 1]; R). The ¯ ¯ are assumed to be continuous realcoefficients e¯, f , g¯ and h valued functions in their respective domains. . the same change of variables from [3] u(x, t) = R Using x e 0 e¯(ξ)dξ u ¯(x, t) that eliminates the reaction term, we consider in the rest of this article the alternate formulation: Z x ut (x, t) = ux (x, t) + f (x)u(0, t) + g(x, y)u(y, t)dy 0 Z 1 (3) + h(x, y)u(y, t)dy,
The authors are with the Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA.
978-1-4799-3271-9/$31.00 ©2014 AACC
x
∀(x, t) ∈ (0, 1) × (0, T ]
5428
x
∀(x, t) ∈ (0, 1) × (0, T ] u(1, t) = U (t), ∀t ∈ (0, T ]
(4)
. with initial condition u(x, 0) = u0 (x) ∈ L2 ([0, 1]; R). The coefficients f , g and h are assumed to be continuous, realvalued functions in their domains (this assumption will be formally stated once these domains are defined). U (t) is the control input. The problem under consideration in this article is to find a feedback gain γ ∈ C([0, 1]; R) such that, under the control law: Z 1 . U (t) = γ(y)u(y, t)dy (5) 0
the origin of the system (3)-(4) is finite-time stable in the topology of the L2 norm. Remark that well-posedness of system (3)-(5) follows readily from [4] (with the addition of a perturbation by a bounded linear operator). To tackle this problem we set to find continuous realvalued kernels p, q, k and l such that the (bounded) transform: Z
equipped with these norms. Furthermore, the chosen k·k∞ norm is equivalent to the usual Euclidean norm. Definition 2: We now define the following spaces: . Xl = C(Tl ; R) . Xu = C(Tu ; R)
(6)
z∈Tl
kskXu
. = sup |s(z)|,
or, equivalently (7)
with IL2 the identity operator on L2 ([0, 1]; R) and Πp,q : L2 ([0, 1]; R) → L2 ([0, 1]; R), has a bounded inverse: Z x u(x, t) = w(x, t) + k(x, y)w(y, t)dy 0 Z 1 + l(x, y)w(y, t)dy
(16)
Note that (Xl , k·kXl ) and (Xu , k·kXu ) are Banach spaces. Assumption 1: The coefficients in (3) satisfy: f ∈ C([0, 1]; R), g ∈ Xl and h ∈ Xu . Definition 3: Define now the space: . X = Xl × Xu
(17)
equipped with the norm:
(18) As defined, (X, k·kX ) is a Banach space. Definition 4: Let us now define the integral operator T : X → X (for A1,1 : Xl → Xl , A1,2 : Xu → Xl , A2,1 : Xl → Xu , A2,2 : Xu → Xu , F1 ∈ Xl F2 ∈ Xu ), for all p ∈ Xl , q ∈ Xu : T
(8)
p q
x
and maps the evolution of system (3)-(4) into the (finite-time stable) target system: wt (x, t) = wx (x, t), ∀(x, t) ∈ (0, 1) × (0, T ] w(1, t) = 0, ∀t ∈ (0, T ]
∀s ∈ Xu
z∈Tu
. . kϕkX = max{kϕ1 kXl , kϕ2 kXu }, ∀ϕ = (ϕ1 , ϕ2 ) ∈ X
x
w(x, t) = (IL2 − Πp,q )[u(·, t)](x)
(14)
equipped with the norm k·kXl (respectively k·kXu ) defined as: . kskXl = sup |s(z)|, ∀s ∈ Xl (15)
x
w(x, t) = u(x, t) − p(x, y)u(y, t)dy 0 Z 1 − q(x, y)u(y, t)dy
(13)
(9) (10)
III. BACKSTEPPING -F ORWARDING CONTROL DESIGN A. Preliminary Definitions
(19)
where: A1,1 [p](x, y) Z x−y . = f (s)p(x − y, s)ds 0 Z yZ σ + h(s, σ)p(σ + x − y, s)ds dσ
Definition 1: Let us define two (closed, bounded) subsets of R2 as follows: . Tl = {(x, y) ∈ [0, 1] × [0, 1], y ≤ x} (11) . Tu = {(x, y) ∈ [0, 1] × [0, 1], x ≤ y} (12)
0
Z
0 y
Z
x−y
g(s + σ, σ)p(σ + x − y, σ + s)ds dσ
+ 0
0
(20) . A1,2 [q](x, y) =
equipped with the norm: . . kzk∞ = max{|z1 |, |z2 |}, ∀z = (z1 , z2 ) ∈ R2
p . =A +F q A1,1 A1,2 p F1 . = + A2,1 A2,2 q F2
Z
1−x+y
f (x − y + s)q(x − y, x − y + s)ds 0
Z
where |·| denotes the absolute value of an element of R (N.B. whenever necessary, we consider R to be equipped with the topology induced by the absolute value metric, or the euclidean norm in R). We should note that (Tl , k·k∞ ) and (Tu , k·k∞ ) are compact in the topology induced by their norms. Hereafter, unless otherwise explicitly stated, we assume Tl and Tu to be
y
Z
1−σ−x+y
g(σ + x − y + s, σ)
+ 0
0
× q(σ + x − y, σ + x − y + s)ds dσ (21) . A2,1 [p](x, y) = −
5429
Z
1−y
Z
σ+x
h(s, σ + y)p(σ + x, s)ds dσ 0
0
(22)
1−y
Z
. A2,2 [q](x, y) = −
y−x
Z
h(s + σ + x, σ + y)
0
0
× q(σ + x, σ + y + s)ds dσ
. A¯2,1 [k](x, y) =
(23) . F1 (x, y) = −f (x − y) −
Z
(24)
. A¯2,2 [l](x, y) =
0 1−y
Z
h(σ + x, σ + y)dσ
(25)
0
in their respective domains. Definition 5: Given functions φ ∈ Xl , ψ ∈ Xu , we define an operator Rφ,ψ : X → X as: Rφ,ψ
k l
k φ . = S φ,ψ + l ψ " # φ,ψ φ,ψ S1,1 S1,2 k φ . = + φ,ψ φ,ψ l ψ S2,1 S2,2
Z
0 0 1−y Z 1−y−σ
h(σ + x, s + σ + y) 0
0
y
g(σ + x − y, σ)dσ
y
f (σ + x − y)l(0, σ)dσ (33) Z0 y Z σ g(σ + x − y, s)l(s, σ)ds dσ −
× q(σ + x, σ + x + s)ds dσ Z 1−y Z 1−σ−y g(s + σ + y, σ + y) −
. F2 (x, y) =
Z
. A¯1,2 [l](x, y) = −
0
0
(34)
× k(s + σ + y, σ + y)ds dσ Z 1−y f (σ + x)l(0, σ + y)dσ 0 Z 1−y Z σ+x g(σ + x, s)l(s, σ + y)ds dσ + 0 0 Z 1−y Z y−x h(σ + x, s + σ + x) + 0
0
× l(s + σ + x, σ + y)ds dσ (35) Z y . ¯ F1 (x, y) = −f (x − y) − g(σ + x − y, σ)dσ 0 Z 1−y . F¯2 (x, y) = h(σ + x, σ + y)dσ
(26)
(36) (37)
0
where:
in their respective domains. y
. φ,ψ S1,1 [k](x, y) =
Z
. φ,ψ S1,2 [l](x, y) =
Z
x
Z ψ(s, y)k(x, s)ds +
0
. φ,ψ S2,1 [k](x, y) =
φ(s, y)k(x, s)ds (27)
y 1
φ(s, y)l(x, s)ds
(28)
ψ(s, y)k(x, s)ds
(29)
x Z x 0
. φ,ψ S2,2 [l](x, y) =
Z
y
Z ψ(s, y)l(x, s)ds +
x
1
φ(s, y)l(x, s)ds y
(30) in their respective domains. Definition 6: Let us now define a third integral operator T¯ : X → X (for A¯1,1 : Xl → Xl , A¯1,2 : Xu → Xl , A¯2,1 : Xl → Xu , A¯2,2 : Xu → Xu , F¯1 ∈ Xl F¯2 ∈ Xu ), for all k ∈ Xl , l ∈ Xu :
T¯
k l
. ¯ k =A + F¯ l A¯1,1 A¯1,2 k F¯1 . = + ¯ A¯2,1 A¯2,2 l F2
(31)
B. Direct Transform Lemma 1: If the operator T , as defined in (19) has a unique fixed point in X (i.e. there exists a unique ζ ∈ X s.t. T ζ = ζ), then transform (6) with kernels p . =ζ (38) q . maps system (3)-(5), with γ(y) = p(1, y), into (9)-(10). . Proof: (Sketch) Condition γ(y) = p(1, y) follows directly from evaluating transform (6) at the boundary x = 1 and using the boundary conditions (4) and (10). Following a similar procedure to that used in standard backstepping, we plug (6) into (9) and (10) and, after integrating by parts some terms and changing the order of integration in some double integrals, we obtain the following set of hyperbolic PIDEs the kernels must verify: Z y px (x, y) + py (x, y) = −g(x, y) + h(s, y)p(x, s)ds 0 Z x + g(s, y)p(x, s)ds (39) y Z 1 + g(s, y)q(x, s)ds, x
where:
∀x, y ∈ [0, 1] s.t. y ≤ x, y 6= 0 Z y qx (x, y) + qy (x, y) = −h(x, y) + h(s, y)q(x, s)ds x Z 1 + g(s, y)q(x, s)ds (40) Zy x + h(s, y)p(x, s)ds,
A¯1,1 [k](x, y) Z y Z x−y . =− g(σ + x − y, s + σ)k(s + σ, σ)ds dσ 0
Z
0 y
Z
−
1−σ−x+y
h(σ + x − y, s + σ + x − y) 0
0
× k(s + σ + x − y, σ)ds dσ
0
(32) 5430
∀x, y ∈ [0, 1] s.t. x ≤ y, y 6= 1
which in turn implies
with boundary conditions: Z
x
p(x, 0) = −f (x) + p(x, y)f (y)dy 0 Z 1 + q(x, y)f (y)dy, ∀x ∈ [0, 1]
kA(ϕ − ϕ)k ˜ X ≤ 2cK (41)
x
q(x, 1) = 0,
∀x ∈ [0, 1]
then the operator T is a contraction. We start by noting that: kT ϕ − T ϕk ˜ X = kAϕ − Aϕk ˜ X = kA(ϕ − ϕ)k ˜ X (46) . Let us denote K = kϕ − ϕk ˜ X , and c defined as in the theorem statement then, after some computations we obtain the following norm estimate: kA(ϕ − ϕ)k ˜ X ≤ max{cK sup (1 + y), cK sup (1 − y)} y∈[0,1]
And therefore, if 2c < 1, T defines a contraction mapping. Corollary 2 completes the first part of the proof. The norm estimate comes from rewriting:
(42)
Integrating (39) and (40) along the level curves of x − y up to the boundary conditions (41) and (42) we obtain a set of integral equations to which the condition of the theorem guarantees a unique solution. Therefore, a suitable direct transform exists. Note that Lemma 1 can be alternatively formulated in terms of the spectrum of the operator A. As a first step toward finding concrete sufficient conditions in terms of the magnitude of the coefficients in equation (3) under which Lemma 1 can be applied we use Banach’s contraction mapping principle, see for instance [10, Theorem 3.1] to obtain the following Corollary. Corollary 2: If the operator T as defined in (19) is a contraction, then transform (6) with kernels p . = lim T n ϑ0 (43) q n→∞ . for any ϑ0 ∈ X, maps system (3)-(5), with γ(y) = p(1, y), into (9)-(10). In particular, if the operator T is a contraction, the spectral radius of A is less than 1 and {1} belongs to its resolvent set. The contraction property depends on the choice of norm for the Banach space and, even if an operator is not a contraction for a given norm, it may be a contraction for an equivalent norm. A similar result holds if there exists n ∈ N such that T n is a contraction. Using the supremum norm already defined in X, we obtain the following sufficient condition for the direct transform to exist: . Proposition 3: If the coefficients in equation (3) verify c = max{sups∈[0,1] |f (s)|, kgkXl , khkXu } < 12 , then transform (6) with kernels p . . = ζ = lim T n ϑ0 (44) q n→∞ . for any ϑ0 ∈ X, maps system (3)-(5), with γ(y) = p(1, y), into (9)-(10). Furthermore kζkX ≤ kF kX /(1 − 2c). Proof: If we can show that there exists C ∈ [0, 1) such that: kT ϕ − T ϕk ˜ X ≤ Ckϕ − ϕk ˜ X , ∀ϕ, ϕ˜ ∈ X (45)
y∈[0,1]
(47)
(48)
ζ=
∞ X
An F
(49)
n=0
and noting that it implies, using (48): kζkX ≤ kF kX
∞ X
(2c)n
(50)
n=0
this, and the condition c
0 and boundary conditions: u(1, t) = U (t)
(60)
vx (0, t) = 0
(61)
v(1, t) = 0
(62)
This system closely resembles the Korteweg-de Vries-like equation presented in [3]. The only two differences (other than notation) are the extra (destabilizing) term au(0, t) and the use of only one boundary to control the full interconnected system (instead of using one boundary of each subsystem). Solving (59) with boundary conditions (61)-(62), and plugging the solution into (58) we obtain a system of the form (3)-(4). We present simulation results for a = 1.25, b = 0.1, c = 0.01, d = 7.5. For these coefficients, a solution can still be found for both systems of integral equations (even though they are larger than the sufficient condition presented in Proposition 9) and therefore the direct and inverse transforms exist and are bounded. Figures 1 (a) and 1 (b) show the obtained direct (respectively inverse) transform kernels for this system. Figure 1 (c) shows the obtained control gain. Figure 2 shows the evolution of the state in open-loop (unstable) and closed-loop (finite-time stable).
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