Feedback Stabilization of Tank-Liquid System with Robustness to Wall Friction

We solve the feedback stabilization problem for a tank, with friction, containing a liquid modeled by the viscous Saint-Venant system of Partial Differential Equations. A spill-free exponential stabilization is achieved, with robustness to the wall friction forces. A Control Lyapunov Functional (CLF) methodology with two different Lyapunov functionals is employed. These functionals determine specific parameterized sets which approximate the state space. The feedback law is designed based only on one of the two functionals (which is the CLF) while the other functional is used for the derivation of estimates of the sup-norm of the velocity. The feedback law does not require the knowledge of the exact relation of the friction coefficient. Two main results are provided: the first deals with the special case of a velocity-independent friction coefficient, while the second deals with the general case. The obtained results are new even in the frictionless case.

of the forces that are developed due to wall friction (see [30]). In the mathematical literature, researchers have used various relations in order to express the effect of wall friction; see for example [4,5,15,17,18]. It is clear that wall friction forces are always present and sometimes their effect is not negligible.
In this paper we solve the feedback stabilization problem for a tank, with friction, containing a liquid modeled by the viscous Saint-Venant system of Partial Differential Equations (PDEs). A spill-free exponential stabilization is achieved, with robustness to the wall friction forces. A CLF methodology with two different Lyapunov functionals is employed. These functionals determine specific parameterized sets which approximate the state space. The feedback law is designed based only on one of the two functionals (which is the CLF) while the other functional is used for the derivation of estimates of the sup-norm of the velocity. The feedback law does not require the knowledge of the exact relation of the friction coefficient. Two main results are provided: the first (Thm. 3.6) deals with the special case of a velocity-independent friction coefficient, while the second (Thm. 3.9) deals with the general case. The obtained results are new even in the frictionless case (Cor. 3.8).
The problem studied in the present work is similar to the problem added in [22] and consequently there are some similarities between the present work and [22] (there are less similarities with [23] since [23] focuses on output feedback laws and numerical aspects). However, there are important differences between the present work and [22]. Indeed, the absence of wall friction in [22] makes the analysis simpler. In the present work the presence of wall friction forces us to use a modified CLF (see Sect. 3.1 below) which is not adequate for the derivation of stability bounds (in sharp contrast with the analysis in [22]). An additional Lyapunov functional has to be used that can provide bounds for the sup-norm of the fluid velocity. However, the use of the additional functional allows us to derive stability estimates in stronger norms (e.g., we provide stability estimates in the H 1 norm and the sup-norm of the velocity while in [22] the stability estimates gave bounds only for the L 2 norm of the velocity; see Rem. 3.7 below).
The paper is structured as follows. In Section 2 we describe the control problem and its main objective. In Section 3 we present the statements of the main results of this work. We also define the state space of the control problem and we present functionals which are essential for the derivation of the stability estimates. This section also includes auxiliary results which are used for the proofs of the main results. Section 4 is devoted to the proofs of all the results that we present in Section 3. Finally, in Section 5 we give the conclusions of this work and suggest topics for future research.
Notation Throughout this paper, we adopt the following notation. * R + = [0, +∞) denotes the set of non-negative real numbers. * Let S ⊆ R n be an open set and let A ⊆ R n be a set that satisfies S ⊆ A ⊆ cl(S). By C 0 (A; Ω), we denote the class of continuous functions on A, which take values in Ω ⊆ R m . By C k (A; Ω), where k ≥ 1 is an integer, we denote the class of functions on A ⊆ R n , which takes values in Ω ⊆ R m and has continuous derivatives of order k. In other words, the functions of class C k (A; Ω) are the functions which have continuous derivatives of order k in S = int(A) that can be continued continuously to all points in ∂S ∩ A. When Ω = R then we write C 0 (A) or C k (A). When I ⊆ R is an interval and G ∈ C 1 (I) is a function of a single variable, G (h) denotes the derivative with respect to h ∈ I.
* Let I ⊆ R be an interval, let a < b be given constants and let u : I × [a, b] → R be a given function. We use the notation u[t] to denote the profile at certain t ∈ I, i.e., (u[t])(x) = u(t, x) for all x ∈ [a, b]. When u(t, x) is (twice) differentiable with respect to x ∈ [a, b], we use the notation u x (t, x) (u xx (t, x)) for the (second) derivative of u with respect to x ∈ [a, b], i.e., u x (t, x) = ∂ u ∂ x (t, x) (u xx (t, x) = ∂ 2 u ∂ x 2 (t, x)). When u(t, x) is differentiable with respect to t, we use the notation u t (t, x) for the derivative of u with respect to t, i.e., u t (t, x) = ∂ u ∂ t (t, x). * Let a < b be given constants. For p ∈ [1, +∞), L p (a, b) is the set of equivalence classes of Lebesgue measurable functions u : (a, b) → R with (|u(x)|) < +∞.
For an integer k ≥ 1, H k (a, b) denotes the Sobolev space of functions in L 2 (a, b) with all its weak derivatives up to order k ≥ 1 in L 2 (a, b).

Description of the problem
We consider a one-dimensional model for the motion of a tank. The tank contains a viscous, Newtonian, incompressible liquid. The presence of viscosity, denoted by µ, is crucial in our approach to the problem. We rely on the viscosity for establishing a region of attraction and we use the viscosity as a gain in the controller on the difference between the boundary liquid levels. The tank is subject to a force that can be manipulated. We assume that the liquid pressure is hydrostatic and consequently, the liquid is modeled by the one-dimensional (1-D) viscous Saint-Venant equations, whereas the tank obeys Newton's second law and consequently we consider the tank acceleration to be the control input.
The control objective is to drive asymptotically the tank to a specified position without the liquid spilling out and having both the tank and the liquid within the tank brought to rest. Let the position of the left side of the tank at time t ≥ 0 be a(t) and let the length of the tank be L > 0 (a constant). The equations describing the motion of the liquid within the tank are for t > 0, z ∈ (a(t), a(t) + L) where H(t, z) > 0, u(t, z) ∈ R are the liquid level and the liquid velocity, respectively, at time t ≥ 0 and position z ∈ [a(t), a(t) + L], κ ∈ C 0 ((0, +∞) × R; R + ) is the friction coefficient that depends on the liquid level and the relative velocity of the fluid with respect to the tank, while g, µ > 0 (constants) are the acceleration of gravity and the kinematic viscosity of the liquid, respectively. Various (empirical) relations have been used for the friction coefficient in the literature: -in [4,18] the authors use the relation κ(h, v) = c f |v|, where c f > 0 is a constant, -in [5] the authors use the relation κ(h, v) = r 0 + r 1 h |v|, where r 0 > 0, r 1 ≥ 0 are constants, -in [15] the authors use the relation κ(h, v) = rh −1/3 (b + 2h) 4/3 |v|, where r > 0 is a constant and b > 0 is the (constant) width of the tank, -in [17] the authors derive the velocity-independent relation κ(h) = 3µc 3µ + 4ch , where c > 0 is a constant.
It is clear that there is considerable uncertainty in the friction coefficient.
The liquid velocities at the walls of the tank must coincide with the tank velocity, i.e., we have: where w(t) =ȧ(t) is the velocity of the tank at time t ≥ 0. Moreover, since the tank acceleration is the control input, we getä where −f (t), the control input to the problem, is equal to the force exerted on the tank at time t ≥ 0 divided by the total mass of the tank. The conditions for avoiding the liquid spilling out of the tank are: where H max > 0 is the height of the tank walls. Applying the transformation where a * ∈ R is the specified position (a constant) to which we want to bring (and maintain) the left side of the tank, we obtain the model:ξ where the control input f appears additively in the second equation of (2.7) and multiplicatively in (2.9). Moreover, the conditions (2.5) for avoiding the liquid spilling out of the tank become: max (h(t, 0), h(t, L)) < H max , for t ≥ 0 ---Condition for no spilling out (2.11) We consider classical solutions for the PDE-ODE system (2.7)-(2.10), i.e., we consider functions ∈ C 2 ((0, L)) for each t > 0, which satisfy equations (2.7)-(2.10) for a given input f ∈ C 0 (R + ).
Using (2.8) and (2.10), we prove that for every solution of (2.7)-(2.10) it holds that d dt for all t > 0. Therefore, the total mass of the liquid m > 0 is constant. Therefore, without loss of generality, we assume that every solution of (2.7)-(2.10) satisfies the equation The open-loop system (2.7)-(2.10), (2.12), i.e., system (2.7)-(2.10), (2.12) with f (t) ≡ 0, allows a continuum of equilibrium points, namely the points where h * = m/L. We assume that the equilibrium points satisfy the conditions for no spilling out (2.11), i.e., h * < H max . Our objective is to design a feedback law of the form , ξ(t), w(t)) , for t > 0, (2.15) which achieves stabilization of the equilibrium point with ξ = 0. Moreover, we additionally require that the "spill-free condition" (2.11) holds for every t ≥ 0. The existence of a continuum of equilibrium points for the open-loop system given by (2.13), (2.14) implies that the desired equilibrium point is not asymptotically stable for the open-loop system. Thus, the described control problem is far from trivial.

The Control Lyapunov Functional (CLF)
Let k, q > 0 be position error and velocity gains (to be selected). Define the set S ⊂ R 2 × C 0 ([0, L]) 2 : We define the following functionals for all (ξ, w, h, v) ∈ S: where δ > 0 and h * = m/L. Moreover, define for all (ξ, w, h, v) ∈ S with v ∈ H 1 (0, L) the functional: where β, γ > 0. We notice that: -the functional E is the mechanical energy of the liquid within the tank. Indeed, notice that E is the sum of the potential energy ( 2 ) and the kinetic energy ( -the functional W is a kind of mechanical energy of the liquid within the tank and has been used extensively in the literature of isentropic, compressible liquid flow (see [21,22] and references therein), -the functional V is a linear combination of the mechanical energies E, W and a Lyapunov function for the tank ( -the functional U is a functional that can provide bounds for the sup-norm of the fluid velocity (due to the fact that v We intend to use the functional V (ξ, w, h, v) defined by (3.2) as a CLF for the system. However, the functional V (ξ, w, h, v) can also be used for the derivation of useful bounds for the function h. This is guaranteed by the following lemma.
Let G −1 : R → R be the inverse function of G and define Then for every (ξ, w, h, v) ∈ S, the following inequality holds where the functions p i : R + → R (i = 1, 2) are defined by the following formulae for all s ≥ 0: Remark 3.2. It follows from (3.9), (3.6), (3.7) and the fact that h * = m/L that p 1 (V (ξ, w, h, v)) > 0 when . Definitions (3.9) imply that p 2 : R + → R is an increasing function while p 1 : R + → R is a decreasing function.
Notice that Lemma 3.1 gives tighter bounds for the liquid level than Lemma 1 in [22]. Moreover, Lemma 1 in [22] can be applied only for the case δ = 1, while here Lemma 3.1 can be applied for all δ > 0.

The state space and useful subsets of the state space
In order to be able to satisfy the conditions for no spilling out (2.11) we need to restrict the state space. This becomes clear from the fact that the set S contains states that violate the conditions for no spilling out (2.11). Define Notice that definition (3.11), the fact that h * < H max and Lemma 3.1 imply that for all (ξ, w, h, v) ∈ S with V (ξ, w, h, v) < R it holds that Therefore, the conditions for no spilling out (2.11) are automatically satisfied when (ξ, w, h, v) ∈ S with V (ξ, w, h, v) < R.
Remark 3.3. It should be noticed that R depends crucially on H max , h * and δ. Obviously, if H max or H max − h * are small then R is low because the control problem becomes more challenging and therefore the initial conditions get more restricted by the inequality V (ξ, w, h, v) < R. The dependence of R on δ is non-monotonic and very complicated. When δ → +∞ then we may have R → +∞ (when The case H max ≤ 2h * is the more challenging case and in this case R becomes restricted when δ → +∞. It should be noticed that R → 0 as µ → 0: this fact shows the crucial role that the viscosity plays in the behavior of the system and in our analysis. Notice that since Lemma 3.1 gives tighter bounds for the liquid level than Lemma 1 in [22], inequality (3.12) is more accurate than the corresponding inequality in [22] (which can be applied only for the case δ = 1).
The state space of system (2.7)-(2.10), (2.12) is considered to be the set X. More specifically, we consider as state space the metric space X ⊂ R 2 × H 1 (0, L) × L 2 (0, L) with metric induced by the norm of the underlying normed linear space R 2 × H 1 (0, L) × L 2 (0, L), i.e., However, in order to construct a robust feedback law we need to approximate the state space from its interior by using certain parameterized sets that allow us to obtain useful estimates: 1) Interior approximations in terms of the state variables: Clearly,X(0) = X(0, +∞) = X and X(ω 1 , ω 2 ) ⊆X(ω 1 ) ⊆ X for all ω 1 ∈ [0, h * ], ω 2 ≥ 0. The physical meaning of the sets X(ω 1 , ω 2 ),X(ω 1 ) is clear: the state h (fluid level) is constrained to take values in [ω 1 , H max ) and in the case of X(ω 1 , ω 2 ) the state v (fluid velocity) is constrained to take values in [−ω 2 , ω 2 ].
2) Interior approximations in terms of sublevel sets of the Lyapunov functionals: Clearly, definition (3.5) implies that X U (r) ⊆ X V (r) for all r ≥ 0. Moreover, inequalities (3.12) imply that See also Figure 1. The following proposition allows us to guarantee an additional inclusion.

Main results
Now we are in a position to state the first main result of the present work, which deals with a special case of the friction coefficient κ; namely, the case where the function κ ∈ C 0 ((0, +∞) × R; R + ) satisfies the following assumption.
(H) There exists a continuous, non-increasing function K : (0, H max ] → R + such that for every ω ∈ (0, h * ] the following inequality holds: Assumption (H) allows us to achieve stabilization in the setX (ω) defined by (3.14) for every ω ∈ (0, h * ]. More specifically, we achieve exponential stabilization in the set X V (r) defined by (3.16) for all r ∈ [0, R) with p 1 (r) ≥ ω (for which X V (r) ⊆X (ω); recall inclusion (3.18)). This is shown by the following theorem.
For example, the velocity-independent friction coefficient κ(h) = 3µc 3µ + 4ch , where c > 0 is a constant, which was derived in [17] satisfies Assumption (H) with K(ω) = 3µc b) It should be noticed that estimate (3.29) Prop. 3.4) provides the following estimate for the sup-norm of the fluid velocity: c) Theorem 3.6 guarantees robust exponential stabilization of the state by means of the nonlinear feedback law (3.27). Here, the term "robust" refers to robustness with respect to friction coefficients that satisfy Assumption (H). d) The control parameters are σ, k, q, δ > 0. It should be noticed that there are only two restrictions for the control parameters: (3.24) which implies that δ must be sufficiently large to overcome friction (see (3.23), (3.24)) and (3.25) which implies that the ratio k/q must be sufficiently small. Definition (3.26) shows dependence of the upper bound for the ratio k/q on r, δ, µ, σ, g, L; consequently, (3.25) is a fundamentally nonlinear gain restriction. The risk of spillage limits the gain on the tank position, even though the gain on the tank velocity is unrestricted. The risk of spillage forces us to be patient in reducing the distance to the target position. e) How large is the set X V (r) for which robust exponential stabilization is achieved? As we noticed above, Proposition 3.5 guarantees that the set X V (r) for r > 0 contains a neighborhood of 0, 0, h * χ [0,L] , 0 (in the topology of X with metric induced by the norm X defined by (3.13)). However the size of the set X V (r) depends on r ∈ [0, R) that satisfies p 1 (r) ≥ ω and on δ, q, k (recall definitions (3.2), (3.16)). The dependence of X V (r) on q, k is clear: the larger q (or k) the smaller the set X V (r). However, the dependence of X V (r) on ω ∈ (0, h * ] (for which p 1 (r) ≥ ω and (3.24) must hold; thus ω affects both r and δ), on δ (through (3.2) and through the dependence of R on δ; see Remark 3.3) is not clear: it is a very complicated, non-monotonic dependence which cannot be described easily. When µ → 0 it holds that R → 0 (recall Rem. 3.3) and consequently (since r ∈ [0, R)) r → 0. Therefore, when the viscosity tends to zero the set X V (r) tends to the singleton {(0, 0, h * χ [0,L] , 0)}. f) It should be noticed that the feedback law (3.27) does not require knowledge of the exact relation that determines the friction coefficient. The feedback law simply requires to know only a velocity-independent upper bound of the friction coefficient (see also (a)). Moreover, the feedback law (3.27) does not require the measurement of the whole liquid level and liquid velocity profile and requires the measurement of only four quantities: -the tank position ξ(t) and the tank velocity w(t), -the liquid level difference at the tank walls h(t, L) − h(t, 0). Theorem 3.6 shows that robustness is achieved by increasing the gain σ (δ + 1) of the total liquid momentum in the feedback law. However, Theorem 3.6 can also be applied to the frictionless case and provides a novel result even in this case. where R > 0 is defined by (3.11). Pick arbitrary σ, k, q > 0 for which (3.25) holds. Then there exist constants M, λ,M ,λ > 0 for which property (P) holds. Corollary 3.8 is an improved result compared to Theorem 1 in [22] due to the following facts: -Corollary 3.8 provides a larger family of feedback laws that stabilize the PDE-ODE system (2.7)-(2.10), (2.12) compared to Theorem 1 in [22], since the feedback laws provided by Theorem 1 in [22] correspond to the feedback law (3.27) with δ = 1, -Corollary 3.8 provides the exponential decay estimates (3.28), (3.29) while Theorem 1 in [22] provides only the exponential decay estimate (3.28).
On the other hand, Theorem 3.6 applies to more regular solutions than Theorem 1 in [22]. More specifically, property (P) holds for classical solutions of system (2.7)-(2.10), (2.12) and (3.27) with Our second main result of the present work deals with the general case of the friction coefficient κ ∈ C 0 ((0, +∞) ×R; R + ) without any further assumption. We achieve stabilization in the set X(ω 1 , ω 2 ) ⊆ X defined by (3.15) for every ω 1 ∈ [0, h * ], ω 2 ≥ 0. More specifically, we achieve exponential stabilization in the set X U (r) defined (3.21)). This is shown by the following theorem.
Pick arbitrary δ > 0 that satisfies Pick arbitrary constants σ, q, k > 0 with Then for every r ∈ [0, R) with where p 1 , R are defined by (3.9), (3.11), there exist constants M, λ, M , λ > 0 with the following property: (P')Every classical solution of the PDE-ODE system (2.7)-(2.10), (2.12) and Remark 3.10 (Remarks on Theorem 3.9). a) Theorem 3.9 guarantees robust exponential stabilization of the state by means of the nonlinear feedback law (3.27). Again, the term "robust" refers to robustness with respect to the friction coefficient.
b) The control parameters are σ, k, q, δ > 0. As in Theorem 3.6, here there are also two restrictions for the control parameters: (3.31) which implies that δ must be sufficiently large to overcome friction (see (3.30), (3.31)) and (3.32) which implies that the ratio k/q must be sufficiently small. Definition (3.33) shows dependence of the upper bound for the ratio k/q on r, δ, µ, σ, g, L; consequently, (3.32) is a fundamentally nonlinear gain restriction. Again, as we remarked in Theorem 3.6, the risk of spillage limits the gain on the tank position, even though the gain on the tank velocity is unrestricted. c) How large is the set X U (r) for which robust exponential stabilization is achieved? Proposition 3.5 and definition (3.5) guarantees that the set X U (r) for r > 0 contains a neighborhood of 0, 0, , where X is defined by (3.13)). The size of the set X U (r) depends on r ∈ [0, R) that satisfies (3.34) and on β, γ, δ, q, k (recall definitions (3.2), (3.5), (3.17)). The dependence of X U (r) on q, k is clear: the larger q (or k) the smaller the set X U (r). However, the dependence of X U (r) on ω 1 ∈ (0, h * ) and ω 2 > 0 (through (3.30), (3.31), (3.34) and (3.35)), on β, γ, δ (through (3.2), (3.5), (3.17) and through the dependence of R on δ; see Remark 3.3) is not clear: it is a very complicated, non-monotonic dependence which cannot be described easily. When µ → 0 it holds that R → 0 (recall Rem. 3.3) and consequently (since r ∈ [0, R)) r → 0. Therefore, when the viscosity tends to zero the set X U (r) tends to the singleton In the proof of both Theorem 3.6 and Theorem 3.9, we are working with two Lyapunov functionals: (3.5). However, the feedback law is designed by using V (ξ, w, h, v) only (it is a feedback law of L g V -type). In other words, the CLF is V (ξ, w, h, v) and the functional U (ξ, w, h, v) is used only for the derivation of estimate (3.29) and the derivation of estimates of the sup-norm of the velocity (used in the proof of Thm. 3.9). Therefore, a question arises: "Why don't we use U (ξ, w, h, v) as a CLF and design the feedback law based on U (ξ, w, h, v)?" The answer is clear: if we design the feedback law (of L g V -type) with U (ξ, w, h, v) as a CLF then the feedback law will contain terms with spatial derivatives of the velocity. Therefore, the measurement requirements of the feedback law will be far more demanding. We avoid the measurement of spatial derivatives of the velocity by using V (ξ, w, h, v) as a CLF. However, since V (ξ, w, h, v) cannot give us the useful estimate (3.29) we also use U (ξ, w, h, v). This situation is something that cannot happen in finite-dimensional systems: a CLF for a finitedimensional system can provide all estimates for the solution. Therefore, in the infinite-dimensional case we may need one functional to play the role of a CLF and a different functional for the derivation of useful stability estimates. This situation has been observed again in [20] in the context of a nonlinear reaction-diffusion PDE, where the CLF could only provide stability estimates for the L 2 norm of the state and different methodologies (Stampacchia's method and an additional functional) were used for the derivation of stability estimates in the sup-norm and the H 1 norm of the state. The advice for the researcher that arises from these studies is clear: -Use as a CLF a functional that can provide bounds for the state norm in a functional space of minimal regularity. Such a CLF will minimize the measurement requirements of the feedback law. -Use different functionals in order to obtain bounds for the state norm in more demanding functional spaces.

Auxiliary results
For the proof of Theorem 1, we need some auxiliary lemmas. The first auxiliary lemma provides formulas for the time derivatives of the energy functionals defined by (3.3), (3.4).
The two next auxiliary lemmas provide useful inequalities for the CLF V defined by (3.2).
Inequality (3.42) provides an estimate of the dissipation rate of the Lyapunov functional for the closed-loop system (2.7)-(2.10), (2.12) with (3.27). On the other hand, inequalities (3.43) provide estimates of the Lyapunov functional in terms of the norm of the state space.
The following two lemmas are technical lemmas which provide useful differential inequalities.
Proof of Lemma 3.12. Let arbitrary (ξ, w, h, v) ∈ X with v ∈ H 1 (0, L) and V (ξ, w, h, v) < R be given. Due to the fact that Using the inequality Since v(0) = v(L) = 0 by virtue of Wirtinger's inequality and (3.8) we have: Combining the above inequality and (3.8) we get Using (4.21), (4.22) and (4.23) we obtain Since p 2 : R + → R is an increasing function and p 1 : R + → R is a decreasing function, it follows from definition (4.25) that Λ : [0, R) → (0, +∞) is a non-decreasing function. The proof is complete.

Concluding remarks
This paper showed that a simple modification of the nonlinear feedback law proposed in [22], namely an increase of the gain of the liquid momentum, can guarantee robustness with respect to wall friction. The constructed stabilizing feedback law does not require exact knowledge of the friction coefficient (for which there is considerable uncertainty). The obtained robustness results are not a surprise: Lyapunov function(al)s are among the most important tools of guaranteeing robustness and the feedback laws were designed by means of the CLF methodology. Therefore, it is justified to claim that robustness is an intrinsic property for the CLF methodology-a well-known fact in the literature of finite-dimensional systems. The results of the present paper confirm this fact for the infinite-dimensional case of systems described by PDEs. The present paper did not study the existence/uniqueness issue for the solutions of the closed-loop system. This is a completely different topic for future research. To this purpose, ideas contained in [24,31] can be used. However, the obtained stability estimates will certainly help the analysis. Another open issue is the construction of numerical schemes for the numerical approximation of the solutions of the closed-loop system. The issue was studied in [23], where we noticed the lack of numerical schemes that can be applied to the closed-loop system. Again, the stability estimates provided in the present work can help the analysis. Even in the preliminary numerical study contained in [23] the time step was selected based on the discretized version of the CLF. We expect that the Lyapunov functionals that were provided in the present work will help the stabilization of the numerical schemes that will be used for the approximation of the solutions of the closed-loop system.