Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention

Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. A characteristic example is nonautonomous and discontinuous systems due to presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists in designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). At last, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid's diffusion.


Introduction
Passivity is an important property in dynamical systems because it gives a sense on the system energy balance ( [2,23]). Roughly speaking, a system is said to be passive if it cannot produce energy on its own, and can only dissipate the energy that is stored in it at any time. Friction is a dissipative mechanism that is ubiquitous in mechanical systems. Although friction may be a desirable property (as in brakes application), it can also lead to limit cycles, undesired stick-slip motion and instabilities. This last phenomenon can be explained qualitatively due to the competition of stored elastic energy and its dissipa-tion via friction. If this stored energy can not be balanced by the frictional dissipation, tan instability will be triggered. This is the case when the frictional force decreases with slip or slip-rate and can be explained through the loss of passivity of the system. The prevention of such instabilities is the main objective in this work.
Due to its energy dissipation nature, friction has been compensated in mechanical systems using passivitybased controllers. The passivity-based control term was introduced in [26] and it has an important role in the control theory with applications to electric motors, power electronics, chemical processes and mechanical systems (see [8,19,26,31]). For the case of totally actuated mechanical systems, one can mention [10] where a LuGre (dynamic) model of friction is compensated with an observer, or [29], where the stabilization of a system with Coulomb friction is analysed using sliding-modes. For the case of a system having less control inputs than degrees of freedom (underactuated system), the Interconnection and Damping Assignment Passivity-based Control (IDA-PBC) presented in [25] was used in [30] and [9] to stabilize systems with dynamic, but not discontinuous, friction models. Furthermore, IDA-PBC presents always the problem of solving partial differential equations (PDEs) on its design, which in some cases may not have a solution.
Despite the attractiveness of passivity concepts, the classical passivity theorems [20,Chapter 6] cannot be used in the case of set-valued frictional systems (like the Coulomb friction), which is the focus of this work. Furthermore, the classical theorems cannot be applied for nonautonomous systems either. There exist some works dealing with the feedback interconnection of multivalued systems using convex analysis (see, e.g., [1,6] and a very recent monograph [7]), yet they do not take into account nonautonomous systems. For this purpose, in this work we extend the classical theorem of passivity related to the negative feedback connection between two passive systems, in such a way to cover the general class of underactuated frictional nonautonomous systems studied herein. This is accomplished by using an invariance-like principle [13,17].
Furthermore, to restore the passivity property of the system, a passivity-based control design is considered. First, stabilization to a domain of zero velocities is obtained, recovering the passivity property by properly designing the underactuated control input. Then, a tracking result over constant references is obtained by augmenting the system with integral action. Finally, dynamics in the actuation are considered and the control is designed to preserve the tracking results. The designed underactuated control is implemented in an earthquake prevention scenario of a seismic fault. This is an important and challenging example of a frictional underactuated system, where the designed control has to be able to dissipate the stored energy slowly, controlling the fast dynamics of an earthquake through a slow diffusion process. Simulations are presented to show how the passivity-based control is able to follow a slow reference instead of dissipating abruptly the stored energy, avoiding an earthquake-like behaviour.
The outline of this work is as follows. The notation and useful definitions and existing theorems are presented in Section 2. The passivity extension for the negative feedback connection between two nonautonomous discontinuous systems, the main theorem of this paper, is presented in Section 3. The frictional underactuated mechanical system description, the link between passivity and the considered friction law and the control objectives are given in Section 4. The structured design of the passivity-based control is detailed in Section 5. The presentation of the fault model and the numerical simulations are shown in Section 6. Finally, some concluding remarks are discussed in Section 7.

Preliminaries
Consider the n-dimensional space ℜ n with the Euclidean norm ||·||. Elements of ℜ n are interpreted as column vectors and (·) T denotes the vector transpose operator. The identity matrix of dimension n is denoted by I n or simply I, if the size can be trivially determined by the context. Let v ∈ ℜ n , be the function sign(·) : ℜ n → ℜ n×n , defined as sign for all i = 1, ..., n and the function |·| : ℜ n → ℜ n is de- Consider the state model given bẏ where An important passivity theorem concerns the negative feedback connection between two passive systems, H 1 and H 2 (Fig. 1). Theorem 1 [20, Chapter 6] If the system H 1 is passive with input e 1 and output y 1 , and the system H 2 is passive with input e 2 and output y 2 , then the negative feedback connection of H 1 and H 2 is passive with input u = [u 1 , u 2 ] T and output y = [y 1 , y 2 ] T .
where f 2 : ℜ n × ℜ ≥0 → ℜ n is piecewise continuous 1 in 1 A function is said to be piecewise continuous in a domain G if it is continuous in G up to a set of measure zero defined by points of discontinuity of the function.
a domain G ⊂ ℜ n × ℜ ≥0 . The above system is nonautonomous and have a set-valued right-hand side (RHS), which are accepted as a basic mathematical model of a discontinuous system (see [24] for more details on discontinuous systems). In the following, the solutions of discontinuous systems like (2) are understood in the Filippov's sense [12].
Theorem 2 (Invariance-like principle) [13,17] Let D ⊂ ℜ n be a domain containing x = 0. Suppose there exists a constant M such that ||f 2 (x, t)|| ≤ M , for almost all (x, t) ∈ D × ℜ. Let V : D × ℜ ≥0 → ℜ be a locally Lipschitz-continuous positive definite function such that Moreover, if all assumptions hold globally and W 1 (x) is radially unbounded, the statement is true for all x(t 0 ) ∈ ℜ n .

Passivity Extension for Nonautonomous Discontinuous Systems
The classical definition of passivity in Definition 1 can not be applied directly to systems in the form of (2) due to the time dependency and the discontinuous RHS (present in the system). For this purpose, we generalize Theorem 1 for systems H 1 and H 2 that can be either time-variant dynamical systems with discontinuous RHS or time-variant discontinuous memoryless functions. Note how the closed-loop model with u 1 = u 2 = 0 takes the form of (2).
Theorem 3 Assume each element of the feedback interconnection of Fig. 1 is passive and satisfies Let D ⊂ ℜ n be a domain containing x = 0 and consider the locally Lipschitz-continuous positive storage function , for all t ≥ 0 and for all x ∈ D, where W 1 (x) > 0 and W 2 (x) > 0 are continuous on D. Choose r > 0 such that B r = { x ∈ ℜ n | ||x|| ≤ r} ⊂ D and let ρ < min ||x||=r W 1 (x). Then, every bounded Filippov solutions of the closed-loop system shown in Fig.  1 with u 1 = u 2 = 0, i.e. a system of the form (2), such that Moreover, if all assumptions hold globally and W 1 (x) is radially unbounded, then the statement is true for all x(t 0 ) ∈ ℜ n .
PROOF. Taking the function V (x, t) = V 1 (x, t) + V 2 (x, t) as storage function of the closed-loop system, its derivative w.r.t. time is written aṡ which results in a classical passivity result of the feedback interconnection. Furthermore, in the case of u 1 = u 2 = 0, the derivative reads asV (x, t) ≤ −W (x) and all assumptions of Theorem 2 are fulfilled. Then, one can obtain the domain in which the system trajectories will be driven when W (x) = 0.
Remark 1 Theorem 3 is an extension of the classical result of the interconnection between two passive systems [20,Chapter 6]. Such extension covers non autonomous dynamical systems and discontinuities in both the dynamical system and the memoryless function. Theorem 1 is recovered then when both systems, H 1 and H 2 , fulfil the necessary smoothness conditions (locally Lipschitz around the origin) requested in the classical result.

Remark 2
Theorem 3 provides the domain in which the trajectories of the closed-loop system will converge to (in contrast to the stability of the origin result obtained for a passive system), which includes the origin.

Underactuated Mechanical System with Coulomb Friction
Consider a n-DOF underactuated mechanical system modelled asδ where δ ∈ ℜ n , u ∈ ℜ n , v ∈ ℜ n , represent the vectors of frictional slips, displacements and velocities (slip-rates), respectively. The state δ(t) represents the accumulated slip and it can not take negative values due to the assumption of δ i (0) = δ 0i ≥ 0 for all i ∈ [1, n]. M ∈ ℜ n×n is the inertia matrix and the termF e (u, v) ∈ ℜ n is the vector of applied forces which here are viscoelastic defined asF whereK ∈ ℜ n×n is the stiffness matrix andH ∈ ℜ n×n is the viscosity matrix. The termF r (δ, u, v,p, t) = [F r1 (δ 1 , u 1 , v 1 ,p, t), ...,F rn (δ n , u n , v n ,p, t)] T is the friction force and is written as follows [23] Fr T is an arbitrary friction function,p ∈ ℜ q is the vector of control inputs and F s = [F s1 , ..., F sn ] T is a vector of static friction coefficients. The static friction counteracts the applied forces below a certain level and, thus, it prevents slip. Note also that q < n, resulting in having more degrees of freedom (DOF) than control inputs.
The set of equilibrium points when t * = 0, i.e., δ * (0), u * (0), v * (0) of the above system is described by where δ 0 ∈ ℜ n , p 0 ∈ ℜ q and F * s ∈ ℜ n are the initial constant vectors of the slip, inputs and static friction, respectively. In the case of F ei (u * i , 0) ≥ F * si for all i ∈ [1, n] in (5),F r (δ 0 , u * , 0, p 0 , 0) = F * s and the system is on the verge of slip. We define this point as the operation point on which the system will be controlled and we shift the system to this point as follows. Let Then we obtaiṅ where This new shifted system is practically the same as (3) except for the new term F * s in the friction function F r (x 1 , x 2 , x 3 , p, t). This term represents the stored potential energy in the system at equilibrium due to the viscoelastic forces and the translation of the states, i.e., F e (u * , 0) = −Ku * = F * s . If this stored energy can not be counteracted by the friction, the system will move abruptly (instability). The prevention of such fast slip behaviour is the main objective in this work.

Passivity and Friction
The functionF (δ, v, p, t) in (5) is modelled as Coulomb friction [23,2,27] and reads as where the term µ(δ, |v| , t) ∈ ℜ n×n is defined as , are friction coefficients. Finally, σ n ∈ ℜ n is a vector containing the normal stresses as σ n = [σ n1 , ..., σ nn ] T , the matrix C p ∈ ℜ n×q is the control coefficient andp ∈ ℜ q is the vector of control inputs.
A plot ofF r (δ, u, v,p, t) defined as (5), (8) According to (7), F * s translatesF r (δ, u, v, 0, t) in the new system (6) (see Fig. 2(c-d)). As a result, the passivity property of the output h( The new shifted friction term (8) can be written as and σ ′ n = σ n − C p p 0 is a vector of constant values. If the control input p ∈ ℜ q is taken into account in (9), the original passivity property could be recovered in the shifted friction term and a stability result of system (6) could be obtained.

Control Objectives
The control objectives are stated as follows: (1) To design the control p in (6),(7),(9) able to make passive the output (2) To design an integral action to the latter control law, obtaining a reference tracking over the output error where C t ∈ ℜ q×n is a matrix to be defined and r 3 ∈ ℜ n is a vector of constant velocity references.
where C h ∈ ℜ q×q , to design the new control input p ∞ ∈ ℜ q capable to reproduce the same results as the ones obtained in objectives 2 and 3.
The first step will allow the friction to recover the lost passivity designing the input p, whereas the second step will allow us to release the stored energy of the system slowly, by choosing the reference r 3 . The final step will recover the first two steps by designing the real control input p ∞ while accounting the slow dynamics of the actuator.
A block diagram of the full passivity-based control design is shown in Fig. 3 and the closed-loop system is illustrated in Fig. 4. The description of every part of such design is explained in the following sections.
The control design will be performed under the next following minimal assumptions for the system (6): A1: The initial condition of system (6) will be the origin:   input [x 1 , x 3 ] T and l δ , l v > 0 assumed to be known constants (see Fig. 2). A6: Matrices C p , C t , C h in (9), (10), (11) have full rank.
Furthermore, C h = C T h .
Remark 3 Assumptions A2-A4 are fulfilled commonly in mechanical systems. Furthermore, µ min always exist due to thermodynamics (energy conservation).

Stabilization of the Frictional System
The feedback interconnection between a mechanical system and a frictional term, i.e. system (6), will be analysed using Theorem 3. Such interconnection can be seen in Fig. 5 and is the same as in Fig. 1 According to the definition of a sector in [20,Chapter 6] and eqs. (7), (9), Assumption A5 leads to Due to the definition of F r (x 1 , x 2 , x 3 , p, t) in (5),(7), the output y 2 when the system is in motion, i.e. x 3 = 0, will be then y 2 = [− |x 3 | , F (x 1 , x 3 , p, t) T ] T described by (9). Its passivity map reads where the sector condition (12) and Assumption A4 for the term µ(x 1 + δ 0 , |x 3 | , t) have been used.
Selecting the control input p as where λ δ , λ v are constants to be designed, the passivity map becomes which results to be passive, i.e. e T 2 y 2 ≥ 0, if the controller gains are chosen as follows Note how the designed control input p in (13) is able to inject passivity into the shifted friction term g(x 1 , x 3 , t).
A strict passivity condition can not be obtained due to the underactuation nature of the system, i.e. C p C T p ≥ 0, but this is not a critical condition for the stability result stated in the next Theorem. (14).

PROOF. Consider the positive definite function
and its time derivative along the trajectories of system (6) aṡ which results to be passive due to Assumption A3. If the controller gains are chosen as in (14), the frictional term is passive and e T 2 y 2 ≥ 0. Consequently, using Theorem (3), every bounded solution x(t) of system (6) (the feedback interconnection between two passive systems with u 1 = u 2 = 0 2n ) converges to the domain The presented stability result is not as strong as the asymptotic (or exponential) stability of the system origin. Nevertheless, recalling the definition of system (6), it results in an increasing evolution of the state x 1 (t) and the impossibility of returning it to the origin once it has started to evolve. Therefore, the obtained stability result is the best that one can obtain for these kind of frictional systems.

Tracking via Integral Control
The latter stability result does not allow the release of the stored energy in the system by letting the system to slip in a controlled manner. For this purpose, tracking will be performed in order to force the system to follow a reference, r 3 , by interconnecting the underactuated mechanical system with an integral extension of the tracking error.

Considering the new integral terṁ
where ξ ∈ ℜ q and y t is the error variable defined in (10). Following a classical integral design (see e.g. [20,Chapter 12]), let us define the tracking error variables as where i = 1, 2, 3 and x i (∞), p(∞), ξ(∞) are the steady state values of the states, the control input and the integral action, respectively.
The error dynamics iṡ due to the fact that r 3 = r 3 (∞), because r 3 is a vector of constant references. The new nonlinear function ∆F (x 1e , x 3e , p e , t) is defined as which has the same characteristics as the term (9). Consequently, the control input p e of the error dynamics in (18)- (21) can be designed as where λ ξ ∈ ℜ >0 is a gain to be designed. The first two terms of the latter control are designed to stabilize the mechanical system, equivalent to the system as in Theorem 4, while the new term includes the integral action to perform the tracking.
Let us study first the passivity map of the output where the definition of b(x 1 , x 3 , t) in (9) and the Assumption A4 were used. Clearly, this output is passive.
Defining the storage function V ξ = Cpξe 0 L(σ)dσ for the system (15). Such function is positive semidefinite due to the passive property of the output L(C p ξ e ) and its derivative reads aṡ if C t is the left pseudoinverse matrix of C p , i.e., C t is defined as in (25). Consequently, matrix C t is full rank as C p is as stated in Assumption A6. The last expression shows how the integral system is passive. Therefore, the feedback connection between the two systems will be passive.
Consequently, using Theorem (3), every bounded solution (x e (t), ξ e (t)) of system (18)- (21) converges to the In order to obtain the domain in the original states, the error x 3e must fulfil the equation Finally, the original control p results from the control (23) and the defintion of errors (16) where one can obtain expression (24) by replacing the steady state control p(∞)

Remark 4
The steady state error tracking using the designed control for the case of a time variant reference, i.e. r 3 = r 3 (t), will not be zero. The presented analysis fits only for constant references. Nevertheless, the resulting error could be improved by choosing slow enough references with low time derivatives, approximating its behaviour to constant references. One can improve also this result by adding more (passive) integrator terms to cover a wider kind of references r 3 (t), as stated in the internal model principle (e.g., [16]).

Actuator Dynamics
So far, the designed control (24) is able to either drive the system (6) states to a given domain E = { x ∈ ℜ 3n | x 3 = 0} as t → ∞, if r 3 = λ ξ = 0, or to perform a tracking over a given velocity reference if r 3 = 0 and λ ξ > 0. If now an actuator dynamics like (11) is considered in the model, p ∞ is the new control input to be designed. For this purpose, consider the control (24) as nominal then one can get the nominal controlp ∞ from (6), (10), (11), (15) and (24) as The time derivative of sign(x 3 ) is equal to zero because we are studying the case when the system is in motion (x 3 = 0).
In order to obtain the control p ∞ able to reproduce the nominal control (27), let us define the next error variablesp leading to the error dynamics from (6), (11), (15) and (27) aṡ Such error system can be seen in Fig. 3 and can be explained as the interconnection of two systems as in Fig. 1: system H 1 is defined as (33) with u 1 = 0 2n , Theorem 6 Every bounded solution (x(t), ξ(t), p(t)) of the closed-loop system (6), (15), (11) approaches to the domain with the nominalp ∞ defined as (27), fulfilling the conditions (14), λ ξ > 0 and matrix C t for the integral action (15) defined as in (25).

PROOF.
The system (29)-(32) is passive with the nominal controlp as shown in the previous tracking analysis. Therefore, the condition e T 2 y 2 ≥V +V ξ + x T 3eH x 3e is fulfilled. Thus, the passivity property must be studied now in system (33).
Defining the positive definite storage function V p = 1 2p T C −1 hp for the system (33), its derivative reads aṡ resulting to be strictly passive and, consequently, the feedback connection between the two systems will be passive.
Finally, to get the domain in which the trajectories will converge, we use Theorem 3 to obtain where the only possibility for the latter domain to be valid is if it takes the form of the one given in Theorem 6.

Earthquake Control
Consider a seismic fault as shown in Fig. 7. In this academic example, the fault is just beneath the surface and its dimensions are 3 × 3 [km 2 ] (x-and z-directions, respectively). The effective normal stress σ ′ n acting on the fault interface is assumed to vary linearly due to the lateral earth pressure. We assume also that the fault is adequately oriented in the tectonic stress regime for slip to occur. In this numerical application, the fault area is discretized into n = N x × N z = 10 × 10 elements. The above physical system can be described mathematically using eqs. (5),(6),(7),(9) where x 1 represents the slip, x 2 the displacement and x 3 the slip-rate (velocity). Several methods in the literature can be used in order to discretize the differential operator representing the underlying continuum elastodynamic problem of seismic slip (e.g., Finite Element Method, Finite Differences, Boundary Element Method, spectral methods, model reduction methods, among others [4], [5], [11] and [22]). In most cases, the resulting discretized equations will finally take the form of (6) and, consequently, the control theory presented in this work can be applied.
The actuator dynamics (11) is also considered, where the control input p ∞ represents the pressure at the peak of four wells injecting fluid to the fault (q = 4), as shown in Fig. 7. The form of eq. (11) corresponds to a finite difference approximation of the diffusion equation, a Partial Differential Equation (PDE). Extension to PDE control could also be explored [21,15,14], but this is out of the scope of the current work. The theorems developed in the previous section can be applied as the diffusion equation remains passive. Then, through the diffusion process according to equation (11), the pressure p affects the fault friction by modifying the effective normal stress σ ′ n according to Terzaghi's principle of effective stress [35]. The control configuration of the wells on the fault can be seen in Fig. 7, where their influence is defined by the definition of matrix C p in the friction term (9).
Furthermore, an even more realistic scenario will be studied where the full state x(t) is not available, but only a measured output of the system (6) as y m = C m x 3 , where y m ∈ ℜ and C m ∈ ℜ 1×n . This single output represents an average velocity over the points of the fault. Therefore, the designed pressure at the fault p = p(x) and, consequently, the designed pressure at the wells, p ∞ = p ∞ (x), have to be now a feedback of the estimated states, i.e.p = p(x) andp ∞ = p ∞ (x), respectively. The design of a high-gain observer for this purpose is shown in Appendix A.
Without a control input, system (6) is unstable, resulting in an earthquake as shown in Fig. 8 (notice the time scale in seconds). It is worth mentioning that very few works are devoted to the control of such systems. In particular, an LQR control was designed to stabilize and perform tracking of an earthquake modelled by a MIMO system in [32], whereas a double-scale asymptotic approach was employed to design a transfer function-based control in [33]. These first applications of control theory to this problem have shown that earthquakes could be controlled, at least from a mathematical point of view, but they have not accounted for underactuation, the discontinuous nature of friction and diffusion. Therefore, the presented theoretical development a more realistic treatment of the problem.
The objective in the sequel is to implement the designed control law (27) and (34) with the integral dynamics (15) and (25), to drive the system states to the domain as t → ∞. If one chooses a slow reference r 3 , this will result in a slow-aseismic response of the system.
The desired reference r 3 is a smooth function reading as r 3 =ṙ(t)I n , r(t) = d max s 3 (10 − 15s + 6s 2 ), (35) where s = t/t op , d max is the target displacement and t op is the operational time of the tracking strategy. The constant d max is the distance the fault slides dynamically in order to reach its sequent stable equilibrium point. For this case, we selected d max = 500 [mm] (approximately two times equal to the seismic slip developed when the system is not controlled) and t op = 360 [days]. The desired total time is considerably larger than the fast slip in the earthquake behaviour (≃ 15 [s]) in order to slowly release and dissipate the seismic energy. Shorter t op can be chosen as well (e.g., of the order of hours) but in this case, the pressure at the tips of the wells, p ∞ , would be very high due to slow dynamics of the diffusion process (see (11)). The characteristic time of the diffusion process (see (11)) depends on the hydraulic diffusivity parameter, which has been taken equal to C h = 2.88 × 10 −7 I (representing injection in a sandstone) and a distance of the injection point to the fault equal to 1.5 [km].

Numerical Results
In order to illustrate the performance of the proposed passivity-based control strategy, simulations have been made based on the shifted system described by (5),(6),(7),(9), (11). Such simulations were performed using the Differential Equations package of Julia [28] and an initial condition x(0) = 0 3n . In particular the TRBDF2 algorithm was used with events for detecting the transition between stick to slip and satisfy (5).
The results are presented in Figs. 9-10. The states now follow successfully a slow reference, dissipating the stored energy aseismically (notice the time scale of days in Figs. 9-10 instead of seconds of the instability Fig. 8). The discontinuous-like behaviour shown in the velocity x 3 is due to the stick-slip motion over the fault, resulting over the fact that Coulomb friction is a set-valued function (see (5), (7), (9)). Nevertheless, the designed control is able to drive the tracking error C t (r 3 − x 3 ) close to zero, using the estimated states from the highgain observer (errors shown in Fig. 10 left and middle plots). Finally, the control signal from the wells p ∞ and the pressure p applied to the fault are depicted in Fig.  10, which show reasonable amplitudes to be used in real actuators.

Conclusions
In this work, we extend the classic theorem for the negative feedback interconnection of passive systems to account for nonautonomous and set-valued (discontinuous) ODEs. This generalization is based on an invariance-like principle and it allows the synthesis of controllers for underactuated mechanical systems with Coulomb friction. Based on this generalization, stabilization of the states to a domain of zero velocities and tracking over constant references, while assuming actuation dynamics, are achieved. The designed control  Fig. 9. Controlled system: The system is tracked aseismically to new (stable) equilibrium state. Note the difference on the time scale (days) with respect to the earthquake-like behaviour described in Fig. 8 (seconds). The same color code was used. injects passivity to (unstable) frictional systems using less control inputs than degrees of freedom. It also need minimum information about the plant, i.e., the minimum bound of the friction coefficient, the belonging sector of the friction law and the coefficient of the actuator dynamics. This in contrast with the IDA-PBC where it is necessary to solve PDEs, or other existing more involved approaches. In order to test the derived control strategy, an earthquake prevention case study is considered. In particular, the unstable dynamic slip of a mature seismic fault is prevented by injecting fluid through four wells located far from the fault. Numerical simulations show the successful tracking of the system output over a reference, despite the presence of the slow dynamics due to diffusion process and uncertainties with respect to the Coulomb frictional rheology, the (visco-)elastodynamic properties of the system and diffusivity of the fluid pressure in the rock. The results were accomplished with minimum measurements and the control signals (pressures) were of acceptable amplitudes for the actuators (pumps). This results in a promising solution for earthquake prevention and control.
Is it worth noticing that the separation principle for nonlinear systems (e.g., [3]) consider systems with sufficiently smooth right-hand sides. Therefore, the analysis of the full closed loop-system (plant, control and highgain observer) with discontinuous RHS presented in this paper, remains as future work.