Energy shaping control of underactuated mechanical systems with fluidic actuation

Energy shaping is a remarkably effective control strategy which can be applied to a wide range of systems, including underactuated mechanical systems. However, research in this area has generally neglected actuator dynamics. While this is often appropriate, it might result in degraded performance in the case of fluidic actuation. In this work we present some new results on energy shaping control for underactuated mechanical systems for which the control action is mediated by a pressurized ideal fluid. In particular, we introduce an extended multi‐step energy shaping and damping‐assignment controller design procedure that builds upon the Interconnection‐and‐damping‐assignment Passivity‐based‐control methodology in a modular fashion to account for the pressure dynamics of the fluid. Stability conditions are assessed with a Lyapunov approach, the effect of disturbances is discussed, and the case of redundant actuators is illustrated. The proposed approach is demonstrated with numerical simulations for a modified version of the classical ball‐on‐beam example, which employs two identical cylinders, either hydraulic or pneumatic, to actuate the beam.


INTRODUCTION
Energy shaping control has been successfully applied to a wide range of systems, including mechanical systems, 1,2 electro-mechanical and power systems, 3 nonholonomic systems described by kinematic models, 4 and recently soft continuum manipulators. 5 The Interconnection-and-damping-assignment Passivity-based control (IDA-PBC) methodology is particularly popular since it provides a physical interpretation of the control action within the port-Hamiltonian framework, and it does not rely on high gains. 6 Initial research on IDA-PBC focused on systems free from disturbances and from physical damping. The effect of damping was then investigated in References 7 and 8, while the effect of dissipative forces was accounted for in Reference 9. Recently, more sophisticated designs were proposed to compensate the effects of matched disturbances 10 and of unmatched disturbances under some structural assumptions. 11 In addition, the effect of input saturation was investigated in Reference 12. Nevertheless, most of the research on IDA-PBC for mechanical systems has neglected actuator dynamics. A notable result in this respect is the study of weakly-coupled electro-mechanical systems presented in Reference 13. Although neglecting the actuator dynamics might be appropriate in some cases, provided for instance that the actuators have higher bandwidth compared to the mechanical system, 14 it might result in degraded performance in the case of fluidic actuation. With the term fluidic actuation we refer to all those cases where the control input acts on the system by means of a pressurized ideal fluid, either liquid or gas. Typical examples are hydraulic actuators, which are ubiquitous in tilt trailers, excavators, and cranes, 15,16 and pneumatic actuators, which are commonly employed in factory automation and in lightweight or compliant robots. 17,18 Fluidic actuation typically consists of a pressure source, supply pipes, a control valve, and a pump producing the control action in the form of a flow rate, and a cylinder transferring the forces to the mechanism. In some applications, including hydraulic tilt trailers and cranes, two or more cylinders can be arranged in antagonistic pairs to provide rotational motion while also resulting in redundant actuation.
From a theoretical viewpoint, fluidic actuation introduces a further challenge in the control of underactuated mechanical systems, since the control input is mediated by the pressure dynamics of the fluid. The study of systems characterized by fluid-structure interaction was first approached with a port-Hamiltonian model in Reference 19. Other notable results in this respect are the passivity-based control of a continuous-stirred tank reactor, 20 and the study of the stability conditions for boundary control in a 1-D spatial domain. 21 Port-Hamiltonian modeling and boundary energy shaping control of a compressible fluid in 1-D were presented in Reference 22. More recently, port-Hamiltonian modeling of two-phase flows was investigated in Reference 23. The passivity based control of a fully actuated double-acting hydraulic cylinder was outlined in Reference 24 by employing a locally linear dynamic assignment. Finally, the energy shaping control of a fully actuated hydraulic drive was presented in Reference 25, however the compressibility of the liquid was neglected. Besides energy shaping approaches, the study of fluid-structure interactions has also motivated research on systems characterized by cascades of partial-differential-equations (PDEs) by employing backstepping transformations and averaging in infinite dimensions. 26 Additionally, the open-loop control of a fluid in a pipe was investigated in Reference 27 by employing a flatness-based approach, while optimal control techniques were employed for the velocity-field control of an autonomous excavator fully actuated by hydraulic cylinders in Reference 16. In summary, the energy shaping control of underactuated mechanical systems with fluid-structure interactions has not been fully addressed to date. In our recent work, 28,29 we have investigated the energy shaping control of soft continuum manipulators with fluidic actuation. Nevertheless, the resulting controllers are only applicable to a narrow class of systems, and they do not include damping assignment, which is an integral part of IDA-PBC.
This work investigates the energy shaping control of underactuated mechanical system with fluidic actuation and presents the following new results.
• An extended multi-step energy shaping controller design procedure for underactuated mechanical systems with fluidic actuation is detailed. The proposed approach comprises potential energy shaping, kinetic energy shaping without imposing additional restrictions on the structure of the inertia matrix, and damping assignment, which was absent in References 28 and 29. In addition, it is applicable in the presence of redundant actuators. As a result, new matching conditions are obtained, while the PDEs that characterize the IDA-PBC for direct actuation are preserved.
• A new nonlinear control law, which applies to fluidic actuators characterized by a nonlinear relationship between volume and position is constructed, thus catering for different arrangements of hydraulic or pneumatic cylinders. Both isothermal liquids and ideal gases are considered.
• Stability conditions are assessed with a Lyapunov approach. The effect of vanishing bounded disturbances is discussed and compared to the baseline IDA-PBC.
• The proposed approach is demonstrated with numerical simulations for a modified ball-on-beam system which is actuated by two identical cylinders. Both hydraulic and pneumatic actuation are illustrated. The controller is compared with a traditional IDA-PBC design that neglects the pressure dynamics, and with an alternative algorithm constructed using a backstepping approach.
The rest of this article is organized as follows: the problem formulation is outlined in Section 2; the main result is detailed in Section 3; simulation results are presented in Section 4; concluding remarks are discussed in Section 5.

PROBLEM FORMULATION
A brief overview of IDA-PBC 6 is provided for completeness. Subsequently, the system model considered in this work is defined.

Overview of IDA-PBC
Consider an underactuated mechanical system with n degrees-of-freedom (DOFs) characterized by the Hamiltonian where the inertia matrix is M(q) = M(q) T > 0, and Ω(q) describes the potential energy. For ease of notation, M and M(q) are used interchangeably throughout the article. The system states are the position q ∈ R n and the momenta p = Mq ∈ R n . The system dynamics expressed in port-Hamiltonian form in the absence of disturbances is The control input is u(t) ∈ R m and the input matrix is G (q) ∈ R n×m , with rank (G) = m < n for all q ∈ R n in case of underactuation. The term I indicates an identity matrix of appropriate dimension, D = D T ≥ 0 is the damping matrix, ∇ q H is the vector of partial derivatives of H with respect to q, and ∇ p H is the vector of partial derivatives with respect to p. The control aim of IDA-PBC corresponds to stabilizing the equilibrium (q, p) = (q * , 0), which can be unstable in open-loop but satisfies the admissibility condition ∇ q Ω(q * ) = 0 (i.e., it is an extremum of Ω), thus it is a regulation problem. The IDA-PBC control law is constructed to achieve the closed-loop dynamics where The potential energy Ω d should be defined such that it admits a strict minimizer q * = argmin (Ω d ), thus the conditions ∇ q Ω d (q * ) = 0 and ∇ 2 q Ω d (q * ) > 0 should be verified. Introducing the pseudo-inverse G † = ( G T G ) −1 G T , the IDA-PBC control law that achieves the closed-loop dynamics (3) is expressed as the sum of an energy shaping component u es , which assigns the closed-loop equilibrium q * , and a damping-assignment component u di , which injects damping in the system through the parameter k v , that is To ensure matching between (2) and (3) on the unactuated states, M d , J 2 , and Ω d should satisfy the set of PDEs where G ⊥ is a full-rank left annihilator of G, that is G ⊥ G = 0 and rank ( G ⊥ ) = n − m. If (5) and (6) are solvable for all (q, p) ∈ R 2n , the control law (4) can be expressed analytically. Computing the time derivative of H d and It follows that the equilibrium (q, p) = (q * , 0) is asymptotically stable when D = 0 provided that the output y = G T ∇ p H d is detectable, since D d = Gk v G T is rank deficient. 6 In addition, asymptotic stability can be concluded if the system has strong dissipation, that is D d > 0 (see Lemma 4.2 in Reference 7). This latter condition can be met for some mechanical systems by setting M d = k T M and k T > 0 provided that D > 0 (see Reference 5 for details). It must be noted that (7) does not contain a negative-definite term in the position q. Consequently, the IDA-PBC control (4) is not inherently robust to disturbances, which has motivated research on more sophisticated designs such as Reference 10.

System model
This work considers a mechanical system with 1 ≤ m < n actuated DOFs indicated as q a = G T q for conciseness, with the control input acting on the system by means of a pressurized ideal fluid, and satisfying the following assumptions.

Assumption 1.
The fluid is isentropic and inviscid, and its bulk modulus Γ 0 is known. In addition, the pressure P of the fluid, its density and speed v are bounded and uniform throughout its volume . Finally, the inertia M 0 of fluid, valves, pumps, and cylinders is negligible compared to the inertia M of the mechanism, that is M ≫ M 0 .

Assumption 2.
The volume of fluid, which includes the supply pipes, only depends on the actuated position q a . In addition, is a known and continuously differentiable function bounded away from zero, that is > 0.
Assumption 3. The PDEs (5) and (6) for the original system (1) with direct actuation are solvable analytically, and In addition, all trajectories of the closed-loop system (3) are bounded, and (q, p) = (q * , 0) is an asymptotically stable equilibrium. Finally, all model parameters are exactly known, and all system states are measurable.
Assuming uniform pressure, uniform density, and uniform speed of the fluid is a reasonable approximation in case of laminar flow, while neglecting the effect of viscosity is appropriate at low speeds. The assumption made on the volume is satisfied by a wide range of cylinder arrangements (e.g., see Section 4). Neglecting the inertia of the fluid and the mass of the moving parts in valves, cylinders, and pumps is a reasonable approximation for hydraulic actuators in tilt trailers, excavators, and cranes. The analytical solvability of the PDEs is a research topic in itself, 3,30 thus it is not investigated further. Nevertheless, the PDEs (5) and (6) are solvable analytically for several canonical examples including the acrobot, 1 the disk-on-disk, 10 the inertia-wheel-pendulum, 31 and the ball-on-beam 6 which is illustrated in Section 4. Finally, the boundedness of the states in closed-loop (3) and the stability of the equilibrium for direct actuation follow from Reference 6.
The bulk modulus of a fluid is defined as For isothermal liquids it is realistic to assume that the bulk modulus is constant. 32 Instead, computing (8) for an ideal gas with polytropic index c yields Γ 0 = cP. The mechanical energy of an isentropic fluid, if its inertia is neglected, corresponds to the internal energy Considering at first an isothermal liquid and solving (8) yields = 0 e P∕Γ 0 . Defining the specific internal energy = Φ yields where 0 = 1, for simplicity. Finally, computing Φ = from (10) yields 33 The pressure dynamics can be expressed as in Reference 34 where the volumetric flow rate u corresponds to the control input (i.e., u > 0 for inflow, while u < 0 for outflow). In practice, the flow rate can be provided by a control valve and a pump in response to a corresponding input voltage. Since the dynamics of pumps and valves is typically much faster than the pressure dynamics, these effects are neglected.
The complete system dynamics can be expressed as where W = H + Φ = 1 2 p T M −1 p + Ω + Φ, and the system states are now the position q ∈ R n , the momenta p ∈ R n , and the pressure P ∈ R m . In case m > 1, the energy of the fluid is computed as Note that the input matrix G does not appear explicitly in (13). Instead, the control input affects the pressure dynamics, and the pressure contributes to the energy of the fluid (11) only with respect to the actuated positions since (q a ) from Assumption 2, thus rank (13) is passive with respect to u and y ′ = Γ 0 ∇ P W, and the product u T y ′ has unit of power. Since the aim of this work is to account for the pressure dynamics of an ideal fluid by building upon the traditional IDA-PBC in a modular fashion, the output y = G T ∇ p H d of (3) remains of interest.
Considering an ideal gas with polytropic index c and computing Φ from (9) yields where 0 > 0 is the initial value of . Computing the total energy W from (14), the complete system dynamics is still given by (13), where Γ 0 = cP. Thus (13) represents the starting point for the controller design with either isothermal liquids or ideal gases.

MAIN RESULT
This section presents an extended energy shaping and damping assignment procedure that builds upon the IDA-PBC (4) to account for the pressure dynamics of an ideal fluid. To streamline the presentation, Section 3.1 outlines the controller design for systems without disturbances or redundant actuators, which is the reference condition for IDA-PBC. 6 Subsequently, the effect of disturbances is discussed in Section 3.2, while the case of redundant actuation is discussed in Section 3.3.

Energy shaping and damping assignment
The control law for system (13) is designed such that the closed-loop dynamics in port-Hamiltonian form becomes where T is a positive definite storage function, and ∈ R m is defined as The terms S ij , with 1 ≤ i,j ≤ 3, are computed according to the following procedure in order for the open-loop dynamics (13) to match the closed-loop dynamics (15) on all DOFs.
Step 1: This step aims to preserve the relationship between the position q and the momenta p. Equating the first rows of (13) and of (15) yields Setting (17). Note that the matrix P ∈ R m×m is a function of , thus it follows from Assumption 2 that it is invertible.
Step 2: This step corresponds to the IDA-PBC energy shaping procedure in Section 2.1, which is modified here to account for the pressurized fluid and for the damping assignment. Equating the second rows of (13) and of (15) yields , the physical damping D on the left side of (18) vanishes. Defining the pseudo-inverse G ⊗ = (G ⊥ G ⊥ T ) −1 G ⊥ and pre-multiplying both sides of (18) by either G ⊗ or by G † while refactoring terms gives the two new matching conditions Equation (19) enforces matching on the unactuated DOFs, and its left-hand side corresponds to the sum of the PDEs (5) and (6). Instead, Equation (20) enforces matching on the actuated DOFs. The term S 23 is thus defined as Substituting (21) = 0, thus preserving the PDEs (5) and (6) as well as their analytical solutions M d , J 2 , and Ω d (see Assumption 3). In addition, (21) verifies (20) thus ensuring matching for the actuated DOFs. In summary, the proposed design procedure yields the new matching conditions (19) and (20) but results in the same PDEs as the traditional IDA-PBC (4), thus it is applicable to the same class of system that was given in Reference 6 in terms of solvability of the aforementioned PDEs.
Step 3: Equating the third rows of (13) and of (15) yields where S 33 is defined as with K i > 0 a constant parameter matrix. Computing the control input u from (22) yields where S 12 , S 13 , S 22 , S 23 , S 33 have been defined previously. Energy shaping of the mechanical system is achieved through the terms M d , J 2 , and Ω d in , while damping assignment is achieved through k v in S 22 and in . Finally, shaping of the internal energy of the fluid (11) or (14) is achieved through the parameter K i . (24). Then the equilibrium (q, p, P) = (q * , 0, P * ) is asymptotically stable for all K i > 0 provided that either D d > 0 or that D d = Gk v G T ≥ 0 and the output y = G T ∇ p H d of the original system (3) is detectable.
indicates the minimum eigenvalue of a matrix, and G T x = y + G T p . Thus p ∈  ∞ and , y ∈  2 ∩  ∞ , while computing the time derivative of p from (15) and of from (16) yieldsṗ,̇∈  ∞ hence y and converge to zero. Asymptotic stability of the equilibrium can be established in a similar fashion to Reference 6 provided that y is detectable (see Theorem 3.2 in Reference 36). ▪ Remark 1. The controller design procedure is applicable to systems with non-constant inertia matrix and non-trivial kinetic shaping. In particular, the design is considerably simplified if k v = 0, J 2 = 0, and M d = k T M as in Reference 29, since would not depend on the momenta p thus resulting in S 13 = 0. While it is possible to account for the inertia M 0 of fluid, valves, pumps, and cylinders within (13), that would result in a different inertia matrix M ′ = M + M 0 . Therefore, the analytical solution of the kinetic-energy PDE (5) would have to be computed on a case-by-case basis, depending on the specific arrangement of the actuators. However, since M 0 only depends on the actuated DOFs, the kinetic-energy PDE would remain solvable analytically for certain classes of systems, including those with underactuation degree n − m = 1 and with M ′ that only depends on q a . 30 Future work shall focus on the analytical solution of the kinetic-energy PDE accounting for the inertia of the fluid. A further direction that could be explored corresponds to employing an algebraic solutions of the PDEs similarly to Reference 3.

External forces
This section investigates the effect of the external forces affine in the momenta p within (13), that is which are accounted for in Step 2 of the energy shaping procedure by introducing an additional vector Λ of closed-loop non-conservative forces, as defined in Reference 11. Thus, the potential-energy PDE (6) can be rewritten as where Ω 0 = Λ T (q − q * ). Again, it is assumed here that both (27) and (28) are solvable analytically for the original system with direct actuation. Since external forces might not be accurately known, for instance due to sensor noise, a further assumption is introduced.
The design procedure in Section 3.1 is modified by redefining the term in (16) as The structure of the control law (24) does not change, however it now contains the constant bound and the vector Λ within . In addition, the measurement error 0 (t) is accounted for in the closed-loop dynamics (15) as (24), where is defined in (29). Then the following claims hold.

• The system trajectories are bounded if
Proof. Computing the time derivative of W d and substituting (15) while accounting for the external forces yieldṡ where the second line has been obtained by using Young's inequality with 0 (t) ∈  2 ∩  ∞ by hypothesis, thus the trajectories of the closed-loop system are bounded. 37 To prove the second claim substitute | 0 (t)| ≤ |∇ p W d | in the first line of (30), which yieldsẆ d ≤ 0 provided that (D d − I) > 0 and K i > 0. Asymptotic stability is concluded employing the same arguments as Proposition 1. A similar claim holds if the disturbances are matched (i.e., = G 0 ) provided that k v > and K i > 0, and that y = G T ∇ p H d is detectable. ▪ Remark 2. If the disturbance 0 (t) is non-vanishing, boundedness of the trajectories of the closed-loop system can be concluded from Lemma 5.3 in Reference 37. Local asymptotic stability of the equilibrium can be concluded for some classes of unknown external forces by introducing in the control law either an integral action 10 or adaptive observers 11 to compensate the effect of . This approach has been illustrated for a specific class of systems in References 5 and 38, thus it is not discussed further. For comparison purposes, accounting for the external forces within the baseline IDA-PBC design in Section 2.1 and computing the time derivative of H d as in (7) yieldṡ It is apparent that (31) lacks the quadratic term in which appears instead in (30). Intuitively, this indicates that explicitly accounting for the pressure dynamics of the fluid has beneficial effects on stability compared to treating this phenomenon as yet another disturbance or to disregarding it altogether.

Redundant fluidic actuation
In engineering practice, multiple pressures P j can be associated to the same actuated DOF. This is the case when two single-acting cylinders with volumes 1 and 2 are employed in an antagonistic pair to achieve bidirectional motion (see Section 4). Thus, the total energy is W = H + Φ 1 ( 1 , P 1 ) + Φ 2 ( 2 , P 2 ) and the closed-loop dynamics (15) becomes where T , and is defined as The terms S 12 , S 22 are defined as in Section 3.1, while S 34 = 0 considering that the pressures in the different actuators do not interact directly. The matching conditions corresponding to (19) and (20) Note that S 13 and S 14 have a similar structure, differing only due to their dependence on P 1 or P 2 . The same property applies to S 23 and S 24 . The pressure dynamics for each hydraulic actuator is given by where Computing the control input from (35) yields Asymptotic stability of the equilibrium can be concluded following the same reasoning of Proposition 1. The general case of multiple redundant actuators, although less common in practice, can be treated in a similar fashion and is omitted for brevity.
Remark 3. In case of pneumatic actuation, the control input can be expressed in terms of the mass flow rate u ′ , rather than the volumetric flow rate u. Thus, substituting u 1 = u ′ R s T are the densities of the ideal gas in either cylinder, and solving (35) with Γ 0 1 = cP 1 and Γ 0 2 = cP 2 respectively yields the control laws where T is the temperature in Kelvin, R s is the specific gas constant, and c is the polytropic index. Note that, differently from (36), the control laws (37) also depend on the temperature.

SIMULATION RESULTS
The ball-on-beam system is revisited here by introducing two identical single-acting cylinders that push against the beam with a sliding contact (see Figure 1). The cylinders form an antagonistic pair and, in case of hydraulic actuation, they are supplied by two volumetric pumps with flow rates u 1 and u 2 . The volumes 1 , 2 of fluid in the cylinders are nonlinear functions of the inclination of the beam q 2 , that is ) .
The pressure of the fluid in the cylinders is denoted by P 1 and P 2 , respectively. The equations of motion of the ball-on-beam with direct actuation, under some assumptions on the masses for simplicity, are given by (2) with the parameters where q 1 is the position of the ball, q 2 is the inclination of the beam, L is half the length of the beam, and g is the gravity constant. 6 In practice, |q 1 | ≤ L and |q 2 | < ∕2 with the proposed constructive solution. The prescribed equilibrium is (q * 1 , q * 2 ) = (0, 0). The IDA-PCB design (4) for direct actuation yields where k p is a positive tuning parameter and the damping assignment is governed by the parameter k v . 6 Since the two cylinders result in redundant actuation, the control law is computed using (36) for isothermal liquids, and using (37) for ideal gases. According to Assumption 1, the inertia M 0 of fluid, valves, cylinders, and pumps is neglected. Additionally, /2 /2 F I G U R E 1 Simplified schematic of the ball-on-beam system with fluidic actuation. Each single-acting cylinder is supplied by a control valve and a pump providing the flow rates u 1 and u 2 . The cylinders push against the beam with a sliding contact. no physical damping is present, that is D = 0. The control laws (36) and (37) employ the parameters (40), which are the solutions of the PDEs (5) and (6) for the ball-on-beam system with direct actuation. 6 The IDA-PBC for direct actuation (4) and a backstepping algorithm (see Appendix A) are employed for comparison purposes.

Hydraulic actuation
Simulations have been conducted in Matlab with an ode23 solver using the model parameters in SI units L = 0.5, g = 9.81, A = 10 −5 , L 0 = 1, Γ 0 = 2 × 10 9 (i.e., corresponding to water, assumed isothermal), with the initial conditions (q 1 , q 2 ,q 1 ,q 2 , P 1 , P 2 ) = (0.45, 0, 0, 0, 2 × 10 6 , 2 × 10 6 ). The time history of the states and of the control input computed with (36), where ∕2, using the tuning parameters k p = 1, k v = 1, K i = 1 is shown in Figure 2 for different values of L 0 . The position converges to (q * 1 , q * 2 ) = (0, 0) in a smooth fashion, and the pressures P 1 and P 2 , in [bar], converge to their initial values at equilibrium. The control input increases with L 0 , while the time history of the position (C) pressure P 1 ; (D) pressure P 2 ; (E) control input u 1 during the initial transient; (F) control input u 2 shows no noticeable difference. This indicates that reducing the dead volume of fluid (e.g., by mounting the control valves near the cylinders) can reduce the control effort, in accordance with engineering practice.
For comparison purposes, the system response with the traditional IDA-PBC controller (4) and the parameters (40) is shown in Figure 3. In this case the tuning parameters were chosen empirically as k p = 50, k v = 100 to obtain a similar transient to that in Figure 2, while u 1 , u 2 were computed as where K 0 = 2 × 10 7 is an additional tuning parameter that accounts for the bulk modulus of the fluid, and u is given in (4). The simulations indicate that neglecting the pressure dynamics in the controller results in an oscillatory response and in larger pressure variations, which are particularly noticeable for a larger L 0 and might lead to loss of contact between the cylinders and the beam (e.g., in case P 1 = 0 or P 2 = 0). Note also that the pressures P 1 and P 2 at the equilibrium vary with L 0 and are larger than with the controller (36) in some cases (e.g., for L 0 = 1), which indicates a higher energy consumption. While a smoother transient can be achieved in this case by reducing k p or increasing k v , this also results in (C) pressure P 1 ; (D) pressure P 2 ; (E) control input u 1 ; (F) control input u 2 a slower response. In addition, simultaneously increasing k p , k v , and K 0 slightly reduces the amplitude of the oscillations on the position. However, using high gains could amplify the effect of sensor noise on the measured states in practice. Conversely, simultaneously decreasing k p , k v , and K 0 results in larger oscillations on the position and in higher peak pressures.
In summary, the new controller (36) yields a smoother response and a reduced magnitude of the control input compared to the baseline IDA-PBC (41), which does not take into account the pressure dynamics of the fluid. This could be beneficial in engineering practice, since a smaller control action could allow employing smaller and less expensive control valves, and smoother movements could reduce vibrations in the mechanism. Nevertheless, the improved performance comes at the cost of having to provide pressure measurements with suitable sensors.

Pneumatic actuation
The simulation results for pneumatic actuation are shown in Figure 4. In this case, the control law is computed using (37), where R s = 287, T = 293, and the polytropic index is c = 1.5, corresponding to adiabatic expansion for an ideal gas. (C) pressure P 1 ; (D) pressure P 2 ; (E) control input u ′ 1 during the initial transient; (F) control input u ′ In this case the internal energy of the gas is given by (14), thus . The area of the cylinders is set to A = 3 × 10 −5 so that the pressures vary within a range representative of industrial air supply (i.e., P 1 , P 2 ≤ 10 bar). The remaining model parameters and the tuning parameters are the same as for the hydraulic actuation (i.e., L 0 = 100 in this case), and the initial conditions are (q 1 , q 2 ,q 1 ,q 2 , P 1 , P 2 ) = (0.45, 0, 0, 0, 6 × 10 5 , 6 × 10 5 ). The system response is similar to that in Figure 2, and the pressures P 1 and P 2 converge to their initial values at equilibrium, in [bar]. However, the control input is considerably smaller, since in this case it corresponds to the mass flow rate of gas into the cylinders.
The effect of the model parameter L 0 is similar to the case of hydraulic actuation, thus it is not shown. Increasing the value of the tuning parameter K i yields a slightly faster response and slightly smaller peak pressures, and it also considerably reduces the control effort during the initial transient (see Figure 4E,F). This can be explained by noting that K i introduces a dissipative effect on the pressure dynamics in (15). Finally, the transient response depends also on the tuning parameters k p and k v in (40) according to the IDA-PBC design (4). Employing the IDA-PBC controller (4) with (40) while computing u ′ 1 and u ′ 2 in a similar fashion to (41) yields similar results to those in Figure 3, which are omitted for brevity. Compared to the ball-on-beam system with direct actuation and with the IDA-PBC (4) (see Appendix B), the proposed controller yields a similar transient response with the same values of k p and k v . The initial overshoot and the slightly longer settling time  Figure 4A for K i = 1 are due to the pressure dynamics and can be reduced by increasing K i (e.g., see K i = 100 in Figure 4A). For comparison purposes, a backstepping algorithm has been constructed based on the IDA-PBC (4) to account for the pressure dynamics of the ideal gas (see Appendix A). Figure 5 shows that the backstepping algorithm (A1) with the tuning parameters k p = 1, k v = 10, K i = 1, chosen empirically to achieve a similar response to that of controller (37), is superior to the baseline IDA-PBC (41). However, the system response is more oscillatory than with the controller (37), which is agreement with our prior work. 28 Increasing the value of K i in the backstepping algorithm also slightly improves the responsiveness, but differently from Figure 4, it does not reduce the control effort. Finally, the effect of a bounded vanishing disturbance acting on the ball, that is = 0 [cos(q 2 ) tanh(q 1 ) 0] with 0 = −0.1, is shown in Figure 6. The new controller (37) yields a similar response to that in Figure 4 with the same tuning parameters (i.e., in this case K i = 100 and = 0 in (29)). Instead, the backstepping algorithm (A1) and the baseline IDA-PBC result in oscillations of increasing amplitude, with the baseline IDA-PBC eventually leading to instability hence confirming the analysis in Remark 2.

CONCLUSION
This article presented some new results on the energy shaping control for underactuated mechanical systems for which the control action is mediated by a pressurized ideal fluid. Building upon the IDA-PBC methodology in a modular fashion, an extended multi-step energy shaping and damping-assignment controller design procedure was outlined for both isothermal liquids and ideal gases. In addition, the effect of bounded vanishing disturbances and the case of redundant actuators were discussed. Simulation results for a ball-on-beam system actuated by two identical cylinders, either hydraulic or pneumatic, indicate that the controller is effective in achieving the prescribed regulation goal, while yielding a smoother transient and a reduced control effort compared to the baseline IDA-PBC. The performance improvement is particularly noticeable in the presence of larger dead volumes of fluid. Further reductions of the control effort, together with moderate improvements in responsiveness, can be achieved by appropriately tuning the additional parameter introduced in the controller design procedure. The proposed controllers also yield a smoother response compared to an alternative backstepping algorithm based on IDA-PBC for direct actuation. The difference in performance is particularly noticeable in the presence of bounded vanishing disturbances.
Future work will aim to extend these results by accounting for the inertia of the fluid in the energy shaping procedure, and eventually for the internal dynamics of different types of actuators, including valves and pumps.