Integral passivity‐based control of underactuated mechanical systems with state‐dependent matched disturbances

This work investigates the energy shaping control of a class of underactuated mechanical systems subject to state‐dependent linearly parameterized matched disturbances. To this end, a new integral interconnection and damping‐assignment passivity‐based design is proposed. The controller design is developed for systems subject to either momenta‐dependent disturbances or position‐dependent disturbances or to both simultaneously. The effectiveness of the proposed approach is demonstrated with numerical simulations on three examples: an Acrobot system with nonlinear friction; a ball‐on‐beam system with constant and position‐dependent disturbances; a disk‐on‐disk system with a complete set of state‐dependent disturbances. In all examples, the proposed controller shows better performance compared to previous implementations.


INTRODUCTION
Underactuated mechanical systems are characterized by fewer actuators than degrees of freedom (DOF), which greatly complicates the control problem.Underactuation can be due to design choices, or it can result from actuator failure. 1 Among the various control methodologies that have been employed for this class of systems, energy shaping approaches have attracted increasing interest.In particular, the interconnection and damping-assignment passivity-based control (IDA-PBC) methodology 2 involves designing the control action such that the closed-loop dynamics preserves the port-Hamiltonian structure of mechanical systems with a new prescribed energy function that has a minimum at the desired equilibrium.Thus IDA-PBC provides a physical interpretation of the control action in terms of system energy.Practical applications of IDA-PBC include magnetic levitation systems, 3 suspension systems, 4,5 and weakly-coupled electro-mechanical systems. 6More recently, IDA-PBC designs have been proposed for underactuated mechanical systems with fluidic actuation, [7][8][9] with high-order actuator dynamics, 10 and with actuator saturation. 11A growing number of research works have investigated the effect of disturbances on underactuated systems by employing various techniques, including nonlinear observers 12 and quadratic programming. 13Additionally, more sophisticated IDA-PBC controllers that achieve robustification through integral actions 14 or nonlinear observers 15 have been proposed.Robust versions of IDA-PBC have been proposed in References 16 and 17 for systems with matched bounded disturbances, and in Reference 18 for constant matched disturbances.In addition, robust IDA-PBC implementations for the inertia-wheel-pendulum system have been presented in Reference 19 considering the effect of bounded parametric uncertainties, and in Reference 20 considering also motion constraints and actuator saturation.Among these approaches, the Integral IDA-PBC (iIDA-PBC) methodology 14 results in an extended closed-loop dynamics characterized by a port-Hamiltonian structure that preserves the energy-based interpretation of IDA-PBC and its stability properties.The initial formulation of iIDA-PBC was limited to systems with a particular structure of the inertia matrix, with constant input matrix, subject to constant matched disturbance, and it employed a change of coordinates. 14Subsequent works have refined this approach by avoiding the change of coordinates, 21 and by extending it to systems with a more general structure of the inertia matrix. 21Recent works have investigated the iIDA-PBC methodology for systems with physical damping 22 and with non-constant disturbances 23 (although not directly compensated by the integral action), and have presented control implementations for the specific case of power converters 24 and other practical examples. 25To the best of our knowledge, however, the iIDA-PBC methodology has so far only been formulated either for constant matched disturbances or to ensure input-to-state stability for time-varying disturbances.The extension of iIDA-PBC designs to systems with state-dependent disturbances is of particular interest in engineering practice, since it could allow compensating the effect of parameter uncertainties such as stiffness (e.g., in suspensions system) or friction, which result in position-dependent forces and momenta-dependent forces respectively.This work investigates the energy shaping control of a class of underactuated mechanical systems with constant input matrix, subject to linearly parameterized state-dependent matched disturbances.To this end, a new control law is constructed by employing the IDA-PBC methodology 2 and by generalizing the design of the integral action originally introduced in Reference 14.To the best of our knowledge, this is the first implementation of the iIDA-PBC methodology that explicitly accounts for linearly parameterized state-dependent matched disturbances.In summary, this work presents the following new results.
1.The design of an integral action for linearly parameterized momenta-dependent matched disturbances, and the corresponding control law.The effect of physical damping is also discussed.2. The design of an integral action for linearly parameterized position-dependent matched disturbances, and the corresponding control law.A comparison with the Immersion and Invariance methodology 26 is presented.3. The controller design for underactuated mechanical systems subject simultaneously to constant matched disturbances, position-dependent disturbances, and momenta-dependent disturbances.
Implementations are presented for three illustrative examples: an Acrobot system, a ball-on-beam system, and a disk-on-disk system.In each example, the proposed controller demonstrates improved performance compared to an iIDA-PBC implementation intended for constant disturbances.
A brief overview of the iIDA-PBC methodology is provided in Section 2 for completeness.The new controller for linearly parameterized momenta-dependent matched disturbances is then outlined in Section 3. The controllers for linearly parameterized position-dependent matched disturbances and for simultaneous position-dependent and momenta-dependent disturbances are discussed in Section 4. The results of numerical simulations for the three illustrative examples are presented in Section 5. Concluding remarks are contained in Section 6.

OVERVIEW OF INTEGRAL IDA-PBC
The dynamics of an underactuated mechanical system with n DOFs and the control input u ∈ R m applied through the input matrix G(q) ∈ R n×m , where rank(G) = m < n for all q ∈ R n , and subject to constant matched disturbances  =  0 ∈ R m , is described in port-Hamiltonian form as where v ∈ R m is the auxiliary control corresponding to the integral action, and D is the physical damping, which is often assumed to be zero.The system states are the position q ∈ R n and the momenta p = M q ∈ R n .The output of ( 1) is y = G T ∇ p H. The mechanical energy of the system, is characterized by the inertia matrix M(q) = M(q) T ≻ 0, and the potential energy Ω(q) > 0. The remaining terms in (1) are the identity matrix I, the vector of partial derivatives of H with respect to q, ∇ q H, and the vector of partial derivatives of H with respect to p, ∇ p H. The controller design aims at stabilizing the prescribed equilibrium (q, p) = (q * , 0).This goal is achieved, in the absence of disturbances (i.e.,  = 0), by using v = 0 if there exist the IDA-PBC control law 2 where The resulting closed-loop dynamics, in the absence of physical damping (i.e., D = 0), is where (4), see Reference 2. The design parameters in (3) and ( 4) are the potential energy Ω d (q), the inertia matrix M d (q) = M T d (q) ≻ 0, the matrix J 2 (q, p) = −J T 2 (q, p), and the constant matrix To achieve the regulation goal, the potential energy Ω d should admit a strict minimizer in q * hence verifying the conditions ∇ q Ω d (q * ) = 0 and ∇ 2 q Ω d (q * ) > 0. The control law (3) exists provided that M d and Ω d verify for all (q, p) ∈ R 2n the partial differential equations (PDEs), also known as matching equations, where G ⊥ is defined such that G ⊥ G = 0 and rank Computing the time derivative of H d along the trajectories of the closed-loop system (4) yields then Thus the equilibrium (q, p) = (q * , 0) is asymptotically stable for all k v ≻ 0 provided that If the system is subject to constant matched disturbances  =  0 ≠ 0, the input matrix G and the matrix M d are constant, and the matrix M is independent of the unactuated coordinates, the integral action design proposed in Reference 14 can be used to compensate the disturbance.Then, the auxiliary control v and the time derivative of the integral action state  take the form where K I ≻ 0 is a tuning parameter, and where z 1 = q, z 2 = p + GK 2 K I ( − ), z 3 = , and  =  0 (k v K I ) −1 .The storage function W d is defined as Computing the time derivative of W d along the trajectories of the closed-loop system (9) yields thus the equilibrium (q, p, ) = (q * , 0, ) is asymptotically stable provided that y 2 = G T M −1 d z 2 is a detectable output of (9). 14otice that (8) is the simplest version of several integral action variations presented in Reference 14. Subsequent versions of iIDA-PBC have avoided the coordinate transformation, 21 and have extended the results to systems with non-constant matrix M d . 27

MOMENTA-DEPENDENT DISTURBANCES
In this section, the controller design is outlined for a class of underactuated mechanical systems only subject to momenta-dependent matched disturbances.The following assumptions are introduced.
Assumption 1.The PDEs ( 5) and ( 6) are solvable analytically with the parameters M d , J 2 and Ω d , where q * = argmin(Ω d ).The matrix G is constant, and D = 0.In addition, (4).All model parameters are exactly known and all system states are measurable.
This assumption defines the class of systems considered in this work and summarizes the requirements of the IDA-PBC methodology. 2In particular, the solvability of PDEs is a major challenge in IDA-PBC which has motivated a number of works including the study of systems with underactuation degree one, 28 the study of kinetic energy shaping for mechanical systems, 29 the algebraic solution of the PDEs, 3 the numerical solution of the PDEs with reinforcement learning, 30 and a constructive method for the explicit solution of PDEs for systems with n = 2, see Reference 31.Nevertheless, the PDEs are solvable for several canonical examples.The requirement on the matrix G has been used in the iIDA-PBC formulation, 14 and it is verified by the examples in Section 5.
Assumption 2. The matched disturbance is defined as  =  p G T f (p), where  p ∈ R is an unknown constant, while f (p) ∈ R n is a globally bounded and integrable function of p, that is there exists a scalar function This assumption defines the specific type of momenta-dependent disturbances considered in this work: the requirement on f (p) to be integrable is in fact restrictive, however it is also a prerequisite for the Immersion and Invariance methodology (see Remark 2).

Controller design
The control law is designed for system (1) to achieve the regulation goal q = q * following a similar procedure to Reference 14 but redefining the integral action .The target closed-loop dynamics proposed in this section is thus The storage function W d is defined as where and  is an unknown scalar constant such that G(k v G T G + Γ 1 )K I = G p , while Γ 1 ∈ R m×m , Γ 1 =  1 I, and the scalar constants  1 > 0, K I > 0 are tuning parameters.The matrices S ij are defined so that the open-loop dynamics (1) matches the extended closed-loop dynamics (10), that is The control input u is given by ( 3), the auxiliary control and the integral state  is defined by the update law The unknown parameters  p and  are not used to compute v and ζ , thus the integral action control is implementable.
Proposition 1.Consider the system (1) that satisfies Assumptions 1 and 2 in closed-loop with the control laws (3) and (13), with the dynamics of the integral state given in (14).The closed-loop dynamics can be written as in (10) with the parameters (12).
Proof.Computing the partial derivatives of W d from ( 11) yields Equating the corresponding rows of (1) and of (10) yields the matching equations Substituting S 12 and S 13 from ( 12) and the partial derivatives of W d from ( 15) into (16a) yields Refactoring terms in the previous expression cancels  and  yielding Substituting S 12 , S 22 and S 23 from ( 12), the partial derivatives of W d from (15), and the equation Refactoring terms in the former expression cancels .Multiplying the former equation by G ⊥ yields the sum of the PDEs ( 5) and ( 6), which are verified by Assumption 1. Instead multiplying the former equation by G † while substituting u from (3) yields the auxiliary control v in (13).
Substituting S 13 , S 23 and S 33 from (12) and the partial derivatives of W d from ( 15) into (16c) yields Refactoring terms cancels the unknown term  and yields the update law (14).▪

Stability analysis
Proposition 2. Consider the system (1) that satisfies Assumptions 1 and 2 in closed-loop with the control laws (3) and (13), where the dynamics of the integral state is given in (14).The equilibrium point (q, p, ) = (q * , 0, ) is asymptotically stable for all k v ≻ 0, ) converges to  asymptotically, and the equilibrium point (q, p, ) = (q * , 0, ) is asymptotically stable for all k v ≻ 0, Proof.Consider the storage function W d defined in (11), which is positive definite by design.Computing the time derivative of W d along the trajectories of the closed-loop system (10) while simplifying skew symmetric terms in S 22 and S 33 yields which is verified for all k v ≻ 0 and Γ 1 =  1 I,  1 > 0, thus all states are bounded, and the equilibrium is stable.It follows from (17) that It follows from LaSalle's theorem (see theorem 3.4 in Reference 32) that the trajectories of the closed-loop system (10) converge asymptotically to the set ∇ p W d ≤ 0 and the projected dynamics of the closed-loop systems (10) is the same as that of (4).Thus, employing a similar argument to Reference 14, it follows that the equilibrium point (q, p, ) = (q * , 0, ) is asymptotically stable provided that y d is a detectable output of (10).If additionally ∇ p W d ≤ 0, and it follows from LaSalle's theorem that the trajectories of the closed-loop system (10) converge asymptotically to the set y d = 0 ∩ ( − F(p) − ) = 0. Thus the equilibrium point (q, p, ) = (q * , 0, ) is asymptotically stable provided that y d = G T M −1 d p is a detectable output of (4).▪ Remark 1. Extending the controller design to systems with physical damping D ≠ 0 requires modifying the term S 22 such that Reference 33 for details.Equating the second rows of (1) and of (10) while extracting the terms dependent on  p and  yields then Redefining S 23 and S 33 as preserves the relationship between  and  p .Thus the auxiliary control v and the time derivative of the integral state  are still given by ( 13) and (14).With this design however, the stability conditions of Proposition 1 would depend on the physical damping D. To highlight this point, we employ the matrix decomposition is the symmetric part and is the antisymmetric part, that is D S = D T S and D A = −D T A .Substituting the new expression of S 33 in (17) yields the stability conditions Gk v G T + D S ≻ 0 and Γ 1 G T G − G T D S G ≻ 0, which might be more stringent than the condition given in Proposition 1, that is k v ≻ 0 and  1 > 0. This point highlights the limits imposed by pervasive damping to the IDA-PBC and the iIDA-PBC methodology (see References 33 and 22 for details).

SIMULTANEOUS LINEARLY-PARAMETERIZED DISTURBANCES
The controller design for the class of mechanical systems defined by Assumption 1 and subject simultaneously to multiple state-dependent matched disturbances is presented in this section.

Controller design for position-dependent and constant disturbances
A different class of disturbances is defined in the following assumption.
Assumption 3. The matched disturbance  ∈ R m is defined as  =  0 +  q G T h(q), where  0 ∈ R m and  q ∈ R are unknown constants, while h(q) ∈ R n is a globally bounded and continuously differentiable function of q such that h ′ (q) = ∇ q h(q) ∈ R n×n .
The closed-loop dynamics is defined in a similar way to (10), that is where the matrices S ij in ( 18) are given by The storage function W d is defined as where and the tuning parameters The integral states are defined by the update laws Proposition 3. The system (1) with Assumptions 1 and 3 in closed-loop with the control law (3), the auxiliary control (21) and the time derivatives of the integral actions (22a,b) yields (18) with the parameters (19).
Proof.Computing the partial derivatives of W d from (20) yields Equating the corresponding rows of (1) and of ( 18) yields the matching equations Substituting S 12 , S 13 , S 14 from ( 19) and the partial derivatives of W d from ( 23) into (24a) yields Refactoring terms in the previous expression cancels  0 ,  1 and the constants ,  yielding , the matrices S 12 , S 22 , S 23 and S 24 from (19), and the partial derivatives of W d from ( 23) into (24b) yields Refactoring terms in the former expression cancels the disturbance  0 and the constant .Multiplying it by G ⊥ yields the sum of the PDEs ( 5) and ( 6), which are verified by Assumption 1. Instead multiplying the former equation by G † while substituting u from (3) yields the auxiliary control v in (21).Substituting S 13 , S 23 , S 33 , S 34 from ( 19) and the partial derivatives of W d from ( 23) into (24c) yields Refactoring the former equation cancels all terms apart from the first row yielding (22a).Substituting S 14 , S 24 , S 43 and S 44 from (19) and the partial derivatives of W d from ( 23) into (24d) yields Refactoring the former equation cancels all terms apart from the first row yielding (22b).▪

Stability analysis
Proposition 4. Consider the system (1) with Assumptions 1 and 3 in closed-loop with the control law (3), the auxiliary control (21) and the time derivatives of the integral actions (22a,b).The equilibrium point (q, p) = (q * , 0) is asymptotically stable, and the quantity  0 +  1 G T h(q) converges to  + G T h(q) asymptotically for all k v ≻ 0, K I 0 ≻ 0, K I 1 > 0, and Proof.Consider the storage function W d defined in (20), which is positive definite by design.Computing the time derivative of W d along the trajectories of the closed-loop system (18) yields where the terms containing J 2 and the terms on the last line of ( 25) are skew symmetric by design.It follows from (19) that the coefficient of the cross term (25) and omitting skew symmetric terms in S 22 , S 33 , S 44 yields thus which is verified for all k v ≻ 0 and Γ 1 =  1 I,  1 > 0, thus all states are bounded and the equilibrium is stable.It follows from (26) that It follows from LaSalle's theorem (see theorem 3.4 in Reference 32) that the trajectories of the closed-loop system (18) converge asymptotically to the set y d = 0 ∩ ( 0 − G T p − ) + ( 1 − p T GG T h(q) − )G T h(q) = 0. Employing a similar argument to Reference 14, it follows that the equilibrium point (q, p, ) is asymptotically stable provided that y d is a detectable output of (4).▪

Simultaneous position-dependent and momenta-dependent disturbances
The controller (21) can be simplified in the absence of the constant disturbance  0 by omitting  0 .Conversely, the design procedure can be extended to include the momenta-dependent disturbance  p .The latter case is formalized in the following assumption for completeness.
Assumption 4. The matched disturbance  ∈ R m is defined as  =  0 +  q G T h(q) +  p G T f (p), where  0 ∈ R m ,  q ∈ R, and  p ∈ R are unknown constants.In addition, h(q) ∈ R n is a globally bounded and continuously differentiable function of q such that h ′ (q) = ∇ q h(q) ∈ R n×n .Similarly, f (p) ∈ R n is a globally bounded and integrable function of p, that is there exists a scalar function The resulting controller design is obtained by combining the auxiliary control laws ( 13) and ( 21), while the integral states are defined by the update laws ( 14) and (22a,b) respectively.For completeness, the specific case of a system subject simultaneously to a constant disturbance, a position-dependent disturbance, and a momenta-dependent disturbance with f (p) = p and h(q) = q is summarized in the following corollary, and it is illustrated in Section 5 for a disk-on-disk system.The case of generic momenta-dependent disturbances and position-dependent disturbances (i.e., f (p) ≠ p and h(q) ≠ q) follows directly from Propositions 1 and 3 hence it is omitted for conciseness.
Corollary 1.The system (1) with Assumptions 1 and 4, where h(q) = q and f (p) = p, in closed-loop with the control law (3) and with the auxiliary control where the integral states  0 ,  1 ,  2 are defined by the update laws has a stable equilibrium point (q, p,  0 ,  1 ,  2 ) = (q * , 0, , , ), where In addition, asymptotically, and the equilibrium (q, p) = (q * , 0) is asymptotically stable for all k v ≻ 0, (4).The proof is given in the Appendix.Remark 2. For comparison purposes, the Immersion and Invariance methodology 26 is employed to estimate the unknown constants  0 ,  q and  p in Assumption 4. The estimation errors are defined as where δ0 , δq and δp are integral states, while  0 ,  q and  p are state-dependent functions.The effect of the disturbances is then compensated by the auxiliary control Computing the time derivatives of z 0 , z q , z p along the trajectories of the open-loop system (1) yields The update laws for the integral states ̇δ 0 , ̇δ q and ̇δ p are thus defined as Substituting ( 33) into (32) yields finally If there exists a scalar function , similar to Assumption 4), then the functions  0 ,  q and  p can be defined as Defining the storage function ) and computing its time derivative while substituting (34) and (35) yields then Consequently, the term converges to zero for all  1 > 0. Substituting the expressions of z 0 , z q and z p from (30) this equates to stating that ) converges to zero.Defining the storage function V ′ = H d + V and computing its time derivative along the trajectories of the system (1) in closed-loop with the control law (3) and the auxiliary control v * given in (31) while simplifying skew symmetric terms yields Employing a Schur complement argument in (37), it follows that V′ ≤ 0 provided that  1 G T Gk v ≻ I 4 .The similarity with the proposed controller is striking: the estimation errors z 0 , z q and z p correspond respectively to ( 0 − G T p − ), ( 1 − p T GG T h(q) − ), and ( 2 − F(p) − ) in Propositions 2 and 4. In addition, combining the auxiliary control v in ( 13) and ( 21) yields the same structure as v * in (31).Nevertheless, the update laws for the integral states  0 ,  q and  p in ( 14) and (22a,b) differ from ̇δ 0 , ̇δ q , ̇δ p in (33), since the latter are computed by considering the open-loop dynamics (1).In summary, the Immersion and Invariance methodology can be employed in conjunction with IDA-PBC, see Reference 15.This however marks a departure from a direct energy-based interpretation since the parameter estimation and the controller design are decoupled, and it can result in more stringent stability conditions (i.e.,  1 G T Gk v ≻ I 4 ).Conversely, the proposed approach combines controller design and integral action, thus preserving the physical interpretation in terms of system energy, and it yields less stringent stability conditions on the tuning parameters (i.e.,  1 > 0 and k v ≻ 0).Note finally that, if multiple disturbances are acting simultaneously, the proposed controller yields a closed-loop dynamics that is not port-Hamiltonian.For instance, in (A1) we have S T 34 − S 43 = 2 Consequently, the Lyapunov derivative (A7) contains a term quadratic in the sum rather than the sum of three quadratic terms.As a result, Corollary 1 concludes that the quantity asymptotically rather than ensuring that each integral action  0 ,  1 ,  2 converges to the corresponding values , ,  respectively.This is exactly the same outcome obtained with the Immersion and Invariance methodology, resulting in the Lyapunov derivative (37), and it is a direct consequence of having multiple disturbances acting simultaneously.

Acrobot system with momenta-dependent disturbance
The Acrobot system 34 consists of an articulated pendulum with a single actuator at the elbow joint q 2 and an unactuated shoulder joint q 1 (see Figure 1).The open-loop dynamics is given by (1) with total energy H = Ω + 1 2 p T M −1 p where Ω = g(c 4 cos(q 1 ) + c 5 cos(q 1 + q 2 )), the input matrix is is the inertia matrix with determinant Δ = det(M) > 0. The terms c 1 , c 2 , c 3 , c 4 , c 5 are constant parameters depending on the size of the links, while g is the gravity constant.The system states are the position q = (q 1 , q 2 ) and the momenta p = M q.The control goal is to stabilize the upright position (q 1 , q 2 ) = (0, 0), which is open-loop unstable.The IDA-PBC controller (3) for the Acrobot system yields 34 where k 1 , k 2 , k 3 , k v are tuning parameters, while the remaining terms are defined as The system is subject to the matched disturbance  = −5 tanh(p 2 ), such that F(p) = 5 log(cosh(p 2 )), thus Assumptions 1 and 2 are both verified.Numerical simulations have been performed in Matlab using an ODE23 solver with the model parameters c 1 = 0.23333, c 2 = 0.53333, c 3 = 0.2, c 4 = 0.3, c 5 = 0.2, g = 9.81, and with the initial conditions (q 1 , q 2 , p 1 , p 2 , ) = (, 0, 0, 0, 0).The tuning parameters for the control law (3) and the auxiliary controller (13)   controller (13) the position reaches the prescribed equilibrium (q 1 , q 2 , p 1 , p 2 ) = (0, 0, 0, 0) in a smooth fashion, while both the control input and the integral state  settle to a constant value within the first ten seconds.Note that the quantity  − F(p) converges smoothly to a constant value, as postulated in Proposition 1, however it does not yield an exact estimate of the parameter  p .This follows from the fact that G T f (p) = tanh(p 2 ) is not bounded away from zero, thus the integral state  does not change further once the equilibrium has been reached.In comparison, the IDA-PBC controller (3) alone fails to reach steady state.Similarly, the iIDA-PBC implementation for constant disturbances fails to stabilize the prescribed equilibrium and yields persistent oscillations affecting the states and the control input.This is expected, since the disturbance violates the assumptions in Reference 14 and introduces a nonlinear negative damping term.The effect of the tuning parameters K I and Γ 1 is illustrated in Figure 3. Increasing K I results in a faster response and a more aggressive control action since this parameter acts as a proportional gain in the auxiliary controller (13).Increasing Γ 1 results in a slower response and a reduced control action since this parameter appears in S 33 (12) serving as a dissipative term in the closed-loop dynamics (10), as highlighted in (17).Increasing either K I or Γ 1 also yields a smaller constant .

Ball on beam system with constant and position dependent disturbances
The ball-on-beam system consists of an unactuated ball moving on an actuated beam (see Figure 4).The equations of motion of the ball-on-beam system, under some assumptions on the masses for simplicity, are given by (1) with the ] , and Ω = gq 1 sin(q 2 ), 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.The control goal consists in stabilizing the equilibrium (q 1 , q 2 ) = (0, 0).The IDA-PCB controller is computed with (3) employing the design parameters 2 where k p and k v are positive constants.The matched disturbance is  = 0.5 − 4q 2 , thus Assumptions 1 and 3 are verified.

F I G U R E 4
Schematic of the ball-on-beam system with key parameters.
Simulations have been conducted in Matlab with an ode23 solver using the model parameters L = 0.5, g = 9.81 with the initial conditions (q 1 , q 2 , p 1 , p 2 ,  0 ,  1 ) = (0.45, 0, 0, 0, 0, 0).The tuning parameters have been set to k p = 1, k v = 0.5, Γ 1 = 1, K I 0 = 1, K I 1 = 1 for illustrative purposes.The system response with different controllers is shown in Figure 5: employing the auxiliary controller (21) the position reaches the prescribed equilibrium (q 1 , q 2 , p 1 , p 2 ) = (0, 0, 0, 0) in a smooth fashion, while both the control input and the integral states  0 ,  1 settle to constant values.Conversely, the IDA-PBC controller (3) yields a noticeable steady state error.The iIDA-PBC, which is obtained here by setting K I 1 = 0, yields persistent oscillations affecting the states and the control input, and consequently the ball moves off the beam (i.e., q 1 > 0.5).

Disk on disk system with simultaneous disturbances
The system consists of an unactuated disks (i.e., disk-2) that rolls without slipping on an actuated disk (i.e., disk-1), see Figure 6.The system states are the angular position of the disks, q 1 and q 2 , and the momenta, p 1 and p 2 , where

F I G U R E 6
Schematic of the disk-on-disk system with key parameters.
p = M q.The dynamics of the system in port-Hamiltonian form is given by (1) with Ω =  2 cos(q 2 ), the input matrix is , where m 1 , m 2 , r 1 , r 2 are the mass and the radius of the disks.The control goal consists in stabilizing the equilibrium (q 1 , q 2 ) = (0, 0).The IDA-PBC controller is computed with (3) employing the parameters 14 ) The tuning parameters are k p , k v , k 1 , k 2 , k 3 , and the system is subject to the matched disturbance  = 0.01 − 0.25q 1 − 0.1p 1 which verifies Assumption 4.
The system response with different controllers is shown in Figure 7: employing the auxiliary controller ( 27) the position reaches the prescribed equilibrium (q 1 , q 2 , p 1 , p 2 ) = (0, 0, 0, 0) in a smooth fashion, while the control input and the integral states  0 ,  1 ,  2 settle to constant values.Conversely, the IDA-PBC controller (3) yields instability, while the iIDA-PBC (i.e., obtained by setting K I 1 = K I 2 = 0) yields persistent oscillations of increasing amplitude, which again is due to the presence of negative damping introduced by  p .

CONCLUSION
This work presented a new controller design for a class of underactuated mechanical systems with constant input matrix, subject to linearly parameterized state-dependent matched disturbances.The controller design is inspired by the iIDA-PBC methodology but proposes a different design of the integral actions.This is achieved by defining a new storage function that bears similarities to the one constructed with the Immersion and Invariance methodology.In this respect, the key advantage of the proposed controller compared to the conventional iIDA-PBC methodology is the ability to reject a wider class of matched disturbances, namely state-dependent.Compared to the Immersion and Invariance methodology, the advantage of the proposed controller is that it preserves the physical interpretation of the control action in terms of system energy, and can yield less stringent stability conditions with respect to the tuning parameters.The simulation results for three illustrative examples, namely an Acrobot system with momenta-dependent disturbances, a ball-on-beam system with constant and position-dependent disturbances, and a disk-on-disk system with a complete set of state-dependent disturbances, indicate that the proposed controller is superior to the baseline IDA-PBC and to the conventional iIDA-PBC.Future work will aim to relax the initial assumptions in an attempt to generalize the results to a larger class of systems and of disturbances, including systems with uncertain control gain and input saturation, and systems with parameter uncertainties. .Note first that the matched disturbance  =  0 +  q G T q +  p G T p verifies Assumption 4.

How
In particular, F(p) = 1 2 p T GG T p.The closed-loop dynamics is thus defined in a similar way to (18), that is where the matrices S ij in (A1) are given by have been chosen as in Reference 34, that is k 0 = −35, k 1 = 0.03386, k 2 = 0.1, k 3 = 0.59073,  = −0.6019,k u = 1, k v = 2, and K I = 1, Γ 1 = 1 for illustrative purposes.The system response with different controllers is shown in Figure 2: employing the auxiliary F I G U R E 1 Schematic of the Acrobot system with key parameters.

[ m 11 m 12 m 12 m 22 ]
is the inertia matrix.The terms m 11 , m 12 , m 22 and  2 are constant parameters defined as m 11 = r 2 1