Robust dynamic state feedback for underactuated systems with linearly parameterized disturbances

This article investigates the control problem for underactuated port‐controlled Hamiltonian systems with multiple linearly parameterized additive disturbances including matched, unmatched, constant, and state‐dependent components. The notion of algebraic solution of the matching equations is employed to design an extension of the interconnection and damping assignment passivity‐based control methodology that does not rely on the solution of partial differential equations. The result is a dynamic state‐feedback that includes a disturbance compensation term, where the unknown parameters are estimated adaptively. A simplified implementation of the proposed approach for underactuated mechanical systems is detailed. The effectiveness of the controller is demonstrated with numerical simulations for the magnetic‐levitated‐ball system and for the ball‐on‐beam system.

The input matrix G ∈ R n × m , with rank(G) = m < n ∀ x ∈ R n , and the matrix functions J, D are assumed continuously differentiable. The matrix G ⊥ is such that G ⊥ G = 0 and rank(G ⊥ ) = n − m. The IDA-PBC control law u aims to achieve the following closed-loop dynamics:̇x The term J d = −J T d ∈ R n×n is the desired interconnection matrix, D d = D T d > 0 ∈ R n×n is the desired damping matrix, while the closed-loop Hamiltonian H d is continuously differentiable and has a strict-minimizer at the assignable equilibrium x * hence H d (x * ) = 0 and 2 H d (x * ) > 0. Definition 1. [9]: x * is an open-loop assignable equilibrium for (1) if the following condition is satisfied: G(x * ) ⊥ (J(x * ) − D(x * )) H(x * ) = 0.
Designing J d , D d , H d and the corresponding control input u requires solving the following matching equations, which is in general a challenging task: To define the algebraic solution of the matching equations (3), we assume without loss of generality that the Hamiltonian can be expressed as H = d + L T x + 1 2 x T Hx + h(x) with d ∈ R, L ∈ R n , H ∈ R n×n , and h(x) : R n → R, where h(x) contains the nonlinear terms of H and is such that h(x * ) = 0, h(x * ) = 0, 2 h(x * ) = 0. Thus, x * = 0 is assignable provided that G(0) ⊥ (J(0) − D(0))L = 0. Definition 2. [9]: the continuously differentiable matrix valued function P ∈ R n × n is an algebraic solution of the matching Equation (3), where is a nonempty open set containing x * , if: In particular, the mapping x → P(x) might not be the gradient vector of any scalar function hence it is in general not a solution of the original PDE (3). Based on the above definition of P, the auxiliary energy function H d (x, ) is defined on an extended state space as: where R = R T > 0 is a free matrix to be defined, ‖w‖ 2 R = w T is the weighted Euclidean norm, and ∈ R n . Assuming x * = 0 without loss of generality, (5) can be expressed as: where h d contains nonlinear terms and is such that h d (0, 0) = 0, h d (0, 0) = 0, 2 h d (0, 0) ≥ 0. It follows from (4) that the matrix in (6) is positive definite hence H d (x, ) is locally positive definite around the desired equilibrium where it has a local minimizer. In addition, if P = P T > 0 and h(x) > 0 ∀ x ≠ x * ∈ R n then (6) is globally positive definite. Defining the terms Φ(x, ) and Ψ(x, ) so that (P(x) − P( ))x = Φ(x, )(x − ) and Ψ(x, ) = (1/2)( P( )x/ ) and computing the partial derivatives of H d (x, ) from (5) gives: According to Reference [9, Proposition 5], system (1) in closed loop with the following dynamic state-feedback, where has a PCH structure (locally) and a locally stable equilibrium at (x, ) = (0, 0) for some K > K = K T ≥ 0 and v = 0, provided that Φ(x, ) = Φ(x, ) T > 0 in some nonempty open neighborhood Ω of the origin, that Ψ T R −1Φ R −1 Ψ < 2(N T + P)D d (N + P) and that R(t) = Φ(x(t), (t)) for all t ≥ 0. Local asymptotic stability is concluded if y d is a detectable output of (1) in closed-loop with (8) and v = − y d .

ROBUST DYNAMIC STATE-FEEDBACK
In this section the dynamic state-feedback (8) is redesigned for the following PCH system with + 1 linearly parameterized additive (matched and unmatched) disturbances f k (x) k ∈ R n : The following assumptions are introduced: indicating the specific disturbance, and index j, 1 ≤ j ≤ n ∈ N, indicating the state. In addition, f k (x) and their first derivatives are bounded, f 0 = I n and f k (x * ) ≡ 0 for k > 0, while k ∈R n are unknown constants.
Notably, the above assumption does not require the disturbances to vanish at x = x * , but for simplicity combines the contributions of all + 1 disturbances at x = x * in the term f 0 0 . Assumption 2. there exists an assignable open-loop equilibrium x * of (9) such that: The equilibrium is unstable if 2 H(x * ) ≤ 0. The equilibrium x * = 0 is assignable if G ⊥ 0 = 0. The first step in the design of the new controller consists in estimating the unknown parameters k using the I&I methodology, 33 as outlined in the following result. (9) under Assumption 1 and define the vectors of estimation errors z k ∈ R n as follows:

Lemma 1. Consider the open-loop dynamics
The termŝk ∈ R n are vectors of estimator states, k ∈ R n are free functions,̃k =̂k + k (x,x) are vectors of adaptive estimates, r k = diag{[r ]} ∈ R n×n are diagonal matrices of dynamic scaling factors r kj ∈ R + . Finally,x ∈ R n are observer states obtained from the following filter with K e ≥ ( ∏ k=0 r k ) 2 + 2 I n ∑ k=0 ( max (r k )‖ k ‖) 2 , where max {r k } indicates the maximum eigenvalue of r k , and k are known functions, while , , ∈ R + are positive constants: Since f k (x) are continuously differentiable by hypothesis, we can write f k (x) T − f k (x) T = e T k for some functions k (x, e) ∈ R n , with e =x − x, ∈ R n . Together with (12) consider the following adaptation law with estimates̃k, with > 1 + , and with > 1/3:̃k Then e and ∑ k=0 (f k (x)z k r k ) are bounded and converge to zero. In addition, r k are bounded ∀k.
Proof: Computing the time derivative of z k we obtain from (11): Substituting (13) with f k (x) T = f k (x) T + e T k yields: Defining the first Lyapunov function candidate W = 1 ≥ 0 and computing its time derivative while substituting (15) gives:Ẇ Defining the second Lyapunov function candidate W ′ = W + 1 2 e T e ≥ 0 and computing its time derivative, wherėe from (9) and (12) gives: Recalling that K e > ( ∏ k=0 r k ) 2 , observing that ≥ 3∕4 for ≥ + 1 and employing a Schur complement argument in (17) confirms thatẆ ′ ≤ 0 if > 1/3. To prove boundedness of r k we define the Lyapunov function candidate U = W ′ + 2 ∑ k=0 max (r k ) 2 ≥ 0. Computing its time derivative from (13) and (17) we obtain: Substituting K e in (18) confirms thatU ≤ 0 and consequently r k ∈  ∞ . In addition, from (17) Notably, the adaptation law (12) and (13) estimates a total of n × ( + 1) different parameters hence it extends the result of Reference 33, where only + 1 parameters would be estimated.   . 34 For the constant component of the disturbances we have f 0 (x) = f 0 (x) = I n and consequently 0 = − x, 0 x = 0,ṙ 0 = 0 in (13). In general, if the functions f k (x) are integrable in x, the adaptation law (12) and (13) can be simplified adopting constant r k and omitting the filter (12) as:̃k Defining the Lyapunov function candidate W 0 = 1 2 ∑ k=0 z T k z k and computing its time derivative while substituting (19) gives in this case:Ẇ which holds true for all > 0. Finally, since according to Assumption 1 the state-dependent component of the disturbance vanishes at the equilibrium, it follows from Lemma 1 that z 0 converges to zero. Consequently, the assignable open-loop equilibrium x * computed with the estimates̃0 in (10) converges to the correct value. The second step in the design of the new robust dynamic state-feedback consists in redefining the partial derivatives of the auxiliary energy function (7) as follows, where we have set R(t) = Φ(x(t), (t)) for all t ≥ 0 as in Reference [9,Proposition 5]: The constant term F 0 and the state-dependent term F 1 (x) are designed according to the following set of one matching equation and two strict-minimizer conditions: The first equation in (22) is of algebraic nature and ensures matching of the disturbances in (9) hence it can be interpreted as a disturbance-matching condition. In particular, subtracting the first equation in (22) from the matching equations for system (9) recovers (3). Consequently, with the proposed controller design, the algebraic solution P of the original matching Equation (3) is preserved in the presence of disturbances. The two remaining conditions enforce a strict-minimizer of H ′ d (x, ) in x = x * and in some = * . It follows from (10) that, employing the parameterization (4) shows that the terms F 0 and F 1 (x) ensure the existence of a strict-minimizer in x = x * even if h(x * ) ≠ 0 and 2 h(x * ) ≠ 0. From (21), the auxiliary energy function in a nonempty set containing the equilibrium x * can be approximated as: If there exist some  x ,  0 ,  1 ∈ R + so that ‖F 0 ‖ <  0 , ‖F 1 (x)‖ <  1 for all ‖x − x * ‖ <  x then there exist some arbitrary constant  ∈ R + so that (23) is locally positive definite.
Finally, the new control law is constructed augmenting the dynamic state-feedback (8) with a disturbance compensation term according to the following result which builds upon. 9 Proposition 1. Consider the PCH system (9) with disturbances under Assumptions 1 and 2. Define the algebraic solution P so that Φ(x, ) = Φ(x, ) T > 0 for all (x, ) in a nonempty set ⊆ R n × R n containing the equilibrium (x * , * ), and let Define the following control law, with F 1 (x), F 0 satisfying conditions (22), and with̃k estimated according to (12) and (13): Then the equilibrium (x * , * ) is locally stable for some then the equilibrium is (locally) asymptotically stable. Proof: Adding the term u * to u in (24) and regrouping common factors results in: Substituting (25) and ∇ x H ′ d from (21) into (9) results in the following closed-loop dynamics: According to Assumptions 1 and 2, x * is an assignable equilibrium of the open-loop system (9), while it follows from (22) that x * is a strict-minimizer of H ′ d . To prove the stability claim, we define the Lyapunov function Computing the time derivative of W ′′ and substituting (17), (21),(26) yields: Rearranging terms in (27) and recalling that J d = −J T d yields: Observing that ) ≥ 1 4 for ≥ + 1 and employing a Schur complement argument in (28) while recalling from Lemma 1 that In particular, ( ∏ k=0 r k ) 2 > I n assigning the initial condition r k (t = 0) = I n and considering thatṙ k ≥ 0 in (13), thus 0 ( ∏ k=0 r k ) 2 > I n for all 0 > 1. Consequently (x * , * ) is a locally stable equilibrium of the closed-loop system.
To prove the asymptotic claim we observe that ∇ x H ′ d ∈  2 , while computing its time derivative and substituting (26)

Corollary 1. Consider the PCH system (9) with disturbances under Assumptions 1 and 2. Define the algebraic solution
Assume that the functions f k (x) are integrable and consider the control law (24) with F 1 (x), F 0 satisfying conditions (22), whilẽk are estimated adaptively with (19). Then the equilibrium (x * , * ) is locally stable for some constants K = K T ≥ 0 and some > 0 such that D d > I n /4.
Proof: Defining the Lyapunov function candidate W ′′ 0 = H ′ d (x, ) + W 0 computing its time derivative and substituting (9), (20), (21), (24) gives: Rearranging terms in (29) gives: which holds true for some K = K T ≥ 0 if > 0, D d > I n /4 and concludes the proof ■. (24) is expressed in a modular form combining the dynamic state-feedback (8), which remains unchanged, and the disturbance compensation term u * . Taking advantage of this structure, different definitions of algebraic solution of the matching equations can be employed. The proposed method is also applicable to the conventional IDA-PBC design that relies on the classical solution of the PDE, and it represents an extension of References 26,35 for multiple additive variable disturbances. In any case, the terms F 1 (x), F 0 contained in u * are computed from (22), which does not include any PDE. Finally, compared with controller (8) an additional parameter is introduced in (24) from (12) to (13). In this respect, different tuning requirements for , D d are presented in Proposition 1 and Corollary 1: ≥ 0 ( ∏ k=0 r k ) 2 > I n , 0 > 1, 0 D d > I n are necessary in the former case; less stringent requirements corresponding to > 0, D d > I n /4 are specified if the functions f k (x) are integrable. In addition, the strict-minimizer condition ∇ 2 x H d (x * , * ) > 0 in (22) provides the opportunity to introduce further tuning parameters within F 1 (x) which can serve the purpose of shaping the transient performance of the closed-loop system (ref. Section 5).

UNDERACTUATED MECHANICAL SYSTEMS
In this section, the definition of algebraic solution is applied to the potential-energy PDE for underactuated mechanical systems in PCH form with multiple linearly parameterized additive disturbances, provided that the kinetic-energy PDE is solvable analytically. The system dynamics in the presence of disturbances is described as: As a result H = d + L T q + h(q) + 1 2 q T H q q + 1 2 p T M −1 p, which corresponds to the expression introduced in Section 2, where h(q) only contains the nonlinear terms of V. The closed-loop dynamics in the absence of disturbances is typically expressed as: where M d = M T d > 0 is the closed-loop inertia matrix to be defined, while J 2 (q, p) = −J T 2 is a free matrix and D d = D T d > 0 is the damping injection matrix. The closed-loop Hamiltonian is defined as H d = 1 2 p T M −1 d p + V d with potential energy V d and has a strict-minimizer in (q, p) = (q * 0 , 0). In the presence of disturbances the system has an assignable open-loop equilibrium if the following equality is verified, where consistently with Assumption 2 we have f p (0) ≡ 0, f q (q * ) ≡ 0: In particular, if G ⊥ 0 = 0 then the open-loop equilibrium q * = q * 0 is assignable and can be stabilized with an appropriate control action. The matching equations corresponding to (3) are termed potential-energy PDE and kinetic-energy PDE and are defined as follows for system (31): Typically D d = Gk v G T with the tuning parameter k v > 0, while the matrix J 2 can be defined as i are free matrices. 1 This design choice allows reducing the number of kinetic-energy PDE to s(s + 1)(s + 2)/6, where s = n − m > 0 is the degree of underactuation. 36 In the particular case of constant inertia matrix M, the kinetic-energy PDE can be solved with a constant M d and J 2 = 0. The following result is, however, not limited to this case, provided that the kinetic-energy PDE is solvable analytically, since the algebraic solution is only defined for the potential-energy PDE (see Section 5.2).
Applying Definition 2 to the disturbance-free system (31), the continuously differentiable matrix valued function P(q) ∈ R n × n is an algebraic solution of the potential-energy PDE (34) if: In general P(q) might not be a solution of the original PDE (34). Similarly to (5), we introduce the auxiliary energy function H d (q, p, ) defined on an extended state space with ∈ R n as: Defining the terms Φ(q, ) and Ψ(q, ) so that (P(q) − P( ))q = Φ(q, )(q − ) and Ψ(q, ) = (1/2)( P( )q/ ) and computing the partial derivatives of H d (q, p, ) gives: Drawing a parallel to controller (8), let us consider an algebraic solution P of the potential-energy PDE (34) such that Φ(q, ) = Φ(q, ) T > 0 in some nonempty open neighborhood Ω of the equilibrium, let R(t) = Φ(q(t), (t)) for all t ≥ 0, and Define the following dynamic state-feedback control law for the disturbance-free version of system (31): Then, according to Reference [9, Proposition 5] the disturbance-free system (31) in closed-loop with (37) has a locally stable equilibrium at (q, p, ) = (q * , 0, 0) for some K = K T ≥ 0 and v = 0 provided that D ′ d > 0. To account for the disturbances in the controller design we redefine the partial derivatives of the auxiliary energy function in (36) as follows, where R(t) = Φ(x(t), (t)) for all t ≥ 0: The constant term F 0 and the state-dependent term F 1 (q, p) should verify the following conditions: From (39), the auxiliary closed-loop energy function in a nonempty set containing the equilibrium (q * , 0, * ) can be approximated as: H ′ d (q, p, ) = H d (q, p, ) + (q − q * ) T (F 0 + F 1 (q * , 0)) + . Finally, the robust dynamic state-feedback controller for underactuated mechanical systems is constructed augmenting the dynamic state-feedback (37) with a disturbance compensation term as stated in the following result. (35) so that Φ(q, ) = Φ(q, ) T > 0 for all (q, ) in a nonempty set ⊆ R n × R n containing the equilibrium (q * , * ), and let Ψ T R −1Φ R −1 Ψ < 2(N T + P)D ′ d (N + P), where R(t) = Φ(q(t), (t)) for all t ≥ 0 and Nq = q V − L. Consider the control law u ′ = u + u * with u defined in (37), F 1 (q, p), F 0 defined according to (39),̃0,̃1,̃2 estimated adaptively with (12)- (13), and u * defined as follows:

Corollary 2. Consider the underactuated mechanical system (31) with disturbances under Assumptions 1 and 2. Define the algebraic solution P of the potential-energy PDE as in
Then the assignable equilibrium (q, p, ) = (q * , 0, * ) with q * defined in (33) is locally stable for some In addition, if f q (q), f p (p) are integrable, the equilibrium (q, p, ) = (q * , 0, * ) is locally stable if > 0 and D ′ d > I n ∕4. The proof follows closely that of Proposition 1 and Corollary 1 employing the Lyapunov function candidate + 1 2 e T e, thus it is omitted for brevity ■.
Remark 3. The algebraic solution (35) only applies to the potential-energy PDE and not to the kinetic-energy PDE, which should be solved in the traditional way. If an analytical solution of the kinetic-energy PDE exists, the result presented in this section is also applicable to systems with nonconstant inertia matrix M and nonconstant input matrix G (see Section 5.2). Limiting the algebraic solution to the potential-energy PDE simplifies the control law (37) and preserves the mechanical structure of the closed-loop system (32) on the extended state space. Notably, while the kinetic-energy PDE can be reduced to ordinary differential equations or to algebraic equations for different classes of systems under some assumptions, 4,37 this is not the case for the potential-energy PDE. Finally, the disturbance compensation controller (40) can in principle be used with different definitions of algebraic solution besides the one proposed in (35). A promising approach in this respect would be to employ directly Definition 2 with the system state x = [ q p ] and the closed-loop dynamics (2), consequently not preserving the mechanical structure of (31) in closed-loop. Employing this method could allow defining an algebraic solution also for the kinetic-energy PDE, which is among the goals of our future work.
Remark 4. Fulfilling the condition 0 D ′ d ≥ I n (or D ′ d > I n ∕4) required by Corollary 2 is typically a challenging task for underactuated mechanical systems since it might not be possible to assign a globally positive definite matrix D ′ d in some cases. 15 Since the result of Corollary 2 is local, a sufficient condition is that D ′ d ≥ 0 globally and that D ′ d > 0 in a nonempty set containing the equilibrium (q, p) = (q * , 0). This condition can be fulfilled in the presence of physical damping provided that locally DM −1 M d > 0. Local stability of the assignable equilibrium could then be concluded with the following argument 34 : (i) the assignable equilibrium is (locally) asymptotically stable for system (31) in closed-loop with controller (37) and (40) if the combined estimation error (z 0 + f q z 1 r 1 + f p z 2 r 2 ) is zero; ii) the combined estimation error is bounded and converges to zero asymptotically if > 1/3, provided that f q , f p and their first derivatives are bounded (ref. Lemma 1).

EXAMPLES: MAGNETIC-LEVITATED-BALL AND BALL-ON-BEAM
In this section, the effectiveness of the proposed approach is demonstrated with numerical simulations on two systems.

Magnetic-levitated-ball system
The magnetic-levitated-ball system 9  . (41) The terms 01 , 11 x 1 represent the matched disturbances on the current, while the terms 02 , 12 x 3 represent the unmatched disturbances and correspond, respectively, to a constant force acting on the ball and to viscous friction. The relationship between ball position and velocity is typically unaffected by external disturbances hence the second element of the disturbance vector is null. The control aim for the disturbance-free version of system (41) is to stabilize the desired equilibrium   (24) is constructed with F 0 , F 1 defined according to (22), which yields: .
In particular, the third element of F 1 is freely chosen asx 2 introducing the tuning parameter which appears in the strict-minimizer condition for H ′ d : The above inequality holds true for somẽ0 2 ,̃1 2 if > − 1. While a constant is considered here for demonstrative purposes, any function (x) > −1 that satisfies (22) can in principle be employed allowing further design freedom.
The numerical simulations employ the following parameters: = 1; m = 1; = 1; g = 9.81; x 2d = 1; c = 0.1; K ≡ 0; = 0; = 2; = 1; = 0 and alternatively = 1.5. In addition, K e = 1.5(r 0 r 1 ) 2 , with r 0 = I 2 and r 1 = . The initial conditions were set to x =x = ≡ 0; r 1 = r 2 = 1 and the disturbances were defined as follows: 01 = 1; 02 = − 5; 11 = 3; 12 = 5. The time history of the system states with = 1.5 and = 0 is depicted in Figure 1 confirming that the ball position reaches the desired value x 2d = 1 in spite of the disturbances. In addition, larger values of result in higher responsiveness in accordance with (43), while the additional states remain at zero with the chosen tuning parameters. For comparison purposes, employing controller (8) without disturbance compensation results in the ball position settling at x 2 = − 1.65 corresponding to a large error. Finally, the estimation errors z I =̃0 1 − 01 + (̃1 1 − 11 )x 1 and z =̃0 2 − 02 + (̃1 2 − 12 )x 3 are bounded and converge to zero asymptotically, while the scaling factors r 1 , r 2 are bounded (see Figure 2). The time histories of the control input and of the Lyapunov function W ′′ are depicted in Figure 3.

Ball-on-beam system
The ball-on-beam system consists of a beam of length 2L hinged at the mid-point and actuated with a torque u, and of a ball with point mass that is free to move above the beam. Making simplifying assumptions on the masses of ball and beam as in Reference 1, the open-loop dynamics in the presence of disturbances consisting of constant forces 01 , 02 and viscous friction 11 , 12 becomes:q 1 + g sin(q 2 ) − q 1q 2 2 = −( 01 + 11q1 ) The position q 1 ∈ (−L; L) is the distance of the ball from the midpoint of the beam, q 2 ∈ is the inclination of the beam from the horizontal, and g is the gravity constant. To express (44) in the PCH form (31) we define the open-loop , which is not constant, the input matrix G = ] and the potential energy V = gq 1 sin(q 2 ).
The control aim for the disturbance-free version of system (44) is to stabilize the equilibrium (q 1 , q 2 ) = (0, 0) corresponding to the horizontal beam with the ball at its midpoint. Note thus that h(0, 0) = 0 and 2 h(0, 0) = 0. In the presence of disturbances, the open-loop assignable equilibrium is computed according to (33) as and exists if |̃0 1 | < g. In this case, since V contains a cross term in q 1 , q 2 , the unmatched disturbances affect the equilibrium of the actuated position q 2 . The dynamic state-feedback controller (37) and the disturbance compensation (40) are employed, while the unknown parameters are estimated using the adaptation law with dynamic scaling (12) and (13) resulting in: 01 = −q 1 ; 02 = −q 2 (L 2 + q 2 1 ); 11 = −q 1q 1 ; 12 = −q 2q 2 (L 2 + q 2 1 ). The closed-loop inertia matrix computed solving the kinetic-energy PDE as in Reference 1 and the algebraic solution of the potential-energy PDE are reported in Appendix A2. The terms F 0 , F 1 (q) are computed according to (39): In particular, the first element of F 1 can be freely chosen and is taken proportional to q 1 introducing the tuning parameter which provides a handle on the transient performance of the closed-loop system and appears in the following strict-minimizer condition: The above inequality holds true for all > − K p /(2L 2 ), since |q * 2 | < ∕2 by hypothesis. The numerical simulations employ the following parameters: L = 0.5; g = 9.81; K p = 1; K v = 1; K ≡ 0; = 0; = 25; = 1; = 1 and alternatively = − 1. In addition, K e = 2(r 0 r 1 ) 2 , with r 0 = I 2 and r 1 = . The initial conditions were set to q = (0.5,0); ≡ 0; p ≡ 0;q = q; r 1 = r 2 = 1 corresponding to the horizontal beam with the ball positioned at its endpoint, while the disturbances were defined as follows: 01 = 2; 02 = 5; 11 = 15; 12 = 10. Employing the IDA-PBC 1 without disturbance compensation the ball moves away from the desired equilibrium and beyond the limit length of the beam. Conversely, with controller (37) and (40) the position converges to the assignable equilibrium q * = (0, −0.205) at a faster rate for larger values of (see Figure 4). The additional states remain at zero with the chosen tuning parameters. In addition, the cumulative estimation errors z I =̃0 1 − 01 + (̃1 1 − 11 )q 1 and z =̃0 2 − 02 + (̃1 2 − 12 )q 2 are bounded and converge to zero asymptotically, while the scaling factors r 1 , r 2 are bounded (see Figure 5). Finally, the time histories of the control input and of the Lyapunov function W ′′ are depicted in Figure 6.

CONCLUSIONS
This work investigated the control problem for underactuated PCH systems with multiple additive linearly parameterized disturbances including matched, unmatched, constant, and state-dependent components. The definition of algebraic solution of the matching equations was employed to construct a new dynamic state-feedback controller which includes a disturbance compensation term, while an adaptation law was designed to estimate the unknown parameters. The proposed approach was detailed for underactuated mechanical systems employing an algebraic solution of the potential-energy PDE. The effectiveness of the controller was demonstrated with numerical simulations of the magnetic-levitated-ball system and of the ball-on-beam system. For the latter, an algebraic solution of the potential-energy PDE was provided. As part of our future work we aim to extend the definition of algebraic solution for mechanical systems to the kinetic-energy PDE, and to validate the results with experiments.

A.1 Magnetic-levitated-ball system
Algebraic solution of the matching equations for the magnetic-levitated-ball system, 9 where: ; and c is a tuning parameter. (A1)

A.2 Ball-on-beam system
Algebraic solution of the potential-energy PDE for the ball-on-beam system. The matrix P and R were approximated with Taylor series for numerical simulation purposes. Notably P ≠ P T thus it is not the gradient of any scalar function. The closed-loop inertia matrix M d , the free matrix J 2 , and the damping matrix D d are defined as in Reference 1. The positive constants K p , K v are tuning parameters. Note that in this case the physical damping D > 0 is present in the form of viscous friction thus D ′ d > 0 locally.