Non-linear PI regulators in control problems for holonomic mechanical systems

ABSTRACT In this paper we propose a solution to the regulation problem of holonomic mechanical systems without velocity measurements by constructing nonlinear PI controllers. An approach based on the Lyapunov functional method is used to analyse the stability of the closed-loop system which has the form of Volterra integro-differential equation. The paper presents the quasi-invariance principle for Volterra nonlinear integral differential equations. Furthermore, we present sufficient conditions that one can use to conclude the asymptotic stability of the closed-loop system via using a Lyapunov functional with semi-definite time derivative. The simulation results of the two-link robot manipulator demonstrate the effectiveness of the proposed PI controller.

One of the approaches to this problem consists in using a state-feedback controller with an observer which reconstructs the lacking velocity signals, see for instance Canudas de Wit and Fixot (1991) and Nicosia and Tomei (1990). As mentioned in Berghuis and Nijmeijer (1993a) the main drawback of such approach is the local stability conditions for the closed-loop system. A different approach to the regulation problem for robot manipulators without velocity measurements consists in using a dynamic position feedback controller via a first-order linear compensator by invoking LaSalle's invariance principle, see for instance Nijmeijer (1993a,1993b), Burkov (1995Burkov ( , 1998Burkov ( , 2009), Loria et al. (1997) and Siciliano and Villlani (1996). However, these results are applicable only for non-dissipative mechanical systems with the potential energy which doesn't depend on time.
In fact the use of dynamic compensators is an adding integral terms in the regulator. Using the representation of integral terms in the structure of control signals as regulators with unlimited after-effect (Anan'evskii & Kolmanovskii, 1989;Andreev, 2009), the motion of CONTACT Olga Peregudova peregudovaoa@gmail.com mechanical systems with these types of regulators can be modeled by Volterra integro-differential equations (Volterra, 1959). Such equations arise in the mathematical modelling of viscoelastic materials (Dafermos & Nohel 1981;Mac 1977;Sergeev 2007b;Volterra, 1959), population dynamics (Britton 1990;Volterra 1959), agedependent epidemic of a disease (El-Doma, 1987), nuclear reactor dynamics (Kappel & Di, 1972). The study of the qualitative theory of Volterra integro-differential equations including the stability problem attracts great attention of numerous researches, see for instance Burton (1983), Grimmer and Seifert (1975) and Sergeev (2007aSergeev ( , 2017 and their bibliographies. In the paper (Anan'evskii and Kolmanovskii 1989) the stabilizability of the rigid body systems was established by using PID controllers and Lyapunov functional method. Nevertheless, the stability analysis for Volterra nonlinear integrodifferential equations still remains an open problem.
In this paper, we show that Volterra integro-differential equations can serve as a powerful tool for mathematical modelling the controlled mechanical systems without velocity measurement. We consider a large class of holonomic mechanical systems under the action of potential, dissipative and gyroscopic forces assuming that the potential energy can depend on time. We present a nonlinear PI controller which solves the regulation problem for such systems.
There are two contributions of this paper. In the first part of the paper new theorems of LaSalle's type on the limit behaviour of the solutions, on asymptotic stability of the zero solution are proved for Volterra integrodifferential equations. The second part of the paper contains the results of solving the regulation problem for a holonomic mechanical system based on the construction of nonlinear PI regulators.

The quasi-invariance principle for Volterra integro-differential equation
Consider a Volterra nonlinear integro-differential equation in the forṁ where x ∈ R n is the phase vector; f and g are the functions defined and continuous in the domains R × D (D ⊂ R n ) and R + × D × D respectively.

Assumption 2.1:
Assume that the function f = f (t, x) satisfies the Lipschitz condition, i.e. for each compact set K 1 ⊂ D there exists L 1 = L 1 (K 1 ) = const > 0 such that the following inequality holds Assumption 2.2: Assume that the function g(t, x, y) satisfies the following conditions: for each compact set K 2 ⊂ D × D the following inequalities hold where L 2j = L 2j (K 2 ) (j = 1, 2).

Consider a family of translates of the function
Using the precompactness property of the family (5), we can find the set of limiting functions (Artstein, 1977) f For (1) define the family of limiting integro-differential equations aṡ Remark 2.1: Equation (7) is the functional-differential equation with infinite delay. The domain of its definition is R × C.
Definition 2.1: Let x = x(t, x 0 ) be some solution of (1) bounded by some compact set K ⊂ D for all t 0. Define a positive limit point p ∈ D and the corresponding positive limit set ω + as follows (Artstein, 1977) The following property of the set ω + holds.
Proof: Let p ∈ ω + be a limit point defined by the sequence t k → +∞, i.e.
Continuing that process further one can find the subsequences t * m → +∞, T * m → +∞ and the function x = ϕ(t) such that the sequence x (m) is a solution of (1), one can get the following equalities Differentiating the equality (8) on t, one can get that the function x = ϕ(t), ϕ(0) = p is a solution of the Equation (7).
Assume that for Equation (1) one can find a Lyapunov functional candidate as where V 1 and V 2 are some nonnegative scalar functions which are defined and continuous differential in the domains R × D and R + × D × D respectively. Assume also that the following estimate holdṡ (10) where W 1 (t, x) and W 2 (τ , x, y) are some non-negative functions defined and continuous in the domains R + × D and R + × D × D. The functions W 1 (t, x) and W 2 (τ , x, y) satisfy in these domains the conditions such as (2), (3) and (4). Hence in particular, for a continuous function x : Consider a family of translates {W τ 1 (t, x) = W 1 (τ + t, x), τ ∈ R + }. We introduce the limiting functions of W 1 as follows The function W * 1 (t, x) is defined in the domain R × D for almost all t ∈ R.
Theorem 2.2: Assume that one can find a Lyapunov functional candidate V = V(t, x t ) bounded for all continuous function x : R + → D whose upper right-hand derivative satisfies the inequality (10). Then, for each bounded by some compact set K ⊂ D solution x = x(t) of Equation (1) the set ω + consists of the solutions of Equation (1) which satisfy the following equalities is monotonically decreasing along the solution x = x(t, x 0 ) due to the condition (10). Therefore, the following holds From the inequality (9) for all T > 0 one can find that Let ω + be a positive limit set and p ∈ ω + be a positive limit point defined by the sequence t k → +∞, x(t k , x 0 ) → p. As in the proof of Theorem 2.1 one can find the solution x = φ(t) of the Equation (7) which passes through the point p, φ(0) = p. Thereafter, for the sequences t * m → +∞ and T * m → +∞ constructed in Theorem 2.1 we have Hence, passing to the limit in (16) for m → +∞ and using (14) we obtain for all t ∈ R and correspondingly W 2 (τ − s, φ(τ ), φ(s)) = 0, s τ . Remark 2.2: Theorem 2.2 presents an invariance principle for Volterra integro-differential Equations (1). (1) the following holds f (0) = 0, g(τ , 0, 0) = 0. Therefore, Equation (1) has a zero solution x(t, 0) = 0.

Assume that in Equation
Using Theorem 2.2 one can easily obtain the following sufficient conditions for the asymptotic stability of the Equation (1). Theorem 2.3: Assume that one can find a functional (9) with a function V 1 (t, x) a( x ) whose upper right-hand derivative satisfies the inequality (10). Assume also that there are no solutions x = φ(t) of the Equation (7) satisfying the following equalities

The regulation problem for a holonomic mechanical system on the base of PI regulator
Consider a controlled mechanical system with n degrees of freedom described by Lagrange equations in the form d dt where q is the vector of generalized coordinates, T = q A(q)q/2 is the kinetic energy of the system with inertial matrix A(q),q = dq/dt, Q(q,q) is the vector of generalized dissipative and gyroscopic forces, Q(q, 0) = 0, Q q 0, = (t, q) is the potential energy, U is the generalized control force, () is the transpose operation. Suppose that included in (18) functions are defined and continuous for all q ∈ R n . Assume that restrictions on the control input U are not imposed.
Represent the Equations (18) resolved with respect tö q in the form The coefficients of the matrix of inertial forces C = (c jk ) are defined by the following equality Consider the stabilization problem of programme positionq Show that the stabilization problem is solved by using a nonlinear PI regulator (see the block diagram in Figure 1) such as where u ∈ R × R n → R is some continuously differentiable function, P : R + → R n×n is some nonnegative matrix function with a derivative ∂P(s)/∂s such that where α(s) > 0 ∀s 0, f : R → R n is some differentiable function which has finite number of the prototypes f (c) in any bounded domain {q ∈ R m : q μ = const}. In other words, there exists finite number of solutions of the equation f (q) = c. Let us make the change of variables such as x = q − q (0) , y = q . Then, the Equations (18) with the controller (22) can be written as where the subscript '1' denotes the functions which are obtained from the corresponding functions included in (21) and (22) as a result of the aforementioned change of variables.
Consider the Lyapunov functional candidate as For the time derivative of the functional (25) due to the Equation (24) we obtaiṅ Accordingly to Theorems 2.2 and 2.3 one can simply obtain the following result.
Remark 3.2: Theorem 3.1 presents the basis for constructing nonlinear PI regulators for solution to the regulation problem for a large class of holonomic mechanical systems.

Example
Consider the regulation problem for a double-link planar robot manipulator (Fantoni & Lozano, 2002). The manipulator (see Figure 2) consists of two absolutely rigid links G 1 , G 2 and two cylinder joints O 1 , O 2 . Both the links can move in the vertical plane only. The mass centre C 1 of the link G 1 is situated on the ray O 1 O 2 . The position of the mass center C 2 of the link G 2 doesn't coincide with the position of the joint O 2 .
Let us introduce the following notation. q 1 is an angle between the horizon line and the ray O 1 C 1 ; q 2 is an angle between the lines O 1 O 2 and O 2 C 2 ; l i is a length of the segment O i C i ; l is a length of the segment O 1 O 2 ; m i is a mass of the link G i ; I i is an inertia moment of the link G i about the axis of the joint O i ; g is the acceleration due to gravity. The kinetic energy of the manipulator is given by T = 1 2 (I 1 + I 2 + m 2 l 2 + m 2 ll 2 cos q 2 )q 2 1 + (I 2 + m 2 ll 2 cos q 2 )q 1q2 + I 2q 2 2 The gravitational moments in the ith joint have the form M 1 = −μ 1 cos q 1 − μ 2 cos(q 1 + q 2 ), where μ 1 = (m 1 l 1 + m 2 l)g and μ 2 = m 2 l 2 g. Choose the reals ν i , p i and α i (i = 1, 2) such that the following inequalities hold Then, using Theorem 3.1, one can conclude that the control moments − p 2 cos q 2 (t) t 0 e −α 2 (t−s) (sin q 2 (t) − sin q 2 (s)) ds (27) globally stabilize the programme positioṅ  (27) were fixed in ν 1 = 7, 4, ν 2 = 1, 5, p 1 = p 2 = 10, α 1 = α 2 = 3. The time interval was chosen as 20 s.
The angular positions q 1 (t), q 2 (t) and programme position coordinates q These graphs show that the controller (27) solves the global regulation problem for the manipulator. The link angular positions q 1 (t) and q 2 (t) converge to the corresponding reference positions. The terms containing μ 1 and μ 2 in the controller (27) compensate the action of the gravitational moments in the programme state (28) of the manipulator. From a practical viewpoint, an advantage of the saturated controller (27) is that it satisfies the input constraints. It should be noted that our controller (27) is given in explicit form and has been designed more easily and simply then the bounded control schemes via dynamic output feedback proposed by Burkov (1995Burkov ( , 2009) and Loria et al. (1997). Moreover, the performance of the controller (27) does not require the integration of additional differential equations.

Conclusion
The paper presents the solution to the regulation problem for holonomic mechanical systems without velocity measurements. Nonlinear PI controller has been proposed which solves this problem. The closed-loop system has been represented in the form of a nonlinear Volterra integro-differential equation. An asymptotic stability property of Volterra equation has been studied by constructing a Lyapunov functional with a semidefinite time derivative. The important advantages of the proposed controller are its explicit form and simple design. Through the new design, the proposed control scheme does not require the on-line integration of the differential equations. Moreover, the proposed control scheme allows the use of uncontrolled forces to solve the regulation problem. The performance of the controller was illustrated via simulation on a two-link planar robot manipulator.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by Ministry of Education and Science of the Russian Federation within the framework of the State task under Grant [9.5994.2017/BP].

Nomenclature/Notation
Let R n be n-dimensional linear real space with the norm x . The symbol R + denotes the positive real semi-axis R + = [0; +∞). The symbol R denotes the real axis R = (−∞; +∞). C is Banach space of continuous functions ϕ : R → R n with the norm |||ϕ||| = sup( ϕ(s) , −∞ < s < +∞}. Denote by a : R + → R + a function of Hahn type, i.e. a(0) = 0 and a is continuous and strictly monotonically increasing. The symbol E denotes the identity matrix E ∈ R n×n . Denote by (f * g)(t) a convolution of two functions f (t) and g(t) defined for t ≥ 0, i.e. (f * g)(t) = t 0 f (τ )g(t − τ ) dτ .