Optimal nonlinear damping control of second-order systems

New nonlinear damping control is proposed for the second-order systems. The proportional output feedback is combined with the damping term which is quadratic to the output derivative and inverse to the set-point distance. The global stability, passivity property, and convergence time and accuracy are shown. The suggested nonlinear damping is denoted as optimal since requiring no design parameters and ensuring a fast convergence without transient overshoots.


Introduction
For the second-order systems, it is understood that linear feedback control [1] pose inherent certain limits in terms of possibility to shape the transient response, exponential convergence of the state trajectories and, as implication, steady-state accuracy of the controlled output of interest. Worth to recall is that the input-output secondorder systems encompass a vast number of practical applications. Input voltage to output speed in the drives, transfer characteristics of different-type RLC circuits, pressureflow dynamics in the fluid transport systems and, finally, motion dynamics of the rigid-body mechanical systems can be mentioned as motivating examples for that.
Nonlinear control methodology addressed, since long, the problem of an efficient feedback shaping, while complexity of the associated analysis and synthesis, availability of the system states, control specification, and type of the system perturbations led to quite different design concepts. Among well-established are the sliding mode control [2], Lypunov redesign [3], backstepping [4], and passivity based control [5], for which details we refer to the basic literature [6,7]. Some examples of the nonlinear feedback stabilization and associated nonlinear damping can be found in e.g. [8,9]. A comparative evaluation of different controllers, benchmarked on a most simple secondorder plant of double integrator, can also be found in [10].
The need to incorporate nonlinear damping in feedback of the second-order systems, especially for improving the stabilizing and convergence properties, has been (empirically) recognized in already former studies in robotics, thus resulting in e.g. nonlinear proportional-derivative controls [11,12]. While the stability proof has been provided for several ad-hoc nonlinear damping strategies, no optimal convergence and trajectories shaping have been so far elaborated. Here it is also worth noting that the convergence Email address: michael.ruderman@uia.no (Michael Ruderman) properties are strongly related to the homogeneity of dynamics vector-field and, as implication, of the feedback map to be determined, see e.g. [13] for use of homogeneity for synthesis in the modern sliding mode control.
In feedback control, it appears that to input energy into a system, through potential field of the output feedback, is more straightforward than to control damping in a meaningful way, ensuring the desired convergence to an equilibrium. For energy shaping in the feedback regulated Euler-Lagrange systems we refer to [14] and seminal work [5].
In this paper, we propose a novel nonlinear damping control of the second-order systems, in combination with the linear output feedback. Using the fact of conservative energies exchange in an undamped (oscillatory) secondorder system, the dissipated energy is shaped in an optimal way with respect to the convergence to zero equilibrium and no transient overshoot independent of the initial state. That way assigned nonlinear damping is quadratic to the output derivative and inverse to the set-point distance, while no free design parameters for the damping term are required. The proposed control is generic and globally asymptotically stable. The principle analysis of the control behavior, provided below, is focused on the unperturbed second-order dynamics, so that the sensitivity and robustness aspects are subject to the future works.

Second order system with state-feedback
Throughout the paper we will deal with the feedback controlled second-order systemṡ where x 1 and x 2 are the measurable state variables, k > 0 is the proportional feedback gain, and D is the control damping we are interested in. Obviously, the system (1), (2) is a classical double-integrator dynamics, for which a vast number of application examples can be found in electrical and mechanical systems and combinations of those.

Optimal linear damping
Using the linear state-feedback damping, the system (1), (2) can be written in a standard state-space form where the system matrix A is Hurwitz, for positive damping coefficients d > 0, and already in the controllable canonical form. It is wort recalling that the state-feedback controlled system (3) is equivalent to the proportional derivative (PD) controller for which an appropriate choice of the feedback gains allow for arbitrary shaping the closed-loop response, either in time t-or in Laplace sdomain. Assuming that k is given (by some control specification) and requiring the control response has no transient oscillations or overshoot, meaning the real poles only, one can assign the linear damping term by solving with respect to d. Here the real double-pole at −λ determines the optimal linear damping, usually noted as critical damping, since for d > 2λ the system behaves as overdamped, while for d < 2λ the system becomes transient oscillating. For any non-zero initial conditions , that can be seen as a setvalue control problem, the trajectories are given by It is obvious that the unperturbed first-order matrix differential equation (3), with two stable real poles, has an exponential converge property, meaning for some β, γ > 0 constants. From the output control viewpoint that means x 1 → 0 for t → ∞.

Optimal nonlinear damping
The proposed nonlinear damping provides the system (1), (2) to bė The single control parameter is the given output feedback gain, while the quadratic damping term yields optimal for all k > 0 values. The solution of (8) is non-singular except in x 1 = 0, while the unique equilibrium (x 1 , x 2 ) = 0 is globally attractive as will be shown below in section 3.2.
The phase portrait of the system (7) 1. One can recognize that the damping rate, and the required control effort, which is ∼ẋ 2 , notably increases in vicinity to x 1 = 0. At the same time, the non-singular solution provides the global convergence to origin within the II and IV quadrants without x 1 zero crossing, thus without transient overshoot of the control response. For showing this, consider the region of attraction in vicinity to the origin. For the steady-state one obtains which results in This can be seen as a trajectories' attractor in vicinity to zero equilibrium. Rewriting (10) as and allowing for the real solution only, results in along which the trajectories converge to zero in vicinity to origin, without crossing the x 2 -axis.

Global stability
Assume the following Lyapunov function candidate which is positive definite for all (x 1 , x 2 ) = 0 and also radially unbounded, i.e. V (x 1 x 2 ) → ∞ as x 1 , x 2 → ∞.
Taking the time derivative and substituting the second state dynamics from (8) results iṅ which implies the origin is globally stable. Since the trajectories do not remain staying on the x 1 -axis when x 2 = 0, that due to non-zero vector field, cf. (8), and proceed towards origin, cf. Fig. 1, the asymptotic stability of origin can also be concluded despiteV = 0 for x 2 = 0, x 1 = 0.

Closed-loop passivity
For analyzing damping properties of the control system (7), (8) we are to demonstrate the passivity of the closedloop dynamicsẋ Here the left-hand side can be seen as a conservative (oscillatory) system plant and the right-hand side as a stabilizing control input u which provides the closed-loop system with a required damping. Recall that for a system (with output y) to be passive, the input-output port power should be greater than or equal to the rate of energy stored in the system self, i.e. uy ≥V . Here the same energy function as the Lyapunov function candidate (13), which is the system's Hamiltonian, is assumed while x 1 is the controlled system output of interest. The above passivity power inequality results in which yields the system passivity condition in the state-space. Based on that it is evident that the system is always passive in the II and IV quadrants of the phase plane, see Fig. 2. Otherwise, the system becomes transiently non-passive for x 2 − x 1 < 0 in the I quadrant and for x 2 − x 1 > 0 in the III quadrant (gray-shadowed in Fig. 2). In those non-passive segments, the level of energy stored in the system increases, this way ensuring the state trajectories always cross x 1 -axis. Following to that, the trajectories fall into the passive segments (II or IV quadrant) of the control attractor to the stable origin.

Convergence time
The asymptotic convergence of the state solutions is ensured byV < 0. In order to ensure the finite-time convergence, one needs to show thaṫ for some positive time constant α > 0. If inequality (18) holds, the finite convergence time t c is bounded by Substituting the Lyapunov function candidate (13) and its time derivative (14) into (18) results in The graphical interpretation of inequality (20) is shown in Fig. 3 by two surfaces, of the energy level and its time derivative. One can recognize that the finite-time conver- gence can be ensured in vicinity to x 1 = 0 and that until certain neighborhood to the origin. Outside of those regions inequality (20) becomes violated, cf. Fig. 3, and the control system (7), (8) features the asymptotic convergence only. Here it is worth emphasizing that, from the applications viewpoint, such partial finite-time convergence can be desired and sufficient, since the convergence to zero equilibrium is inherently restricted by some finite resolution of the sensors used for the feedback control.

Comparative study
Two feedback control systems described by (1), (2) are compared: one with the linear damping D l = dx 2 and one with the proposed nonlinear damping D nl = x 2 2 |x 1 | −1 sign(x 2 ). The convergence of the state trajectories is shown in Fig. 4, for the [x 0 1 , x 0 2 ] = (1, 0) initial values and output feedback gain assigned to k = 100. The optimal (critical) linear damping factor, cf. (4), is d = 20.
Next the convergence of the controlled output is logarithmically shown in Fig. 5 for both, the linear and nonlinear damping. It can be seen that the nonlinear damping control reaches quadratically some low boundary of the steady-state accuracy, comparing to the linear decrease (on the logarithmic scale) of the linear damping control.
Finally, the output convergence and state trajectories in the phase plane are shown in Fig. 6 for the nonlinear

Conclusions
This brief has proposed the novel nonlinear damping control for the second-order unperturbed systems with an output feedback. The control is claimed to be optimal since it does not require any additional parameters and provides a fast (exponentially quadratic) convergence without transient overshoots. The global asymptotic stability, passivity, and finite-time convergence until certain neighborhood of the stable origin have been explored. An improved performance has been demonstrated comparing to the linear optimally (critically) damped controller.