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This paper addresses a leader–follower formation control problem for four-legged robots under discrete-valued input constraints. Four-legged robots are more suitable for rough terrain missions than wheeled robots because the tip positions of their legs can be changed depending on the terrain. However, it is difficult to control these robots through continuous-valued inputs because they are steered by switching between specific movements (i.e., discrete-valued signals). Motivated by this fact, we have proposed controllers that achieve fixed formations of four-legged robots using discrete-valued inputs, but moving formations, which are necessary for some applications, have not been considered yet. Herein, we present a solution to the above leader–follower formation control problem based on the combination of PD-like formation controllers and dynamic quantization. We further introduce a performance index to evaluate the difference between two systems whose inputs are quantized and unquantized and analyze the index for the feedback system with the presented controllers. As a result, an upper bound of the performance index is derived as a function of the system parameters. This is useful to evaluate the impact of the quantization on the behavior of the feedback system and to provide a theoretical guarantee of the stability of the system.

This paper considers the scalable control of power systems, i.e., control with the constraint that controllers of all generators are the same. Imposing this constraint reduces the time and effort required to construct large-scale power systems because we can obtain controllers of all generators merely by designing one controller. Power system control with this constraint has not been discussed in existing studies where controllers of generators have been assumed to be separately designed and implemented. The problem addressed here is to find the same controllers for stabilizing an equilibrium point of the resulting feedback system and improving the performance in terms of the time response. As a solution to this problem, we present controllers to increase the damping forces of generators uniformly. Then, we prove that the equilibrium point of the resulting feedback system is stable under certain conditions. Further, we present a design method of the controller gain to minimize a performance index for evaluating the time response of the resulting feedback system. The proposed controllers and design method are verified via numerical simulation.

This article considers tractors traveling on rough terrain and studies the effects of the road surfaces on their behavior. Clarifying these effects is useful for their stable traveling on rough terrain, which contributes to the automation of tractors. We select the H2 norm as a performance index to quantify the behavior of tractors and attempt to analyze the norm of a system corresponding to a tractor traveling on rough terrain. To improve the accuracy of the analysis, we modify the system by focusing on the relationship between the H2 norm, white noise, and the characteristics of the road surface. Subsequently, the H2 norm of the resulting system is derived as a function of the parameters, including the road roughness. Numerical examples demonstrate that the analysis result based on the modified system accurately captures the behavior of the tractor compared with the result based on the original system.

This paper studies scalable control of power systems, i.e., control with the constraint that controllers of all generators are the same. This control framework is useful to reduce the cost of constructing large-scale power systems because we can obtain controllers of all generators merely by designing one controller. The problem addressed here is to find the same controllers stabilizing an equilibrium point of the resulting feedback system and improving the performance in terms of the time response. As a solution to this problem, we present controllers to uniformly increase the damping forces of generators. We then show that an equilibrium point of the resulting feedback system is stable under certain conditions. In addition, we present a design method of the controller gain for improving the performance of the resulting feedback system in terms of the time response.

In this paper, we concern swing down control of the Acrobot which is a 2-link planar robot with a single actuator driving the second joint, whose control objective is to stabilize the Acrobot to the downward equilibrium point with the two links in the downward position for all initial states of the Acrobot with the exception of a set of Lebesgue measure zero. To achieve this control objective, we design a nonlinear controller by using nonnegative linear feedback of the sine function of the angle of the second joint in addition to the negative linear feedback of its angular velocity. By analyzing globally the solution of the closed-loop system consisting of the Acrobot and the presented controller and focusing on the equilibrium points of the closed-loop system and their stability, we prove that the control objective is achieved provided that some conditions on two control gains are satisfied. We design the two control gains such that the real parts of the dominant poles of the linearized model of the closed-loop system around the downward equilibrium point are minimized. We provide simulation results for two Acrobots to show the effectiveness of the presented controller.

This paper presents a theoretical analysis of the small-signal stability of a power system in which a synchronous generator and a photovoltaic (PV) generator supply power to an infinite bus. The problem considered here is to investigate the existence of the equilibrium points of the system and their stability. In terms of this problem, by focusing on the condition to be satisfied by the equilibrium points and appropriately using the intermediate value theorem, we derive a sufficient condition on the magnitude of the PV current for the existence of the equilibrium points. The condition is given as inequalities with respect to the system parameters. These inequalities show that, if the power system from which the PV generator is removed has equilibrium points, then equilibrium points exist also in the original system as long as the PV current is small. Moreover, we analyze the stability of the equilibrium points and show that the equilibrium points found under our existence condition are asymptotically stable. These results imply that, when the PV current is below a certain level, the existence of the asymptotically stable equilibrium points is preserved even though the PV generator is introduced.

For an n-link underactuated planar revolute robot connected to a fixed base in a horizontal plane, in this paper, we study an open problem: Under what actuator configuration is the robot moving in constantly rotating frame with all its links stretching straight out linearly controllable and observable? Different from the existing result for the robot with one unactuated joint, we prove that such linear controllability and observability are guaranteed for all its possible physical parameters, if and only if its first joint is active (actuated). Furthermore, when its first joint is passive (unactuated), we show that for any of its physical parameters, the robot is linearly uncontrollable and unobservable regardless of whether any of its rest joint(s) is active or not. We validate the above results via a physical 3-link underactuated planar revolute robot and a 4-link planar model of a gymnast on the high bar in existing references. This paper presents new insight of the linear controllability and observability for an n-link underactuated planar revolute robot moving in constantly rotating frame in a horizontal plane.

This paper presents an algorithm estimating the regions of attraction of power systems based on the Lyapunov function approach where a sublevel set of a Lyapunov function for a target system is used as the estimate. In particular, we focus here on the algorithm based on sum of squares (SOS) programming, which has been recently proposed, and aim to develop a simpler algorithm for the practical use. For this aim, we present an algorithm overcoming the difficulty of the SOS programming problem addressed in the existing study, i.e., the bilinear constraints, in a simpler way. In the proposed algorithm, two SOS programming problems are iteratively solved, and the number of the problems solved at each iteration is reduced to half of that in the existing algorithm. In addition, we theoretically analyze the proposed algorithm, and show the convergence under certain conditions. The performance of our algorithm is demonstrated by numerical examples.

This paper aims to characterize and classify the synchronization phenomena in the system consisting of multiple identical metronomes placed on an elastically fixed cart by using describing function (DF) method, which deals with a discontinuous model of the torque driving the metronome in terms of its angle and angular velocity. Different from a previous study on the uniform amplitude synchronization of all metronomes in the system, this paper analyzes uniform and non-uniform amplitude synchronization. By investigating and using the relation between the motion of the cart (common base) and the multiple metronomes which was not studied in the previous study, this paper presents a new DF solution of the system to characterize and classify the synchronization phenomena in the system by the following two types of synchronization: One is the inherent synchronization, for which the cart is still. The frequency and amplitudes of all synchronized metronomes are the same as those of a single metronome hanging from a fixed base. The other is the non-inherent synchronization, for which the cart moves in synchronization with all metronomes. It includes in-phase uniform amplitude synchronization and non-uniform amplitude synchronization. The new characterization and classification of the synchronization is validated via numerical investigation to the system with two, three, and four metronomes.

This paper analyzes the transient stability of power systems with uncertain parameters. More precisely, we suppose that the inertia constants and damping coefficients of generators are uncertain, and consider the problem of checking the stability of an equilibrium state for all possible values of the uncertain parameters, and estimating the intersection of the region of attraction for those parameter values. By using a polytope representation for the uncertain parameters, we present a method to solve the problem. In the presented method, we solve a sum of squares (SOS) programming problem based on the polytope representation. It is proven that if this SOS programming problem is feasible, then our method provides a solution to the analysis problem of the transient stability.

This paper analyzes the transient stability of power systems with uncertain parameters. More precisely, we suppose that the inertia constants and damping coefficients of generators are uncertain, and consider the problem of checking the stability of an equilibrium state for all possible values of these uncertain parameters, and estimating the intersection of the region of attraction for those parameter values. By using a polytopic representation for the uncertain parameters and a concept from the field of robust control, we present a method for solving this problem. In the presented method, we solve a sum of squares programming problem with constraints imposed on systems where the values of the uncertain parameters correspond to the vertices of a polytope. We prove that, if the sum of squares programming problem is feasible, our method solves the analysis problem of robust transient stability. A numerical example demonstrates that our method correctly analyzes transient stability despite the parameter uncertainties, whereas an existing method gives an incorrect analysis result owing to the uncertainties.

This paper addresses the synchronization problem of the non-uniform Kuramoto oscillators, i.e., the Kuramoto oscillators with non-homogeneous damping coefficients and coupling strengths. As a solution to this problem, we present a new necessary condition for the synchronization of the $n$ oscillators, and show that it is tighter than an existing necessary condition proposed in Dodier and Bullo (SIAM Journal on Control and Optimization, vol. 50, no. 3, 2012). For the oscillators of cases of n = 3,4, and 5, our numerical simulation investigations of 10000 systems for each case with the parameters generated randomly show that the conservativeness of the existing condition is reduced considerably by our newly presented condition.

For multi-link planar robots moving in the vertical plane with a last active joint, this paper studies how to design angular momentum based stabilizing controllers with a large region of attraction for the upright equilibrium point (UEP) of the robots, where all the links are upright. First, for an $n$ -link planar robot, the presented stabilizing control in this paper contains a term compensating the gravitational term related to the last joint and a linear feedback of $2n$ variables, which are the angular momentum of the robot about the first joint, its first-time derivative, its second-time derivative, and the other $(2n-3$ ) variables to be designed such that the Jacobian matrix of these $2n$ variables is nonsingular at the UEP. Second, as an application, regarding the three-link case, this paper proves that the Jacobian matrix is nonsingular at the UEP regardless of all the mechanical parameters if the three variables to be designed are chosen to be one of the nine combinations of any two joint angles from three joint angles and any one joint angular velocity from three joint angular velocities of the robot. Finally, the simulation results of a three-link robot are presented to show that the regions of attraction of the obtained nine controllers are bigger than that of the conventional linear state-feedback controller with the state variables being the angles and angular velocities of the robot. The new results obtained in this paper include the existing results for the two-link case as a special case.

For the Acrobot (a two-link planar robot moving in the vertical plane with a single actuator at the second joint), this paper studies how to design angular momentum based stabilizing controllers with a large region of attraction for its upright equilibrium point, where all the links are upright. First, in addition to the angular momentum of the Acrobot about the first joint, its first-time derivative, and its second-time derivative, this paper shows how to design a variable from the non-singularity of the Jacobian matrix of these four variables. This paper presents a necessary and sufficient condition such that the Jacobian matrix is nonsingular regardless of all the mechanical parameters, and proposes two parameterizations of the variable to be designed and two types of stabilizing controllers using these four variables. Finally, the simulation results are presented to show that the regions of attraction of the proposed controllers are much bigger than those of the linear state-feedback controllers having the same poles of the linearized system of the closed-loop system.

This paper aims to analyze the synchronization phenomena in the system consisting of n identical metronomes on a hanging plate by using the describing function (DF) method, which deals with the driving torque as a discontinuous function of the angle and angular velocity of each metronome. Different from a previous study on the synchronization of the system with the first harmonic of the angle of each metronome having the same amplitude, this paper analyzes the synchronization without any assumption on the amplitudes of n metronomes. By investigating the relation between the motion of the hanging plate (common base) and n metronomes, this paper characterizes the synchronization phenomena by the following two types of synchronization: One is the inherent synchronization, whose frequency and amplitude of all synchronized metronomes are the same as those of a single metronome on a fixed base. The other is the non-inherent synchronization, which includes the in-phase synchronization with the same amplitude, and the synchronization with different amplitudes. The analytical results are validated by numerical simulation for three metronomes about the synchronization with the same or different amplitudes.

This paper analyzes the stability of a power system in which a synchronous generator and a photovoltaic (PV) generator supply the power to an infinite bus. The infinite bus corresponds to a large system to which the two generators are connected. The problem considered here is to investigate the existence of the equilibrium points of the system and their stability. For the problem, we derive a sufficient condition on the magnitude of the PV current for the existence of the equilibrium points. In addition, we analyze the stability of the equilibrium points, and show that the equilibrium points found under the derived condition are stable. These results clarify the impact of the penetration of the PV generator on the existence of the stable equilibrium points of the system.

This paper addresses a problem of finding distributed controllers for the robotic mass games whose purpose is to let mobile robots organize themselves into a formation displaying a given grayscale image. As a solution to this problem, we develop three-mode hybrid controllers by combining ideas of the conventional coverage control and the halftone image processing. The roles of the three sub-controllers are to remove extra robots from the mass game field, to gather them unobtrusively, and to organize player robots into a formation displaying the image, respectively. The performance of the proposed controllers is demonstrated by numerical simulations with the standard images.