We introduce a new hybrid control strategy, which is conceptually different from the commonly used synergistic hybrid approaches, to efficiently deal with the problem of the undesired equilibria that precludes smooth vectors fields on $SO(3)$ from achieving global stability. The key idea consists in constructing a suitable potential function on $SO(3)\times \mathbb{R}$ involving an auxiliary scalar variable, with flow and jump dynamics, which keeps the state away from the undesired critical points while, at the same time, guarantees a decrease of the potential function over the flows and jumps. Based on this new hybrid mechanism, a hybrid feedback control scheme for the attitude tracking problem on $SO(3)$, endowed with global asymptotic stability and semi-global exponential stability guarantees, is proposed. This control scheme is further improved through a smoothing mechanism that removes the discontinuities in the input torque. The third hybrid control scheme, proposed in this paper, removes the requirement of the angular velocity measurements, while preserving the strong stability guarantees of the first hybrid control scheme. This approach has also been applied to the tracking problem on $SE(3)$ to illustrate its advantages with respect to the existing synergistic hybrid approaches. Finally, some simulation results are presented to illustrate the performance of the proposed hybrid controllers.
We consider the problem of designing a state feedback control law to achieve nonovershooting tracking for feedback linearisable multiple-input multiple-output nonlinear systems. The reference signal is assumed to be obtained from a linear exosystem. The design method adopts known methods for the nonovershooting tracking of linear systems and extends them to the output regulation of feedback linearisable nonlinear systems. The method accommodates arbitrary initial conditions and offers design choice of the tracking convergence speed.
University of Quebec in Outaouais（乌塔韦魁北克大学）；University of Groningen（格罗宁根大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）
In this paper we present a hybrid feedback approach to solve the navigation problem of a point mass in the n-dimensional space containing an arbitrary number of ellipsoidal shape obstacles. The proposed hybrid control algorithm guarantees both global asymptotic stabilization to a reference and avoidance of the obstacles. The intuitive idea of the proposed hybrid feedback is to switch between two modes of control: stabilization and avoidance. The geometric construction of the flow and jump sets of the proposed hybrid controller, exploiting hysteresis regions, guarantees Zeno-free switching betw…
Probabilistic Data Association-Feedback Particle Filter for Multiple Target Tracking Applications
Coordinated Science Laboratory and the Department of Mechanical Science and Engineering at the University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校协同科学实验室与机械科学与工程系）
The proliferation of neural networks in safety-critical applications necessitates the development of effective methods to ensure their safety. This letter presents a novel approach for computing the exact backward reachable sets of neural feedback systems based on hybrid zonotopes. It is shown that the input-output relationship imposed by a ReLU-activated neural network can be exactly described by a hybrid zonotope-represented graph set. Based on that, the one-step exact backward reachable set of a neural feedback system is computed as a hybrid zonotope in the closed form. In addition, a neces…
In this paper, we present a global state feedback tracking controller for underactuated surface marine vessels. This controller is based on saturated control inputs and, under an assumption on the reference trajectory, the closed-loop system is globally asymptotically stable (GAS). It has been designed using a 3 Degree of Freedom benchmark vessel model used in marine engineering. The main feature of our controller is the boundedness of the control inputs, which is an essential consideration in real life. In absence of velocity measurements, the controller works and remains stable with observer…
We use online convex optimization (OCO) for setpoint tracking with uncertain, flexible loads. We consider full feedback from the loads, bandit feedback, and two intermediate types of feedback: partial bandit where a subset of the loads are individually observed and the rest are observed in aggregate, and Bernoulli feedback where in each round the aggregator receives either full or bandit feedback according to a known probability. We give sublinear regret bounds in all cases. We numerically evaluate our algorithms on examples with thermostatically controlled loads and electric vehicles.
We obviate the use of observers for the purpose of output feedback tracking control of Lagrangian systems and solve some long-standing yet well-documented open problems. As often implemented in control practice, we replace unavailable derivatives with approximate differentiation. Our contribution consists in establishing uniform global asymptotic stability in closed-loop, for Lagrangian systems without dissipative forces (friction) using only position feedback. Firstly, for fully-actuated relative-degree-two systems, the controller is reminiscent of passivity-based controllers for robot manipu…
This paper proposes a general framework for constructing feedback controllers that drive complex dynamical systems to "efficient" steady-state (or slowly varying) operating points. Efficiency is encoded using generalized equations which can model a broad spectrum of useful objectives, such as optimality or equilibria (e.g. Nash, Wardrop, etc.) in noncooperative games. The core idea of the proposed approach is to directly implement iterative solution (or equilibrium seeking) algorithms in closed loop with physical systems. Sufficient conditions for closed-loop stability and robustness are deriv…
This paper applies geometric PID control for asymptotic tracking of a desired trajectory by a hoop robot in the presence of disturbances and uncertainties. The hoop robot, consisting of a circular body rolling without slip along a one-dimensional surface, is a planar analog of a spherical robot. A variety of coupled mechanical system may be used to actuate the hoop robot. This paper specifically considers two different actuators, one a simple pendulum and the other an internal cart. The geometric PID controller requires the plant to be a mechanical system, and the hoop robot does not satisfy t…
This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time-invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time-varying) disturbances in the dynamical sys…
This paper addresses the sensitivity analysis for hybrid systems with discontinuous (jumping) state trajectories. We consider state-triggered jumps in the state evolution, potentially accompanied by mode switching in the control vector field as well. For a given trajectory with state jumps, we show how to construct an approximation of a nearby perturbed trajectory corresponding to a small variation of the initial condition and input. A major complication in the construction of such an approximation is that, in general, the jump times corresponding to a nearby perturbed trajectory are not equal…
We study output reference tracking of systems with high relative degree via output feedback only; this is, tracking where the output derivatives are unknown. To this end, we prove that the conjunction of the funnel pre-compensator with a minimum phase system of arbitrary relative degree yields a system of the same relative degree which is minimum phase as well. The error between the original system's output and the pre-compensator's output evolves within a prescribed performance funnel; and moreover, the derivatives of the funnel pre-compensator's output are known explicitly. Therefore, output…
We extend results on backstepping hybrid feedbacks by exploiting synergistic Lyapunov function and feedback (SLFF) pairs in a generalized form. Compared to existing results, we delineate SLFF pairs that are ready-made and do not require extra dynamic variables for backstepping. From an (weak) SLFF pair for an affine control system, we construct an SLFF pair for an extended system where the control input is produced through an integrator. The resulting hybrid feedback asymptotically stabilizes the extended system when the synergy gap for the original system is strictly positive. To highlight th…
A novel approach to design the feedback control based on past states is proposed for hybrid stochastic differential equations (HSDEs). This new theorem builds up the connection between the delay feedback control and the control function without delay terms, which enables one to construct the delay feedback control using the existing results on stabilities of HSDEs. Methods to find the upper bound of the length of the time delay are also investigated. Numerical simulations are presented to demonstrate the new theorem.
The design of an invariant tracking control law for the kinematic car driving on a sphere is discussed. Using a Lie group framework a left-invariant description on SO(3) is derived. Basic geometric considerations allow a direct comparison of the model with the usual planar case. Exploiting the Lie group structure an invariant tracking error is defined and a feedback is designed. Finally, one possible design of an invariant asymptotic observer is sketched.
We present a numerically tractable formulation for computing the optimal control of the class of hybrid dynamical systems whose trajectories are continuous. Our formulation, an extension of existing relaxed-control techniques for switched dynamical systems, incorporates the domain information of each discrete mode as part of the constraints in the optimization problem. Moreover, our numerical results are consistent with phenomena that are particular to hybrid systems, such as the creation of sliding trajectories between discrete modes.
In this work, we consider the output tracking and disturbance rejection problems of a mono-tubular heat exchanger process and a novel finite-dimensional output feedback regulator is developed. In the proposed output feedback regulator design, measurements available for the regulator do not belong to the set of controlled outputs. In other words, design emphasizes that other than controlled output is used as input signal to the regulator. The proposed output feedback regulator with only plant measurement ym(t) can realize the exosystem state estimation, disturbance rejection and reference signa…
