Output regulation of linear systems with bounded continuous feedback

This paper studies the classical problem of output regulation for linear systems subject to control constraint. The asymptotically regulatable region, the set of all initial conditions of the plant and the exosystem for which output regulation is possible, is characterized in terms of the controllable region of the antistable subsystem of the plant. Continuous output regulation laws, of both state feedback type and error feedback type, are constructed from a given stabilizing state feedback law. It is shown that a stabilizing feedback law that achieves a larger domain of attraction leads to a feedback law that achieves output regulation on a larger subset of the asymptotically regulatable region. A feedback law that achieves global stabilization on the asymptotically controllable region leads to a feedback law that achieves output regulation on the entire asymptotically regulatable region.     

On the stabilization of decentralized control systems

This paper considers the problem of stabilizing a linear time-invariant multivariable system by using several local feedback control laws. Each local feedback control law depends only on partial system outputs. A necessary and sufficient condition for the existence of local control laws with dynamic compensation to stabilize a given system is derived. This condition is stated in terms of a new notion, called "fixed modes," which is a natural generalization of the well-known concept of uncontrollable modes and unobservable modes that occur in centralized control system problems. A procedure that constructs a set of stabilizing feedback control laws is given.     

关于最小分散镇定结构的研究

本文考察线性时不变多变量系统的分散镇定问题,揭示了局部控制站间的通信与消除固 定模间的内在联系,并由此把求最小(最经济)分散可镇定结构问题转化成一个显式的特殊0-1 规划问题,导出了一种求最小分散可镇定结构的有效算法.     

不确定非完整动力学系统的时变自适应控制

研究了具有未知惯性参数非完整动力学系统的镇定问题,对于一类非完整系统,给出了一种新的时变自适应律镇定律,不同于其它文中的控制律,该镇定律不是高增益的,文中也讨论了一般不确定非完整动力学系统的镇定问题,证明了时变周期镇定律的存在性,一个简单的例子说明了如何应用文中结果设计镇定律,仿真结果表明了本文所提设计方法的有效性。     

Generic nonlinear stabilization of systems with matching algebraic structure

The paper demonstrates that using algebraic methods for the construction of time varying stabilizing controls for general controllable systems which are affine in the control is not only computationally feasible, but delivers generic feedback laws. A single feedback control law can be stabilizing for all systems which have the same algebraic structure and also for systems that can be adequately approximated by this structure. The systems considered are not limited to those whose controllability Lie algebra is nilpotent or even finite dimensional. The stabilizing controls are constructed by the help of an open-loop control problem on an associated Lie group which is posed as a trajectory interception problem in the logarithmic coordinates of flows.     

Nonlinear feedback control with global stabilization

Hamilton-Jacobi-Bellman theory is shown to provide a unified framework for nonlinear feedback control laws for special classes of nonlinear systems. These classes include Jurdjevic-Quinn type systems, as well as minimum phase systems with relative degree {1, 1, ..., 1}. Several examples are given to illustrate these results. For the controlled Lorenz equation, results obtained by Vincent and Yu are extended. Next, for spacecraft angular velocity stabilization with two torque inputs, a family of nonlinear feedback control laws that globally asymptotically stabilize angular velocity is established. Special cases of this family of control laws include generalizations of the locally stabilizing control laws of Brockett and Aeyels to global stabilization as well as the globally stabilizing control laws of Irving and Crouch and Byrnes and Isidori. Finally, the results are applied to spacecraft angular velocity stabilization with only one torque input. These last results extend control laws given by Outbib and Sallet.     

关于最小分散镇定结构的研究

本文考察线性时不变多变量系统的分散镇定问题,揭示了局部控制站间的通信与消除固定模间的内在联系,并由此把求最小(最经济)分散可镇定结构问题转化成一个显式的特殊0-1规划问题,导出了一种求最小分散可镇定结构的有效算法。     

Robust control of a general servomechanism problem: The servo compensator

The robust control of a general servomechanism problem, which is an extension to the results of [1], is considered in this paper. Necessary and sufficient conditions, together with a characterization of all robust controllers which enables asymptotic tracking to occur, independent of disturbances in the plant and perturbations in the plant parameters and gains of the system, are obtained. A new type of compensator, introduced in [1], called a servo-compensator which is quite distinct from an observer is shown to play an essential role in the robust servomechanism problem. It is shown that this compensator, which corresponds to an integral controller in classical control theory, must be used in any servomechanism problem to assure that the controlled system is stabilizable and achieves robust control; in particular, it is shown that a robust controller of a general servomechanism problem must consist of two devices (i) a servo-compensator and (ii) a stabilizing compensator. A study of the stabilizing compensator is made; in particular, it is shown that a new type of stabilizing compensator called a complementary controller, may be used together with the servo-compensator to form a robust controller for the servo-mechanism problem.A study of the case when perturbations in the robust controller are also allowed is then made; this leads to the Strong robust servo limitation theorem which imposes a fundamental limitation on the ability of practical servomechanisms to regulate a system.     

Solving 1D Conservation Laws Using Pontryagin’s Minimum Principle

This paper discusses a connection between scalar convex conservation laws and Pontryagin’s minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax–Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin’s minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions.     

Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems

A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to higher order dynamics. This is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one. We also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective. Our construction does not invoke certainty equivalence, nor requires a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. Finally, it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form, yielding new stabilizing control laws. We illustrate the method with several academic and practical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application.     

Discontinuous stabilization of nonlinear systems: Quantized and switching controls

In this paper we consider the classical problem of stabilizing nonlinear systems in the case the control laws take values in a discrete set. First, we present a robust control approach to the problem. Then, we focus on the class of dissipative systems and rephrase classical results available for this class taking into account the constraint on the control values. In this setting, feedback laws are necessarily discontinuous and solutions of the implemented system must be considered in some generalized sense. The relations with the problems of quantized and switching control are discussed.     

Stabilizing switching design for switched linear systems: A state-feedback path-wise switching approach

This paper addresses the problem of switching stabilization for discrete-time switched linear systems. Based on the abstraction-aggregation methodology, we propose a state-feedback path-wise switching law, which is a state-feedback concatenation from a finite set of switching paths each defined over a finite time interval. We prove that the set of state-feedback path-wise switching laws is universal in the sense that any stabilizable switched linear system admits a stabilizing switching law in this set. We further develop a computational procedure to calculate a stabilizing switching law in the set.     

Fundamental design limitations of the general control configuration

The theory of fundamental design limitations is well understood for the case that the performance variable is measured for feedback. In the present paper, we extend the theory to systems for which the performance variable is not measured. We consider only the special case for which the performance and measured outputs and the control and exogenous inputs are all scalar signals. The results of the paper depend on the control architecture, specifically, on the location of the sensor relative to the performance output, and the actuator relative to the exogenous input. We show that there may exist a tradeoff between disturbance attenuation and stability robustness that is in addition to the tradeoffs that exist when the performance output is measured. We also develop a set of interpolation constraints that must be satisfied by the disturbance response at certain closed right half plane poles and zeros, and translate these constraints into generalizations of the Bode and Poisson sensitivity integrals. In the absence of problematic interpolation constraints we show that there exists a stabilizing control law that achieves arbitrarily small disturbance response. Depending on the system architecture, this control law will either be high gain feedback or a finite gain controller that depends explicitly on the plant model. We illustrate the results of this paper with the problem of active noise control in an acoustic duct.     

Heterogeneous control and qualitative supervision,application to a distillation column

This paper addresses the following control problem: a continuous plant (P) is to be controlled via heterogeneous control law. The heterogeneous control laws apply even in the presence of incomplete knowledge of the system. This approach is based only on the knowledge of the physical boundaries of system input and output, and the maximum rate-of-change of the input. The qualitative values of the input, output, error and respective derivatives in a closed loop are used to recognise any characteristic dynamics such as: dead-time, damping, negative gain, inverse response and oscillations. The control actions are the result of qualitative operations concerning the magnitude and the derivative of error and a numeric operation to calculate the values of system input. Hence, the numeric and qualitative parts can be considered as a new approach to define a recurrent control law. These results are illustrated in a current application for the start-up and disturbance recovery of a distillation column.     

Stabilization of linear autonomous systems of differential equations with distributed delay

Consideration is given to the problem of optimal stabilization of differential equation systems with distributed delay. The optimal stabilizing control is formed according to the principle of feedback. The formulation of the problem in the functional space of states is used. It was shown that coefficients of the optimal stabilizing control are defined by algebraic and functional-differential Riccati equations. To find solutions to Riccati equations, the method of successive approximations is used. The problem for this control law and performance criterion is to find coefficients of a differential equation system with distributed delay, for which the chosen control is a control of optimal stabilization. A class of control laws for which the posed problem admits an analytic solution is described.     

Uniting local and global controllers for uncertain nonlinear systems: beyond global inverse optimality

This paper provides a solution to a new problem of global robust control for uncertain nonlinear systems. A new recursive design of stabilizing feedback control is proposed in which inverse optimality is achieved globally through the selection of generalized state-dependent scaling. The inverse optimal control law can always be designed such that its linearization is identical to linear optimal control, i.e. optimal control, for the linearized system with respect to a prescribed quadratic cost functional. Like other backstepping methods, this design is always successful for systems in strict-feedback form. The significance of the result stems from the fact that our controllers achieve desired level of ‘global’ robustness which is prescribed a priori. By uniting locally optimal robust control and global robust control with global inverse optimality, one can obtain global control laws with reasonable robustness without solving Hamilton–Jacobi equations directly.     

Performance improvement of the PD-based bilateral teleoperators with time delay by introducing relative D-control

In a force-reflecting bilateral teleoperator with a time delay, teleoperator stability is a serious problem. We have studied a bilateral teleoperator system with a time delay. We obtained stable conditions using proportional derivative-based (PD-based) control law. In this paper, PD-based control law is further studied. First, we study a PD control law with relative damping gain and its stabilizing effect that previously has not been studied quantitatively. A stable condition is derived with this PD-based controller with relative damping gain. Next, teleoperator performance by the PD control law with relative damping is evaluated and compared to PD control laws with only grounded damping using transparency analysis with a hybrid matrix. We showed that, the performance of the PD-based controller can be improved by introducing relative damping gain into the controller. As a controller design example, numerical simulations and 1-DOF experiments were conducted. Finally, peg-in-hole experiments and performance evaluations in realistic multi-DOF environments were conducted to demonstrate performance improvements by introducing the relative damping. A controller design that guarantees both stability and performance was achieved by iterating stable gain setting and performance evaluation.     

Solvability conditions and solutions to perfect regulation problem under measurement output feedback

The problem of perfect regulation is to design a family of control laws for a given plant such that the resulting overall closed-loop system is internally stable and its controlled output can be reduced to zero arbitrarily fast from any initial condition. Such a problem was heavily studied by many researchers in the 1970s and early 1980s. However, to the best of our knowledge, all of the earlier results deal only with the problem under full state feedback. In this paper, we solve the long-standing problem of perfect regulation via measurement output feedback for general linear time-invariant multivariable systems. In particular, we derive necessary and sufficient conditions under which the problem of perfect regulation via measurement output feedback is solvable for general systems, and, under these conditions, construct two families of feedback laws, one of full order and the other reduced order, that solve the problem.     

Finite-time Attitude Control: A Finite-time Passivity Approach

This paper studies the finite-time attitude control problem for a rigid body. It is known that linear asymptotically stabilizing control laws can be derived from passivity properties for the system which describes the kinematic and dynamic motion of the attitude. Our approach expands this framework by defining finite-time passivity and exploring the corresponding properties. For a rigid body, the desired attitude can be tracked in finite time using the designed finite-time attitude control law. Some finitetime passivity properties for the feedback connection systems are also shown. Numerical simulations are provided to demonstrate the effectiveness of the proposed control law.