Static State Feedback Law Research Articles

Integration of inventory control into the port‐Hamiltonian framework for dissipative stabilization of chemical reactors

AbstractThis work integrates passivity‐based inventory control with the port‐Hamiltonian framework to design stabilizing feedback laws for lumped parameter chemically reacting systems. A static state feedback law with admissible controls is developed using an inventory‐related storage function as a closed loop Hamiltonian function. The control gain matrix is derived from the interconnection and damping matrices of the resulting Hamiltonian representation. Consequently, the proposed controller globally and exponentially stabilizes the system at a desired set‐point (including the open loop unstable equilibrium point) without restrictions on the reaction kinetics. Numerical simulations illustrate the application of the theory to a nonisothermal, continuous stirred tank reactor with the steady‐state multiplicity. The proposed approach is compared with the thermodynamic availability‐based control approach in terms of amplitude and variation rate to show its performance and effectiveness.

Multistable Energy Shaping of Passive Linear Systems with Hybrid Mode Selector

Abstract This paper presents a novel control strategy for stable linear time–invariant systems operating with a finite number of set points. Inspired by the theory of passivity-based control, the proposed method aims at simultaneously and asymptotically stabilize all the desired working modes by means of a static nonlinear state feedback law. An asynchronous external signal is then employed to trigger a hybrid controller in order to switch between the different working modes. The proposed approach is validated by means of simulations performed on the ubiquitous mass-spring-damper system.

Constrained Generalized Continuous Algebraic Riccati Equations (CGCAREs) Are Generically Unsolvable

In this letter, we show that a constrained generalized continuous algebraic Riccati equation (CGCARE), arising in the context of a singular infinite-horizon linear quadratic regulator (LQR) problem, is generically 1 unsolvable. We achieve this result in three steps: first, we provide a set of necessary and sufficient conditions for the solvability of a CGCARE. Then, using these conditions, we show that the determinant of a certain matrix pencil being identically zero is necessary for a CGCARE to be solvable. Finally, we show that this necessary condition is generically false. It is well-known that CGCARE solvability is a necessary and sufficient condition for a singular infinite-horizon LQR problem to admit a solution that is implementable as a static state-feedback control law. Thus, our main result reveals that a singular infinite-horizon LQR problem generically disallows solution by a static state-feedback law.

On path following control of nonholonomic port-Hamiltonian systems via generalized canonical transformations

This paper proposes a constructive design method of a static state feedback law which makes a nonholonomic port-Hamiltonian system follow a desired path. A generalized canonical transformation connects two port-Hamiltonian systems through a pair of a feedback and a coordinate change. This paper clarifies how a generalized canonical transformation connects the plant nonholonomic port-Hamiltonian system with an error system. Stabilizing the path following error system allows one to derive a constructive path following control law for the nonholonomic port-Hamiltonian system. Finally, an example shows a concrete design procedure of the proposed method.

On the Consensus of Homogeneous Multiagent Systems With Positivity Constraints

This paper investigates the consensus problem for multiagent systems, under the assumptions that the agents are homogeneous and described by a single-input positive state-space model, the mutual interactions are cooperative, and the static state-feedback law that each agent adopts to achieve consensus preserves the positivity of the overall system. Necessary conditions for the problem solvability, which allow us to address only the special case when the state matrix is irreducible, are provided. Under the irreducibility assumption, equivalent sets of sufficient conditions are derived. Special conditions either on the system description or on the Laplacian of the communication graph allow us to obtain necessary and sufficient conditions for the problem solvability. Finally, by exploiting some results about robust stability either of positive systems or of polynomials, further sufficient conditions for the problem solvability are derived. Numerical examples illustrate the proposed results.

Necessary and Sufficient Conditions for Global External Stochastic Stabilization of Linear Systems With Input Saturation

We consider the problem of global external stochastic stabilization for linear plants with saturating actuators, driven by a stochastic external disturbance, and having random Gaussian-distributed initial conditions. The objective is to control such plants by a possibly nonlinear static state feedback law that achieves global asymptotic stability in the absence of disturbances, while guaranteeing a bounded variance of the state vector for all time in the presence of disturbances and Gaussian distributed initial conditions. Results for continuous-time open-loop critically stable plants as well as for a chain of integrators have been obtained before. The goal of this technical note is to extend this result to critically unstable plants. We view our contribution in this technical note as a critical step in solving the LQG control problem for linear systems subject to saturated inputs which is a research problem with a long history in our field.

Realization of full column rank precompensators using stabilizing static state feedback

In some control problems, it is convenient to use a precompensator to alter the transfer function of the system such that the transfer function of the compensated system has a specified property. Then, if possible, the action of the compensator on the system is realized by a static state feedback law applied to the system. When the compensator is square and non-singular, the state feedback is regular and the problem has already been solved. Non-square compensators, however, result in non-regular state feedback. In this paper, necessary and sufficient conditions are presented for the compensated system to be stabilizable by such a non-regular static state feedback law.

On global external stochastic stabilization of linear systems with input saturation

We consider the problem of global external stochastic stabilization for linear plants with saturating actuators, driven by a stochastic external disturbance, and having random Gaussian-distributed initial conditions. The aim of this stabilization is to control such plants by a possibly nonlinear static state feedback law that achieves global asymptotic stability in the absence of disturbances, while guaranteeing a bounded variance of the state vector for all time in the presence of disturbances and Gaussian distributed initial conditions. Results for continuous-time open-loop critically stable plants have been obtained before. This paper shows how to extend this result to plants consisting of a chain of integrators. We view our contribution in this paper as a first critical step in solving the LQG control problem for linear systems subject to saturated inputs which is a research problem with a long history in our field.

A non-linear tracking control scheme for an under-actuated autonomous underwater robotic vehicle

This paper proposes a model based trajectory tracking control scheme for under-actuated underwater robotic vehicles. The difficulty in stabilizing a non-linear system using smooth static state feedback law means that the design of a feedback controller for an under-actuated system is somewhat challenging. A necessary condition for the asymptotic stability of an under-actuated vehicle about a single equilibrium is that its gravitational field has nonzero elements corresponding to non-actuated dynamics. To overcome this condition, we propose a continuous time-varying control law based on the direct estimation of vehicle dynamic variables such as inertia, damping and Coriolis & centripetal terms. This can work satisfactorily under commonly encountered uncertainties such as an ocean current and parameter variations. The proposed control law cancels the non-linearities in the vehicle dynamics by introducing non-linear elements in the input side. Knowledge of the bounds on uncertain terms is not required and it is conceptually simple and easy to implement. The controller parameter values are designed using the Taguchi robust design approach and the control law is verified analytically to be robust under uncertainties, including external disturbances and current. A comparison of the controller performance with that of a linear proportional-integral-derivative (PID) controller and sliding mode controller are also provided.

2313 外乱が作用した場合における旋回クレーンシステムの振動抑制(GS1-2 一般セッションII,GS1 一般セッション)

In this paper we propose a neuro-controller (NC) for suppression of load swing with disturbance in a crane system rotaring around the vertical axis. As in a nonholonomic system, the traditional control method using a static continuous state feedback law cannot stabilize the load swing. It is necessary to design a time-varying feedback controller or a discontinuous feedback controller. Previous research had been successful in constructing the suppression controller of the load swing with a single initial rotation angle when disturbance appeared. In this paper, the performance of the NC optimized by genetic algorithm will be examined with three initial rotation angles.

NUMERICAL STATIC STATE FEEDBACK LAWS FOR CLOSED-LOOP SINGULAR OPTIMAL CONTROL

Singular and non-singular control trajectories of agricultural and (bio) chemical processes may need to be recalculated from time to time for use in closed-loop optimal control, because of unforeseen changes in state values and noise. This is time consuming. As an alternative, in this paper, numerical, nonlinear, static state feedback laws are developed for optimal control on the singular arc that can be applied in closed-loop without the need for iteration. The efficacy of these laws is demonstrated in an example.

Transient stabilization of multimachine power systems with nontrivial transfer conductances

We provide a solution to the long-standing problem of transient stabilization of multimachine power systems with nonnegligible transfer conductances. More specifically, we consider the full 3n-dimensional model of the n-generator system with lossy transmission lines and loads and prove the existence of a nonlinear static state feedback law for the generator excitation field that ensures asymptotic stability of the operating point with a well-defined estimate of the domain of attraction provided by a bona fide Lyapunov function. To design the control law we apply the recently introduced interconnection and damping assignment passivity-based control methodology that endows the closed-loop system with a port-controlled Hamiltonian structure with desired total energy function. The latter consists of terms akin to kinetic and potential energies, thus has a clear physical interpretation. Our derivations underscore the deleterious effects of resistive elements which, as is well known, hamper the assignment of simple gradient energy functions and compel us to include nonstandard cross terms. A key step in the construction is the modification of the energy transfer between the electrical and the mechanical parts of the system which is obtained via the introduction of state-modulated interconnections that play the role of multipliers in classical passivity theory.

Transient Stablization of multimachine power systems with nontrivial transfer conductances

In this paper we provide a solution to the long-standing problem of transient stabilization of multi machine power systems with non-negligible transfer conductances. More specifically, we consider the full 3n-dimensional model of the n-generator system with lossy transmission lines and loads and prove the existence of a nonlinear static state feedback law for the generator excitation field that ensures asymptotic stability of the operating point with a well-defined estimate of the domain of attraction provided by a bona fide Lyapunov function. Our derivations underscore the deleterious effects of resistive elements which, as is well-known, hamper the assignment of simple “gradient” energy functions and compel us to include non-standard cross terms. A key step in the construction is the modification of the energy transfer between the electrical and the mechanical parts of the system that play the role of multipliers in classical passivity theory.

Stabilization of nonlinear processes with input constraints

This work deals with nonlinear processes which, in the absence of input constraints, can be locally stabilized by a linearizing static state feedback law. Such systems, under unconstrained linearizing control laws, possess a cascaded structure between a (asymptotically or exponentially) stable nonlinear subsystem and an exponentially stable linear subsystem, which allows for a straightforward stability analysis. In the presence of input constraints, however, this cascaded structure breaks down to an interconnection between two nonlinear subsystems; analyzing the stability of such interconnections is a rather cumbersome task, that typically results in conservative estimates of regions of stability. In this article, we present an analysis framework for the local stabilization of such processes and the estimation of regions of closed-loop stability in the presence of input constraints. The proposed approach entails: (i) specifying a region in state–space where the closed-loop system behaves effectively as a cascade and asymptotic stability can be guaranteed in the presence of constraints, provided that the states of the system remain in this region for all times; and (ii) constructing invariant sets within this region that qualify as regions of closed-loop stability. A detailed case study is carried out on a polymerization reactor example and the desirable features of the proposed methodology are aptly illustrated.

Dynamic Robust Output Min–Max Control for Discrete Uncertain Systems

Min---max control is a robust control, which guarantees stability in the presence of matched uncertainties. The basic min---max control is a static state feedback law. Recently, the applicability conditions of discrete static min---max control through the output have been derived. In this paper, the results for output static min---max control are further extended to a class of output dynamic min---max controllers, and a general parametrization of all such controllers is derived. The dynamic output min---max control is shown to exist in many circumstances under which the output static min---max control does not exist, and usually allows for broader bounds on uncertainties. Another family of robust output min---max controllers, constructed from an asymptotic observer which is insensitive to uncertainties and a state min---max control, is derived. The latter is shown to be a particular case of the dynamic min---max control when the nominal system has no zeros at the origin. In the case where the insensitive observer exists, it is shown that the observer-controller has the same stability properties as those of the full state feedback min---max control.

Feedback control strategies for a nonholonomic mobile robot using a nonlinear oscillator

Among control problems for mobile robots, point-to-point stabilization is the most challenging since it does not admit designs with smooth static state feedback laws. Stabilization strategies for mobile robots, and nonholonomic systems generally, are smooth, time-varying or nonsmooth, time-invariant. Time-varying control strategies are designed with umdamped linear oscillators but their fixed structure offer limited flexibility in control design. The central theme of this paper lies in use of nonlinear oscillators for mobile robot control. Large numbers of qualitatively different control strategies can be designed using nonlinear oscillators since stiffness and damping can be functions of robot states. We demonstrate by designing two fundamentally different controllers for two-wheeled mobile robot using two variants of a particular nonlinear oscillator. First controller is dynamic and generates smooth control action. Second controller is almost-smooth and time-invariant. While first controller guarantees global asymptotic stability for any desired posture of robot, second controller is stable, and converges robot from almost any posture to desired posture. The only gap in posture space is unstable equilibrium manifold of measure zero. For both control strategies we mathematically establish stability and convergence of mobile robot to desired posture. Simulation results support theoretical claims. ©1999 John Wiley & Sons, Inc.

Partial model matching via static feedback (the multivariable case)

The problem of partial model matching is studied for the case of multi-input/multi-output (MIMO) linear time-invariant systems. Using a regular static state feedback law, the necessary and sufficient conditions for the problem to have a solution are established, the general analytic expressions of the controller matrices are derived. For the case of single-input/single-output systems sufficient conditions are derived for the solution of the partial model matching problem with simultaneous stabilizability. For the same type of systems the necessary and sufficient conditions for the solution of the partial model matching problem with simultaneous regulation of the free response of the closed loop system are established. Finally, the problem of partial model matching, via regular static measurement output feedback, is solved for the case of MIMO systems.

Robust regional stabilisation of an electropneumatic actuator

Sufficient conditions are derived to solve the problem of robust regional stabilisation of linear systems with nonlinear, uncertain structure and uncertain order. The problem of increasing the speed and accuracy of uncertain spool-valve actuators is studied. The robust regional stabilising control technique is applied to compensate for the large variations in the uncertainties of the electro-pneumatic actuator. The problem of the robust stability of the electro-pneumatic actuator is proved to always be solvable, via an independent of the actuator's uncertainties static state feedback law depending only upon the stability margin. The closed loop response remains satisfactory for all values in the uncertainty domain.

A general approach to input-output pseudolinearization for nonlinear systems

A nonlinear system is said to be input-output pseudolinearized if its family of linearizations about constant operating points has input-output behavior that is independent of the particular operating point. The problem of constructing static state feedback laws that achieve input-output pseudolinearization for a general class of nonlinear systems is considered. A generalization of the well-known structure algorithm to the case of parameterized families of linear systems plays an important role.

Block Decoupling of Generalized State Space Systems

The problem of input-output block decoupling of generalized state space systems, via a regular static state feedback law, is studied for the first time. The necessary and sufficient conditions for the problem to have a solution are established. The necessary and sufficient conditions for block decoupling with simultaneous stabilizability, and the necessary and sufficient conditions for block decoupling with simultaneous arbitrary assignment of the system poles are established. © 1997 Elsevier Science Ltd.