Static State Feedback Control Law Research Articles

Adaptive robust bearing-based formation control for multi-agent systems

The problem of bearing-based formation control with disturbance rejection has been studied by quite a few papers under various assumptions on the disturbances such as constant disturbances, trigonometric polynomial disturbances or bounded uniformly continuous disturbances with known bounds. In this paper, we further study the same problem with the disturbances being bounded piecewise continuous time functions. We first consider the case where the bounds of the disturbances are known. We show that the problem can be solved by a distributed static state feedback control law. Then we further consider the case where the bounds of the disturbances are unknown. We manage to solve this problem by introducing the adaptive control technique to deal with the unknown bounds of the disturbances. Our results include all the existing results on the bearing-based formation control with disturbance rejection as special cases. Moreover, we present a sufficient condition on the initial values of the closed-loop system for each case that prevents the collision.

Leader‐following consensus of a class of heterogeneous uncertain multi‐agent systems with a distributed model reference adaptive control law

SummaryThis paper considers the leader‐following consensus problem of a multi‐agent system whose agents have heterogeneous uncertain dynamics in homogeneous state dimensions. Each agent utilizes a distributed model reference adaptive control (MRAC) law to deal with uncertainties. The nominal part of the MRAC is a distributed static state feedback control law. The reference model is the leader‐following consensus problem of reference agents with homogeneous linear time‐invariant dynamics. This problem becomes an N‐player graphical differential game under a given cost function. The reference agents utilize distributed static state feedback controllers that constitute a Nash equilibrium solution to the graphical differential game. This paper provides the conditions for the distributed MRAC to guarantee that each agent asymptotically tracks the corresponding reference agent; consequently, the multi‐agent system solves the leader‐following consensus problem as the reference model does. These conditions yield a straightforward design method for the MRAC. A numerical example demonstrates an application of the proposed approach to a multi‐robot system with nonlinear dynamics.

A Data-Driven Approach to the L₂ Stabilization of Linear Systems Subject to Input Saturations

This paper revisits the L2 stabilization problem for linear systems subject to saturating input and energy-bounded additive disturbance from a data-driven point of view. The backbone of the results resides in writing the stability conditions in a compact dedicated structure allowing to exhibit LMI conditions. Based on the assumption that the data collection is informative, the finite L2-gain stabilization of the closed-loop system is addressed by considering a static state-feedback control law. Then, the L2-gain of the closed loop and an inner-approximation of the basin of attraction of the origin for the disturbance-free closed-loop system are characterized. The potential of the method is discussed along the treatment of an academic example.

Reducing conservatism in robust stability analysis and stabilisation of arbitrary switched linear systems

Parametric modelling of uncertainties has been considered for switched linear systems in the past few years to improve the efficiency of robust stability analysis. The main goal of this paper is simultaneously alleviating conservatism and computational effort for robust stability analysis and stabilisation of arbitrary switched linear systems with uncertain parameters. For this purpose, stability conditions are revised and some constraints are relaxed. First, an improved weighted stability bound is proposed which is larger than previous results. Since the obtained stability bound depends on the chosen Lyapunov function, an optimisation problem is also given to determine its maximum value. Then, using the stability results, robust stabilisation of parametric uncertain switched linear systems under arbitrary switching is investigated. In this regard, a design method is presented to offer a static state feedback control law. The conservatism of the proposed control law is reduced by relaxing the constraint on upper bound of a matrix variable. Finally, illustrative examples are included to demonstrate the effectiveness of the proposed method.

Port-Hamiltonian Flight Control of a Fixed-Wing Aircraft

This brief addresses the problem of stabilizing steady, wing level flight of a fixed-wing aircraft to a specified inertial velocity (speed, course, and climb angle). The aircraft is modeled as a port-Hamiltonian system and the passivity of this system is leveraged in devising the nonlinear control law. The aerodynamic force model in the port-Hamiltonian formulation is quite general; the static, state feedback control scheme requires only basic assumptions concerning lift, side force, and drag. Following an energy-shaping approach, the static state feedback control law is designed to leverage the open-loop system’s port-Hamiltonian structure in order to construct a control Lyapunov function. Asymptotic stability of the desired flight condition is guaranteed within a large region of attraction. Simulations comparing the proposed flight controller with dynamic inversion suggest it is more robust to uncertainty in aerodynamics.

Output regulation of linear systems via sampled-data control

In this paper, we study the output regulation problem for linear systems by the aperiodic sampled-data control approach. Suppose the problem is solvable by a continuous-time static state feedback control law and a continuous-time dynamic output feedback control law. Then, we first show that, if the exogenous signal is constant, then the output regulation problem is solvable exactly by both the sampled-data static state feedback control law and the sampled-data dynamic output feedback control law. If the exogenous signal is time-varying with bounded derivative, then we further show that the problem is solvable practically by both the sampled-data static state feedback control law and the sampled-data dynamic output feedback control law.

State and output feedback control for constrained discrete-time nonlinear systems

This paper presents a new systematic controller design approach for disturbed nonlinear time-varying systems subject to input and state constraints in discrete-time case. The Takagi–Sugeno (TS) formalism and fuzzy Lyapunov framework are used to handle the time varying parameters and control input saturation. Moreover, the proposed control method is based on L2 criterion which results in two different control design procedures. Both non-parallel distributed compensation (non-PDC) static state feedback control law and dynamics output feedback controller are proposed. To ensures the closed-loop system stability with respect to the given saturation constraints on the control input different optimization problem are also formulated in terms of linear matrix inequality conditions which can be solved efficiently with available solvers. The proposed techniques are illustrated through numerical examples.

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.

Output Regulation in Differential Variational Inequalities using Internal Model Principle and Passivity‐Based Approach

AbstractWe consider the problem of designing state feedback control laws for output regulation in a class of dynamical systems where state trajectories are constrained to evolve within time‐varying, closed, and convex sets. The first main result states sufficient conditions for existence and uniqueness of solutions in such systems. We then design a static state feedback control law using the internal model principle, which results in a well‐posed closed‐loop system and solves the regulation problem. As an application, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Local input-to-state stabilization and ℓ ∞ -induced norm control of discrete-time quadratic systems

Summary This paper addresses the problems of local stabilization and control of open-loop unstable discrete-time quadratic systems subject to persistent magnitude bounded disturbances and actuator saturation. Firstly, for some polytopic region of the state-space containing the origin, a method is derived to design a static nonlinear state feedback control law that achieves local input-to-state stabilization with a guaranteed stability region under nonzero initial conditions and persistent bounded disturbances. Secondly, the stabilization method is extended to deliver an optimized upper bound on the l ∞ -induced norm of the closed-loop system for a given set of persistent bounded disturbances. Thirdly, the stabilization and l ∞ designs are adapted to cope with actuator saturation by means of a generalized sector bound constraint. The proposed controller designs are tailored via a finite set of state-dependent linear matrix inequalities. Numerical examples are presented to illustrate the potentials of the proposed control design methods. Copyright © 2014 John Wiley & Sons, Ltd.

Model Matching Via Stabilizing Static State Feedback

The problem of stable exact model matching of linear systems by static state feedback is considered. Based on the effect that a static state feedback produces on the unobservable/controllable part of the closed-loop system, we provide the set of all static state feedback control laws that solve the exact model matching. Moreover, necessary and sufficient conditions for the existence of static state feedback that solves the model matching with stability are presented, along with a method to determine such a required state feedback law.

Local Stabilization of Markov Jump Nonlinear Quadratic Systems

This paper investigates the problem of local stabilization of Markov jump nonlinear quadratic systems. A method is presented for the synthesis of a static nonlinear quadratic state feedback control law that ensures the local exponential mean square stability of the zero equilibrium point of the closed-loop system in some polytopic region of the state-space with a guaranteed region of stability inside this polytope. The proposed control design is tailored in terms of linear matrix inequalities together with convex optimization to achieve an enlarged stability region. A numerical example is presented to illustrate the application of the stabilization method.

Design of a Model-based Controller with Temperature Feedback for Laser Cladding

Laser cladding, also known as direct metal deposition, is an additive manufacturing technique for the production of freeform metallic parts. High quality parts can be created with the use of feedback control systems which stabilize the melt pool during the cladding process. Current laser cladding control systems are based on low order, empirical models of the process, which have low dynamic accuracy and limit the performance that can be achieved. In this paper, a control system based on a physical heat conduction model of the melt pool dynamics is presented. The control structure consists of a static linear state feedback control law designed using pole placement and combined with a PI controller. The controller is able to regulate the melt pool size by modulating the laser power using a number of surface temperature measurements as the feedback signal. Simulation results using a detailed finite element model show that the controller has good tracking behavior and disturbance rejection properties.

Pole placement in non connected regions for descriptor models

Abstract This paper proposes a method to compute a static state feedback control law which achieves a non strict pole assignment on a continuous or a discrete descriptor system. The specification on the closed-loop poles is given in terms of D -stability, i.e. in terms of a pole-clustering region D . In this work, the clustering region is possibly non path-connected since it can result from the union of disjoint and non symmetric subregions. Such a choice, which is original when the design of linear descriptor systems is concerned, is made possible by a technique that enables a partial pole placement via aggregation. The distribution of the finite poles in various subregions can be chosen. Some key steps of the procedure are addressed through strict LMI (Linear Matrix Inequalities). This is an extension of a previous work from conventional to descriptor models. Therefore, not only stability is to be ensured but also, because of infinite poles, regularity and causality or impulse freeness (the whole of those properties being termed admissibility).

Stabilisation of nonlinear chemical processes via dynamic power-shaping passivity-based control

It is well known that energy-balancing passivity-based control is stymied by the presence of pervasive dissipation. To overcome this problem in electrical circuits, some authors have used power-shaping techniques, where stabilisation is achieved by shaping a function akin to power instead of energy. Some extensions of the techniques to general nonlinear systems, yielding static state-feedback control laws, have also been reported. In this article, we extend these techniques to dynamic feedback control and apply them to nonlinear chemical processes. The stability analysis is carried out using the shaped power function as Lyapunov function. The proposed technique is illustrated with two nonlinear chemical process examples.

Issues on global stabilization of linear systems subject to actuator saturation

This paper reexamines the classical issue whether linear or nonlinear static state feedback control laws are needed for globally stabilizing linear systems subject to actuator saturation.

Two-degree-of-freedom output feedback controllers for discrete-time nonlinear systems

A discrete-time, model-based output feedback control structure for nonlinear processes is developed in the present work. The structure makes use of a closed-loop observer, while at the same time it guarantees that the overall feedback controller possesses integral action. An algebraic transformation is applied on the observer states to insure that the input/output gain of the observer matches the model upon which the static state feedback control law is based. The resulting control algorithm is a two-degree-of-freedom control law, in the sense that the output and the set point are processed in different ways. The control structure is shown not only to have the same properties as the standard model-state feedback structure, but also that it emerges from a model algorithmic control framework. Finally, a simulation example using an exothermic CSTR operating at an open-loop unstable steady state is used to evaluate the closed-loop performance of the proposed method.

Pole placement in a union of regions with prespecified subregion allocation

This paper proposes a method to compute a static state feedback control law which achieves a non-strict pole assignment. The specification on the closed-loop poles is given in terms of a clustering region. The originality of this work is the choice of the region which can result from the union of disjoint and non-symmetric subregions. Such a choice is made possible by a technique that enables a partial pole placement via aggregation. The distribution of poles in various subregions can be chosen. The proposed conditions are formulated through an LMI approach and some robustness aspects are also considered from an analytical point of view.

Two-degree-of-freedom output feedback controllers for nonlinear processes

In this work, a nonlinear output feedback control algorithm is proposed, in the spirit of model-state feedback control. The structure provides state estimates using a process model, the measured output, and the residual between the model output and the measured output. These estimates will track the process states at a rate determined by a set of tunable parameters. An algebraic transformation of the state estimates is incorporated in the control structure to ensure that the input/output gain of the observer matches the model upon which the static state feedback control law is based. The transformed states are then used in the control law. This leads to a controller of minimal order possessing integral action. The control structure is shown to have the same properties as the standard model-state feedback structure. The resulting algorithm is a two-degree of freedom control law, in the sense that the control action is not a function of the error only, but the output and the set point are processed in different ways. Finally, a simulation example using an exothermic CSTR operating at an open-loop unstable steady state is used to demonstrate the closed-loop performance of the proposed method.

Feedback passivity of nonlinear discrete-time systems with direct input–output link

This paper is devoted to the study of the feedback passivity property in nonlinear discrete-time systems. The relative degree and zero dynamics of the non-passive system are related to the feedback passivity of the system. Two main results are presented. First, some relative degree-related properties of passive systems in general form are stated. Second, sufficient conditions in order to render a multiple-input multiple-output (MIMO) system passive by means of a static state feedback control law are obtained.