Saturated Linear State Feedback Research Articles

Event‐triggered constrained control of positive systems with input saturation

SummaryThis paper addresses the problem of event‐triggered stabilization for positive systems subject to input saturation, where the state variables are in the nonnegative orthant. An event‐triggered linear state feedback law is constructed. By expressing the saturated linear state feedback law on a convex hull of a group of auxiliary linear feedback laws, we establish conditions under which the closed‐loop system is asymptotically stable with a given set contained in the domain of attraction. On the basis of these conditions, the problem of designing the feedback gain and the event‐triggering strategy for attaining the largest domain of attraction is formulated and solved as an optimization problem with linear matrix inequality constraints. The problem of designing the feedback gain and the event‐triggering strategy for achieving fast transience response with a guaranteed size of the domain of attraction is also formulated and solved as an linear matrix inequality problem. The effectiveness of these results is then illustrated by numerical simulation.

Global stabilization of the linearized three-axis axisymmetric spacecraft attitude control system by bounded linear feedback

In this paper, the three-axis attitude stabilization of the axisymmetric spacecraft with bounded inputs is studied. By constructing some novel state transformations, saturated linear state feedback controllers are constructed for the considered attitude control system. By constructing suitable quadratic plus integral Lyapunov functions, globally asymptotic stability of the closed-loop systems is proved if the feedback gain parameters satisfy some explicit conditions. By solving some min–max optimization problems, a global optimal feedback gain for the underactuated attitude stabilization system is proposed such that the convergence rate of the linearized closed-loop system is maximized. Numerical simulations show the effectiveness of the proposed approaches.

Global Stabilization of Linearized Spacecraft Rendezvous System by Saturated Linear Feedback

The problem of global stabilization of the circular orbit spacecraft rendezvous system subject to actuator saturation is studied in this brief. Based on the linearized Clohessy–Wiltshire equations, a class of saturated linear state feedback controllers is designed by utilizing Lyapunov functions that are frequently used in the absolute stability theory. The global stability of the closed-loop system with the proposed saturated linear feedback is proved by providing explicit conditions on the parameters in the feedback gains. Optimal linear feedback gains are also discussed. The most significant advantages of the proposed controllers are that the controllers are linear and the global stability of the resulting closed-loop system can be guaranteed. Numerical simulations show the effectiveness of the proposed approaches.

Magnetic attitude control of bias momentum spacecraft by bounded linear feedback

This paper addresses the three-axis magnetic attitude control of small spacecraft with momentum bias configuration by bounded linear feedback. Necessary and sufficient conditions are derived to guarantee that the linearized dynamics and kinematics are stable in the Lyapunov sense. Explicit solutions to some Lyapunov equations associated with the considered open-loop system are obtained. Based on the explicit solutions to the Lyapunov equation, both explicit saturated linear state feedback controller and function observer based output feedback controller are designed. Globally asymptotic stability of the closed-loop systems is proved by constructing explicit Lyapunov functions. Simulations show the effectiveness of the proposed approach.

Robust control of uncertain multi-inventory systems via linear matrix inequality

We consider a continuous time linear multi-inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of ε-stabilising the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which ε-stabilisability is possible through a saturated linear state feedback control. All the results are based on a linear matrix inequalities approach and on some recent techniques for the modelling and analysis of polytopic systems with saturations. Numerical simulations are provided.

Robust control in uncertain multi-inventory systems and consensus problems

We consider a continuous time linear multi–inventory system with unknown demands bounded within ellipsoids and controls bounded within polytopes. We address the problem of ∈-stabilizing the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which ∈-stabilizability is possible through a saturated linear state feedback control. The idea of this approach is similar to the consensus problem solution for a network of continuous time dynamic agents, where each agent evolves according to a first order dynamics has bounded control and it is subject to unknown but bounded disturbances. In this context, we derive conditions under which consensus can be reached. All the results are based on a Linear Matrix Inequalities (LMIs) approach and on some recent techniques for the modeling and analysis of polytopic systems with saturations.

Attraction Region of Planar Linear Systems with One Unstable Pole and Saturated Feedback

The bifurcation of the attraction region for planar systems with one stable and one unstable pole under a saturated linear state feedback is considered. The attraction region can have either an unbounded hyperbolic shape or be bounded by a limit cycle. An analytical condition, under which either of these boundary shapes occurs, is given with a formal proof. This condition is based on the relationship between the stable and unstable manifolds associated with secondary saddle equilibrium points, whose presence is caused by the saturation on the input.

Stability of current-mode control for DC–DC power converters

DC–DC power converters are switched devices whose averaged dynamics are described by a bilinear second-order system with saturated input. In some cases (e.g., boost and buck–boost converters), the input output dynamics can be of nonminimum-phase nature. Current-mode control is the standard strategy for output voltage regulation in high dynamic performance industrial DC–DC power converters. It is basically composed by a saturated linear state feedback (inductor current and output voltage) plus an output voltage integral feedback to remove steady-state offset. Despite its widespread usage, there is a lack of rigorous results to back up its stabilization capability and to systematize its design. In this paper, we prove that current-mode control yields semiglobal stability with asymptotic regulation of the output voltage.

Stabilization of exponentially unstable linear systems with saturating actuators

We study the problem of stabilizing exponentially unstable linear systems with saturating actuators. The study begins with planar systems with both poles exponentially unstable. For such a system, we show that the boundary of the domain of attraction under a saturated stabilizing linear state feedback is the unique stable limit cycle of its time-reversed system. A saturated linear state feedback is designed that results in a closed-loop system having a domain of attraction that is arbitrarily close to the null controllable region. This design is then utilized to construct state feedback laws for higher order systems with two exponentially unstable poles.

Some new results on closed-loop stability in the presence of control saturation

Some new sufficient conditions for determining global asymptotic stability of an equilibrium point under control saturation are presented. For an open-loop stable system, we show that there exists a class of linear time invariant state feedback matrices which globally stablizes an equilibrium under control saturation. In addition, we show that for open-loop unstable plants, an equilibrium can never be globally stabilized using a certain class of saturated linear time invariant state feedback. Finally, we extend the stability results to nonlinear systems whose nonlinearities are globally Lipschitz.

Semiglobal stabilization of multi-input linear systems with saturated linear state feedback

For multi-input linear systems with eigenvalues in the closed left-half comlex plane, we address the problem of stabilization by a linear state feedback subject to saturation. Using an eigenvalue-generalized eigenvector assignment technique, we prove that such systems can be semiglobally stabilized with a saturated linear state feedback. Based on this result, we propose an algorithm to calculate an ε-parametrized family of state feedback gain matrices that semiglobally stabilize the system. Two examples are used to illustrate the results.

A new design of constrained controllers for linear systems

A new method is presented to find stabilizing saturated linear state feedback controllers for linear continuous-time and discrete-time systems. A controller of this type was satisfactorily tested on board a submarine as a depth regulator.